Лекции и упражнения по многосеточным методам

!"# $% &'""# ...

Author: Ольшанский М.А.

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& )**+ ,$

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S T

$( A ∈ Rn×n 6 ,(! #% b ∈ Rn 6 ,(' -%" x ∈ Rn 6 -%" , n ,-%'0 % A 8(# ("1 $% "8%#" % det A = 0 <%% -"%- #%' A 9%-" ,-!% "% "%" (" ,(& "%"" -",% %# S T

! ",-"" #%' A 0""" ,-%' #%"( 3 %8% O(n3 ) 2#%&0 " (! ! S T >1 ! #%"%&! " "% ## (! ! 8"1 0 %# S106 8" ,-%'0T -",790 - '0 ('0 ,(&0 4" #"$0 &!0 #%"(' (! 8 "$" S , ,(" %"&"%7T 0" (! ! S T !-!7%! 8" -%'# " (79# &# %" -%" x ∈ Rn &%" % (!

"%"" ,-%" '-" 21 4 =%"# & % %'0 & 0"(% x %"&" ("%%"&" % 8 x ##" "$"%7 4" #"$0 " !0 A !-!%! , " #% % #"$ =#%' A -' 7 <("-%" "! #"1 ! %" #%' -%" ""-! "%-!79! %1 "'0 #%"("- !-!%! "%"%" U(-"V " 6 " "% "(!%! "&%-"# -'0 =#%"- #% " %"$" (! , =%" " %8%1 ! 0% #" #!% %"" -' =#%' 0 "" - #%

?#"%# 8,"-' %"' #%"( - (79# -( /% ,( -%" xi #%-#' # 8 ,-%"# -%" x 7 ,(& S T[ " i ≥ 0 !-1 !%! ("#[ 0"(# "-" S(#! &%" &T 81 xi+1 , ""%"! W (xi+1 − xi ) = b − A xi ,

i = 0, 1, 2, . . . .

S T

/%& &%"8' ,"-% S T "80"(#" "(% 1 (79* \ @&" 8 x0 S# x0 = 0 -" #"$0 1 &!0 6 0"" -'8"T[ ! " # ! ! " $ ! !% " # &$ ' " ' !

5

] 4"#"$%7 #% W S#' "%" " -$( #' (&' -'8" 6 "" % W -" (&" #%T[ A % "%"- S# ,% xi ("%%"&" 0""1 # 8 # x b − Axi ≤ ε (! "%""$" , -'8"$" ε > 0 "%"" "#' · T ?" -"#"$%" #%' W 8(% -( ", % W W −1 "$( ,'-7% "8"--%# 4'& xi+1 ,("# xi ,'-7% ?,"% ei := x − xi # ( ,-%'# # ,-%1 '# 8 # ,"-# -" #' ,%1 "-' &%"8' "8 8' 8, 7 (# $"-"% &%" #%"( "# "8 %#%! 7* ei → 0,

i → ∞.

X%" "( ,-% "% -'8" "%" "#' 3"-"1 !% &%" #%"( #% $"#%&7 0"(#"% ",%# q ∈ [0, 1) "%"' ,'-7% (" % "%"! "# "8 "9%! # &# - q−1 ,* ei+1 ≤ qei ,

i = 0, 1, 2, . . . .

F# # q %# 8'% S&T 0"(%! #%"( ! %"$" &%"8' (% " 0"(#"% #%"( "," #"%% ""%"1 "%""# ("-%-"!% "8 +, S T S T ,7&# &%" -%" "8 ("-%-"!% -"# ""%"7 ei+1 = (I − W −1 A) ei ,

i = 0, 1, 2, . . . ,

S T

,( ( I "8",&% (&7 #% % S := I − W −1 A , S T "%-%%-! , ,# "8 1 (" % #% ,- 6 #%"( S T $" "-% &%" xi+1 ("-%-"!% ""%"7 xi+1 = S xi + g,

i = 0, 1, 2, . . . ,

S BT

$( g 6 -%" "%"' ,-% "% -" &% b " ,-% "% xi 4" "8%"* (" ""%" -( S BT (! "-#%"$" %""$" #%"( % %"$" &%" , xi = x (% xi+1 = x %" #% S , S BT 8(% #% %

- =%"# ,( -%%%! !( %"# " 0"(#"% #1 %"( S T "%"' !-!7%! !#'# (%-# ""%"! (! "8 S T " & "## "" "( =#%'0 2%"- , " $8' / ( -$" (! ",-"'0 =#%"- a, b $8%"- -("- "%%- " !'# ",-(# (·, ·) "## -%-" " |(a, b)| ≤ (a, a) (b, b) = a b. S T + (! 78'0 -9%-'0 a b -%-" ^$ 1 2

|a b| ≤

1 2

ε 2 1 |a| + |b|2 , 2 2ε

∀ ε ∈ (0, ∞).

S T

4#! -"%-# #% (# $"-"% &%" #%&! "# · -%"" "#" · v (! 78" #%' A ∈ Rn×n A =

sup

0=x∈Rn

Axv . xv

S T

/" !" "$ 2"'# ,"# "# "(1 !#! - S T % ,'-%! "%"" "#" #%' A 4 !0 8(% ","-%! %"" #%&' "#' "1 " (' #18" -%"'# % - =%"# & #1 7% #%" (79 "%' (%-! ""%"! S T S#' 8(# "% ( "$( , "%% !" 6 #%! --( #%&! -%"! "#T* Ax ≤ Ax, ∀ A ∈ Rn×n , x ∈ Rn , S CT AB ≤ AB,

∀ A, B ∈ Rn×n .

8",&# &, AT %",7 #%' A* AT = {aji } (! A = {aij } 4 & %(%"$" -("- !"$" "1 ,-(! "&-(" -'"!%! (Ax, y) = (x, AT y),

∀ x, y ∈ Rn , A ∈ Rn×n .

% A ,'-%! ##%&" A = AT . "#" &" λ = λ(A) ∈ C !-!%! "8%-'# ,1 &# ψ ∈ Cn , ψ = 0 "8%-'# -%""# #%' A Aψ = λψ.

/",-"! #% A ∈ Rn×n #% 8" n ,&'0 "#'0 "8%-'0 ,& ##%&" #%' - "8%-' ,&! -9%-' " %"$" (! ##%1 &" #%' 9%-% ," A = QDQ−1 , S T $( D 6 ($"! #% =#%' "%"" % "8%-1 ' ,&! #%' A Q 6 "%"$"! #% % Q 6 -'" (! #% %! &%" QT = Q−1 <%"8' Q 6 "81 %-' -%"' A $# "-# #" " ,% &%" - Rn #" " -'8% "%""#"-' 8, , "8%-'0 -%"1 "- ##%&" #%' 4 S- "89# & "#'T "8%-' ,&! A "8,7% % #%' _` (A) N& ρ(A) := max |λ| λ∈sp (A)

,'-7% #%' A .%9 -- 8",&# &, x2 -("- "# -1 %" x = (x1 , . . . , xn ) #"%# % "#' x1 =

n

|xi |,

x∞ = max |xi |. 1≤i≤n

i=1

/" % &%" "" ('# #%&'# "## !-!7%! a. A2

=

A1

=

b. c.

A∞

=

ρ(A AT ),

max

1≤j≤n

max

1≤i≤n

n i=1 n

|aij |, |aij |.

j=1

( " & )# ! " " "! " "

! "- % a. (" % & (71 9 ""%"! (! ",-"" ##%&" #%' B ,-%' *

λmax =

sup

0=x∈Rn

(Bx, x) , (x, x)

λmin =

inf

0=x∈Rn

(Bx, x) . (x, x)

S 5T

! "- %"- ] A "(8% & -%" x %" &%" -' "( ,"# sup - -" &% S T (% "80"(#7 #%&7 "# (! B (" % S CT (! =%0 "#

.%9 -) "

% -%-" 3(*

A22 ≤ A∞ A1

∀ A ∈ Rn×n

S T

@"# A2 "$( ,'-7% "#" A > ρ(A2 ) = ρ2 (A) %" (! ##%&" A ## A2 = ρ(A) /-(# (- "%' %"#' " 0"(#"% %""$" #%"(

% --

S ! ! S ≤ q < 1 ! " #$%& " ' ! x0 ' ! b ei ≤ q i e0 ,

i = 0, 1, 2, . . .

S T

! ! !'( ! ) !

+, S T "(%-" (%

ei = Sei−1 ≤ Sei−1 ≤ qei−1 .

/# -"(% S T > q < 1 %" 1 -! &% - " S T %#%! 7 i → ∞ 78" &" "8 e0 *! " # "! # ! ! # +

B

% -) * #$%& !" x

0

" " ρ(S) < 1 + " S i = O ip−1 ρ(S)i−p+1 ,

"

i → ∞,

" p , ! ! ! - S ) ! >"# (", - PQ (

"("1 -" "#" 2"#' #%' #" " % # - PQ

> x ,-%" %" "8 ei ,-% O #%! ,#! " (! S %" - -%- ei ≤ Si e0 ,& ei #" " "% %"&"%7 (" -&' &" "8 e0 +& , ,-%'0 - " % -& " &%- 8 ! xi #" % $"-"% -%" ri = b − A xi ,'-#' ! O xi 6 %"&" %" ri = 0 - "89# & , ""%"! S"-% $"aT A ei = ri

(% x − xi ≤ A−1 b − A xi ?#"%# (79 -"%-" #% (# % A > 0 S A ≥ 0T (Ax, x) > 0 S""%-%%-" (Ax, x) ≥ 0T (! ",-""$" -"$" -%" x ∈ Rn > A > B A − B > 0 % A > 0 SA ≥ 0T 8(# ,'-% # & N#%# &%" , ""1 %" "("% #%' (% "" %"% =#%"- $-" ($" " (% "" %"% "%%"% "%'0 =#%"- ?-" , "1 " %"% =#%"- #%' A (% -""89 $"-"! A > 0 .%9 T-+ ! ",-"" #%' A "-%* \T #% A A 6 ##%&! "%%" "(![ ]T A = A0 + An $( #% A0 = 12 (A + AT ) 6 ##%&! An = 12 (A − AT ) 6 ""1##%&! % An = −ATn [ 8" %"$" (A x, x) = (A0 x, x),

∀ x ∈ Rn .

/"" %' ##%&' #%' "8,7% - ' #% ! %" #%' A (79 !" "1 ,-( "# "%" "(' (x, y)A

:=

(Ax, y),

xA

:=

(x, x)A2 ,

1

∀ x, y ∈ Rn , ∀ x ∈ Rn .

%- ( (79 %"#' -'$!(% $"#",(" " "1 ("8%! (! (",%%- 0"(#"% "%"'0 "%'0 %1 "'0 #%"("-

% -+ :;0<= A = A

W > 12 A " #$%& " ' ! · A T

ei+1 A ≤ qei A ,

>0

1

q = (1 − 2δ1 δW −2 ) 2 ,

" δ = λmin (A) δ1 = λmin (W0 − 12 A) W0 = 12 (W + W T )

) ! +, ""%"! S T "(! "#' · A

"&#

ei+1 2A = (ASei , Sei ) = (A(I − W −1 A)ei , (I − W −1 A)ei ).

8",&# v = W −1 Aei %"$( ",! ##%&"% A #"1 # % "( ""%" 1 ei+1 2A = ei 2A − 2((W − A)v, v). 2

S T

$" "-% S T &%" W0 = W0T S BT (! 78"$" v /"=%"# ((W − 12 A)v, v) = ((W0 − 12 A)v, v) , ""%" ?=! S 5T (% ((W0 − 12 A)v, v) ≥ δ1 v2 ># "8,"# "&# (W v, v) = (W0 v, v)

1 2((W − A)v, v) ≥ 2δ1 v2 . 2

S T

< ($" %""' -(-" ei 2A

= (Aei , ei ) = (W v, A−1 W v) ≤ A−1 W v2 ≤ A−1 W 2 v2 2 2 −1 = λmax (A−1 )W 2 v2 = λ−1 W 2 v2 , min (A)W v = δ

%" %

S T

v2 ≥ δW −2 ei 2A .

S T 6 S T -#% S T -% ei+1 2A ≤ (1 − 2δ1 δW −2 )ei 2A .

@ (% !( #"- 8,"-'0 %"'0 #%"("- 4%% -- %"( * W = w−1I "%"'# w > 0 "%"' % ,'-7% .%9 -3 /("" # &%" _` (A) ⊂ R+ " % &%" #%"( "%" % 0"(%! (! 78"$" &"$" 1 8 ! x0 w = A−1 (! ",-"" "#' · . <& "-% &%" ρ(A) ≤ A (! 78" "#' A

4%% -) %"( . * W = bc\d (A) N( ( bc\d (A)

6 #% "&! , A ,#" =#%"- - $-" (1 $" .%9 -1 /("" # &%" A #% %"$" ($"1 " "8(* |aii | >

n

|aij |,

i = 1, . . . , n.

j=1, j=i

" % &%" #%"( e"8 0"(%! +",% "# , !

4%% -+ ?#"%# (79 ,"

#%' A*

A = D + L + R,

$( D = bc\d (A) L (R) 6 -! S-!T %$"! &% A % #% "&! ,#" - A -0 =#%"- $-1 " ($" ( S"(T %"( / 0 "(!%! -'8""# W = L + D

>"# "#"$% # %"-% 0"(#"% #%"( 31 1 N(! ",-"" #% A = AT > 0. %-1 %" A > 0 %" D > 0 < ($" %""' - S BT ("" %"#' W > 12 A -"" W0 > 12 A. ! #%"( 3 1 N(! W = L + D % RT = L #1 #* W0 = 12 (W + W T ) = 12 (R + L + 2D) X%" -&% "80"(#" "- - %"# * 1 1 1 W0 − A = W0 − (D + L + R) = D > 0. 2 2 2

.%9 -5 /%

⎛

α A=⎝ β 0

β α β

⎞ 0 β ⎠. α

! 0 ,& α β \ %"( e"8 0"(%!f ] %"( 3 1 N(! 0"(%!f @" ,#%# &%" #%"(' e"8 3 1 N(! #"$% 8'% ,' - 8" =""#" (! -'& 2"#* \ %"( e"8* Dxi+1 + (L + R)xi = b.

] %"( 3 1 N(!* (L + D)xi+1 + Rxi = b.

.%9 -> /(""

# &%" " "8( "%"# q* q|aii | >

|aij |,

A

i = 1, . . . , n,

#% %"$" ($"1 q ∈ [0, 1).

j=1, j=i

" % &%" (! #%' % #%"( 3 1 N(! S = (L + D)−1 R -(- " S∞ < q.

+ ("-%" #%"( N(! 0"(%! ",%# 0"(1 #"% q C

! ",-""$" -%" x ""%-%%-79$" -1

%" y = S x % ""%" 0 -!,'-79 L y + D y + R x = 0. S T

! "&! " y∞ < qx∞ #"%% k17 %" %#' S T - ("" &%" (! y∞ = maxi |yi | #1 ## - -" &% ("%$%! (! i = k Sxx∞ < q (! ",-""$" -%" x = 0 8(% ∞ ",&% " S∞ < q " "(7 "#' #%' S T

- & ##%&" #%' A #% % #%"( 3 1 N(! ##%& 6 "8"--% W ##%& +"$( 8'-% ("8" #"%% ##%1 &' -% #%"( 3 1 N(! =%" (" - 1 (79# # 4%% -3 ( %!

" / 0 "%"% - "("-%"# -'" (-0 $"- 81 ,"-"$" #%"( 3 1 N(! !#" "8%" #1 ,-%'0 X%" =--%" "("-%"# #1 7 "8"--% "%-'0 , ($" -" %$"" &% A ,%# , ($" -" %$"" &% #" ,! xi 0"(# xi+1 , ""%" 1

(L + D)xi+ 2 + R xi

=

b,

i+ 12

=

b.

i+1

(D + R)x

+ Lx

.%9 -2 /"-% &%" ##%&' #%"( 3 1

N(! #" % 8'% , - -( 8,"-"$" %""$" #1 %"( S T "8"--%# W = (L + D)D−1 (D + R). S CT +0"(% , "(! ##%&"$" #%"( 3 1 N(! -' ! xi+1 &, xi b -"",% ""%"!# S"-% 0T −L(D + L)−1

= D(D + L)−1 − I,

(D + R)−1 R

= I − (D + R)−1 D.

! ##%&" #%' A ## R = LT 4 =%"# & , S CT -(" &%" (! ##%&"$" #%"( 3 1 N(! W = WT 4%% -1 O%%-'# "8,"# #" " --% #% 1 - #%"( 3 1 N(! /"&79! #%"( "% "9" ,- Kgh S"% $ KiAAj__ckj gkjl hjm\n\ocpqT /"8"--% W - #%"( Kgh #% -( W =

1 (D + ω L), ω

$( ω 6 #% "$&" #%"( 3 1 N(! %"# ",-"!% "-% (79 %- (* (! %#' #% A = AT > 0 #%"( Kgh 0"(%! ω ∈ (0, 2) 78"# &"# 8 78" -" &%

.%$" % A ,'-%! 1 aij ≤ 0 (! 78'0 i, j : i = j A "8%# - =#%' A−1 "" 1 %'

' #% -# "! - %" %"1 '0 #%"("- % #7%! "%' %- (! "8 0 0"(1 #"% (! ! %# - r1#%# S# PQT /% A !-!%! r1#% /% - =#%' #% W −1 W − A "%%' %"$( #%"( S T 0"(%! /("" # &%" #% W "& , A %# ,#' "%"'0 =#%"- aij , i = j =#%' wij % &%" aij ≤ wij ≤ 0 %"$( #%"( S T 0"(%! <-"%-" #%' A 8'% r1#% (% - "%""# #' -"%-" #""%""% ,"%" 0#' "#1 79 - - &%'0 ",-"('0 ""%""% "(" -'"!%! -" #"$0 %790 &- ""8" ","- 0# -'""$" "!( %"&"% /"=%"# (1 - !0 #' 8(# "%--%! =%"# -"%- #%1 5

! "9! ," ! - ("# $2 "" # - S T 89"% ,%%"- "%! 8(% % %1 "' #%"( S T #" % 8'% , 8, "8"--1 %! " --(! "-" #%' Aˆ = W −1 A "-" -" &% bˆ = W −1 b " %"$" A = AT > 0 W = W T > 0 %" Aˆ !-!%! #""! " "" %" "(" #% "%"%" !'0 ",-( (·, ·)W (·, ·)A #" $" "-% &%" W = I.

ˆ y)A = (x, Ay) ˆ A (Ax, ˆ x)A > 0 (Ax,

∀ x, y ∈ Rn , ∀ 0 = x ∈ Rn .

N#%# &%" ##%&' #%' #" " #%-% &%' & #""! '0 #% "%"%" -1 ("- !"$" ",-(! 4 %' -"%- #1 #%&'0 #% " "%"'0 $"-"" -' "%"'# #' ","- -' (! #""! '0 #% "%"%1 " ",-"'0 !'0 ",-( /"=%"# (! - "1 $" &! ##%&" #%' A - -(' 1 ,%%' -(-' S- ""%-%%-790 "#0T (! #%"( %-'# ##%&'# "8"--%# 4-(# - S T #%' ,-!9 "% "# %* τi ∈ R, i = 0, 1, 2, . . . xi+1 − xi = τi (b − A xi ), i = 0, 1, 2, . . . . S T 8'&" #%' τi --"(!%! (! "! 0"(#"% #%"1 ( <9%-% ("-"" #"$" ""8"- -'8" τi ?#"%# "%"' ?#"%# & & A = AT > 0 ("" # &%" ,-% 2"#! " $0 %* _` (A) ∈ [m, M ] O1 ",%! "%"!' #% τi = τo (! -0 i > 0 %" & -'8" 8(% τo = 2(m + M )−1 %-%" , -! (! "8 S T "&# ei+1 ≤ I − τ Aei .

> 6 ##,"-% ei+1 %" ""8# #1 #,"-% -' #" % - -" &% -'8! τ 1

y6 y = 1 − τ0 λ q0 s r λ −1 τ

0

? * %#' #% (! #%"( "%" % &# "8,"# @# ,-%" &%" (! %" "#' -'"!%! I − τ A = max |1 − τ λ|. λ∈ sp (A)

% _` (A) ,-%" &%" " ( % "%, [m, M ] "=%"# ,#'# !-!%! -'8" τo = \ld min

τ ∈R

? τo =

max |1 − τ λ| .

λ∈[m,M ]

2 m+M

"7%"-" =%"# I − τo A ≤ qo =

M −m . M +m

S 5T

O "%,%! "% -'8" τ "%"!'# ,2"-% "&%-" % #%"( S T # k %" & #%"- τi , i = 0, 1, . . . , k − 1 "%"' ##,% k17

"8 ek % #" % 8'% ( X%"% 8" ,-% "% -'8"$" ( k +, ""%"! (! "8 S T "&1 # ek = (I − τk−1 A) · · · · · (I − τ0 A)e0 .

"$&" &7 "%"!'# τ "80"(#" % "%#,"" ,(&* {τi }i=0,...,k−1 = \ld min (I − τk−1 A) . . . (I − τ0 A) τ ∈R i

\ld τmin ∈R

=

max |((I − τk−1 λ) . . . (I − τ0 λ))|

_` (A) 4 "8%-' ,&! A % - =#%' #" %- _` (A) #"$% 8'% ,-%' "(" # ,-%' 0 $' /"=%"1 # ('(97 "%#,"7 ,(& ,#!# (71 97

max |(1 − τ0 λ) . . . (1 − τk−1 λ)| . S T τi=0,...,k−1 = \ld min τ ∈R λ∈[m,M ] i

λ∈

i

@8" τi 79 ,(& S T ,-% S# # P QT* -&' τi−1 !-!7%! "!# #"$"& F8'- %1 k %- [m, M ]* τi−1 =

1 2

π(2i + 1) m + M + (M − m) cos , i = 0, . . . , k − 1. 2k

S T

! =%"$" "%#"$" 8" #%"- -(- " √

max |(1 − τ0 λ) . . . (1 − τk−1 λ)| ≤ 2

λ∈[m,M ]

√ k M− m √ . √ M+ m

N& - -" &% "-% -0 ",% 0"(#"% #%"( S T , k % &8'-# 8""# #%"S T

! , 0"(#"% %"'0 #%"("- ! 1 %# ##%&" #% A 8" "(0"(% " "81 %7 #% A−1 >&'# #"# !-!%! (79 /("" # &%" (! -0 x ∈ Rn γ1 x2 ≤ (Ax, x), S T 2 −1 x ≤ γ2 (A x, x) S BT

"%"'# "%%# γ1 > 0 γ2 > 0 ! ##%&" A =% "-! =--%' "# "8%-' ,&! _` (A) ∈ [γ1 , γ2 ] N#%# % &%" -%"" "- #" % 8'% " - -( Ax2 ≤ γ2 (Ax, x).

F%"8' "&% " %7 "# #%' %1 S = I − τ A -'# (79 ""%"!* Sx2 = (I − τ A)x2 = x2 − 2τ (Ax, x) + τ 2 Ax2 ≤ x2 − 2τ (Ax, x) + τ 2 γ2 (Ax, x) = x2 − (2τ − τ 2 γ2 )(Ax, x) ≤ x2 − (2τ − τ 2 γ2 )γ1 x2 = (1 − 2τ γ1 + τ 2 γ2 γ1 )x2 .

/" "(7 "#' S = sup x=0

1 Sx ≤ (1 − 2τ γ1 + τ 2 γ2 γ1 ) 2 , x

"=%"# -'8" #% τ = γ2−1 "&#

S ≤

1−

γ1 . γ2

S T

N& - -" &% S T "-% -0 ",% 0"1 (#"% %""$" #%"( S T W = I ",-"" #% A ("-%-"!79 "-!# S T S BT " 7 ("-%-"%' " (! "8%-'0 ,& #%' A " (! "8%" #%' -" #"1 $0 %&0 ,(&0 ,-%' ,-%' "# U$8"V 4%% ,(& "%#, τi "%%%- "("8" 2"# ( , "%0 #%"("- 0"(% %" τi (! i1" % &%" (! ,("$" xi "# -!, ri+1 = b − Axi+1 Si+1T1" % ## $# "-# (! =%"$" #%"( "80"(#" -'8% τi = \ld min b − Axi+1 . τ >0 i

F%"8' "&% -' (! #"%# Axi+1 − b2

τi

$"(" (! -'&

= A(xi − τi ri ) − b2 = ri − τi Ari 2 = ri 2 − 2τi (Ari , ri ) + τi2 Ari 2 .

B

## =%"$" -' ! ("%$%! τi =

(Ari , ri ) Ari 2

S T

4 =% (! -(-' - "89# & #1 #%&" #% <""% 0"(#"% =%"$" #%"( - #1 #%&"# & 0 &# #%"( "%" % "1 %#'# τo - & ##%&" A 0 &# ,1 ! - S T %-%" #"%# -" ""%"1 "%""# ("-%-"!7% -!, Sri = b − Axi T /"-% &%" "" "$&" -"# ""%"7 S T (! "8"* ri+1 = (I − τi A)ri , i = 0, 1, 2, . . . . S T +, $" ,7&# (! #%"( "%" % "%#'# #%"# ri+1 = (I − τo A)ri ≤ I − τo Ari .

4 %" -#! - #%"( "%" % "(8%! τo %" &%" ##,%! "# I − τo A - -" &% " -'1 8" S T ##,% "(%-" ri+1 @# -%1 &" "%" ""%" "%"" -!,'-% "8 Sei = x − xi T -!, i1" %* Aei = ri . S CT +, S CT !#7 (% A−1 ri ≤ ei ≤ A−1 ri .

># "8,"# %# -!, 7 -&% %# "8 7 "8""% " %"$" #%"%& ",1 % 0"(#"% U- -!,0V S% "%" rr i → ∞T U- "80V S%" "%" " (! "8"T 6 "("-' 4#! ("$" #%"( S T "'# 1 #%# "$" S T @"(%-! ,! #%"( " 2"## S T S T %8% (-0 #" #%' -%" ("# %""# $ 6 -" #" %1 ("#" " (" #%"( #" " % "$&-1 "(" " #" ! #%' -%" ("# i+1 i

$ N(-! "-' -%" z i = A ri #" # % 2"#1 S T -' ""%"! S T S T τi

(z i , ri ) , z i 2

=

S T

S 5T S T /#&%" &%" -'&!0 " 2"## S T 1 S T -! &% b %#' S T &%-% %"" - 0" ( 1 &" -!,* r0 = b − A x0 xi+1 ri+1

= =

xi + τi ri , ri + τi zi .

4" #"$0 &!0 8" =22%-' %"' #%"(' #"1 $% 8'% "%"' ##,"-% "# -!, ($" 8" "(0"(!9 2" -! "( %1 "' #% ,(-! "%"'# "("%%-"# "%"" "(!%! - " % "%""# (1 % 8 7 /8 (" " ##,1 "-% 2" =%"# "("%%- @# (! %# "" %" "(" ##%1 &" #% A "(0"(!9# -'8""# !-!%! ##,! -(%&"$" 2" f (x) = (Ax, x) − 2(b, x). S T

%-%" , ""%"! (A−1 (Ax − b), Ax − b) = (Ax, x) − 2(b, x) + (A−1 b, b)

-(" &%" ##,! f (x) =--% ##, 21 " "% -!, Ax − bA "%"' "&-(" ("%$% ### %#' S T G"" "% "& #%"(' ##, 2" ""-' "("%%-0 '"- 4 =%"# & %"%! "("-%"% -" '0 "("%%Vk %! &%" −1

dim(Vk ) = dim(Vk−1 ) + 1.

#" Vk = Vk−1 ⊕ _`\q{Ak r0 },

V0 = _`\q{r0 },

k = 1, 2, . . . ,

$( r0 6 &! -!, _`\q{ } "8",&% 7 "8""&1 ># "8,"# Vk = _`\q{r0 , Ar0 , . . . , Ak r0 }, V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Rn . Vk "% ,- k 1" ! 2 ! ""-1 " #% A -%""# r0 O - ," r0 " %# "8%-'0 -%""- #%1 ' A - "=22%' ," ! "%&' "% ! %" Vn "-(% " -# "%%-"# Rn (" n1"# $ 8 "-(% # - ("" "8 "%%1 %- "8" "$! @ % "(" "("8' #%"(' #%-7% %"' (" %"&' /"0"1 (% =%" " # " (-# &#* -"1-'0 -! "8" "$! ,'-%! %"&"% "%"! "(1 "%%- '"- % %"&"% ! ,(& #1 #,[ -"1-%"'0 &%" ,#"% n "& - 81 xk (! "%""$" k n #% ("-%-"%7 %"&"% "8" -'(# & A = AT > 0 / ,("# &"# 8 x0 UtV1" 8 8(# "(!% xk = arg min f (x), $( Vk (x0 ) := {x : x = x0 + v, v ∈ Vk }. x∈V (x ) k

0

4 =%"# & #" " ",% &%" 0" ( 8

xi ""-" ##, 2" S T "("-1

%"% "("%%- '"- %8% !-"$" "%"1 ! 0! S# - -( 8,T #0 "("%%- #" % 8'% ," - -( -"$" %0&"$" ""%1 "! 8 !* xi+1 = αi xi + (1 − αi )xi−1 − βi ri ,

i = 1, 2, . . .

&'# ""%"#* x1 = x0 − β0 r0 ,

β0 =

(r0 , r0 ) . (r0 , Ar0 )

/#%' αk , βk "(!7%! (#& , -%μk δk βk−1 αk

= =

(rk , Ark )/(rk , rk ) (rk , rk )

−1 −1 = μk − δk δk−1 βk−1 = βk μk

S T

" % 8'% ("," &%" -! &% - S T "%& "% !

' #%"( "& ,- #%"( " ! S#' 8(# ","-% "89"%8%7 88-%1 EM "% $"$" Epquid\oj Ml\bcjqoT %"( #" % 8'% , - "0 2"#0 @"%"' , 0 "8(7% 8"1 %"&-"%7 -!7 "8" "$! ($ 8" =""#&' (! -'& S# # P QT "%#1 &" "%%%- "8" "$! EM #%"( 8' %"&" %#' Ax = b 8" &# , n % (1 " ##" 8 &%" ("%$%! , #"$" #1 &" % /"=%"# EM #%"( "8'&" ",7% %"' /",% $" #%"%&" 0"(#"% %1 " #%"( S T &8'-# 8""# #%"- " %"&" (! "8 -'"!%! (79! " - "# · A * e A ≤ 2 k

√ √ M− m √ √ M+ m

k

e0 A ,

$( m = λmin (A) M = λmax (A) % !! -0!! $1 ' % #%' A "%"' ,% (! , EM #%"(

! ##%&'0 ,""('0 #% 21 " S T # "(0"(% ?,#" %%$ 8'" 8' '1 %%! ##,"-% 718" "# -!, # -1 ("-* f (x) = Ax − b2 ?,! %0 #%"("- S# 1 # - PQ " "!"$" #%"( MrhvK 6 "8"891 "$" #%"( ##'0 -!,"T -7&% - 8! #%&"1 -%"' ",-(! #% A "( "%"$",1 (" % "%"! "80"(# (! "&! "1 %"$""$" 8, - "("%%- '"- " 7

C

, & # A > 0 " A < 0 "

(! ##%&'0 ,(& =%"% " #" % 8'% -( ""%# -'# ""%"!# #%"( "! '0 $(%"- %8% 0! -0 -!," r0 , . . . , ri−1 (! 1 0" (! "-"$" 8 ! xi %#"% S##,"" -"%-"T #%"("- ""1 -'0 "("%%-0 '"- #" % 8'% -' - (79# -( F, Pk "8"1 ,&# #" %-" #"$"&"- Pk (x) % k -9%-'# "=22%# %0 &%" Pk (0) = 1 ! #%"( "! '0 $(%"- -(-" ek A = min Pk (A)e0 A , Pk ∈Pk

(! S"8"89"$"T #%"( ##'0 -!," rk = min Pk (A)r0 , Pk ∈Pk

%( - &%"% (% (! #%"( "! '0 $(1 %"

max Pk (λ) r0 . rk ≤ min S BT P ∈P λ∈sp (A) k

k

0"(#"% #%"("- ""-'0 "("%%-0 '1 "- - ,&%" % "(!%! %"# #%' A - &%"% $" $# U"%"%7V "" ! "81 %-'0 ,& @# A #% m ,&'0 "81 %-'0 ,& S%%-" "%&'0 "% 0T %" - "%%%- "8" "$! #%"( (% %"&" 8" &# , m % O '%%! "0% U""%7V -7 2"# - EM #%"( (! ##%&'0 #% A %" (%! "1 %-"-% -"%-"# ##, /#"# %"$" "(0"( % #%"( "%"$"'0 <"! '0 3(%"- S1 !%! 88-% 6 wcEMT %-" %"&" "%" #%"( !-!%! (79 -"%-" " EM #%"1 ( (! ##%&" A ! -!," "&790! - " % EM #%"( -'"!%! (rk , rj ) = 0,

∀ j = 0, . . . , k − 1,

"(" EM #%"( ## %"" %## ##%&'# "" %" "('# #%# wcEM #%"( "%"' ## - 8" "89# & (%! "8&-% "%"$"1 "% -!, rk -!,# %#' S-"%-" 8"%"$""%T* (rk , ˜rj ) = 0,

∀ j = 0, . . . , k − 1,

$( ˜rj 6 -!, "%"' 8' 8' "&' # %0 -'0 ""%" (! ! "! " ,(&* A∗ x ˜ = b.

N#%# &%" A∗ "-(% AT (! -("- !"$" "1 ,-(! ! A = A∗ S- &%"% (! ##%&'0 #%T wcEM #%"( "-(% #%"("# "! '0 $(%"- +,-%' -%' wcEM #%"( "%"' %87% !-1 '0 -'& AT ( , -# "!'0 -%"- S""-"# " & & %"&-"% "7 "8" "$!T 6 #%"( wcEM_o\] (" ' 4 ( 4"%"# P5Q (" 8'" "%#&" =%"% #%"( "8&-1 % ##, "$"18" "(0"(!9$" 2" "=%"1 # #% "" #""%"7 0"(#"% 6 "( #! "8 #" % #%! "("# -",%! 3",(" 8" " #%"(0 ""-'0 "("%%-0 '"- "! '0 $(%"- #" " ,% # , &8"- P Q PQ 4 "!( -"&" 2"# - 1 ( -(' $"%"-' (! , "#7% 1 $"%#' #%"("- MrhvK wcEM_o\]

.%9 -0 \ /"%"% %"' " ""%1

-%%-79 ##, 2" f (x) , S T #%"("# "$" * $(

xi+1 δi

= =

xi − δi ∇ f (xi ), \ld min f (y), y = xi − δ∇f (xi ). δ∈R

* ! !" )# " ! ' -./0 ' # " 1 !! " # '

" " & " '

5

] N% " - 2"# %879 %"" "( "1 7 #" ! #%' -%" (" % @0" ( δi "$&" 0" (7 τi - #%"( ##'0 -!,"

! -'"! ] -'% -" ""%" (! -!,"

! 4 =%"# ,( "%"# # #' #"%# ""-' #%"1 ( "&'0 =#%"- 6 $-"$" %#% - !0 (1 ,&"$" (! -(! &"$" ! (221 "$" -! 7 %#' $8&0 - +," &%! "&! ("8" "%"- (221 " ,(& "" (! ! "(" %! -'-"1 ("# (%" ,(& "# U8,"%V =%0 (-0 ,(& "! &% %" -( - -( -#% ,!# # 4 %" -#! %"! ,"8! #"("# # !-!%! 2(#%"# (! 1 , #"$"%"&'0 #%"("- " "("8" #%"("# "&'0 =#%"- #" " ","#%! - &" $ <! PQ - 8" "-#'0 ,(!0 P5Q

! "

8(% ","-%! %(%' "8",&!* L2 = L2 (0, 1) 6 "%%-" %$#'0 S" 8$T 2 1 v(x) ,('0 (0, 1) "&'# %$"# 0 v 2 (x) dx[ %1 $ 01 u(x) v(x) dx ,(% L2 (0, 1) !" ",-([ C k (0, 1) 6 "%%-" 2 k , '-" (221 #'0 (0, 1)[ Hk = Hk (0, 1) 6 "%%-" %$#'0 (0, 1) 2 #790 ",-"(' "!( 0, . . . , k , L2 (0, 1)[ - #"$"#1

# ! !"%

"# & #7%! --( - #' ",-"(' (" "!(1 k -7&%"[ H10 = H10 (0, 1) 6 "%%-" 2 , H1 (0, 1) -'0 7 !0 "%, <!" ",-( - H1 (0, 1) S - H10 (0, 1)T ,(%! -%-"# (u, v)H 1 :=

1

u(x) v(x) + u (x) v (x) dx.

0

/"%%- L2 H !-!7%! $8%"-'#

! 2 , H10 (0, 1) -(-" -%-" R(0* 1

H10

1

2

12

u dx 0

1 ≤ 2

1

2

12

(u ) dx 0

.

S T

+, -%- R(0 (% &%" "%%- H10 (0, 1) #" " --% (79 !" ",-( (u, v)1 :=

1

u (x) v (x) dx,

0

&# "#' · H · 1 8(% ! ! 4 #"$"#1 "# & % -(- " L2 1"#' 2 , H10 &, L2 1"# $(% 1

+% - &%- # #"%# "("#" - (22,* −(k(x)u ) = f

(0, 1),

u(0) = u(1) = 0

S T S T

(# ("$% &%" k(x) 6 "&"1'-! (0, 1) 2! 1 ≤ k(x) ≤ K. #" # S T ",-"7 ! & 2 $ $ $ $ &

# # 3+4 ( · · & !" H# " a b " " ! c C # ! c va ≤ vb ≤ C va

" "$ " v H

27 v(x) ∈ H10 (0, 1) "%$# "&" -%-" " &%!# "% 5 (" $"(! -'# $&'# "-!# (! v(x) 0"(# ""%"7*

1

k(x)u v dx =

0

1

0

f v dx ∀ v(x) ∈ H10 (0, 1).

S CT

R! u , H10 (0, 1) ("-%-"!79! S CT ,'-%! S"8"89'#T # S T 1 S T O %$ - 1 -" &% S CT ,(% ' "$&' 2" H10 (0, 1) %" S CT 9%-% (%-" " %"1 $" S CT u(x) "=22% k(x) % $( 2 S# u ∈ C 2 (0, 1), k ∈ C 1 (0, 1)T %" #" " "-1 % &%" u(x) !-!%! % # S T 1 S T - "# S"8'&"#T #' .%9 --* /% f (x) ∈ L2(0, 1) ",% -(1 -"% " (! ! u(x)

1

2

12

u dx

+

0

1

12

2

(u ) dx 0

≤

1

2

f dx 0

12

S T

""%" ("" # &%" k(x) ∈ C 1 (0, 1) ",% &%"

1

2

(u ) dx 0

12

≤C

1

2

f dx 0

12

S B5T

! "- S T -"","-%! -(-"%7

S CT (! &%"$" &! v = u -%-"# "* |(a, b)| ≤ ab $( (·, ·) 6 !" ",-( - L2 (0, 1) · 6 "# # "" (#! 4"","-%! % -%-"# R(01 S T &" " & "

5!%6" 7 " a(u, v) 8 $"% "$ H# ! a(v, v) ≥ cv2 " "$ " & H# f (u) 8 " !) "# u ∈ H# "

a(u, v) = f (v)

∀ v ∈ H.

! "- S B5T & -"","-%! ε1-%-"# S T "&% , S T -",-(! "8 &% - -(% %1 $! S5 T* (1 − ε)

1 0

1 (u ) dx − ( − 1) ε 2

1 0

2

2

(k ) (u ) dx ≤

1

f 2 dx

0

S B T

"#"97 %$"-! " &%!# -%- R1 (0 "-%

1

2

12

(u ) dx 0

1

12 1 2 ≤ (u ) dx . 2 0

4"","-%! =%# -%-"# % "$&# k ",-""# - -'8" ε &%"8' "% , -7 &% S B T S T S B5T !-!7%! ""-'# % ,-%'# , 8, ,! u(x) "# (! ! ,1 (& S T 6 S T N#%# &%" (! -(-"% S B5T "1 ("8" (""%" ("" U$("%V ('0 - ,(&* k(x) ∈ C 1 (0, 1)

# $

4 &%- 8 "$" #%"( ! S T 1 S T 1 #"%# #%"( "&'0 =#%"- "-" (! #%"( "&1 '0 =#%"- % 8! "%"- ,(& S CT 4'8# "%%-" Uh ,#"% "1 #79 "%%-" H10 (0, 1) %" &%" Uh ⊂ H10 (0, 1) F%" "#%! "( "-# U Uh "#% H10 (0, 1)V %% "1 !% ", , ! <#' "%" # Uh 6 "&"1' '-' 2 [0, 1] "%"%" ,8! [0, 1] N -'0 "%,"- (' h /"%8# %1 -'"! $&'0 "- vh (0) = vh (1) = 0, ∀vh ∈ Uh . "&"1=#%'# # ,(& S T 1 S T ,'-%! 2! uh , Uh ("-%-"!79! -%-

B

0

1

k(x)uh vh

dx = 0

1

f vh dx

∀ vh (x) ∈ Uh .

S BT

6

5

ψh1

h

-

n

2h

6

5

ψhi

(i − 1)h ih (i + 1)h

-

n

? * ,' 2 (! "&"1'0 =#%"- > Uh 6 "%%-" "&" ,#"% %" " 1 ! S BT #" % 8'% -( 7 %#' $8&0 - @ % =%" #" " (% -'8- - Uh "1 8" 8, @# % {ψhi }, i = 1, . . . , N − 1 6 8, - Uh , 2 -( ,"8 "$" ψhi (x)

⎧ ⎨

=

− (i − 1), x ∈ [(i − 1)h, ih], − hx + i + 1, x ∈ [ih, (i + 1)h], ⎩ &. 0, x h

4 "% S BT "%"%" vh ("%%"&" "%1 8"-% -'" SN − 1T1"$" -! -( 0

1

k(x)uh (ψhi )

dx = 0

1

f ψhi dx

∀ i = 1, . . . , N − 1

S BT

O S BT -'"!%! %" S BT % #% #%" ? uh 8(# % - -( ," ! " 8,'# 2!# /% zh 6 -%" "=22%"- {zi } ," ! uh " 8, {ψhi } * zh ∈ RN −1 uh =

N −1

zi ψhi .

S BBT

i=1

/"(%-# ," S BBT - S BT <""%" S BT -"1 (%! %# '0 $8&0 - S< T $( ,-%'# !-!%! -%" "=22%"- zh ∈ RN −1 * Ah zh = fh . S BT X#%' #%' -! &% -'&!7%! " 2"## Ah = {aij }i,j=1,...,N −1 , fh = {fi }i=1,...,N −1 , 1 1 1 1 aij = k(x)(ψhi ) (ψhi ) dx, fi = f ψhi dx. h 0 h 0

S BT

/"!- #%8790 #" % h1 ( !'# ",-(!# "!# &% ", ?#"%# - RN −1 !" ",-( "# * y, z :=

N −1

h yi zi , y := y, y1/2

S BT

i=1

@"-" "%%-" 8(# "8",&% &, Rh " "%&%1 ! "% RN −1 "(# !"$" ",-(! ?-%-" S BBT ,(% ,"#"2,# Ph : Rh → Uh # ( "%%-"# 2 Uh Rh "%%-"# "=22%"- - ," 2 " 8, / =%"# "%"'# "%%# c C ! ( h -'"" cz ≤ Ph z0 ≤ Cz, S BCT $( · 0 6 L2 1"# /"!- #" %! h - "( 1 !"$" ",-(! S BT "80"(#" (! -'"! 1 -%- S BCT A "%%# =--%"% ,-!9# "% h 4 "89# & =%"% #" % ,#" "" % -'# "9( "%! ψhi mes(supp (ψhi )) 4 -"7 "&( "!-1 #" % h1 ( !'# ",-(!# - "(1 =#%"- #%' Ah -" &% fh "80"(#" (! %"-! -# ","$" ""%"! S BT , 1 ! 4 "89# & =%"% #" % 8(% - ""%-%1 %-" mes(supp (ψhi ))−1 <""%" , (79$" 1 ! "%"" #" % % %%-" S BT (! "1 (! #%' Ah 8(% %-" ","-%! - %" #"$"%"&'0 #%"("-

.%9 --- 8(% &%" (! ",-"'0 z, y ∈ R

h

uh = Ph z, vh = Ph y -'"!%! Ah z, y =

1

0

k(x)uh vh dx.

S BT

4%% -5 /% k(x) = 1 %"$( #% Ah , S BT #% -( S-' =#%' %"!% %0 ($"!0 ,#"% (n − 1) × (n − 1)T* ⎛

2 ⎜ −1 1 ⎜ ⎜ Ah = 2 ⎜ h ⎜ ⎝ ... 0

−1 2

0 −1

... 0

0 ...

−1 0

2 −1

⎞ 0 ... ⎟ ⎟ ⎟ ⎟. ⎟ −1 ⎠ 2

4%% -> ?#"%# 9 , - S T (! k(x) = 1 " ($# -'# "-!# u(0) = 0,

∂u (1) = a. ∂x

S 5T

<% #%"( "&'0 =#%"- #%" %" ,(1 & "%"% - ,( & -" 8" 2"#"- @ =%"% , - &%- ""-"$" "%%- -'8# -#%" ¯ 1 2 , H1 (0, 1) ("-%-"!790 H10 (0, 1) "%%-" H "-7 0* u(0) = 0 /" %$"-! " &1 %!# - ('(9# # "&# -%-"

1

k(x)u v dx = 0

1

f v dx + a v(1) 0

¯ 1. ∀ v(x) ∈ H

S T

R! u , H¯ 1 ("-%-"!79! -%- S T !-!%! 8'# # -! S T -'# "-!# S 5T "-! @# - %"& x = 1 ",'-7%! &%' - %$1 "# ""%" S T 4 &%- Uh #" " -,!% "%%-" "&"1'0 2 [0, 1] -'0 7 - %"& x = 0 @ -"# " 0 "- %-%! &%" ""%-%%-% ("8-7 9

"(" 8," 2 6 U"1#(V - %"& x = 1 %1 Ah , S BT #% -( S-' =#%' %"!% %0 ($"!0 ,#"% n × nT* ⎛

2 ⎜ −1 1 ⎜ ⎜ Ah = 2 ⎜ h ⎜ ⎝ ... 0

−1 2

0 −1

... 0

0 ...

−1 0

2 −1

⎞ 0 ... ⎟ ⎟ ⎟ ⎟. ⎟ −1 ⎠ 1

4%% -2 4 &%- 9 "("$" # #"%# (-1

#7 ,(& /"* % 27 u(x, y) ("-%-"!71 97 -7 −

∂2u ∂2u − 2 =f ∂x2 ∂y

- Ω = (0, 1) × (0, 1),

u|∂Ω = 0.

S T

#" ! S T ",-"7 27 v(x, y) , H10 (Ω) %$! " &%!# "&# 87 "%"- ,(&* 1 % u ∈ H10 (Ω) ("-%-"!797 -%- 0

1

0

1

∂u ∂v ∂u ∂v + dx dy = ∂x ∂x ∂y ∂y

1

1

f v dx dy, 0

0

∀ v ∈ H10 (Ω).

S T

"$&" ,(& (22, #%"( "&'0 =#%"- "1 %"% - ,# $8%"- "%%- H10 (Ω) - S T "&1 "#" "%%-" Uh ! "%"! Uh ,"8# "8% =#%' %$" =%" "," /% Uh 6 "%%-" -0 "&"1'0 S% '0 ("# %$"&T 2 '-'0 -" - "81 % Ω e" &%" 78! 2! , Uh ""%7 "(!%! -"# ,&!# - -0 %$""- ,8! N1 ## - % -% 9 -% "8% -' #18" "8,"#[ # =%" (" 4 1 &%- i1" 8," 2 -",## 27 , Uh #1 797 ,& 1 - i1" - 0 -" -0 "%'0 -0 C

⎧ ⎨ h

C

B

Uh Uh

⎩ h

? * ?,8 "8% #! - (! "&"1 '0 =#%"-

% Ah "(!#! ""%"# S BT ", ⎛

A1 1 ⎜ ⎜ D Ah = 2 ⎜ h ⎝ 0

D A2

0 D

D

⎞

⎟ ⎟ ⎟. ⎠

S BT

An

$( 8" A1 D !-!7%! #%# n × n #7% -(* ⎛

4 ⎜ −1 ⎜ ⎜ Ai = ⎜ ⎜ ⎝ 0 ...

⎞ ... 0 ⎟ ⎟ ⎟ ⎟, ⎟ −1 4 −1 ⎠ 0 −1 4 −1 0 4 −1

⎛

−1 ⎜ 0 ⎜ ⎜ D=⎜ ⎜ ⎝ ...

0 −1 0 ...

... 0

−1 0

⎞

... ⎟ ⎟ ⎟ ⎟. ⎟ 0 ⎠ −1

$" "%" -'8" Uh 6 ,8% "8% Ω =#%1 ' -(% #"%% - 2 1 ("# =#% ,8! '-' -" - "8% 78! %! 2! "- "(",&" "(!%! -"# ,&!1

# - -0 ,8! % Ah #% -( S BT 8"1 # ⎛

−1 8 ... −1 0

8 ⎜ −1 1⎜ Ai = ⎜ 3⎜ ⎝ 0 ...

⎛ ⎞ ... ⎜ 0 ⎟ ⎟ ⎜ ⎟, D = 1 ⎜ ⎟ 3⎜ ⎝ −1 ⎠ 8

0 −1 ... 8 −1

−1 −1 0 ...

−1 −1 ... −1 0

0 −1 ... −1 −1

⎞ ... 0 ⎟ ⎟ ⎟. ⎟ −1 ⎠ −1

<(79 -",-9! "("#" ,(& U2"-'-%V &%" ",&% "!% U"%%-" Uh

"%%-" H10 (0, 1)V .%9 --) /",% &%" (! ",-"" 2 v(x) ( 9 H2 (0, 1) ∩ H10 (0, 1) -(- "

1

inf

vh ∈Uh

0

(v −

vh )2

dx ≤ h

2

1

(v )2 dx.

0

S T

/"- S T #"

" "9%-% - % =%* ! ,(" v(x) -'8% v¯h %7 &%" v¯h (ih) = v(ih), i = 0, . . . , N /"&% "7 " % (! ",-""$" "%, [ih, (i + 1)h], i = 0, . . . , N − 1 ",%

(i+1)h

(v − v¯h )2 dx ≤ h2

ih

(i+1)h

(v )2 dx.

ih

! =%"$" #% %"# " (#* (%! %"& x0 ∈ (ih, (i + 1)h) %! &%" v¯h (x0 ) = v (x0 ) -"","-%! 2"#" @7%" 1 8 (! f (x) = v¯h (x)−v (x) [ih, (i+ x 1)h] "%"%" %"& x0 * f (x) = f (x0 ) + x f (x) dx 4'-% 1 1 0

0

(v − v¯h )2 dx ≤ h2

(v )2 dx,

0

##! "' " "$&" "-!%! &%" (! ",-"" v(x) ∈ H10 (0, 1) -'""

inf

vh ∈Uh

B5

0

1

2

(v − vh ) dx ≤ h

2

0

1

(v )2 dx.

S T

, (79$" ! ",'-% &%" 1 "&" #%"("# "&'0 =#%"- !-!%! ,#1 '# 8 # 7 (22" ,(& .%9 --+ /% u 6 S CT u(x) ∈ H2(0, 1) % uh 6 #%"("# "&'0 =#%"- /",% &%"

1

0

(u −

uh )2

2 2

dx ≤ K h

1

(u )2 dx.

0

",%%-" #"

S T

% 8'% "-(" - "" 1 $"- /"-% "%"$""% "8 ",-"" vh , Uh *

1

0

k(x)(u − uh )vh dx = 0 ∀ vh ∈ Uh .

",% -%-"

0

1

(u −

uh )2

dx ≤ inf

vh ∈Uh

0

1

S CT

k(x)(u − uh )(u − vh ) dx.

4"","-%! -%-"# " "" , ! 4 (79# ($%! "-% &%" (""%'0 ("" !0 k(x) -(- (""1 %! " 8,"% u uh - L2 1"# & &# "# · 1 X%" #"$" " ('(90 .%9 --3 /% k(x) ∈ C 1(0, 1) -'"' "-! ! ",% "

1

0

(u − uh )2 dx ≤ C h2

0

1

(u − uh )2 dx,

S T

$( C 6 "%% ,-!9! "% u uh h 4"","-%! -%-"# S "-% $"aT

0

1

(u−uh )g dx =

0

1

k(x)(u −uh )(vg −vh ) dx

∀ g ∈ L2 (0, 1), vh ∈ Uh ,

$( vg 6 S CT -" &%7 f = g -""1 ,"-%! (! g vg "" S B5T , ! 5 "" S T -%-"# a = supb=0 (a,b) b B

" # $

4 #%-#"# # #' 8(# ("$% &%" - ,(& S T 1 S T ! ! %#' S BT #% Ah %" - # #" " =22%-1 " #% !#' SU%"&'VT #%"(' % #%"( 31 (" (! 8" " '0 ,(& # (-#'0 %0#'0 % #%"(' "%#' %-%" $" "-% &%" -'&%! " "% #%"( 3 (! 1 ! %#' #% , # C - O(n ) 2#1 %&0 " (! '0 % (! 8'0 =1 #%"- $( n 6 &" ,-%'0 - %# S T %#! " "% - O(n) 4 %0#"# & , - % n # ( "%#" " "%7 " "%7 !#'0 #1 %"("- 9 8" / 8""# & ,-%'0 "80"(#' %"' #%"(' k(x) = 1

3 2

%&' () * !

?#"%# "( 8,"-' %"' #%"( , x S1 # T 6 #%"( e"8 ! ("8%- (""%" --(# - # "' #% w /% z0 1 ,(" (! i = 0, 1, 2, . . . #' -'&!# zi+1 = zi − wD−1 (Ah zi − fh ); S 5T @"## &%" D = diag(Ah ) 6 ($"! #% "%-1 ! , =#%"- $-" ($" Ah

%"( S 5T =--% (! ("$" # #%"( "1 %" % S- -' =#%' D "("-'T* zi+1 = zi −

wh2 (Ah zi − fh ); 2

S T

0"(#"% S T #" " (% ,! % Ah @"(1 %-" "-" 8 (#! &%" Ah ψ k = λk ψ k ,

(! B

ψ k = sin(πk jh), λk = 4h−2 sin2 (πk h2 ), k = 1, . . . , N − 1.

S T

N#%# &%" λmin ≤ π2 λmax = O(h−2 ) <("-%" ",1 % 0"(#"% S T "%#"# w S,& q0 - S 5TT #" " "% &, 1 − O(h2 ) "&%-" % "80"1 (#'0 (! ! %#' %"&"%7 ε O(h−2 ln ε−1 ). 4'-"(* (! #'0 h #%"( 0"(%! "& #("[ 6 "%#! " "% -'& SO(h−1 ) (! (" ,(&T ,-("#" ("%$%! > #"%# %"" U-'""&%"%'V "8%-' 2 % % ψk (! "%"'0 k ≥ N/2 = 2h1 N#%# &%" (! %0 k -(-" 2 = 4 sin2

4'8# I−

w = 1/2

wh2 2 Ah

##*

π ≤ h2 λk ≤ 4. 4

- S T >"$( (! #%' %

|λk (Sh )| = |1 −

h2 k 1 λ (Ah )| ≤ , 4 2

k ≥ N2

Sh =

S T

@"## &%" #% % "%-%%- , ,# "81 - #%"( S#S TT "=%"# " S T ",&% &%" -'1 "0 $#"0 #%"( e"8 #% ""% 0"(#"% 01 &# 5 $# "-# -%" "8 ," % 8, , "8%-'0 2 ψk %" "=22%' -'"1 0 Sk ≥ N2 T $#"0 "97%! (" % # &# - (- , <("-%" " "0 % -( -'"0 $#" - "8 eh = zih − zh 8(% ,&% > 8"0 k 2 ψk 8'%" ,#1 !7%! S"7%T %" $"-"!% &%" "8 U$ -%!V ># "8,"# (%-!%! ,#" (79! %%1 $! 0" (! ! S BT ,(" %"&"%7* G$ @"" U$ -790V % % e"8[ G$ @"" 01%" SfaT % (! "(-! ,1 "&%"%'0 "#"% - "8 % (! #! "1 =22%"- ψk k < 2N1 [ B

G$ O %8#! %"&"% ("%$% # "# -!, - 9 - %" % $ 4 &"# S$"#%&"#T #"$"%"&"# #%"( $ ,'-%! " $8" %

& !*

- "%'0 " - '0 ,#&!* \ O "8 eh = zih − zh 6 $(! 2! %" " #" % 8'% 0""" "#"- % ## "1 &%-"# ,"- S0"%! #' #" # ,% - !-"# -( eh # - # 2% 9%-"-! %" 2 1 "#79T[ ] 8 eh !-!%! # -! Ah eh = dh ,

$(

dh = Ah zih − fh .

S BT

O #' (# eh %" #' (# zh = zih − eh <1 %# S BT % %(" % 0"(7 S BT (" #! -" -# (" ,#& \ "'%#! 1 % e2h eh " S $"# 2hT ! ,(& S BT $8" %* S T

A2h e2h = d2h .

O p : R2h → Rh 1 "%" %" "-" 8 1 zi+1 7 zh 8(% h S T

zi+1 = zih − pe2h h

<""%" S T ,'-%! % "%'%'# -""'*

"

9

$ # e 8 ! ! ) & R ! ! % h h " !) & Uh # " ! # # !%

) ! ! " &" " ! !

: & # & " "& $ % " & 7 ;

! %" !% )# " $ ! <

BB

\ -'8% "%" "(" ! pf ] -'8% "%" ! r : Rh → R2h &%"8' "(% -!, $8" % d2h = r dh f A ,(% " A2h f 4 "-'0 "8",&!0 ""%" S T #" % 8'% 1 " i zi+1 = zih − pA−1 S T h 2h r(Ah zh − fh ). <(79 $2 "%-&% -""' \ 6 A

' & !*

"$( < "& - ,%% #! #%"( "&'0 =#%"- "%"' "(" ! "%"-! %%-1 " -'8% (79# "8,"#

! #%"( "&'0 =#%"- %&" !-!%! %! "$( U2h ⊂ Uh $( U2h 6 "&"1=#%" "%%-" "1 "%-%%-79 $8" %

4%$"9

> "%" "(" ! p¯ , U2h - Uh "(!%! %1 -"* p¯(u2h ) = u2h %" "(" - "%%-0 "=221 %"- p : R2h → Rh "(!%! "#"97 ,"#"2,#"S# x T Ph : Rh → Uh P2h : R2h → U2h * P

P −1

h 2h p : R2h → U2h ⊂ Uh → Rh .

># "8,"#

S CT

! "&"1'0 =#%"- (%- "%" "(" 1 ! p "," B $( ("# i1"# , % "1 "%-%%-% i1' "=22% -%" , Rh 4 "89# & p !-!%! S, "T #% "%"7 #" " -'&% ,! Ph P2h B p := Ph−1 P2h .

5

h p

6

2h

6

h 1 2

5

...

2h ] 1 2

6

2h

1 2

x ] 1 2

6

]

1 2

1 2

...

6

x

? B* %" "(" ! (! "&"1'0 X $1 8" % &%7

4%%& ?#"%# L2 1"7 "%%-0 "&'0 r¯ : Uh → U2h /" "(7 r¯ ,(%! -%-"# (¯ rvh , u2h )L2 = (vh , u2h )L2

∀ vh ∈ Uh , u2h ∈ U2h .

=#%"S T

X%" "" (% "& "%" r : Rh → R2h (79# "8,"# /# S T - "%%-0 "=22%"-* −1 rPh−1 vh , P2h u2h R2h = Ph−1 vh , Ph−1 u2h Rh

∀ vh ∈ Uh , u2h ∈ U2h .

" ""%" ,(% r∗ = Ph−1 P2h $( r∗ 1 "%" "!1 ' r % r·, ·R = ·, r∗ ·R . ># "8,"# &%'-! S CT "&# r = p∗ S 5T 2h

h

<""%" S 5T % S - %"T &%" 8%! U-"V (! "(! "%" ,("# "%" "(" ! 4 #%-#"# # - %#0 #% $" "-% &%" r = 12 pT , $( 2%" 12 "!-!%! , &% #" % h 2h - "( ·, ·R ·, ·R ""%-%%-1 " <0#%&" (%- "%" "," O1 - %"& x = 0 ,(' "-! 0 27 uh (x) %" "%" r - ,(%! B h

2h

h

5 3 4

r ? 2h

h

?

1 4

5

...

2h 1 4

1 2

^ ?

1 4

x 1 4

1 2

^ ?

1 4

...

2h

1 4

1 2

^ ? x

? * %" "%"-! (! "&"1'0 X &%" % $87

.%% ,%!8 A 2h

! -'8" "%" $8" % A2h 9%-% (- "(1 0"( 3 ,(& S T "(%-" "#%! $8" % % - "%%- U2h > (! #"%"$" # A2h 8(% ,(-%! ""%"#

A2h z, yR2h =

0

1

k(x)u2h v2h dx,

∀ z, y ∈ R2h , u2h = P2h z, v2h = P2h y. (!# A2h

/ 3

%""-

",-( %0 "1

A2h := r Ah p.

.%9 --1 /",% &%" (! #"- #! #1

%"( "&'0 =#%"- , x "&"$" -'8" r p % " 2"## S CT 1 S 5T !#" "(0"( "(0"( 31 (7% "("-' "%" A2h . >&'# %!# "$( =% (- "(0"( (7% ,&1 ' #%' $8" % !-!7%!* %$' 0"1 ( "=22%"- #%' - #%"( "&'0 =#%"-'&!7%! %"&" "#"97 -(% &%" 8'-% ,1 8 " - & - #'# "=22%# # # B[ - 2"#"- (%" ,(& ("8-!1 7%! &' ,-!9 "% ,8! "8% # # %"$" B

#%"( - x B [ (! "# (22" ,(& ",%! #%"( "&'0 =#%"- #%"( "1 &'0 ,"% "&'0 "8Z#"- +

*

> "$( - "80"(#' "#"%' "(' "#1 # &%" (-0%"&' #%"( #" " #"%% %1 "' 6 " ,("# 8 7 zkh (79 8 zk+1 -'&!%! - (- =% 4"1-'0 -'"!%! "" h S"89# &"# ν T $ - # " #%"( e"8 4 S BT "-" "# %"&" 8 ˜zk ("-%-"!% -%- ˜zk = S ν zkh + g ν , S T $( Sh 1 #% $ -79 % > (! #%"( e"8 - # # Sh = I − wh2 Ah gν 6 -%" ,-!9 %"1 0"(%! " "% -" &% %#' fh 4"1-%"'0 zk+1 h ,%% " ˜zk $8" % &%" - S T #" % 8'% ," 2

zk+1 = ˜zk − pA−1 zk − fh ). h 2h r(Ah ˜

<",! S T (%

¯ h zk + M ¯ h g ν − pA−1 r fh = M ¯ h zk + g¯ν , zk+1 =M h h h 2h

$( M¯ h 6 #% % (-0%"&"$" #%"(*

S T 4$" ( %# % " $8" % (" ν $ -790 % +, "89 %" %"'0 #%"("- S# x T (% &%" (! 0"(#"% (-0%"&"$" #%"( "80"(#" -'"1 ! -%- |λ(M¯ h )| < 1 "%%"&'# "-# !-!%! " M¯ h < 1 - "%"" "%"" "# .%9 --5 /" % &%" (!−1-'8" S 5T "%" " $8" % Kh = I − pA2h rAh !-!%! "%"1 "# % Kh = Kh2 <("-%" #% %&"$" #'1 -'"!% (- 8" "("-%'0 $"- " $8" % ¯ h = (I − pA−1 rAh ) S ν . M h 2h

BC

,

% - -* #

! #%-#"$" #"("$" # ,('0 r p S# B T 0"(#"% (-0%"&"$" #%"( #" " "-% !#'# -'&!# "#"97 , R @ #' "-# &%" "8%-' -%"' ψk ψN −k , S T "8,1 7% -%" "%%-" "%"%" #" ! M¯ h <("-%" M¯ h -"(%! - 8, , "8%-'0 -%"" 8"&"1($""# -( $( 8" #7% ,#"% 2×2 8",&# =% 8" &, M¯ h(k) , k = 1, . . . , N2 − 1 "##* N 6 &%" #' (# &%" 8" #7% -(* ¯ (k) = M h

s2k s2k

c2k c2k

c2ν k 0

0 s2ν k

,

N

¯ ( 2 ) = 2−ν , M h

S T

$( s2k = sin2 πk h2 , c2k = cos2 πk h2 4! $ -790 %1 " $8" % ,"1 -'""&%"%1 ' &% % 0""" -(' , %"! M¯ h(k) > -' 8" ," ! S T ""%-%%-% " $8" % 4 =%"# 8" #' "=22% s2k #" 7%! U$(V "8%-' -%" ψk % % (! "%"'0 k ≤ N2 @"8""% "81 %"% (" -'# 8""# ," ! S T ""%-%%-% $ -79# %!# @ #' "=22% s2ν k % #" 7%! U8'%"1"79V "8%-' -%" % % (! "%"'0 k ≥ N2

" " "&% %"&' " @"(%-' -'&1 ! (7%* (k)

2 2ν ¯ ) ∈ {0, s2k c2ν λ(M k + ck sk }, k = 1, . . . , h

N − 1, 2

(N )

¯ 2 ) = 2−ν . λ(M h

N#%# &%" s2k ∈ [0, 0.5] (! k = 1, . . . , N2 − 1 /"=%"# ¯ h )| |λ(M

≤ ≤ =

max (x(1 − x)ν + (1 − x)xν )

x∈[0,0.5]

max (x(1 − x)ν ) + max ((1 − x)xν )

x∈[0,0.5]

1 ν+1

1−

1 ν+1

ν

x∈[0,0.5]

+

ν+1 1 . 2

B

+ #' "&# |λ(M¯h )| ≤

1 2

|λ(M¯h )| ∼

1 1 e ν

ν → ∞.

! " %" "#' #%' % ("8" -"1 ","-%! -%-"# 3y( S T* 2 2ν 2 2ν 2 2ν M¯h 2 ≤ M¯h ∞ M¯h 1 ≤ max (s2k c2ν k + ck sk )·2 max{sk ck ; ck sk } . k

"$&" " "8%-'0 ,& "&# M¯h ≤

1 2

√ M¯h ∼

2 1 e ν

ν → ∞.

># "8,"# (! ("$" #"("$" # "# #%1 ' % (-0%"&"$" #%"( - "%& "% "#' #%' 8,"-'0 % "-%! ,-#" "% ha N-#"% "% & $ - -( O(ν −1 ) 0% (! #"$"%"&'0 #%"("-

4%&% :->+= > ψk ψN −k 6 "8%-' -%" Ah , S T #% $ -79 % Sh - I − h4 Ah %" 2

h k S ν ψ k = (1 − sin2 (π k ))ν ψ k = c2ν k ψ , 2 h N −k . S ν ψ N −k = (1 − sin2 (π (N − k) ))ν ψ N −k = (1 − c2k )ν ψ N −k = s2ν k ψ 2

># "8,"# #7 S ν {ψk , ψN −k } ""%-%%-% -' 8" ," ! S T ?#"%# % (%- #%' " $8" %1 "8%-' -%" ψk (! ",-""$" k = 1, . . . , N − 1 5

+",! %$""#%& ""%"! "-!#* Ah ψ k k rψ m

4 2 k s ψ , h2 k

=

(! m = 1, . . . , N2

= c2k sin(πk m 2h),

A−1 2h sin(πk m 2h)

=

[p sin(πk m 2h)]j

=

− 1;

h2 −2 −2 c s sin(πk m 2h); 4 k k (! j = 2n ψjk . (c2k − s2k )ψjk (! j = 2n + 1

! "%" " $8" % "&#* ! 0 (! j = 2n −1 k (I − pA2h rAh )ψ j = ; j = 1, . . . , N −1. (2s2k )ψjk (! j = 2n + 1 N#%# (79 ""%"!* ψjk =

−ψjN −k ψjN −k

(! (!

j = 2n j = 2n + 1

;

j = 1, . . . , N − 1.

<("-%" #" # ,%* k 2 k 2 N −k (I − pA−1 . 2h rAh )ψ = sk ψ + sk ψ

% s2k = c2N −k %" N −k (I − pA−1 = c2k ψ k + c2k ψ N −k . 2h rAh )ψ

+, "(0 (-0 ""%" #' -(# &%" -%" ψk ψ N −k (%-%" "8,7% -%" "("%%-" (! ¯ "%" (I − pA−1 2h rAh ) ("-%" (! Mh 4 8, =%"$" "("%%- , "8%-'0 -%""- {ψk , ψN −k } #' #" # ,% (I −

pA−1 2h rAh )

"

k

ψ ,ψ

# N −k T

=

s2k s2k

c2k c2k

ψk ψ N −k

.

' "& -' #" % S8"T - ," S T ! &! k = N/2 ,#%# &%" c2k = s2k = 12 /"=%"# M¯ h( ) = 2−ν N 2

% $

@ 7 "1 # !-!%! %#' '0 $8&0 - Ah zh = fh , S T "&" - ,%% (%, (22" ,1 (& #%"("# "&'0 =#%"- X%" #" % 8'% ,(& (21 2, - /" , ('(9 $-' $ #' -%%!%! - !0 ", 4 ('(9 $- 8' #"% (-0%"&' #%"( % "%""$" "%"! - %"# &%" " "0 8,"-'0 % (! %#' S T %" ,(& (! ,-%" "8 $8" % SA2h e2h = d2h T X% ,(& %"&" /"1 =%"$" -'&!" 8 ˜eh = p e2h "8 eh &%" % "0"( "! 8 ! 7 &%" % @8" %("# - -'&%"# =% (-0%"&"$" #%"( 6 =%" ,(& $8" %1 N#%# &%" "&#" ˜eh 6 %"" 8 eh ("-%" % %'0 & 0"(% e2h %"&" ' #" # "$&%! 0" (# 8 ! ˜e2h e2h

! 8 "$" ! %#' $8" % ## - -"7 "&( "" S"89# &"# γ T % (-0%"&1 "$" #%"( 0"(! - # % 9 8" $87 *" "%"% - "-%" (" "(' 1 -" " 8(% ("%$% % #'# "&%-"# ,"SU#! $8!VT "%"" %# - #" % 8'% 1 %"&" # "#"97 #%"( 3 R"#,# (7 #"$"%"&"$" #%"( /% ,( 1 0! "&"1=#%'0 "%%-* U0 ⊂ U1 ⊂ · · · ⊂ Uk ⊂ · · · ⊂ Ul = Uh .

ST

/"%%-" U0 ""%-%%-% #" $8" % <""%-%%-1 79 "%%- "=22%"- -!,' "%"# "1 %"-! "(" !* p

p

p

p

p

→ → → → → R0 ← R1 ← . . . ← Rk ← . . . ← Rl = Rh . r r r r r

/("" # &%" "%"' Al = Ah Ak , k = 0, . . . , l − 1 ,(1 ' S&%' " -# , x T / ('0 %'0 ν &" $ -790 % ("# "- γ &" -'0 -',"-"- #%"( ("# "- - 0 ST "( %! #"$"%"&"1 $" #%"( ! %#' Ah zh = fh "(!%! -1 " "(" znew = M GM (l, zold , fh ) N( zold 6 &" 8 zh znew 6 "&" "-" 8 <# "( " ¯z = M GM (k, z, b){ 4 k = 0 %" 5 ¯z = A−1 0 b S%"&"

#" $8" %T ( -'1 0"( , "('[ 5 Sk > 0T z0 = z zi+1 = zi − Wk−1 (Ak zi − b) i = 0, . . . , ν − 1 Sν 8,"-'0 % 6 $ -T[ d = r(Ak zν − b) S"%"- -!, $87 %T[ y0 = 0 S&" 8 "8T[ B

B 4'"!# yi+1 = M GM (k − 1, yi , d) (! i = 0, . . . , γ − 1 Sγ % #"$"%"&"$" #%"( $8" %T ¯z = zν − p yγ S"! 8 ! zν $8" %T }

(" -'" "(' znew = M GM (l, zold , fh ) !-!%! "(" % #"$"%"&"$" #%"( 4$" -'"!7% ,(1 " "&%-" % " "# -!, Ah znew − fh %% # ,("$" ε /# %* zold = 0[ /" Ah zold − fh ≥ ε -'"!%* znew = M GM (l, zold , fh ) zold = znew

. / 0

/% ", &%" ,#" -'8% γ = 1 γ = 2 =%"# % %"%& ("' S( - !0T / γ = 1 "( %! #%"( "& ,- :1 γ = 2 ;1 ? 7%7% =% ,-1 ! <%"& ",'-7% (& ('0 "("$" %"&"$" "-! ($" 3","%! %"& ",&% &%" -'1 &! - ("# & 6 $ -! "0"(!% "("# "- 4 '0 " !0 "& "% %1 D1 "%"' "#"# # ( :1"# ;1 "# 4 # (! ! ,(& $8" % "( -','-% "( , 8! "%"# "( , :1 S" T* ¯z = M GM F (k, z, b){ 4 k = 0 %" 5 ¯z = A−1 0 b ( -'0"( 5 Sk > 0T

, "('[

z0 = z zi+1 = zi − Wk−1 (Ak zi − b) i = 0, . . . , ν − 1 Sν

d = r(Ak zν − b)

-T[

$ 1

S"%"- -!, $87 %T[ y0 = 0 S&" 8 "8T[

7 %"&' "-

*

R

zν z old -

K l

U

z new `

-

K l

U

^

`

-

K l

5 v l 1 "! ` 1 "(" K 1 $ -! v 1 %"&" U

-

`

? * :1

5

zold-zν

K l

W -

K l

W K-

l W v- `

KW -

-

? * ;1

z new ` W W -

W -

znew `

zold -

K l

W

-

W

5

W

-

W

-

W

W

-

? * D1 B

y1 = M GM F (k − 1, y0 , d) SD1 $8" %T y2 = M GM (k − 1, y1 , d) γ = 1 S:1 $8" %T[

¯z = zν − p y2

S"! $8" %T

}

>("#"% $"%# 8 "# ,(& !-!%! "%#" &" 2#%&0 " (! ("% ! ,(" %"&"% ,-% " "% & ,1 -%'0 ?#"%# -"" %("#"% "("$" #"$"1 %"&"$" #%"( /% nk 1 &" ,-%'0 k1"# %"&"# "- 8"1 ,&# Cn := inf nk /nk−1 . ST k=1,...,l /("" # &%" "( %! $ -! % -'"1 "%"-! "(" ! k1"# "- %87% O(nk ) " " "- -'"!%! (! %# -",1 790 - #%"( "&'0 =#%"- (! - - &%'0 ",-"('0 (! U,#'0V $ -790 % %0 #%"( e"8 N(! 8",&# &, Nmg %("#1 "% "("$" #"$"%"&"$" #%"( $" "-% 1 (79 %- ( S (! 4T .%9 )- /% κ = γCn−1 %"$( \ κ < 1 %" Nmg = O(nl ), ] κ = 1 %" Nmg = O(nl ln nl ),

κ > 1 %" Nmg = O(κl nl ). /"-% &%" Nmg ≈ c nl (1 + κ(1 + κ(1 + κ(. . . )))) -$" l -"

! # -! (22, , ('(9 $-' Cn = 2 :1 "(% - \ \ ;1 - ] .%9 )) % " "% D1 (! # "(1 "#"$" -! (22,

&' &'

*

?#"%7 - ('(90 %0 "7 $ -! &%" ,'-7% " ! S`lj_sppozcqdT % " -'"!%! ( " $8" % <$ -79 % #" " -'"!% " " $8" % %"$( " ,'-7%! " ! S`p_o_sppozcqdT F%" - #"$"%"&"# #%"( -'"!%! ($ - % "%$ -

4%% )-

znew = M GM V (l, zold , fh ) ¯z = M GM V (k, z, b){ k = 0! "# ¯z = A−1 0 b! $ k > 0

%# z0 = z! z1 = z0 − Wk−1 (Ak z0 − b) & '$ (# d = r(Ak z1 − b) & ) '$ *# y0 = 0 &+ ) ) '$ ,# y1 = M GM V (k − 1, y0 , d) & ) '$ -# z2 = z1 − p y1 & ) '$ .# ¯z = z2 − Wk−1 (Ak z2 − b) & '$ }

C

/% ("# %"&"# "- -'"!%! ν1 ($1 - ν2 "%$ - % % #"$"%"&1 "$" #%"( Ml (ν1 , ν2 ) #" " "(% -"* Mk (ν1 , ν2 ) =

Skν2 (I

M0 (ν1 , ν2 ) = 0, SBT γ ν1 −1 − p(I − Mk−1 )Ak−1 rAk )Sk , k = 1, . . . , l. ST

/"-# ""%"! SBT 1 ST "#"97 ( " k @"## &%" #% % 6 =%" #% "%-%%-! , ,# -%" "8 (" % ! k = 0 SBT -(-" % #" $8" % %# 1 %! %"&" % "8 , "( U%7V "89%! - " /"-# ST (! ",-""$" k > 0 Sk 1 #% $ -1 790 % #" % Skν Skν ""%-%%-7% -'"1 7 ν1 $ -790 % (" " $8" % ν2 $ -790 % " @(# #% % (! " $8" % /% dk−1 = r(Ak zik − bk ) "! -!, k 1"$" Sk − 1T1' "1 - $( zik 1 8 7 %#' Ak zk = bk " ν1 ($ - @ Sk − 1T1"# "- # 8 " %# Ak−1 zk−1 = dk−1 . ST /% zk−1 = A−1 k−1 dk−1 %"&" ST #' (# γ %1 S"-T #"$"%"&"$" #%"( Sk − 1T1"# "- (! ! ST &# Mk−1 6 #% =%0 % @(1 " %# "8,"# 8 "8",&# &, zγk−1 >"$( (! "8 -(-" 1

2

γ zk−1 − zγk−1 = Mk−1 (zk−1 − z0k−1 ),

% z0k−1 = 0 S# $ - "( "(' M GM (k, z, f )T %" γ zk−1 − zγk−1 = Mk−1 zk−1 ,

$(

zk−1 = A−1 k−1 dk−1 .

<("-%" γ γ −1 −1 zγk−1 = A−1 k−1 dk−1 − Mk−1 Ak−1 dk−1 = (I − Mk−1 )Ak−1 dk−1 .

@(" 8 zγk−1 "(" %! k1' "- -'&!%! "-" 8 zi+1 k * zi+1 = zik − p zγk−1 k

/"(%-!! "&" -' (! zγk−1 ## γ i zi+1 = zik − p(I − Mk−1 )A−1 k k−1 r(Ak zk − bk ),

&%" #" " % γ i zi+1 = (I − p (I − Mk−1 )A−1 k k−1 r Ak )zk + gk ,

$( -%" gk "% zi ,-% ># "8,"# #% % " $8" % #% -( γ I − p (I − Mk−1 )A−1 k−1 r Ak .

<",! #%# (1 "%$ - (% ST /"& "" (! "#' %"$" ( #%1 ' % 6 ""-" ""8 (",%%- 0"(#"% %1 ""$" #%"( ! #"$"%"&"$" #%"( =% ,(& 1 " "%% -!, # ( $" #% % #% %1 (-0%"&"$" #%"( ¯ k (ν1 , ν2 ) 1 #% % (-0%"&"$" 6 ! /% M #%"( %"$( ¯ 1 (ν1 , ν2 ), M1 (ν1 , ν2 ) = M ST ¯ k (ν1 , ν2 ) + S ν p M γ (ν1 , ν2 )A−1 r Ak S ν SCT Mk (ν1 , ν2 ) = M k−1 k k k−1 (! k > 1

",%%-" "(%-" (% , ST (%-1 ! S T (! #%' (-0%"&"$" #%"(* 2

1

¯ k (ν1 , ν2 ) = S ν2 (I − pA−1 r Ak )S ν1 M k k k−1

& '

4 =%"# ,( 8(% (",' " ",% 0"(#"1 % #"$"%"&"$" #%"( %#%# &%" & :1 ;1"%87% "%("$" #"%! 5

1 - /

/("" # (79 "-! "%" $ -!* (! -0 k = 1, . . . , l -0 ν 8" "%""$" ν¯ ST cxk−1 ≤ p xk ≤ Cxk−1 ∀ x ∈ Rk−1 ν Sk ≤ Cs . S 5T "%"'# "%%# c, C Cs ,-!9# "% k 4 (1 "" &%" Pk : Rk → Uk 6 ,"#"2,# "%%# =-1 -%"% ,-!9# "% k S# S BCT - # , x T -%- ST -'"!7%! -%"#%& "- S 5T -'1 "!7%! # (! -0 0"(!90! $ -790 %1 >"# %- (% &%" $"-"! "9"

! " 7 ?#"%# & "%%%-! "%$ -* ν2 = 0

% )- ! ! #%8&

#%$9&

! " !! ¯ k (ν, 0) ≤ η(ν) M

- η(ν) ! ( k η(ν) → 0 ν → ∞ " " ν¯ > 0 ! ν ≥ ν¯

" γ ≥ 2 Mk (ν, 0) ≤

γ η(ν). γ−1

S T

" # %"# (! &! γ = 2 S;1T

",%%-" 8(% 8,"-%! -"# ""%" SCT <& ,#%# -(-"% ""%"* Skν = I, % ν2 = 0, S T pR →R ≤ Cp - ST, S T −1 ν ν ν ν ¯ pAk−1 r Ak Sk = Sk − (I − pA−1 k−1 r Ak )Sk = Sk − Mk (ν, 0). S BT +, ST S 5T S BT (% ) !

2

k−1

k

−1 ν ν A−1 k−1 r Ak Sk ≤ CpAk−1 r Ak Sk ¯ k ≤ C(Skν + M ¯ k ) ≤ C(Cs + 1). = CSkν − M

S T

4 "(# -%- #' ","- &%" M¯ k ≤ 1 (! ("%%"&" 8"0 ν 8",&# ξk := Mk (ν, 0) . (",%%- 6 "&% " -0 S T* ξk < 2η(ν) (! -0 k /"-# "#"1 97 ( " k 4 ""- ( 8(% & k = 1

%-%" -%-" ST ("" %"#' (! #% M¯ k (7% ξ1 ≤ η(ν).

> , SCT ,7&# (! k > 1 ξk

≤

2 ν ¯ k (ν, 0) + p Mk−1 M (ν, 0)A−1 k−1 r Ak Sk

2 ν ≤ η(ν) + p Mk−1 (ν, 0) A−1 k−1 r Ak Sk

[S 5T

2 S T S T] ≤ η(ν) + Cp ξk−1 C(Cs + 1)Cs . +## -" ""%"

S T /("" # &%" " ξk−1 ≤ 2η(ν) (", 4",## ν¯ %" &%" η(ν) ≤ 4C1 (! ν ≥ ν¯ %"$( "&# #" ξk ≤ 2η(ν) /"( (% , S T -%- η(ν)+C∗ 4η 2 (ν) ≤ 2η(ν) -(-"$" (! η(ν) ≤ 4C1 "$&" #" " (",% 0"(#"% ν2 = 0 .%9 )+ ",% %- ( %"#' (! γ > 2 2 ξk ≤ η(ν) + C ∗ ξk−1 , k = 2, 3, . . . , l.

∗

∗

%* &'

6! ! " ! 3 9%-% 2! η(ν) : %! &%" η(ν) → 0 ν → ∞ Ak Skν ≤ η(ν)Ak

∀ k ≥ 0.

R+ → R+

S T

R! η(ν) ,-% "% k 6! !

* (! "%"" "%%' CA > 0 −1 −1 A−1 ∀ k ≥ 1. S CT k − pAk−1 r ≤ CA Ak CA ,-% "% k +#% #%" %"# " 0"(#"% (-0%"&"$" #%"(

% )) ! ! " !

! " ( ! ν¯ > 0 ! ν ≥ ν¯ ! ¯ k (ν, 0) ≤ CA η(ν). M

) !

",%%-" (% , -%-

S T

¯ k (ν, 0) = (I − p A−1 r Ak )Skν = A−1 − p A−1 r (Ak Skν ) M k−1 k k−1

@"# -" &% "-# &, ",-( "# -'1 "( "8# #!# S T S CT "&# S T $"(! -"%- $ -! -! &% S T #"1 % 8'% ( $"(" # - &%"% # , &% -'8" ("%%"&" 8""$" ν X% " !-!%! ("%%"&'# "-# 0"(#"% #%"( 6 ! <-"%- $ -! "# "-! ST S 5T $%7% 0"(#"% ;1 ,-!97 "% & %"&'0 "- X%" ",&% &%" " "% "(" %1 S"("$" T #%"( "%# % """ "&%- ,-%'0 #" #" % κ < 1 $( κ "(" - %" "89! " "% #%"( "81 0"(#! (! ("% ! 78" (! ! %#' % "%# 0 /%%- #" % Ak Ak −1 - -'0 &%!0 S T S CT "8!,%" " !-!%! U%(V %" #"$"%"&'0 #%"("- % #%"(""$& #' @2"#" "8Z! ,- -"%- #" % 8'% 1 (79#* Ak -" !-!%! #% (%, (21 2"$" "%" 1"$" "!( /"=%"# # Ak -%" x (" " U-'(!%V - x -'" $ $#" 6 % -,!% -%"" ",-"(" "% f (x) = ∞ m=0 {am sin(mx) + bm cos(mx)} -&% 8"7%" ,& "=22%"- m1'0 $#"0 - m2 , Ak Skν x ",&% &%" " #! ν $ -790 % -( -'"0 $#" - Skν x # %"%" 2 η(ν) - $ -79# -"%- S CT "%1 #%# &%" %&'# -("# η(ν) !-!%! η(ν) = O

1 ν

η(ν) = O

1 √ ν

.

S5T

−1 (! "#' ,"% A−1 k − pAk−1 r ",&% &%" "1 %"' (-0 "(0 %"&'0 "-!0 k k − 1 8, % ,(& $8" % {t1 {0""" "#% ,(& 8" &%" % {t{

% - . ' * #

/("" # &%" #% Ah , S T !-!%! ##%&" "" %" "(" 6 %&! %! #1 #%"( "&'0 =#%"- (! (%, #""!1 '0 (22'0 ,(& ># ,(&# !-!7%! # -%&-! -! (22, ,(& /" , (79$" $2 +% (! #%' #" &%" % Al := Ah #' (1 "$# Al = ATl > 0 @"## &%" "( -%-" " "(7 ",&% (Ax, x) > 0 ∀ x = 0 , 0"(#"% :1 ("8" "-% - U=$%&"V "# #" "# ,(-#" "%""# Al 1

· Al := Al2 · .

N((# "%"' "(" ! "%"-! "%" $8" % "&# "8,"#* r = p∗ , Ak−1 = r Ak p. S T <-"%- $ -! "# 8(% ("8" "%8"1 -% - #"$" ,#"# -( <-"%-" $ -!* % #% 8,"-'0 % t1"# %"&"# "- #% -( Sk = I − Wk−1 Ak ("" # &%" Wk = WkT ≥ Ak . ST = " # "

)" &% $ & # " > ? & & &! ) # ' ! & ' & '# 2 6 % ) Al &! &" ! ' " " " !'

B

<-"%-" "# "%8# - -( S-% S CTT −1 −1 A−1 . ST k − pAk−1 r ≤ CA Wk >" &%" 8,"-' % "- ST "8(7% -"%-"# $ -! (""!79# -"%-" ST %"-%! !" "1 -'"! (790 .%9 )3 /",% &%" , ST ST (% %-1 ( %"#' " 0"(#"% (-0%"&"$" #%"( /",% &%" , ST (% Ak Skν ≤

c Wk . ν+1

4 &%- "(, # ## SBT 4 #%"( e"8 S #%"#T Wk = w1 bc\d (Ak ) "1 -7 ST #" " -$( ("-%-"% -'8! #% w ("1 %%"&" #'# "%% CA , ST 8(% -""89 $"-"! ,-% "% w .%9 )1 /" % &%" (! # -! (21 2, k(x) = 1 '0 "&'0 =#%"- , $-' #%"( e"8 ("-%-"!% ST (! 78"$" w ∈ (0, 12 ] /(# ""-" %"# " 0"(#"% :1

% )+

! ! #%%$& , %%:& ; ν1 = ν2 = ν/2, ν = 2, 4, . . . ν ν CA , Ml ( , )Al ≤ 2 2 CA + ν

SBT

" CA #%%:& ) ! / (",%%- %"#' ( - !0

# "("8!%! "" "%'0 %- ( "%"' #' "8 - (797 ## "%"'# 8(# ","-%! -" 8, '0 "$"-""

)- < A A = 1 7 , ! n × n 1

2

A = AT > 0 · , "

1

1

> M A = A 2 M A− 2 ? @ A ≤ c ! A ≤ c I ? A A1 ≤ A2 ! B A1 B ∗ ≤ B A2 B ∗ ? B CD (AB) = CD (BA)? E W = W T > 0 " A ≤ W −1 A ≤ W −1

/"-# %- ( j > "8 #%1 ' A W ("-%" W −1 ##%&' %" -%-" A ≤ W −1 (! %" "#' " "(7 =-1 -%" -%- ##' "8%-' ,&! S0 ("" !0 - "8%-' ,&! "" %1 'T ) !

λmax (A) ≤ (λmax (W ))

−1

.

##%&'0 #% - "8%-' ,&! -9%-1 ' (! 78"$" λ(W ) ∈ _` (W ) "8%! -& λ(W )−1 !-!%! "8%-'# ,& W −1 "8""% S"-%aT /"=%"# λmax (W )−1 = λmin (W −1 ) /"&# λmax (A) ≤ λmin (W −1 ). ST

! "- %- (! j # "80"(#" (! ",-"1 "$" -%" x ∈ Rn (",% (Ax, x) ≤ (W −1 x, x). ST @-%-" ST (% , ST "" (Ax, x) ≤ λmax (A) x2 ,

λmin (W −1 ) x2 ≤ (W −1 x, x).

X% " - -"7 "&( $" "-% ",! ,"1 −1 # S T (! ##%&'0 #% A = Qa Da Q−1 = a W −1 Qw Dw Qw $( Qa Qw 6 "%"$"' #%' Da Dw 6 ($"' #%' "%"!9 "8%-'0 ,& A W −1 /"( -%-" -(-" - ST .%9 )5 /"-% %- (! ##' \ 6 b

/"(" # (",%%-" %"#' 4 S T -%-" ! =%" ( "1 /" %- (7 \ ##' -(-"

Al = ATl -&% Al−1 = ATl−1 /"(" &# Ak = ATk (! -0 k = 0, . . . , l 1

−1

Mk Ak = Ak2 Mk Ak 2 ,

$( · 1 %! "# %"' 1

− 12

Mk := Ak2 Mk Ak

S 8(% "-" T #""! ' @# %8%! ("1 ,% " Ml ≤ CC+ν "%"! - %- (! ] ##' (% , -%A

A

1

− 12

0 ≤ Ml = Al2 Ml Al

≤

CA I. CA + ν

ST

" # ST "-(! (7 " k ! =%"$" "-# 1 (79 %- (* % 0 ≤ Mk−1 ≤ ξk−1 I < I, SCT %"$( 0 ≤ Mk ≤ max {(1 − ξ)ν (ξk−1 + (1 − ξk−1 )CA ξ)} I. ST ξ∈[0,1] /("" # &%" (" %- ( ("," %"$( "1 " % ξk−1 = CC+ν %" ### - -" &% ST ("%$1 %! ξ = 0 - CC+ν &%" (",'-% ST " ( - ""- "%"" k = 0 S"## &%" M0 = 0 "=%"# SCT ,-("#" -'""T %%! "-% &%" , SCT (% ST 4'&!# " "(7 ST A

A

A

A

Mk

1

ν

ν

− 12

2 = Ak2 Sk2 (I − p (I − Mk−1 )A−1 k−1 rAk )Sk Ak

=

1

ν

−1

1

1

1

ν

−1

2 2 2 2 (Ak2 Sk2 Ak 2 ) (I − Ak2 p (I − Mk−1 )A−1 k−1 rAk ) (Ak Sk Ak ).

%' -% "8" "8,#* 1

ν

− 12

Ak2 Sk2 Ak

1

1

ν

= (I − Ak2 Wk−1 Ak2 ) 2 ,

1

1

2 Ak2 p (I − Mk−1 )A−1 k−1 rAk = 1

−1

1

−1

−1

1

2 2 2 2 (Ak2 pAk−1 ) (I − Ak−1 Mk−1 Ak−1 ) (Ak−1 rAk2 ).

/"&# ν

S5T

ν

Mk = (I − Xk ) 2 (I − p (I − Mk−1 )r )(I − Xk ) 2 ,

$( 1

1

Xk = Ak2 Wk−1 Ak2 ,

1

−1

2 p = Ak2 p Ak−1 ,

−1

1

2 r = Ak−1 r Ak2 .

> p = (r )∗ , %" , S5T -(" &%" Mk−1 = (Mk−1 )T -1 &% Mk = (Mk )T ># "8,"# "-!%! ##%&"% Mk ( ν2 , ν2 ) (! -0 k +, S5T SCT "&# S-%-" (% , %- (1 ! A ##' #% B = (I − Xk ) p T ν 2

Mk

ν

ν

≤ (I − Xk ) 2 (I − (1 − ξk−1 )p r )(I − Xk ) 2 = =

ν

ν

(I − Xk ) 2 ((1 − ξk−1 )(I − p r ) + ξk−1 I)(I − Xk ) 2 ν ν (I − Xk ) 2 ((1 − ξk−1 )Qk + ξk−1 I)(I − Xk ) 2 ,

$( 1

1

1

1

−1 −1 2 2 2 Qk = Ak2 (A−1 k − p Ak−1 r)Ak ≤ CA Ak Wk Ak = CA Xk .

/"( -%-" (% , -"%- "# ST % j ##' /"=%"# Mk ≤ (I − Xk ) ((1 − ξk−1 )CA Xk + ξk−1 I)(I − Xk ) . S T −1 +, ST -'%% _` (Wk Ak ) ∈ [0, 1] " %- (7 b #1 #' _` (Wk−1 Ak ) = _` (Ak Wk−1 Ak ) = _` (Xk ) ∈ [0, 1], &%" -#% S T (",'-% ST ν 2

ν 2

1 2

1 2

" & ' !

/## "&' ,%%' (! (",%%- 0"(#"1 % :1 ;1"- #"$"%"&"$" #%"( (! 0" (! 1 8 "$" "&"1=#%"$" ! ,(& /"* −Δ u = f - Ω, u|∂Ω = 0, Ω ∈ R2 , ST C

$

$( Δ := 2i=1 ∂x∂ Ω 6 "$&! "8% " %"$" &%"8' ,8 % $"#",(0 %0&0 (% 8(# &%% &%" Ω 6 #"$"$" <8! "%"- ,(& "%"% - 0" ( u ∈ H10 (Ω) ("-%-"!79 -%- a(u, v) = (f, v) ∀ v ∈ H10 (Ω), ST $( 2

2 i

a(u, v) :=

2 ∂u ∂v d x, Ω i=1 ∂xi ∂xi

(f, v) := Ω

f vdx

∀ u, v ∈ H10 (Ω).

/("" # &%" ,( %# -" '0 "&"1=#%1 '0 "%%U0 ⊂ U1 ⊂ · · · ⊂ Uk ⊂ · · · ⊂ Ul ⊂ H10 (Ω)

$!'# ,#&# % "&"1""#'# % -' d 2!# vk ∈ Uk /("" # &%" ""%1 -%%-79 #% (%, ("-%-"!% "-7* c0 2−k ≤

hk ≤ c1 2−k , h0

c0 , c1 ,-!% "% k "8% Ω %! &%" (! f ∈ L2 (Ω) 1 u ( % H2 (Ω) uH ≤ c(Ω)f 0 . SBT @"## &%" · 0 "8",&% L2 "# /"( "- "1 % ,- H2 1$!"% S(! "("#" ,(& "$&1 ! " 8' - 5 - $- T O "8% Ω !-!%! #"$"$""# %" "- H2 1$!"% -'"1 !%! - & "%%%-! -0"(!90 $"- (# ("$% &%" "%%- Uk "#7% H10 - "8'&"# #'* (! ",-"" v ∈ H10 ∩ H2 (Ω) (%! vk ∈ Uk %! &%" S# T v − vk 1 ≤ c hk vH ST 2

2

=) ! & " ") "

" k

N( ( (! 2 , H10 (Ω) 8(# ","-%! "#" 2 12 ∂v 2 v1 = dx . ∂xi Ω i=1

R" ·1 !-!%! "#" - -%- R(0 v0 ≤ C(Ω)v1

∀ v ∈ H10 (Ω).

ST

/% f ∈ L2 (Ω) uk ∈ Uk 1 "&"1=#%" ,(& "%"! "#% ST - "%%- Uk * a(uk , vk ) = (f, vk ) ∀ vk ∈ Uk ,

ST

%"$( , ST ST (% S(",%%-" "$&" 1 !# B (! "("#" ,(&T* u − uk 0 ≤ c h2k uH2 ,

u − uk 1 ≤ c hk uH2 .

SCT

$"(! SBT "&# u − uk 0 + hk u − uk 1 ≤ c h2k f 0 ,

ST

$( u1 ,(& ST

% Al #" % "(!%! , ""%"! Al z, yRl = a(Pl z, Pl y)

∀ z, y ∈ Rl ,

SB5T

-! &% fl , ""%"! fl , yRl = (f, Pl y) ∀ y ∈ Rl , $nk = h2k i=0 zi yi @"## &%" Pk : Rk → Uk

SB T

6 %1 $( z, yR %-' ,"#"2,# # ( "%%-"# "=22%"- Rk "%%-"# "&"1=#%'0 2 Uk "(" 1 "%"- ,(7%! -%-# k

p = Pk−1 Pk−1 ,

r = p∗ .

SBT

> - & "&"1'0 =#%"- , # C - x (! -'&! ,&! r uh -" -%# , $8" % 5

⎧ ⎨ hk ⎩

0

1 8

1 8

1 8

1 4

1 8

1 8

1 8

0

6 -' (1 9 $8" % #" % ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ hk−1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

6 -' (1 9 %"" #" %

? B* G8" (! "%" "%"-! r #" % $87 - & "&"1'0 =#%"- %8%! "##"-% ,&! uh -,!%' - ("# , "(0 # -# ",'# B @# -,!%" -"#" ,8 $"# hk (! #" % hk−1 = 2hk (! $8" N&! p u2h - "% =#%"- -'&!1 7%! (79# "8,"#* , ( % $8" % %" ,& - # "0!%![ ( % %" 8%! ! %"!! " (-# ,&!# - ,0 $8" % "" '0 %"# 8 %$" ,8! U 2hV

%*

! "- -"%- "# 6 & 1 % , "%%- "=22%"- Rk - "%%-" "&"1 =#%'0 2 Uh $( -"","-%! -"%-"# "&"1 =#%"$" ! "#"-% (221 " ,(& /"%"# "80"(#" "-% "8%' 0"( Rk /",! "(# "#' #%' "%" -"1

%-" "# S CT # - -( −1 (A−1 k − pAk−1 r)fk ≤ CA Ak −1 fk

∀ fk ∈ Rk ,

SBT $( Ak zk = fk Ak−1 zk−1 = r fk /"" ,( %%-' ,"#"2,# # ( Rk "%%-"# "&'0 =#%"- Uk %" "- SBT ("8" -% " "#' ,"% "&"1 =#%'0 U"(0V %"&'0 "-!0* % Fk = (Pk∗ )−1 fk , Fk ∈ Uk uk ∈ Uk uk−1 ∈ Uk−1 6 ! ,(& zk − p zk−1 ≤ CA Ak −1 fk ∀ fk ∈ Rk ,

a(uk , φ) = (Fk , φ) a(uk−1 , φ) = (Fk , φ)

∀ φ ∈ Uk ∀ φ ∈ Uk−1 ,

%"$( -(-' -%- zk = Pk−1 uk ,

p zk−1 = Pk−1 uk−1 .

%-%" 8$"(! ,%% ! ,-%" &%" (! 78"$" k ≤ l #% Ak−1 := rAk p ,(%! ""%"1 # SB5T (! z, y ∈ Rk /"=%"# -" -%-" (% "(%-" , SB5T SB T -%"" , SB5T SB T "&1 −1 ∗ = Pk−1 (Pk−1 )−1 r = (Pk∗ )−1 -('0 ""%" pPk−1 > SBT 8(% ("-% , " C uk − uk−1 0 ≤ c CA Ak −1 Fk 0 .

SBBT

$( "%%' c C "!- , "" S BCT "#' ,"1 #"2,#"- Pk Pk−1 @,-#"% =%0 "%% "% k (% , 0 ("" % ,#& @"# ,"% uk − uk−1 "-%! &, U8,"%V uk uk−1 7 (22" ,(&* uk − uk−1 0 ≤ u − uk 0 + u − uk−1 0 ,

$( u 6 ,(& a(u, φ)

=

(Fk , φ) ∀ φ ∈ H10 (Ω).

> -"",#! "" ST - Uk % - Uk−1 * u − uk 0 u − uk−1 0

≤ c h2k Fk 0 , ≤ c h2k−1 Fk 0 ,

% hk−1 ≤ c hk %" SBT %" ",% &%" h2k ≤ c Ak −1 ! =%"$" -"",#1 ! U"8%'#V -%-"# (! "&"1=#%'0 2* vk 1 ≤ c h−1 ∀ vk ∈ Uk . SBT k vk 0 "%% c - SBT ,-% "% -( "&'0 =#%"- S"% %1 ""#"-T " ,-% "% k /"&# (! ",-""1 $" y ∈ Rk * uk − uk−1 0 ≤ c h2k Fk 0 .

−2 2 2 Ak y, yRk = a(Pk y, Pk y) = Pk y21 ≤ ch−2 k Pk y0 ≤ c0 hk yRk .

> #% Ak ##%& %" -%-" Ak y, yRk ≤ c0 h−2 k y2Rk

- ""%" S 5T -&% " λmax (Ak ) ("-1 %" %7 "# S- λ(Ak ) "" %'T* Ak ≤ c h−2 SBT k . > -%-" SBT #" " % uk − uk−1 0 ≤ c Ak −1 Fk 0 .

<-"%-" "# (! ,(& /" (","

! (",%%- 0"(#"% :1 -"%-" "1 # %8%! - -( ST O - &%- $ -790 % -'8 #%"( e"8 #%"# %" Wk = w1 Dk ! "- ST ("%%"&" "-% &%" Wk ≤ cAk Dk ≤ cAk #% w ,-% "% k @" Dk = bc\d (Ak ) "=%"# -$( -(-" Dk ≤ Ak /"1 - " Wk ≤ cAk (! $ - ##%&'# #1 %"("# 3 1 N(! "%" " - " (79$" $2

%* &'

%' ,"9&/

4 ##%&"# & %(%'# %#%"# (! "-1 -"%- $ -! !-!%!

)) B 0 ≤ B = B

"

B(I − B)ν ≤

) !

1 e(1 + ν)

≤ I,

T

SBCT

∀ ν ≥ 0.

> B = B T , %"

B(I − B)ν = max{|λ(1 − λ)ν | : λ ∈ _` (B)}. λ

/" "-7 λ(B) ∈ [0, 1] "&# B(I − B)ν = max x(1 − x)ν . x∈[0,1]

R! f (x) = x(1−x)ν ("%$% 8"$" ,&! - x0 = 1 1 1+ν . $" 8(%! &%" f (x0 ) ≤ e(1+ν) . /"-# $ -79 -"%-" (! #%"( e"8 1 #%"# ##%&"$" #%"( 3 1 N(! /% φki 6 i1! "! 8,! 2! , Uk # %! # C - x %"$( ($"' =#%' #%' Ak #" " "% 0"(! , "(! −2 −2 k k k (Dk )ii = (Ak )ii = h−2 SBT k a(φi , φi ) = hk φi 1 ≥ c1 hk . −1 2 +",! Dk−1 = maxi (Dk )−1 ii ≤ c1 hk SBT "&# (! c2 = c0 c−1 1

Ak ≤ c2 Dk−1 −1 .

/"" # - #%"( e"8 w = c1 %"$( (! Wk %- (! j ##' (% 2

=

1 w Dk

-

S5T

! ##%&"$" #%"( 3 1 N(! - &%- $ 1 - ## S# S CTT 0 ≤ Wk−1 Ak ≤ I.

Wk = (Lk + Dk )Dk−1 (Dk + LTk ) = Ak + Lk Dk−1 LTk ≥ Ak ,

B

&%" (",'-% S5T (! ##%&"$" 3 1 N(! S5T !-!%! $ -79# -"%-"# (! (",1 %%- 0"(#"% :1 /"-# &%" Wk ≤ c Ak S T

! #%"( e"8 S T 8'" %"-" - " ('(91 $" $2 ! ##%&"$" #%"( 3 1 N(! S T (% , (790 ( (! 8,! 2! - Uk #% -" & "% "&'# ,-1 !9# "% k &"# ($0 8,'0 2 X%" &" "-(1 % "&%-"# -'0 =#%"- - ""%-%%-79 %" #%' Ak /"=%"# ⎛

nk

Lk 2 ≤ Lk 1 Lk ∞ = ⎝max j

≤

c max(Ak )2ij i,j

⎞⎛

|(Ak )ij |⎠ ⎝max i

i=j+1

i−1

⎞

|(Ak )ij |⎠

j=1

2

≤ cAk .

ST

<("-%" Wk = Ak + Lk Dk−1 LTk ≤ Ak + Lk 2 Dk −1 ≤ cAk .

/" # &%" -'"!%! $ -79 -"%-" "%"" %8%! (! , ;1 #" (" # &%" Ak Skν ≤

c Ak . ν+1

ST

8",&# B = Wk− Ak Wk− %"$( _` (B) = _` (Wk−1 Ak ) ⊂ [0, 1] > Sk = I − Wk−1 Ak %" 1 2

Ak Skν

1 2

1

1

= Wk2 B(I − B)ν Wk2 ≤ Wk B(I − B)ν c 1 Wk ≤ Ak SBT ≤ e(1 + ν) 1+ν

># "8,"# "-' "80"(#' "-! (! %"# 1 ("-%" 0"(#"% :1 A "("-'# &"#

1 "%$ - 0"(#"% ;1 &"# 1 "%$ - 8"# "%""$" ν¯ > 0 (! "&"1 =#%" "# ,(& /" (",' .%9 )> /" % &%" #%"( "%" % "1 -!# A w , B "8(% -"%-"# $ -!

%' ,"9&/

4 & Ak Wk 6 ##%&' #%' %" #1 # "(0"(% (! "- -79$" -"%- 8,"1 -'0 % > "8%"!% ( (! $ - 1 # ",%! #%"( 3 1 N(! S ##%,"-'T #%"( Kgh 4 ##%&"# & "(0"(!9# %#%"# #" % % (79! ## (%- ,

)+ :;30<= B ∈ R

n×n

!

B ≤ 1 ! F" !

! !

(I − B)(I + B) ≤ 2 ν

ν+1

2 , πν

ν = 1, 2, 3, . . . .

ST

",%%-" ##' #" " "&% - PQ <(%-# #1 #' !-!%! %"# %--79! $ -79 -"%-" (! %"'0 #%"("- #% % Sk = I − Wk−1 Ak .

/"$! - ## B = I − 2Wk−1 Ak "-!! &%" %"# -'8" Ak Skν =

1 2

ν+1

Wk (I − B)(I + B),

"&# (797 %"#

% )3 ! ! I − 2Wk−1 Ak ≤ 1 Wk ≤ CAk ,

ST ST

" ! ! " !'( ! !

Ak Skν

≤C

2 Ak . πν

SCT

@ =% %"# 8(% # (! (",%%- $1 -79$" -"%- #%"( #%"( Kgh "%"'# #%1 "# (! %#' -",79 "# -! /" '# "&'# =#%# +, ("1 ,'0 "" SBT SBT (% ST

(A2k x, x) ≤ c (Dk Ak x, x).

"$&" , ST SBT (% S5T

(Lk Ak x, x) ≤ c (Dk Ak x, x).

?#"%# #%"( Kgh #%"# ω > 0 % %1 #%"( k1"# "- #% -( Sk = I − ω(ω Lk + Dk )−1 Ak .

/" "-7 %"#' B (" "-% I − 2ω(ω Lk + Dk )−1 Ak ≤ 1.

S T

! =%"$" ",-(# -'( (",%%- %"1 #' "&# (! ",-""$" -%" x v = (ωLk + Dk )−1 Ak x -%-" (I−2ω(ω Lk +Dk )−1 Ak )x2 = x2 −2ω((A−1 k (ω Lk +Dk )−2ω I)v, v).

" # &%" -%"" $#" "" %" (! ("%%"&" #1 '0 " ,-!90 "% k ,& ω /",-"(! 9 "( ,#1 v = Ay "&# &%" ("%%"&" ",% (((ω Lk + Dk )Ak − 2ω A2k )v, v) ≥ 0.

/"( (% ("%%"&" #"# ># "8,"# (","

ω

, S5T ST

(I − 2ω(ω Lk + Dk )−1 Ak )x2 ≤ x2 .

>8 * <0"(#"% #%"( ν1 = ν2 = 1 $ -!# e"8 "&%-" %"&'0 "- Sl + 1T B :1 | B | B | B | B ;1 5| 5| C 5| C | C γ } 5| 5| C | C | C / 0 )

>8 * <0"(#"% #%"( ν1 = ν2 = 1 $ -!# 3 1 N(! "&%-" %"&'0 "- Sl + 1T B :1 | | | | ;1 B| | | | γ } B| | | | / 0 )

%( (% S T " "(7 "#' 4%"! "80"(#! - %"# " (ω Lk + Dk ) ≤ CAk

"&-( % Ak ≥ Dk ≥ c1 Lk > , %"#' 1 (% $ -79 -"%-" #%"( Kgh - %" "#

2

4 =%"# $2 #' -(# ,%%' &%"- "#"97 #"$"%"&"$" #%"( ,(& /" - -(% 4 &%- "# -'8 "2"#' "&' =#%' S% Uh ⊂ H10 (Ω)T* "&"1' (! ,8! Ω %$"1 "&"18' (! ,8! -(% "- C

>8 * <0"(#"% :1 Sm}BT $ -!# e"8 (! ,&"$" & (1 "%$ - ν1

5 B 5 ~ B C B C 5 5 C F" % (! 8'0 =#%"ν2

B 5 C

""%-%%-% $ % h0 = 14 " &%! % "1 & ,8# =#%"$" %$" B #0 %$" %# "(! (0 %"& %"" "$&1 " -(%# ># "8,"# ,& l = 3 - %80 "1 "%-%%-% hl = 321 @"## &%" l 6 "# #"$" -0$" %"&"$" "-! 4 &%- $ - ",%! #%"( e"8 1 #%"# #%"( 3 1 N(! 4 #%"( e"8 ,& 1 ""$" #% ω -'8" -'# 5C %"' "(" ! "%"-! %"!%! "&# "8,"# +% "(" 7%! (" %0 " " "%" %1 " "#' -!, "# -" &% %"-" # 10−9 -"# &"# 8 * z0 = 0[ -'"!% zi = M GM (l, zi−1 , fh ), i = 1, 2, . . . S# x T (" %0 " " k =0

Al zi − fl < 10−9 . fl

4 %80 -"(%! "80"(#" "&%-" %1 S"-T #%"( (! ("% ! ,(" %"&"% /"1 &' ,&! "(%- (7% ""-" -"%-" #"$"%"&1 "$" #%"( 6 0"(#"% ,-% "% $ % S& %"&1 '0 "-T ' -(# &%" 0"(#"% ;1 & &# :1 (" ;1 %8% #" - , 8"

>8 B* <0"(#"% ;1 Sm}BT $ -!# e"8 (! ,&"$" & (1 "%$ - ν1

5 B 5 ~ 5 C C 5 C B 5 C C F" % (! 8'0 =#%"ν2

-'& %"&'0 "-!0 "## k = l − 1, . . . , 1 &# :1 ! 01#" ,(& /" =%" ",&% -& %("#"% "(" % #" -1 & γ (% &! 0"(#"% 6 -% &" %1 -" 1" 1 %"0 %8 89 87( "%"% %"# &%" $ -! 3 1 N(! -"(!% & 0"(1 #"% " -7 " $ -!# e"8 4"&# (% "#% &%" $ -! e"8 - 8"%- &- # %("y# &# $ -! 3 1 N(! 0""" 1 -7%! &%0 #"$"""'0 "#7%0 >8' B ,&7% ,-#"% 0"(#"% :1 ;1 "% ,#! & (1 "%$ - 8 1 %& &-%-%' %"# -'"!7%! $1 -! (" " $8" % " (% , %" 0"(#"% &%! -&# & $ 1 -790 % > $ -! !-!7%! -" #" %("#" &%7 #"$"%"&"$" #%"( %" (% -'8% "89 &" $ - ν1 + ν2 "# 8"# 4 # # "%#'# 8(% 2−4 $ -790 % ("# "-

C5

" (! ! ( ,(& "%"% - 0" ( ! %#' '0 $8&0 - #% A (! "%"" ("" # A = AT > 0* S T

A z = b.

F, η "8",&# " &" "8"-"% #%'

A*

η = Apqb (A) := AA−1 =

λmax (A) . λmin (A)

> (! #%' , # - x S' "&' =#%' (! -! (22,T "&# , S T η=

cos2 ( 12 π h) = O(h−2 ) sin2 ( 12 π h)

h → 0.

ST

/ "%%%- (""%" 2"# " %% 1 % #%' A ,-%" , ,( "(# , &0 %"'0 #%"("- !-!%! #%"( "! ' $(%"- C

%"( %8% (" % "("$" (-0 S- ,-#"1 % "% ,T # "%" A "%""# -%"1 * zi+1 = zi + αi (zi − zi−1 ) + β i (Azi − b), ST i i #%' α β -'&!7%! - " % ST ! "8 -(-"* √

η−1 k z − z ≤ 2 √ z − z0 , η+1 k

k = 1, 2, . . . .

SBT

O η 1 "0"(% - ST #'0 h %" "# "8 8'-% #(" /("" # &%" - # "! #%! "%" B = B T > 0 %" &%" Apqb (BA) = η¯ η S"## &%" Apqb (AB) = Apqb (BA)T (! ",-""$" -%" x #" " =22%-" -'&% Bx >"$( -#%" S T # %# ¯ b ¯ = Bb AB ¯z = b, z = B ¯z. BAz = b, ST %" AB #""! - !"# ",-( (B·, ·) \ "%" BA - (B −1·, ·) +, ,( # ,-%" &%" "1 %" B ,'-%! "8"--%# 6 - 8,"-"# #%"( S T B = W −1 %"( "! ' $(%"- #' ST 8(% "8&-% 0"(#"% - "# ·B ·B &# ",%# 0"(#"% " -7 SBT −1

z − zk B ≤ 2

√

k η¯ − 1 √ z − z0 B , η¯ + 1

k = 1, 2, . . . .

ST

4 ,( B -( $"%# EM #%"( "8-1 -%# B %--79 0"(#"% - ·B 1"# −1

& #* ! -

/% (! ! %#' S T ,(' % -(* zi+1 = M zi + N b,

ST

@ # ) Bx ! " % " Ax

C

$( M 6 #% % ?,#'# %8"-# "%"" &%"% -'"!%! (! -0 %"'0 #%"("- #"%1 '0 - ,( !-!%! %" &%" %#' z = A−1 b !-!%! "(- " %"&" (! ST* z = M z + N b.

X%" #" " % A−1 b = M A−1 b + N b.

4 ",-""% b "( -&% A−1 = M A−1 + N

SCT /% M = Ml 1 #% % S"("$" T #"$"%"&1 "$" #%"( (! ! S T /("" # &%" #"$"%"&' #%"( 0"(%! M ≤ ξ < 1 /"%"# "8"--% B : Rl → Rl (79# "8,"#* (! ",-""$" x ∈ Rl ,((# -" (! -'&! y = Bx #" -%" y !-1 !%! 8 # 7 %#' A¯y = x "&'# " -'"! k % S"-T #"$"%"&"$" #%"( ! &'# 8 # y0 = 0 : N = (I − M )A−1 .

y = M k y0 + N x = N x.

/",! (%-# SCT "&#* ST ?-%-" ST % (! %&0 ("- 1 (! 2%&"$" -'&! Bxa R%&" -'&1 Bx "%"% - -'" k "- #"$"%"&"$" #%"( B = N = (I − M k )A−1 ,

k = 1, 2, . . . .

/("" # &%" B ",%! "8"--% >"$( BA = (I − M k ) k

+ξ ) . Apqb (BA) = I − M k (I − M k )−1 ≤ (1 (1 − ξ k )

C

/"( -%-" (% , I − M k ≤ I + M k ≤ I + M k ≤ (1 + ξ k )

(I − M k )−1

= sup

≤ sup y=0

x=0

(I − M k )−1 x y = sup x (I − M k )y y=0

y y ≤ sup ≤ (1 − ξ k )−1 . k y − M y y=0 (1 − M k )y

<("-%" ""% 0"(#"% #%"( "! '0 $(1 %"- #"$"%"&'# "8"--%# #" " "% S# STT ) 1+ξ k √ −1 η¯ − 1 1 − 1 − ξ 2k 1−ξ k ) √ = < ξk . = k 1+ξ k η¯ + 1 ξ +1 1−ξ k

/"=%"# ","- #"$"%"&"$" #%"( (! ,(! 1 "8"--%! #" % 8'% -'$"( "$( ("8 &# &%- #"%"!%'0 % 4 =%"# & #%"( "!1 '0 $(%"- ,'-7% ! %!# #"$"1 %"&' #%"( ! %!# ! #"$"%"&"$" #%"( #"%"$" ( #%"( "! '0 $(%"- &%- -0 % - 8"%- &- "80"(#

" )$ $

?#"%' - ('(9 $- #"$"%"&' #%"( "$1 ( ,'-7% ! #"$"%"&'# #%"("# ! 8"%- U%%-'0V "# #""! '0 =1 %&0 ,(& " "8&-% ,-#7 "% & %"&1 '0 "- ""% 0"(#"% (" #%"( ""8" ;1 "& =22%-' (! -! #"$""1 "'0 "#7%0 % &%' %"&'0 "-!0 ",-"(!%! "("-%" ! -'& #"$""1 "'0 "#7%0 &%" ",'-%! 8" =22%-'# ! #"$"%"&' #%"( " %"$" ((%-' #%"(' CB

#"$% 8'% 8" ("8' S %"& ,! ,T &# #1 %%-' (! ,(& ",790 "" $979! S,#&#'T % 4"" " "" $9790! %0 8(% "8 (%! - " $-' +",! ("" ! "8",&! , $2 ,1 ((# "( %7 ((%-"$" #%"( (! ! %#' Al zl = fl "#"97 -',"- "(' znew = AM GM (l, zold , fl ) $( zold 1 &" 8 zl * ¯z = AM GM (k, z, f ){

4 k = 0 %"

5 ¯z = A−1 0 f S%"&" #" $8" %T ( -'0"( , "('[ 5

S""- -!, $87 %T[ z0 = z zi+1 = zi − Wk−1 (Ak zi − f) i = 0, . . . , ν − 1 Sν 8,"-'0 % S$ -TT[ y = 0 S&" 8 "8T[ B -'"!# y¯ = AM GM (k − 1, y, d) S#"$"%"&' #%"( (! ! ,(& $8" %T[ ¯z = zν − θp y¯ S"! 8 ! zν $8" %T } $( θ > 0 1 #% N#%# &%" $ B ,-% "% $ #" % -'"!%1 ! " ># "8,"# $ -79 % -0 %"&'0 "-!0 #"$% -'"!%! " ((%-' #%"( 7%"- 0#" <0"(#"% ((%-"$" #%"( %8% -"$" -'8"1 #% θ "80"(#" ,& θ , #" % 8'% ,-%" /"=%"# ((%-' #%"( - 8"7%"# 8"%- &- ",7% "8"--% - #%"( "! 1 '0 $(%"- θ = 1 &%" ",-"!% -%"#%& -'81 % "' #%' +("- 0"(#"% %1 0 % 8(% "-(" ! =%"$" # "("8%! %"! %"'0 #%"("- " "("%%-0 C d = r(Ak z − f)

5

zold

l

K

U

l

K

U

l

U

K v

znew ` ` ` -

K 1 $ -! v 1 %"&" l 1 "! ` 1 "("

? * . ((%-"$" #%"(

.%9 +- /",% &%" #% % (-0%"&1 "$" ((%-"$" #%"( #% -(

−1 ν Ma = I − (θ pA−1 2h r + (I − Sh )Ah )Ah ,

S 5T

$( Sh 6 #% $ -790 % 0 +, S 5T -(" &%" ((%-' #"$"%"&' #1 %"( - "%& "% #%%-"$" !-!%! %"&'# Sh < 1 ν → ∞ -&! & $ - "8!,1 %" -(% &7 0"(#"% #%"( 8(% %"&'# %"" - & θ ,-% "% ν θ(ν) → 0 ν → ∞.

"" %

-*

O9 , #"%# %# '0 $8&0 -1 A z = b, S T $( b z ( % "%%- V = Rl /% ,(" ,"1 V - ## "("%%- S "8!,%" !#7aT* V=

l i=0

C

Vi ,

Vi ⊂ V,

S T

% (! 78"$" v$ ∈ V 9%-% S#" % 8'% (%-"T ," v = li=0 vi , vi ∈ Vi @ 6 -% %1 "' #%"( (! ! S T 7 %# - "("%%-0 89 (79* %# S T "1 %! "("%%- $( S8 "T 7%! 1 %#' # ,#"% /" =%"$" U"8%!V 2"#1 ! " -0 Vi (! "&! "-"$" 8 ! 7 V (# "%"' "%"-! Pi , Qi : V → Vi "1 %" "("%%- Ai : Vi → Vi (79# "8,"#* (Qi v, vi ) = (v, vi ),

(Pi v, vi )A = (v, vi )A ∀ v ∈ V, vi ∈ Vi , (Ai zi , vi ) = (Azi , vi ) ∀ zi , vi ∈ Vi .

%" Ai #" " #%-% "7 "%" A "("%%-" Vi %-%" , "(! "%""(% (! ",-"'0 v ∈ V vi ∈ Vi (Ai Pi v, vi ) = (A Pi v, vi ) = (Pi v, vi )A = (v, vi )A = (A v, vi ) = (Qi A v, vi ).

4 ",-""% v vi "&# Ai Pi = Qi A.

/!#'# (%-# S T S T !-!%! -%-" Ai zi

=

bi ,

S T S BT

$( zi = Pi z, bi = Qi b. - S BT ,'-7% ! ! /("" # &%" %# S T 8 " %! %1 "'# #%"("#[ - "%""# z0 6 ,-%" 8 7 z r0 = A z0 − b 6 -!, e = z0 − z 6 ,-%! "8 4%" "8 ("-%-"!% -7* A e = r0 .

O ("" % &%" e (" %" %"&" 0"(%! z = z0 −e (# % 8 e (79# "8,"# ?#"%# - (! "8 "("%%- Vi * Ai ei = Qi r0 , $( ei = Pi e. C

4" #" % &%! % &%" -! "("%%-0 ,%(%" % %"&" (" ("" # &%" -",1 #" " 0"(% 8 ˆei ei ˆei = Ri Qi r0 ,

$( Ri 6 8 A−1 i 4 ,"#'0 %#0 #" " 1 ,% &%" Ri 6 "8"--% Ai >"$( U"8!V 1 8 ! "8 "("%%-0 "-" 8 7 0"(# - -( z 1 = z0 −

l

ˆei ,

i=0

&%" #" % 8'% " "8"-' %"1 ' #%"(* z1 = z0 − B(A z0 − b) S T "8"--%# B=

l

Ri Qi .

S T

i=0

+- 4

! Ri B #:$G& ! ) ! <#""!

"% (% , (790 1 -%- (! ",-"'0 y, v , V* l l (By, v) = ( Ri Qi y, v) = (Ri Qi y, Qi v). i=0

S T

i=0

> " ("" 7 Ri -'"!%! (Ri Qi y, Qi v) = (Qi y, Ri Qi v) %" " "(7 "&# #""! "% B % (By, v) = (y, Bv) /"(%-!! y = v - S T "&# "%1 %"% (Bv, v) ≥ 0

! "" %" "("% "%" "-% &%" (Bv, v) = 0 -&% v = 0 %-%" % (Bv, v) = 0 %"$( , S T y = v "" %" "("% Ri (% CC

&%" Qi v = 0 ∀i 4",## "18" ," v = Vi . /"&# (v, v) =

l i=0

(v, vi ) =

l

$l i=0

vi , vi ∈

(Qi v, vi ) = 0.

i=0

># "8,"# A = AT > 0 Ri = RiT > 0 %" (! 1 ! %#' S T (%-!%! ,#'# (79 ", % :?-=* * " ! A ! ! B

4%% +- ?#"%# V = R

," $nRn - ## "("#'0 "%"$"'0 "("%%- Rn = i=1 span{¯ei } ,( e¯i 6 (&' 8,' -%" %"$( n

Qi v = vi e¯i ,

$(

vi 6 i1! "#"% -%" v (A¯ ei , e¯j ) = aij (¯ ei , e¯j ) = aij δji "=%"#

! #%'

A

"&#

Ai = aii .

O "" % Ri = A−1 = a−1 i ii %" #%"( " "("1 %%-0 S T 8(% "-(% #%"("# e"8 4%% +) /% ,( %# -" '0 "&"1=#%1 '0 "%%- U0 ⊂ · · · ⊂ Ul % V = Rl Vi = Ri $( Ri 6 "%%- "=22%"- "&"=#%'0 2 , Ui i1"# %"&"# "- S# x T 4" Vi ⊂ V 8(# "#% - #' -" "&"1 =#%'0 "%%-* Vi = Ri ∼ Ui ⊂ Ul ∼ R = V,

=#% vi ∈ Vi 8(# #%-% - V =#% pl pl−1 . . . pi+1 vi N( pi 6 "& "%" "(" ! , Ri−1 - Ri S('(9 $- ( i "!T "& "%" "%"-! 8(# ""%-%%-" "8",&% &, ri /" "(7 "&# (Qi v, vi )Vi

=

(v, pl pl−1 . . . pi+1 vi )V = (p∗i+1 . . . p∗l−1 p∗l v, vi )Vi

=

(ri+1 . . . rl−1 rl v, vi )Vi .

C

<("-%" Qi = ri+1 . . . rl−1 rl ,

Ql = I.

8!,%'# =#%"# #"$"%"&"$" #%"( !-!7%! $ -! /% Si 1 #% $ -790 % i1 "# "- @# Si = I − wDi−1 Ai

(! $ - e"8 4 "-'0 "8",&!0 Si (%-% -%"0 , Vi 4 x 8'" "-" &%" # ν %1 (! ! %#' #% Ai #" % 8'% 2"#1 " ," #" -" &% %#' #% (I − Siν )A−1 i > $" -(% &%" #%"( " "1 S T 8(% =--% #%"( "%" % "(1 # "# !" #"$"%"&"$" #%"( - &%- 1 "8"--%! S -'# &'# 8 # θ = 1T / =%"# -(-" 2"#" -%-" Ri = (I − Siν )A−1 S CT i . /"%"' %# "8,"# "8"--% #% 1 ,- wL " # -%""-* wl\s]mj L\_Ac\t i PBQ O$" "8'&" ",7% "8"--% - #%"( "! 1 '0 $(%"- ("8! ,! =%"$" #%"( - %"# & (%-'0 %0 8(% "%(" #"% - ,( >"! #%"( " " "("%%-0 "("8%! # (! (",%%- 0"(#"% #%"( "!1 '0 $(%"- ((%-'# #"$"%"&'# #%"("- - 1 &%- "8"--%! / ( &# #' (# ,&7 0"(#"% #%"( 1 " " ""%" #"%# #%"( "("-%1 " " "("%%-0

-*

4 "%& "% #%"( " " - #%"( "("1 -%" " $ -! "("%%-0 ,%# 5

"! "("%%- "0"(!% "("-%" %1 %! V0 /% ,(" zold ∈ V, %"$( (79 8 znew 0"(%! , l + 1 $* "" # z0 = zold -'&!# zi+1 = zi − Ri Qi (A zi − b),

i = 0, . . . , l.

S T

> znew = zl+1 /"(# , "$ #"$"%"&'# #%"("# ! =%"$" "8",&# Ti = Ri Qi A 4 S T -(-" Ti = Ri Ai Pi +% - S T -'"!7%! "("-%" "=%"# #%1 % #%"( "("-%'0 " #% -( S5T

Ms = (I − Tl )(I − Tl−1 ) . . . (I − T0 ).

.%9 +) 4 (""

!0 , # (",% &%" #%"( "("-%'0 " =--% :1 #%%-"$" #"$"%"&"$" #%"( ν1 = 0 ν2 = ν S%"" "%$ -!T /",% -%-" #%' % S5T #%1 ' % :1 Mi = Mi (0, ν) (! "%"" #' ,# M0 = 0,

Mi = Siν (I − pi (I − Mi−1 )A−1 i−1 ri Ai ), i = 1, . . . , l.

S T

! =%"$" ("8" % S T "(%-- Mk = I − Bk Ak , $( 2"#" ,# Bk = (I − Mk )A−1 k #" ! "&-1 ! -%-" - Pk "80"(#" "&% -" ""%" "%" Pk − Bk Ak Pk ",! =%" 1 -" ""%" S CT ",% &%" #% % Ml (0, ν) = Pl − Bl Al Pl

#" % 8'% , " 2"# S5T / (",%%- 81 (% ",' % (79 ""%"! S0 (" "-1 %aT* rk Ak = Ak−1 Pk−1 ,

Pk−1 Pk = Pk−1 ,

Pl = I.

% - -*

""% 0"(#"% #%"( "! '0 $(%"- "8"--%# B #" % 8'% "( "" & "8"-"% #%' AB ! #%"( " "1 "("%%-0 " Apqb (AB) (%! (79 %"#" " 8(% # (! (",%%- ,1 -!9 "% & %"&'0 "- 0"(#"% EM #%"( (1 (%-'# #"$"%"&'# "8"--%# &"# -! /"

% +-

A Ri ,

$ '" ! v ∈ V ( ! v = i vi , vi ∈ Vi , K0 ! l

(Ri−1 vi , vi ) ≤ K0 (Av, v).

ST

i=0

Ti = Ri Qi A = Ri Ai Pi , ' !! y, v ∈ V K1 ! l

(Ti y, Tj v)A ≤ K1

i,j=0

l

⎞1/2 1/2 ⎛ l ⎝ (Ti v, v)A ⎠ (Ti y, y)A

i=0

F" AHIB (BA) :=

λmax (BA) λmin (BA)

ST

j=0

≤ K0 K 1

$

8",&# T = BA = i Ti ,#%# &%" - ##' "%" T = BA !-!%! #""! '# "" %" "('# - !"# ",-( (·, ·)A ("-%" - $" "8%-' ,&! -9%-' "" %' ! "- λmax (BA) ≤ K1 ,# " ) !

T v2A =

l

(Ti v, Tj v)A ≤ K1 (T v, v)A ≤ K1 T vA vA .

i,j=0

' ","- ST y = v +, " ,7&# T vA ≤ K1 . 0=v∈V vA

|λ(T )| ≤ T A = sup

@"## &%" 78! "%"! "# #%' "-% -1 0 %' ( > "-# " λmin (BA) ≥ K0−1 /" "-7 %"1 #'$(! ",-""$" v ∈ V #" " -'8% ("#",7 v = i vi , vi ∈ Vi %7 &%" " ST -'" 4""1 ,#! =%# - (79 "& -%- (v, v)A

=

l i=0

=

l

(vi , v)A =

l

(vi , Av) =

i=0

=

(A−1 Ri−1 vi , Ri Ai Pi v)A ≤

l i=0

1

1 (Ri−1 vi , vi ) 2 (Ti v, v)A2

≤

l

l

(vi , Ai Pi v)

i=0

",# ##%&"% A Ri

i=0

≤

(vi , Qi Av) =

i=0

i=0 l

l

1

1

(A−1 Ri−1 vi , vi )A2 (Ri Ai Pi v, v)A2 12 l

(Ri−1 vi , vi )

i=0

12 (Ti v, v)A

i=0

1

K0 vA (T v, v)A2 .

/"& v2A ≤ K0 (T v, v)A "%( (% " , λmin (T ) - &%- v -,!% ""%-%%-79 "8%-' -%" +

3

3"-"!% &%" (! "("%%- U1 ⊂ U U2 ⊂ U $( U 1 $1 8%"-" -("-" "%%-" -'"" " 1 -%-" "1G- |(u1 , u2 )U | ≤ γu1 U u2 U

∀ u1 ∈ U1 , u2 ∈ U2

SBT

"%"" γ ∈ [0, 1) N#%# &%" "8'&" -%-" " -$( $%% -'" SBT γ = 1 (" - -",#" 1 " &%" γ < 1* (! "%"$"'0 "("%%- # γ = 0 4& arccos γ ,'-7% 9 $"# # ( "("1 %%-# (! K1 , ST #" " "&% (",'-! 1 -%- % '0 -%- "1G- (! "("1 %%- Vi Vj , ," ! S T ! =%"$" "8,#

$#' - -" &% ST ",! "(!# #%1 S"%""-T Ti Qi * (Ti y, y)A

=

(ATi y, y) = (ARi Qi Ay, y) = (Ri Qi Ay, Ay)

= =

(Ri Qi Ay, Qi Ay) = (Ri−1 Ri Qi Ay, Ri Qi Ay) (Ri−1 Ti y, Ti y).

> Ti y ⊂ Vi , %" ST 8(% ("-% , " l

(vi , vj )A ≤ K1

i,j=0

l

⎞1/2 1/2 ⎛ l ⎝ (Rj−1 vj , vj )⎠ (Ri−1 vi , vi )

i=0

j=0

ST (! ",-""$" 8" -%""- vi ∈ Vi , vj ∈ Vj i, j = 0, . . . , l > ("" # &%" (! ",-"'0 vi ∈ Vi vj ∈ Vj -'"" (vi , vj )A ≤ γij (Ri−1 vi , vi )1/2 (Rj−1 vj , vj )1/2

ST

$( "%%' γij ,-!% "% -'8" 2 , Vi Vj +, "%% γij "%-# ##%&7 #% G = {γij } ,#1 "% (l + 1) × (l + 1). /% ρ(G) 6 %' ( G

! ##%&" #%' %" "#' -(-" G = ρ(G) J! ! #:%K& ! K1 = ρ(G) 1 %-%" (! ",-"'0 x, y ∈ Rl+1 -'"!%! l

γij xi yj = (Gx, y) ≤ Gxy = ρ(G)xy

i,j=0

= ρ(G)

l

⎞1/2 1/2 ⎛ l ⎝ x2i yj2 ⎠ .

i=0

j=0

ST

ST "&%! ##"-# ST " -# i, j 1 ## ST A xi = (Ri−1 vi , vi )1/2 xj = (Ri−1 vj , vj )1/2

! "&! U0""V " K1 # ("%%"&" ("1 ,% ST "%%# γij %# &%" ρ(G) ,-% "% hl "% & %"&'0 "- l B

"* & ' ! @ # ,(& /" (" # ,-!97 "% hl l 0"(#"% #%"( "! '0 $(%"- ((%-'# #"1 $"%"&'# #%"("# - &%- "8"--%! SwL11 "8"--%T ?#"%# $ -! e"8 "-! % $!"% ,8! 8(# ("$% % - ,( ,8 S T 8%! - #%"( wL SVk = Rk T /("" # &%" "&"1=#%' "%%- Uk "%"!% , "&"1'0 "&'0 =#%"- "%"%1 " ,8! Tk "8% Ω %$"

#' 8(# "% ( 0 - "8",& L2 1 "#' +, "%% 8(% !" "$( &, · "8",&%! L2 1"# 2 "$( -("- "# -%" F, · τ 8(# "8",&% L2 1"# 2 " "("8% τ +

4 K1

" # -& " K1 ! #%"( e"8 Rk = Dk−1 , Dk 6 ($" Ak ("-%" -(-" S# ,( T* 2 c1 h−2 k vk

c3 vk 2A

≤ (Rk−1 vk , vk ) ≤

(Rk−1 vk , vk )

2 ≤ c2 h−2 k vk ,

SCT ST

(! ",-"'0 vk ∈ Vk , k = 0, . . . , l "%%# c0 , c1 , c2 ,-!9# "% k vk

" # (797 -"#"$%7 ##

+)

0 ≤ k ≤ m ≤ l " ! u ∈ Uk , v ∈ Um ! ! ! hm |(∇u, ∇v)| ≤ c ∇u h−1 S5T m v. hk ) ! ! k = m %- ( (% , -%- " "8%"$" -%- (! "&"1=#%'0 2 S# x T /% k < m -",## ",-"' =#% τ S%1 $"T %$! Tk , diam(τ ) ∼ hk /"-# S5T

=%"# %$" % "-# &%" |(∇u, ∇v)τ | ≤ c

hm ∇uτ h−1 m vτ . hk

S T

"! " S T "-&% $"87 S5T %-1 %" ##! S T " -# %$"# ,8! Tk "&# |(∇u, ∇v)|

* * * * hm * * = * (∇u, ∇v)τ * ≤ ∇uτ h−1 m vτ * * hk τ ∈Tk τ ∈Tk 12 12 hm −1 2 2 ≤ h ∇uτ vτ hk m τ ∈Tk τ ∈Tk hm = ∇uh−1 m v. hk

/(# (",%%- S T ?,"8# v ## v = $( "&"1! 2! v0 , Um %! &%" v0 "-(% v -" -0 ,0 ,8! Tm ( 90 ∂τ S$ %$" τ T - 0 - "%'0 ,0 Tm >"$( v1 = 0 ∂τ #' %$! " &%!# ## v0 + v 1

(∇u, ∇v1 )τ = (Δu, v1 )τ = 0,

τ - "% u τ /% S 6 SU$&!VT &% τ $( v0 "%& "% ! %"$(

Δ|u = 0

(∇u, ∇v0 )τ = (∇u, ∇v0 )S ≤ ∇uS ∇v0 S .

" % u 6 ! τ %" "#"%' ∇u !-!7%! "1 %%# τ ∇u2S =

mess(S) hm h k hm ∇u2τ = c 2 ∇u2τ = c ∇u2τ . mess(τ ) hk hk

4 "8%"$" -%- "&# % −1 ∇v0 S ≤ ch−1 m v0 S ≤ chm vτ .

<("-%" "! " S T (",*

(∇u, ∇v)τ = (∇u, ∇v0 )τ = (∇u, ∇v0 )S ≤ c

hm ∇uτ h−1 m vτ hk

$"(! ,"#"2,## # ( Uk Rk = Vk "(1 7 #%' A " S5T -&% ""%-%%-797 " - "%%- "=22%"- "&"1=#%'0 2 #"

(vk , vm )A ≤ c

hm vk A h−1 m vm , hk

∀ vk ∈ Vk , vm ∈ Vm , k ≤ m.

"#8! ST SCT ST "&#

(vk , vm )A ≤

hm −1 −1 (R vk , vk )1/2 (Rm vm , vm )1/2 , hk k

' (", " ST γij = c min

+ hi , hj

hj hi

ST

k ≤ m.

, .

/" "-7 %$!7 -'"!%! hh ≤ c( 12 )i−j @" %' ( |ρ(G)| #%' G #" " "1 % &, ∞1"#* i

j

ρ(G) ≤ G∞

√ 2 2 . = max |γij | ≤ c √ i 2−1 j=1 l

> K1 ≤ ρ(G), %" " K1 ,-!9! "% hl l (", +

4 K0

/% ,( ",-"! 2! v ∈ V := Rl (! %1 8%! % ," %" &%" ST -'"" /(# - "%%-" "&'0 =#%"- Ul ⊂ H10 (Ω) #"%# ""%-%%-797 u ∈ Ul u = Pl v (# "(0"(!9

," (! u " -#! "8%" - "%%- "1 =22%"- /% uk ∈ Uk 6 H1 1"%"$"! "! u Uk % (∇uk , ∇ψ) = (∇u, ∇ψ) ∀ ψ ∈ Uk . ST / ("9'0 ("" !0 %$!7 "81 % Ω %! &%" -'"" "- H2 1 $!"% SBT ## 81@ S# 8,"-7 %% " #%"( "&1 '0 =#%"- PQ B (! "("#"$" &!T 8 % "" u − uk ≤ chk u − uk 1 . SBT ?#"%# % (79 8" 2 u¯k ∈ Uk : u ¯ 0 = u0 ,

u ¯k = uk − uk−1 , k = 1, . . . , l,

ul = u.

+## ," u=

l

ST

u ¯k .

k=0

> Uk−1 ⊂ Uk , %" ST -&% (! -0 k = 1, . . . , l (∇uk−1 , ∇ψ) = (∇uk , ∇ψ)

∀ ψ ∈ Uk−1 .

ST

<("-%" "$&" SBT #' ## m < k

ST - ST "&#

(∇¯ uk , ∇¯ um ) = 0 ∀ k = m.

SCT

uk − uk−1 ≤ chk−1 uk − uk−1 1 ≤ c1 hk uk − uk−1 1 .

" %"$" "(%-!! "%"$""%

ψ = u ¯m

ST #" " % ¯ uk ≤ chk ¯ uk 1 ,

k = 1, . . . , l.

/("" # &%" h0 ∼ O(1) "=%"# (! u¯0 ∈ U0 " ¯ u0 ≤ h0 ∇¯ u0 (% "(%-" , -%- R1 (0 4",-9! - "%%- "=22%"- S"$! , "

$ R0 = A−1 0 #

$ ! &

C

vk = Pl−1 u ¯k

$( Pl 6 ,"#"2,# # ( V Ul # x T $ "&# ," v = li=0 vi (! "%""$" -'"!%!* vk ≤ chk vk A , k = 0, . . . , l.

ST

<("-%" - -"%- "%"$""% SCT "%"" - "%%-0 Vm Vk #% -( (vk , vm )A = 0 ∀ k = m "&# v2A =

l

vk 2A .

k=0

#!# ST v2A =

l

vk 2A ≥ c

k=0

l

2 h−2 k vk

k=0

/"(!! " SCT -&% v2A ≥ c

l

2 h−2 k vk ≥ c

k=0

l

(Rk−1 vk , vk ).

k=0

"%%' K0 ,-!9! "% hl l "& ># "8,"# A 6 #% "&"1=#%" (1 %, -! /" B 6 "8"--% "1 &' ## ((%-"$" #"$"%"&"$" #%"( %" Apqb (BA) ≤ C "%"" "%%" C = K0 K1 ,-!9 "% $ % "&%- %"&'0 "- /"( "81 &-% h1,-#7 0"(#"% #%"( "! '0 $(1 %"- (! ! %#' Az = b "8"--%# B. +

2

4 =%"# $2 -"(!%! ,%%' &%"- "#"97 ((%-"$" #"$"%"&"$" #%"( (! ,(& /" - -(1 % 4 &%- "# -'8 "&"1' "&' =#%' (! ,8! Ω %$" ?,8 %"" % - x

>8 * N-#"% 0"(#"% ((%-"$" #%"( 1 #"%"!%" % "% #% θ N& θ 5 5 5 5C 5 5 % C C 5 C >8 * %#' #% θ (! ((%-"$" #%"( #"%"!%" % "&%-" $ - ν B C θopt 5C 5 5 5 5 4 &%- $ - ",%! #%"( 3 1 N(! %"' "(" ! "%"-! -'8# "&1 # ' 8(# %%"-% ((%-' #%"( - &%- #"%"!%'0 % S"$&" #%%-"# #1 %"( - x T % - &%- "8"--%! - #%"( "1 ! '0 $(%"- 4 "8"0 &!0 % "(" 7%! (" %0 " " "%" %" "#' -!, "# -" &% %"-%! # 10−9 -"# &"# 8 S# x TT / #"% ((%-"$" #%"( - &%- #"%"1 !%'0 % - 7 " $% "' 1 #% θ S# ,( T 4 %8 ,&%! 0"(#"% #1 %"( - ,-#"% "% ,#! θ 4(" &%" ( 8"" "%" "% "%#"$" ,& #" % -% 91 %-"# 0(7 0"(#"% +, %8' 0""" ,#%1 " &%" ,& "%#"$" θ ,-% "% &%- S- %& "% &T $ - S# % ,#& - " ,( T 4 %80 B --%! 0"(#"% ((%-"$" #1 %"( #"%"!%" % "%#'# θ #%"( "! '0 $(%"- "8"--%# - -( ((1 55

>8 * F" % ((%-"$" #%"( " $ -1 !# 3 1 N(! θ = θopt , %8' F" %"&'0 "- ν B B B B5 B B B B >8 B* ((%-' #%"( "8"--% - #%"1 ( "! '0 $(%"- θ = 1 F" %"&'0 "- ν B B %-"$" #"$"%"&"$" #%"( - "(# & θ = 1 " " 8(%! - #9%- 1"$" "(0"( 6 0" %("#"1 % "(" % #%"( "! '0 $(%"- %8% #1 % (! ("% ! "("-" %"&"&% "80"(1 #"% -'8" θ &,% @0"(% "(%- ( (,1 ! %" ,-#"% " 2%" 0"(#"% "% & %"&'0 "-

4%8!"&"' $ .%/9' ,%$&

@ -"(%! $"%# EM #%"( (! ! %#' 6 ##%&' "8--% ! ,! ""- #""! "% "%" B A - 1 !"# ",-( (B −1 ·, ·) @ x0 6 &" 8 1 xk 6 k1" 8 7 zk , pk 6 -"#"$%' -%" %# "%"- % #" " "" % "%"1 5

B Ax = B b $( B

%" # -!, (! ,("$" ε* rk /r0 ≤ ε. 4'&% r0 = b − Ax0 ,

! k = 0, 1, 2 . . . -'"!%

z0 = Br0 , p0 = r0

αk = (rk , zk )/(Apk , pk ), xk+1 = xk + αk pk , rk+1 = rk − αk Apk , zk+1 = B rk+1 , βk = (rk+1 , zk+1 )/(rk , zk ), pk+1 = zk+1 + βk pk .

"+ , !' ( @# ,-%" &%" "( ((%-"$" #"$"%"&"$" #%"( #" " ,% - %#0 #%"( " "("1 %%-0* unew = uold − Br = uold −

l

SB5T

Ri Qi r,

i=1

$( r 1-!, r = A uold − b O #' #!# wL1"8"--% " $ -1 !# e"8 $8" % # ,(& %"&" %" 1" $#" - SB5T #" " % Br =

A−1 0 Q0 r

+

Pl−1

nk l (k) (rh , φi )

(k) φi (k) (k) k=1 i=1 a(φi , φi )

,

SB T

$( Pl : Rl → Ul rh = Pl r 6 -!, - "%%- "&"1 =#%'0 2 nk 6 &" 8,'0 2 k1"# 1 6 8,' 2 k1"# "- a(·, ·) 6 %"&"# "- φ(k) i 8! 2"# "&"1=#%" "%"- ,(& ! # ,(& /" a(ψ, φ) = (∇ψ, ∇φ). @"## &%" #%"( "&'0 =#%"- #% Ak = aij -%" - -" 5

&% %#' b = bi 0"(!%! , ""%" (k)

(k)

(l)

Ak e¯i , e¯j Rk = a(φj , φi ),

b, e¯i Rl = (f, φi ).

/("" # &%" % #" % ,#&%! "(""("*

k 1"# "- ("8-!7%! ,' % xi "## i = nk−1 + 1, . . . , nk %"$( #% #' (79! #"(2! SB T* ⎛ ⎞ l (k) (rh , φi ) (k) ⎟ −1 ⎜ φi ⎠ , SBT Br = A−1 ⎝ 0 Q0 r + Pl (k) (k) , φi ) (k−1) a(φi k=1 (k) φi

=φi

$( #' -- "8",&* (k)

φi

(k−1)

=φi

nk−1

:=

(k)

i=1; φi

+ (k−1)

=φi

nk

.

i=nk−1 +1

@ k1"# "- - SBT -0"(!% %"" 8,' 2 "%"1 !9! ! S(! k1"$" "-!T ,# ,# "(# # % - "-' 8,' 2 @ k1"# "- -'&1 ! ",-"(!%! %"" - "("8% $( ,#&%! %1 -" - "8% Ω X%" #" % 9%-" "%% -'&! 4 &%- -! #"%# # "$( (-#" "8% % 0"( k1"$" k + 11' 1 %"&' "- ,#&%! - "("8% -" " "9( 1 2 "% "9( "("8% $( ,#& % 0"1 ( k − 11"$" "-! k1' >! %! &%" -%&%1 ! % "$( (22" ,(& 1 " ,#!%! - "%"" "("8% # """ $' /"=%"# % (%%! &%"8' -",#" " 8'" 0""" 1 8,% - %0 "8%!0 ! (" "$" # %(" "&%% &%" "&%-" ,"- k1"# "- 8(% $ -" (1 + 32 ki=1 2i−1 ) n0 $( n0 6 &" ,"- #" $8" %1 "%% , ST Cn → 2 l → ∞ N&% & #"$"%"&' #%"( , $-' %% 8'% "%#'# " -'&%" " "% (! γ ≥ 2 ((%-' #%"( -( SB5T "%""# B , SBT 78" "("8" %1 %$ ,#& % 8(% #% "%#7 " "% 5

<9%-7% -%' #%%-"$" #%"( ""81 ' (! &%"- (%-'0 %0 # PBCQ "8,"7 %%7 P Q " -!# ((%-"$" #%%-"$" #%"("- 4 ! P CQ # B U"91 %!V ( #% -(* ⎛

Br =

A−1 0 Q0 r

+

Pl−1

⎝

l

nk

(k)

(rh , φi )

k=1 i=nk−1 +1

(k)

(k)

a(φi , φi )

⎞

(k) φi ⎠ ,

SBT

&%" ""%-%%-% -'8" Vk = Pk−1 (Uk − Uk−1 ) - #%"( "1 "("%%- @ "-"# t1"# "- &% "0"1 (% %"" - "-'0 " -7 t1 1"# "-# ,0 4 =%"# & 2"# &%"- "& "% ("8 (! 1 -! 8,' 2 , SBT #" " #"-% ,% SBT - -(*

Br =

A−1 0 Q0 r

+

Pl−1

nl

i=n0 +1

(r, φˆi ) ˆ φi a(φˆi , φˆi )

.

SBBT

! "("#"$" &! '0 X %% φˆi 7%1 "- 4 (79# $2 # (-#" ,(& /1 " #' " # &%" #%"( 0&0 8,"- (% "81 "--% B "%#' U %"&"%7 (" "$2#&"$" #" %!V ,

% - # ! #

(! wL1"8"--%! , 0"(#"% #%"1 ( "! '0 $(%"- "8"--%# ""- 0&0 8,"- #" % 8'% "-( - #0 %" #1 %"( " " "("%%-0 ' "&# " "%% K0 K1 , %"#' 4 ("" (%,7 , ,( B #"%1 # ,(& /" A 6 #% %"% #%"( "&'0 =#%"- #" #" % $"# (%, hl /% 5B

6

φˆ0

k=0

-

n

5 6

k=1

φˆ1

φˆ2 -

n

5 6

k=2

φˆ3

φˆ4

φˆ5

φˆ6 -

n

5 ? * +0& 8,

5

6

(2)

φ0

(2)

φ1

(2)

φ2

(2)

φ3

(2)

φ4

(2)

φ5

(2)

φ6

k=2 -

n

5 ? * @0& 8,

"8"--% B ,(%! "$" #%"( 0&0 8,"- S2"# SBBTT %"$( cond(B A) ≤ K0 K1 , $( K0 = O(| ln hl |), K1 = O(1). SBT /" -7 wL1"8"--%# " SBT ,-1 % "% h "(" "$2#& "% & "8"-"% ,&%7 &%%! ##" U"V , ("8%-" &%"-

",%%-" SBT "-"(%! " %" 0# &%" "1 & " (! wL1"8"--%! - ,( B N1 # B 2"#" - -( B=

l

Ri Qi .

SBT

i=1

Qi 6 "%" "%"-! "%%-" Vi = Pi (Ui − Ui−1 ) V0 = P0 U0 > "%%-" V ('-%! - !#7

##*

V = V1 ⊕ · · · ⊕ Vl ; −1

Ri = [diag(Ai )] Vi

$( Ai 6 #% %"% "%%-

N#%# &%" Ui − Ui−1 ⊂ Ui "=%"# Vi ⊂ Pi Ui ! ",1 -""$" 8" -%""- vi ∈ Pi Ui vj ∈ Pj Uj , i, j = 0, . . . , l " ST 8' (", - x B S"## &%" - #%"1 ( wL Vi = Pi Ui T ! #%"( 0&0 8,"- " 5

ST ("-%" ST 8(% -(- %" "1 %%" K1 = O(1) &%" (! #%"( wL /"&# " (! K0 , ST ? (! , x B #"$% 8'% "-%"' (""-" % 2 u¯i , ,"1 ! ST #"$% ( % "%%-# Ui − Ui−1 ("-%" vk = Pi u¯i #" % ( % Vi - #%"( 0&0 8,"- ?#"%# (%-" ," (! ",-"" v ∈ V* v=

l

vk ,

vk ∈ Vk .

SBT

k=0

(# "%" Πk : U → Uk "#"97 -%- Πk u = u -" -0 %"&'0 ,0 k1"# %"&"# "- Πk u ∈ Uk 9 ,'-7% ,"-'# %"!%"# (! u ∈ U ' -"",#! (79# -"%-"# ,"-"$" %"!%* (! ",-"" φ ∈ H1 -'"" φ − Πk φ ≤ c hk φH . SBCT /% u = Pl v uk = Pk vk (! -%""- , ," ! SBT %"$( uk = Πk u − Πk−1 u N#%# % Πk−1 uk = 0 $"(! -"%- SBCT "&# uk = uk − Πk−1 uk ≤ c hk−1 uk 1 ≤ c1 hk uk 1 , SBT uk = Πk u − Πk−1 u = Πk u − Πk−1 Πk u ≤ c1 hk Πk u S5T 1. @# "("8%! (79! " (! "&"1=#%'0 21 S" -(- - (-#"# &T* uL ≤ C(Ω) | ln hl | uH ∀ u ∈ U. S T @ ,(& "% uk &, u1 4 S5T # ("%1 %"&" "% Πk u1 &, u1 ! =%"$" ,# Πk u21 = ∇(Πk u)2 . ST 1

1 2

∞

1

τ ∈Tk

> Πk u ! ("# %$"& ,8! τ ∈ Tk %" $" "-% &%" ∇(Πk u) ≤ max Πk u, ST τ 5

% Πk u = u - -0 τ %" max Πk u ≤ max u. τ

τ

SBT

' ### " ST6SBT ",! &%"# #%1 8"-! S T ("# %$"& τ "&# Πk u21 ≤ c | ln

hk hl | ∇ u2 = c ln u21 hk hl τ ∈Tk

$"(! S5T "&# l

2 2 2 h−2 k uk ≤ c (ln hl ) u1 ,

k=0

&%" (! ""%-%%-790 -%""- (% l

2 2 h−2 k vk ≤ c (ln hl ) (A v, v).

k=0

4"#! &%" - & ,(& /" S -"#" %T -'"!%! diag(Ak ) ≈ h−2 k I "&# l

(Rk−2 vk , vk ) ≤ c (ln hl )2 (A v, v).

k=0

ST K0 = c (ln hl )2 (", ># "8,"# "8"--% B "%"' " #%"1 ( 0&0 8,"- (% (! (-#" ,(& /" " &" "8"-"% -( cond(B A) ≤ c (ln hl )2 .

4 %0#"# & ,-#"% "$&" " "% h 8" #%&!

5C

4 =%" $- 8(% #"%" ","- #"$"%"&'0 #1 %"("- (! 0" (! 8 "$" ! "%"'0 ,(& -",790 - &%"% - " !0 -!,'0 #"(1 "-# (- ! ("% $,"- X% ,(& #7% ""1 8"% - "%"'0 "(%-" # "%1 0 #"$"%"&'0 #%"("- -$( (% ("-%-"%' ,%%' S- "%& "% # -! /" , ('1 (90 $-T ! &! ,&%7 %8%! ","- '0 "#"% S'0 $ - "1 %""- 0"( "("$" %"&"$" "-! ($"T <(% "%#%% &%" (! #%-#'0 ,(& - %"!9 -#! "%1 %%-% "" %"$" #%#%&" "8""- U0""V S2"- =%"$" "!%! 8(% ", T 0"(#"% #"$"1 %"&"$" #%"( - %" -#! % =22%-"% -'1 & #" % ",%! -" ("-%-"%" 5

* &! -

/("" # &%" "%" L(ε) (22" ,(& ,1 -% "% #% ε ∈ (0, 1] N(& L(ε)u = f - Ω, (u) = g ∂Ω SB T ,"-# " ! ( "%" L(0) #% (1 $" % &# L(ε) 4 =%"# ,( 8(% #"%' (79 #' =%&0 - - &%'0 ",-"('0 1 "$" "!( 4%% 3- - 1(22, −εΔu + u = f - Ω, u|∂Ω = 0. SBT 4 =%"# # L(0) #% -" "!(" 4%% 3) ,"%"" - (22, −ε

∂2u ∂2u − 2 =f ∂x2 ∂y

- Ω,

u|∂Ω = 0.

SBT

4 =%"# # L(0) %% 8'% =%&# 4%% 3+ - "-1(22, −εΔu + a1

∂u ∂u + a2 =f ∂x ∂y

- Ω,

u|∂Ω = 0.

SBBT

4 =%"# # L(0) #% -' "!(" (""(' -' "-! 0 - SBT1SBBT -,!%' &%- # -",#" "%"- ($0 S@# #'0T -'0 "- L #M&3 "$"%"&' #%"( ! %# 1 '0 $8&0 - "#790 $!"1 -",#97 ,(& 8(# ,'-% ! S"% $ NH@OCPT $" ",% 0"(#"% #" " "% "%"1 " "%%" q < 1 ,-!9 "% $ % & %"&'0 = " "" !'

% # # 3+4

5

- ,&! ε , (0, 1] =%"# &" 2#%&0 " ("# "- % (" " ,-% "% ε 8'% """" & ,-%'0 @"80"(#"% - "%"1 -'0 #"$"%"&'0 #%"("- (! ,(& % SBT 6 SBBT -",% # - -'&%" $("(#* N(& % SBT -",% #"% !-'0 0# " -# (! &"$" ! 8"&0 -1 S# P QT 4 =%"# & ε ∼ ντ $( ν 6 "=22% (22, τ 6 $ (%, " -# f "( % "#"-' !-" &' -! ,"%"' % % % -'%!%' - "("# , 1 - 8'-7% "80"(#' (! "# 1 #790 #7% "$&' " /"$&' " 6 S8"!T &% "8% """ $' - "%"" 1 #% 8" &%' ",-"(' - "%""# 1 - "8'&" "%"$""# $ - /1 " &%" "80"(#" -"#"$%! ,(& 1 "#"-" ,"%"" % U=--%"V "# SBT -"#" % ,"%"1 ! % U-'%!%V - - " y / =%"# ε = 2 hx $( hx Shy T 6 ( =#% % -(" " x S"1 hy "%-%%-" yT - % SBBT -",% &% (- (1 "% $,"- # - "- S &% %#%' "%" ("% $,T 4 =%"# 1 & (a1 , a2 ) 6 -%" ""% "%" u 6 " (1 ! %#%' ε 1 "=22% %""-"("% /"1 ##" =%"$" SBBT !-!%! , =%&" &% -! #"#%"- (- ! ("% S$,T &1 %" % -"#"$%" ,(& 4 =%"# & u 6 -%"12! SBBT 6 %# , 10 S- (-#"# &T - ε 6 "=22% #%&" -!,"% 4" -0 %0 #0 ,& ε #" % 8'% "!( (1 ' % #'#* "!( 10−3 − 10−6 # /"=%"# " (1 %! &%" U0""V #"$"%"&' #%"( (" ("-%-"!% -"%- -"%

+

&' - & #

/("" # &%" ,(& SB T "#"- (-0 %"&1 '0 "-!0 "%"' ""%-%%-7% "&"1=#%'# "1 %%-# Uh U2h /% uh u2h 6 ! #%"("# X ,(& SB T %"$( (! "&"$" -'8" "%""- "1 %"-! "(" ! r p -"%-" "# S CT =--%" " S# x T uh − u2h ≤ CA Lh (ε)−1 f ∀ f ∈ L2 (Ω), SBT $( Lh (ε) 6 #% %#' $8&0 - , #%"( "&'0 =#%"- uh −u2h (% , 0"(#"% "&"1=#%"$" ! (22"# " "#' (221 "$" ! &, -7 &% -(* uh − u ≤ C1 (ε)h2 uH , SBT uH ≤ C2 (ε)f . SBT O #"%# # ,"%""$" -! (22, %" "&# Lh (ε)−1 = c h2 CA ∼ C1 (ε)C2 (ε) 8 "%%' C1 C2 ,-!% "% ε &# CA := ζ(ε) → ∞ ε → 0. /% η(ν, ε) 6 2! , -"%- $ -! S T $( ν 6 &" $ - ( ShT #%"( -&% 1 (79 %8"- η(ν, ε) ! 78"$" ,("$" q ∈ (0, 1) 9%-% ν¯ > 0 ,-!9 "% ε %" &%" η(ν, ε)ζ(ε) ≤ q < 1, ∀ ν > ν¯. SBCT

$# "-# $ -79 % (" ' 8'% "%"1 ' % &%" " U"#7%V 0( -"%- "1 # ε → 0 4 &%"% ζ → ∞ %" , SBCT (% η(ν, ε) → 0 ε → 0, %j $ -79 % 7% -'1 " (7 ,(& S"&%T %"&" N-#"% CA "% ε #" " '%%! "8% "-!! -"1 %-" "# - ($" "# S"%&" "% (%"$" "$ L2 1"#'T (" %"$( -"%-" $ -! "81 0"(#" (",% - "-" "# &%" 8'-% " " " "1 8"-% &% CA -'8! p r "&# "8,"# 2

2

- ,-#"% "% "%" Lh (ε) @ =%"# % "&' ("1 -%-"%' &' ,%%' (! ,(& SBBT "(" ! 0"(#"% %0 #%"("- %(" " %"1 $" (! -0 %"- $!"1-",#9'0 ,(& "&"1 =#%'0 0# (%! "%"% $ -! "%"' "81 &-7% %"&" ,(& ε = 0 ,'-%! # 1 # ,(& 1(22, "-1(22, &%" =%"# & #" % 8'% "%" -' #"$"%"&' #%"( " "%" (! & - "%" -1 '0 #"$"%"&'0 #%"("- (% -",#" "% &! $1 -790 % "-'# %#%"# ,( !-!7%! "' IJ SHIJT ," ! #%"(' 8"&"$" % =%"# ,&%7 - '# !-!%! ! #! ,"- % S8,'0 2T %"!9# #"#% ,-%' (79 ,%%' " ,(&# % SBT 1 SBBT* ! SBT ,-%" &%" #%%-' #"$"%"&1 ' #%"( !-!%! -'# ((%-' !-!1 %! SPBQ PQT ! SBT ,-%' -' #%%-' ((%-' #"$"%"&' #%"(' ",7% HIJ 8"&' #%"( 3 1 N(! e"8 - &%- $1 - SPQ PQ PQ PQT ! SBBT &' =#%' ",'-7% -1 7 0"(#"% #"$"%"&'0 #%"("- 2 a1 (x, y) a2 (x, y) ("%%"&" $( "&"1=#%! "#! -'8%! '# "8,"# 4 &1 %- $ - "8'&" -'8%! HIJ #%"( #%"( 3 1 N(! " " # ,"- S,-#"% "% a1 , a2 T %% ,%$-79! ,&1 ' %' #! #"$"%"&'0 #%"("- (" ,(& "8 # # P5Q PCQ PBQ PBQ P5Q PBQ P Q PQ @ %"!9 #"#% -! 0"(#"% #%#%& %"$" (", %"" (! -# &%'0 !" !& "

&-* (! "%"!'0 a1 , a2 -"#" % <1 & 0#' - "&'0 ,"%!0 -"$" "!( ,"8 - P Q "&'0 =#%"- - PBBQ ?#"%# "%"' =#%' "%"! -'0 #"$"%"&'0 #%"("- +

% - & #&

4 =%"# $2 8(% (", -! 0"(#"% :1 ;1"- #%%-"$" #%"( (! -! 1 (22, −εΔu + u = f - Ω, u|∂Ω = 0. SBT - #0 , ('(90 $- 8! "%"- ,(1 & "&%! #" # SBT ",-"7 27 v , H10 (Ω) %$"-# "&"$" -%- " &%!#* ε Ω

2 ∂u ∂v dx + uvdx = f v d x. ∂xi ∂xi Ω Ω i=1

<87 2"#"- ,(& ("8" ,% - -(* % 21 7 u , H10 (Ω) ("-%-"!797 ""%"7 a(u, v) = (f, v) ∀ v ∈ H10 (Ω), SB 5T $( 8! 2"# a(·, ·) "(!%! a(u, v) := ε Ω

n ∂u ∂v dx + u v d x, ∂xi ∂xi Ω i=1

∀ u, v ∈ H10 (Ω).

4 #%"( "&'0 =#%"- $8%"-" "%%-" H10 (Ω) ,#!%! "%%-"# "&"1=#%'0 2 Uh ⊂ H10 (Ω) /"%"- ,(& "%"% - 0" ( uh ∈ Uh ("1 -%-"!79 -%- a(uh , vh ) = (f, vh ) ∀ vh ∈ Uh . SB T ! 2"# a(·, ·) !-!%! ##%&" 221 ' ,(& ##%&'# 8'# 2"## 8(# B

,'-% #""! '# % -",79! - #%"( "&'0 =#%"- SB T ##%&" a(·, ·) 8(% #""1 ! " 8%# -# - " -"%-" "%"$""% "81 #%"( "&'0 =#%"- "%%- Uh %-%1 " -%-" SB 5T -'"" - &%"% (! v = vh "=%"# , SB 5T SB T (% a(u − uh , vh ) = 0 ∀ vh ∈ Uh . SB T <-"%-" "%"$""% SB T 8(% , ","-%! (# <%# '0 $8&0 - -",% , SB 5T - ,%% -'8" S"&"$"T 8, - Uh ! , 0"(#"% #"$"%"&"$" #%"( ! =%" %#' 7&1 -'# #"#%"# !-!%! " "8 "&"1=#%"$" ! , ##' B 4 =%"# $2 ( - $- c, c1 , . . . C, C1 , . . . "8",&7% "%"' 8"7%' "%%' ,-!9 "% ε $ % "# %"&"$" "-! "$"-"" "%-"

3-

Ω

'

! Q %: u(x, y) uh (x, y) , ! #R8& " u − uh ≤ c

-ε .−1 + 1 f , h2

SB T

) ! 8",&# "8 eh = u−uh &%'-! "1 - "%"$""% a(eh , vh ) = 0 (! -0 vh ∈ Uh "&# eh 20 ≤ a(eh , eh ) = a(u, eh ) = (f, eh )0 ≤ f 0 eh 0 ,

%# "8,"#

eh 0 ≤ f 0 .

4 %" -#! (! ",-"" vh ∈ Uh ## εeh 21 + eh 20

SB BT

= a(eh , eh ) = a(u − vh , eh ) = ε(u − vh , eh )1 + (u − vh , eh ) ≤ εu − vh 1 eh 1 + u − vh 0 eh 0 1

1

≤ (εu − vh 21 + u − vh 20 ) 2 (εeh 21 + eh 20 ) 2

/"&# SB T 4 &%- vh -",## 27 , Uh %7 &%" -'"' %(%' "#"' "* u − vh 0 ≤ c h2 uH u − vh 1 ≤ c huH ># vh !-!%! # (·, ·)1 1 "! u # SCT +",! "7 " u2 ≤ c ε f 0 S(",%%-" #" " % - PBQT "&# εeh 21 + eh 20 ≤ εu − vh 21 + u − vh 20 .

2

2

εeh 21

+

eh 20

h2 ≤c ε

<("-%" εeh 1 ≤ c

h ε

h2 1+ ε

f 20 .

1 h2 2 . 1+ ε

SB T

> ",# $#% (-"%-"% 0" 1 # B /% w ∈ H10 (Ω) %! &%" a(w, v) = (eh , v) ∀ v ∈ H10 (Ω).

> eh ∈ L2 %" #' ,# &%" w ∈ H2 (Ω) w2 /% wh 8(% (·, ·)1 1" w Uh >"$( eh 20

≤ εc eh 0

= a(w, eh ) = a(w − wh , eh ) ≤ εw − wh 1 eh 1 +w − wh 0 eh 0 ≤ c (ε hw2 eh 1 + h2 w2 eh 0 )

h2 ≤ c heh 1 + eh 0 eh 0 . ε

+",! SB BT SB T #' "&# (! &! eh 0

h2 ε

≤1

h2 ≤ c(heh 1 + f 0 ) ε

1 h h2 h2 2 ≤ ch f 0 + c f 0 1+ ε ε ε

SB T

2

≤ c hε f 0 .

SB BT -#% SB T "" min{1, hε } ≤ ( hε -&% SB T 2

2

+ 1)−1

%- ( ##' - "&% 8" U"&-("V "" "# Ak S# 0" " (! -! /"T

c1

ε ε + 1 ≤ A ≤ c + 1 k 2 h2k h2k

SB CT

(% -"%-" "# "%%" CA ,-!9 "% hk #' 8(#! &%" "% $ - e"8 #%"# (% -' #%%-' #"$"1 %"&' #%"(

%-%" (! ($"" &% Ak ## " ε

Dk−1 ≤

c3 , Ak

% , SB CT (% Dk−1 = (min(Ak )ii )−1 ≤ c−1 1

i

−1 ε c2 + 1 ≤ Ak −1 2 hk c1

! $ - e"8 #%"# Wk = ωDk %"$( (! "(1 0"(!9$" ω _` (Wk−1 Ak ) ⊂ (0, 1] SB T 6 #% Ak ##%& % ##%& 8! 2"# a(·, ·) > #" " "-% $ -79 -"%-" Ak Skν ≤ c

1 Ak , ν+1

ν = 1, 2, . . . .

SB5T

(! #%' $ -790 % Skν = I − Wk−1 Ak ! =%"1 $" "8",&# B := Wk− Ak Wk− N#%# &%" B ##%&' _` (B) ⊂ (0, 1] " %"$" 1 2

1 2

SB T /" ## B(I − B)ν ≤ (ν + 1)−1 > Wk 6 #%81 "-! ($"! &% Ak %" Wk ≤ cAk "%1 %" c ,-!9 "% k ε <("-%" SB5T -'"!%! @"(%-" , SB T (% $ -79 -"%-" (! :1 1

1

Ak Skν = Wk2 B(I − B)ν Wk2 ≤ Wk B(I − B)ν .

## B SB CT SB TSB5T -&% -'"! -"%"# $ -! "%%# ,-!9# "% ε &%" !-!%! ""-" (! (",%%- -" 0"1 (#"% :1 ;1 "- #"$"%"&"$" #%"( #%" ,(& SB T /"9' (! "$&' # ,(& /" "("8" ,"8"# - ,( +

567 ' 567 &'

4 ('(9# $2 %" !" &%" (! "%"! -1 "$" #"$"%"&"$" #%"( (! -! 1(221 , ("%%"&" "$&%! %# "%'# "#"%# $ -! e"8 "& "(" "1 %"- /"8# ,7& - "- -"%- 1 "# $ -! (!! ""8" -# ,-#"1 % "%% "% #% ε (" (! =22%-"$" ! ($0 ,(& #"$% "("8%! 8" " ' "#"%' %1 HIJ $ -! " "%"'0 "(% & - =%"# $1 2 /",-"! "8%#! #% #" % 8'% S(%-1 '# "8,"#aT ," - ",-( A = L U, SBT $( L 1 %$"! U 1-0%$"! #% "(1 ,#-%! &%" "8 #%' #"$% #% -' =#%' $-" ($" "( S(T %' L U % #7% "8%' O ("" % &%" (! #%' A ," SBT ,-%" %" %#' A z = b #" " (% , (- "("-%'0 $* -"1-'0 % -"#"$%' -%" ˜z , L ˜z = b,

-"1-%"'0 % , U z = ˜z.

8 $ $" -'"!7%! "#"97 !#"$" 0"( #%"( 3 R%& 0" ( ," ! SBT =--%" #1 %"( 7& 3 %-%" "8",&# A(1) = A. C

X#%' #%' A(k) 8(% "8",&%! &, a(k) ij . ?#"%1 # A(2) = L1 A(1) , $( ⎛

1

⎜ − ⎜ ⎜ · L1 = ⎜ ⎜ ⎜ · ⎝ (1) an1 − (1) (1) a21 (1) a11

a11

⎛ ⎜ ⎜ A(2) = ⎜ ⎝

(1)

a11 0 · 0

0

...

1

0

...

0 ·

1 ·

· ·

0 ⎟ ⎟ ⎟ 0 ⎟ ⎟, · ⎟ ⎠

0

...

0

1

(1)

a12 (2) a22 · (2) an2

0

⎞

0

(1)

... ... · ...

a1n (2) a2n · (2) ann

⎞ ⎟ ⎟ ⎟, ⎠

aij = aij − ai1 [a11 ]−1 a1j , i, j = 2, . . . , n /"-%"!! "( (! $" -"$" 8" #%' A(2) % ( 0"(# (2)

(1)

(1)

(1)

(1)

," SBT $(

−1 −1 L = L−1 1 L2 . . . Ln−1 ,

U = A(n) .

X#%' A(n) 0"(!%! "("-%"* (s+1)

aij

⎧ (s) ⎪ ⎨ aij 0 = ⎪ ⎩ a(s) − a(s) [a(s) ]−1 a(s) 11 ij i1 1j

(! i ≤ s, (! i > s, (! i > s,

#% L #% -( ⎛

⎜ ⎜ ⎜ L=⎜ ⎜ ⎝

1

0

0

...

1

0

...

·

·

(1)

a21

(1)

a11

·

(1)

(2)

an1

an2

a11

a22

(1)

(2)

...

·

(n−1)

an1

(n−1)

an−1,n−1

0

j ≤ s, j > s,

SBT

⎞

⎟ 0 ⎟ ⎟ . · ⎟ ⎟ ⎠ 1

O A 6 ##%&! #% %" 8'-% ("8" % ,"1 - -( A = (L + D)D−1 (L + D)T , SBBT

$( D 6 ($"! #% =#%# dii = aii(i) L 6 %$"! #% -" $-" ($"7 /("" # &%" #% A , ! % #% O(n) -'0 =#%"- " 7 #%' L U , SBT #"1 $% "%!% =%" -"%-" +#" % "8%"!% ( #%# , # C " %"$" $"%# 0" (! LU ," ! #% "%#" 2#%&" " "% 6 #1 %"( 7& 3 " - O(n3 ) @87(# "%"" (% ",'# #"% IJ1," ! #%" , '# #%# -",79# - %790 ,1 (&0 !-!%! %" &%" (! %0 #% #"$ =#%' , L U #"$% ",%! U8,V 7 /--! =% =1 #%' 7 #' "&# #%' L˜ U˜ % &%" "-! #% M = L˜ U˜ U8,V A " %"$" %#' - #%# L˜ U˜ #" " % "#"97 #%"( 3 , O(n) 2#%&0 (%- "=%"# -'& M −1 v (! 78"$" -%" v "%#" " " "% 4 =%" !-!%! &# &%"8' #"%% #% M "8"--1 % " " (!%! &%" % znew = zold − M −1 (A zold − b) SBT 8'%" 0"(!%! " # 0""" $ -7% "81 /("" # &%" #% A #" " (%-% - -( ˜U ˜ − N = M − N. A=L SBT ?," SBT ,'-7% IJ SHIJT % SBT 6 HIJ $ -# <%(%' $"%# 0" (! L˜ U˜ ,(%! -'8""# P ! ! % P 6 8" ("- Si, j T

#" %8"- P 6 -7& ($"'0 =#1 &%'-%! - SBT %"" %"- Si, iT > =#% a(s+1) ij (i, j) ∈ P O card(P ) = O(n) %" 2#%&! " 1 "% -'&! M = L˜ U˜ #! M −1 % O(n) (1 # , %&'0 -'8""- (! P !-!%! 8" %0 (1 "- (! "%"'0 =#%' 0"(" #%' "%&' "% !* (1) aij = 0 - %% " #% ,- HIJ -'# ,"1 # "8",&%! HIJS5T 5

<(79! ## ",'-% &%" ," SBT (&1 " S% "# #%' N #T %" $ -! SBT =221 %-' % "%% U!%!V - ""%" η(ν) = O(ν −1 ) (! 2 η , -"%- $ -! # S5T 8'-% "1 """ N

3) A = A

T

> 0, M = M T > 0,

A = M − N ! !'(' '

S = I − M −1 A = M −1 N α > 2 δ > 0 ! A + αN N S

"

!

A S ν ≤ Cs δ max

SBT SBCT SBT

≥ 0, ≤ δ, ≤ Cs ,

1 1 , (1 + )(α − 1)2−ν ν−1 α−1

) ! 8",&# X := M "& -"'0 -%-*

− 12

NM

1

/ ν = 2, 3, . . . .

− 12

SB5T ,# (-

1

A ≥ 0 ⇔ M − N ≥ 0 ⇔ I − M−2 NM−2 ≥ 0 ⇔ I ≥ X A + αN ≥ 0

1

1

M + (α − 1)N ≥ 0 ⇔ (α − 1)M − 2 N M − 2 ≥ −I X ≥ −(α − 1)−1 I.

⇔ ⇔

8",&# θ = (α − 1)−1 , "&'0 -%- (% _` (X) ∈ [−θ, 1] /"-# " SB5T <& #"%# "& -%-* AS ν

=

(M − N )(M −1 N )ν = (N − N M −1 N )(M −1 N )ν−1

= N (I − M −1 N )(M −1 N )ν−1 1

1

1

1

= N M − 2 (I − M − 2 N M − 2 )M 2 (M −1 N )ν−2 (M −1 N ) 1

1

1

1

1

1

1

= M 2 M − 2 N M − 2 (I − M − 2 N M − 2 )(M − 2 N M − 2 )ν−2 · 1

1

1

·(M − 2 N M − 2 )M 2 1

1

= M 2 X(I − X)X ν−2 XM 2 .

<("-%" 1

1

AS ν = M 2 X(I − X)X ν−2 XM 2 1

1

≤ (I − X)X ν−2 M 2 X 2 M 2 = (I − X)X ν−2 N M −1 N ≤ (I − X)X ν−2 N S ≤ Cs δ max |(1 − x)xν−2 | x∈[−θ,1]

≤ Cs δ max{

1 , (1 + θ)θν−2 }. ν−1

.%9 3- :;-1<= @% HIJS5T ," , SBT (! #%'

⎛

4 ⎜ −1 A=⎜ ⎝ −1 0

−1 −1 9 0 4 9 0 4 −1 −1

"%%" N

⎞ 0 −1 ⎟ ⎟. −1 ⎠ 4

"&%' "%-% (! "%%* ⎛

0 ⎜ 0 N =⎜ ⎝ 0 0

+ +

0 0

1 4

0

⎞ 0 0 1 0 ⎟ 4 ⎟. 0 0 ⎠ 0 0

& #* &

@&# #%"( "&'0 =#%"- (! -! SBT ?1 (! "$&" # -# /" 1 (22, "&# (797 2"#"-* % uh ∈ Uh ("-%-"!797 ""%"7

∂uh ∂vh ∂uh ∂vh + dx = ε ∂x ∂x ∂y ∂y Ω

Ω

f vh dx

∀ vh ∈ Uh .

SB T

+(# & "$( "8% 6 (&' -(% Uh 6 "1 %%-" "&"1'0 2 "%"%" -"#1 " % /("" # &%" $ % h - M1 /#1 # ,' % - (79# "!(* (h, h), (h, 2h), . . . , (h, (M − 1)h), (2h, h), . . . , (2h, (M − 1)h), . . . .

% ""%-%%-79 %#' $8&0 - 81 (% #% 8"&"1%0($"' -( ⎛

A1 ⎜ I¯ ⎜ 1 ⎜ Ah = 2 ⎜ h ⎜ ⎝ 0

I¯ A2

I¯

I¯

⎞

0

I¯ AM −1

AM −2 I¯

⎟ ⎟ ⎟ ⎟. ⎟ ⎠

SBT

$( 8" Ai I¯ !-!7%! #%# (M −1)×(M −1) ,(7%! ⎛

2(1 + ε) ⎜ −1 ⎜ ⎜ Ai = ⎜ ⎜ ⎝ 0

⎞

−1 2(1 + ε)

−1

−1

0

2(1 + ε) −1

−1 2(1 + ε)

⎟ ⎟ ⎟ ¯ ⎟ , I = −ε I. ⎟ ⎠

%#%# &%" %(%! 1%"&&! "&"1,"%! "1 #! -! -"(% %" %# %1 %- #%&" 2"# , %# '0 - # 8(% ("8" ","-% % "&"1=#%" 0#' N( 8" #% -(* ⎡

0 1 ¯ A(ε) = 2 ⎣ −ε h 0

−1 2(1 + ε) −1

⎤ 0 −ε ⎦ . 0

@AB ,"9&/

! #%' %"% Ah , SBT "&# HIJ ," /## HIJS5T #%"( "&# ," SBBT $( L 6 (" & # !)# & &"' % !# &# "& # !! $ " " A$" & " # !! & &" % " &" ! !) & &

!) ' &"' ! 9 $" ! ) # ! & !) &"' ! B &"

! ) $" A$" ! % " " $" P " ) " ' % CDE &"

!! %$"! &% A(ε) S8, $-" ($"T =#%' ($"" #%' D -'&!7%! " 2"#* ⎧ 2(1 + ε), ⎪ ⎪ ⎨ 2(1 + ε) − ε2 /d(i, j − 1), d(i, j) = ⎪ 2(1 + ε) − 1/d(i − 1, j), ⎪ ⎩ 2(1 + ε) − ε2 /d(i, j − 1) − 1/d(i − 1, j),

i = j = 1, i = 1, j > 1, i > 1, j = 1, i > 1, j > 1,

SBT $( d(i, j) 6 =#% ($"" #%' D "%-&79 ,1 &7 %"&" 2 - , (jh, ih) F%"8' -'&% "%%" N #" # #%' - HIJ ,1 " X% "7 ("8" "-% -"","-- ,1 7 #% - -( 8""- > # (L + D)D−1 = LD−1 + I ""%-%%-% 8" ⎡

0

P¯1 = ⎣ − di,j−1 0

D + LT 8"

⎤ 0 0 ⎦, 0

0 1

ε

1 − di−1,j

⎡ 0 −1 1 P¯2 = 2 ⎣ 0 di,j h 0 0

⎤ 0 −ε ⎦ . 0

/#" # ,8-! P¯1 % $#'0 /"&# (! h2 P¯1 P¯2 ⎡

0 ⎣ 0 0

−1 di,j 0

⎤ ⎡ 0 ⎢ −ε ⎦ + ⎣ 0

ε di,j−1

−ε 0

0 2

ε di−1,j

0

0

⎤

⎡

0 ⎥ ⎣ 0 0 ⎦+ 0 0

/"&# (! U"%%V N , SBT 8" ⎡ 1 ¯ = ⎣ N h2

ε di,j−1

0 0

0 0 0

0 0

0 1

di−1,j

−1

0 0

⎤

⎦.

ε di,j−1

⎤ ⎦

ε di−1,j

< "#"97 ""%" SBT $" "-% (! " ( " √ dkl ≥ 1 + ε + 2ε. SBBT B

>"$( "- SBCT -'"" δ = h1 ε /"- "-! √ 2(1+ε+ 2ε) SBT α = 1+ε ("-" %0& #' "# "%'! %79$"! &%%! PQ "- SBT -'"!%! Cs = 1 % 2

S = M −1 N ≤ M −1 N ≤ D(L + D)−1 2 N < 1

/"" Ah = -"%- "&#

c h−2

SBT

%" (! 2 η , $ -79$"

η(ν, ε) = c ε max{

1 , θν−2 }. ν−1

SBT

<$ -79 -"%-" (" " 8'% (""" -"%-"# 1 "#* (! 2 ζ , SBCT #" " "-% " ζ(ε) ≤ Cε ! ""%' ," ! (",%%-" 8(% 1 -(" A

,"9&/ 8"' $ ! C$"/

! #%' Ah , SBT #"%# ," 8"&"1 ($"7 S8"&"1T -7 -7 %$"' #%' Dh Lh LTh * ⎛ Dh =

1 ⎜ ⎝ h2

A1 0

0

⎞ ⎟ ⎠,

AM −1

⎛

0 ¯ ⎜ I 1 ⎜ Lh = 2 ⎜ h ⎝ 0

0 0

I¯

⎞

⎟ ⎟ ⎟. ⎠

0

?#"%# 8"&' "$ ##%&"# #%"( 3 1 N1 (! , # B +% - 8"&"# #%"( -'"!7%! "$" 2"## 1

(Lh + Dh )xk+ 2 + Lh xk

=

b,

k+ 12

=

b,

(Dh +

LTh )xk+1

+ Lh x

k = 0, 1, 2 . . . .

> 8" Ai 6 =%" %0($"' #%' %" "( %1 ! #%"( S"(" $ -T #% "%#7 " "% "%"%" & ,-%'0* O(M 2 ) 2#%&0 "1

> - C "-!%! &%" ##%&' 8"&' #%"( 3 1 N(! #" % 8'% , - -( 81 ,"-"$" %""$" #%"( S T "8"--%# B = (Lh + Dh )Dh−1 (Dh + LTh ).

SBT

! "- $ -79$" -"%- "%#! #1 % ## B % % #% -( S = I−B −1 A 4 "8"1 ,&!0 ##' B ## M = B S = M −1 N $( N = B − A

! %"$" &%"8' "&% -' (! N ,#%# &%" B = (Lh + Dh )Dh−1 (Dh + LTh ) = Ah + Lh Dh−1 LTh .

+%

SBCT /"-# "-! SBT6SBT , ##' B 4"1-'0 "&1 -("* N > 0 "=%"# "- SBT -'"!%! 78"# α ≥ 0 4"1-%"'0 ,#%# &%" (! "8%-'0 ,& #1 #%&'0 8""- Ai -'"!%! " λ(Ai ) > 2ε h−2 "=%"1 # Dh−1 ≤ 12 ε−1 h2 "# %"$" Lh = εh−2 /"&# N = Lh Dh−1 LTh .

N ≤ Lh 2 Dh−1 ≤

ε . 2h

># "8,"# "%% δ , SBT #" " "" % -" 1 −2 @" % - SBT "-!# "- SBT 2 εh Cs = 1 > ## B "8&-% -"%"# $ -! -(* Ah sν ≤

1 ε . 2h2 ν − 1

- & HIJ $ - $ -79 -"%-" (! ##%&"$" 8"&"$" #%"( 3 1 N(! (" " 8'% (""" -"%-"# "#* (! 2 ζ , SBCT "81 0"(# " ζ(ε) ≤ Cε A

&& ..%D

"$&" , ,(& /" -7 1(21 2, 7&-'# #"#%"# !-!%! " L2 "#' "8

"&"1=#%"$" ! &, L2 "# -" &% -1 ! f ! %"$" &%"8' "&% "80"(#7 " (! ζ(ε) ("%%"&" "-% (797 " (! "8* u − uh ≤

h2 f . ε

SBT

/"-"(# %(%' (! S# SB T (! 1 # -! 1(22,T ! 2"# ""%1 -%%-79! -7 SBT #% -(

a(u, v) =

ε Ω

∂u ∂v ∂u ∂v + dx. ∂x ∂x ∂y ∂y

! "8 eh = u − uh ##

6 6 6 6 6 ∂eh 62 6 ∂eh 62 6 6 = a(eh , eh ) = a(u − vh , eh ) 6 6 ε6 + 6 ∂x 6 ∂y 6 6 6 6 1 6 6 6 1 6 6 6 ∂eh 62 6 ∂eh 62 2 6 ∂(u − vh ) 62 6 ∂(u − vh ) 62 2 6 +6 6 6 6 6 . ε6 ≤ ε6 6 6 6 6 ∂x 6 + 6 ∂y 6 6 ∂x ∂y

4 ",-""% vh , Uh "&#

6 6 6 6 6 6 6 2 62 6 2 62 2 62 6 ∂eh 62 6 ∂eh 62 6∂ u6 6∂ u6 6 ∂ u 2 6 6 6 6 6 6 6 6 6 ε6 6 ∂x 6 + 6 ∂y 6 ≤ c h ε 6 ∂x2 6 + 6 ∂y 2 6 + 6 ∂x y 6 .

SBB5T N( #' ","- "#"' -"%- "&"1=1 #%'0 "%%- - -(* 6 6 6 2 6 6 2 6 6∂ u6 6 ∂ u 6 6 ∂(u − vh ) 6 6 6 6 6 6 ≤ ch 6 inf 6 6 ∂x2 6 + 6 ∂x y 6 , 6 vh ∈Uh ∂x 6 6 6 2 6 6 2 6 6∂ u6 6 ∂ u 6 6 ∂(u − vh ) 6 6 6 6 6 6 ≤ ch 6 inf 6 6 ∂y 2 6 + 6 ∂x y 6 . 6 vh ∈Uh ∂y

SBB T SBBT

/"- =%0 "%'0 "" #" " % # - PQ @"80"(#" "% SBB5T % SBT ! =%"$" "%87%! "' " (! ! u &, L2 1"# -" &% f @&# %"$" &%" -",-(# -%-" SBT - -(% "1 %$# $" " - "8% Ω /"&# f 2 =

ε Ω

∂2u ∂2u + 2 ∂x2 ∂y

2

dx.

SBBT

$"(! "(""('# -'# "-!# "%" $"#%&1 " 2"# Ω "&# %$! " &%!# Ω

∂2u ∂2u ε 2+ 2 ∂x ∂y

2

6 2 62 6 2 62 2 2 6∂ u6 6∂ u6 ∂ u∂ u 6 6 6 6 dx = ε 6 2 6 + 6 2 6 + 2ε dx 2 2 ∂x ∂y Ω ∂x ∂y 6 2 62 6 2 62 6∂ u6 6 6 ∂2u ∂2u 6 + 6 ∂ u 6 + 2ε dx. = ε2 6 6 ∂x2 6 6 ∂y 2 6 Ω ∂xy ∂xy 2

+",! =%" ""%" SBBT 0"(# 6 2 62 6 2 62 6 2 62 6∂ u6 6∂ u6 6∂ u6 2 6 6 6 6 6 ε 6 2 6 + 6 2 6 + 2ε 6 6 ∂xy 6 = f . ∂x ∂y

SBBBT

2

> SBBBT SBB5T (7% 6 6 6 6 2 6 ∂eh 62 6 ∂eh 62 6 6 ≤ c h f 2 6 6 ε6 + 6 ∂y 6 ∂x 6 ε

SBBT

4 -%- % R(0 - Ω = (0, 1) × (0, 1)* 6 6 6 ∂v 6 6 v ≤ 6 6 ∂y 6

∀ u ∈ H10 (Ω),

" SBBBT (",'-% "!(" 0"(#"% #%"( "&'0 =#%"- O( √hε ) (" (! (",%%- -" 0"1 (#"% #"$"%"&"$" #%"( "#"97 -"%- $ -! "# # "80"(# "!(" 0"(#"% O( hε ) F%"1 8' &% "!(" 0"(#"% (" %8#"$" ## $1 #%' (-"%-"% "$&" %# &%" 8' ","-' , #%"( "&'0 =#%"- (! -! 1 (22, #" (! "8 eh ∈ L2 (Ω) ∩ Uh #"%# 27 w ∈ H10 (Ω) ∩ H2 (Ω) "%"! !-!%! # ,(& 2

−ε

∂2w ∂2w − = eh ∂x2 ∂y 2

- Ω,

w|∂Ω = 0.

SBBT

>"&" %# "8,"# (! ! u ## 6 2 62 6 2 62 6 2 62 6 6∂ w6 6 6 6 6 + 6 ∂ w 6 + 2ε 6 ∂ w 6 = eh 2 . ε2 6 6 ∂x2 6 6 ∂y 2 6 6 ∂xy 6

C

SBBT

4 %" -#! #" ! SBBT eh %$! " Ω ,%# " &%!# ",! "%"$""% "8 78" 2 wh , Uh -%-" " "&# eh 2 = a(w, eh ) = a(w − wh , eh ) 6 6 6 6 1 6 6 1 6 6 6 ∂w − wh 62 6 ∂w − wh 62 2 6 ∂eh 62 6 ∂eh 62 2 6 6 6 6 6 6 6 6 +6 inf +6 . ≤ ε6 wh ∈Uh 6 ∂x 6 ∂y 6 ∂x 6 ∂y 6

> SBBT -"%-" "# SBB T 1 SBBT (7% h eh 2 ≤ c √ f h ε

6 6 1 6 6 6 6 6 ∂ 2 w 62 6 ∂ 2 w 6 6 ∂ 2 w 62 2 6 6 6 6 6 ε6 6 ∂x2 6 + 6 ∂y 2 6 + 6 ∂xy 6

@" SBBT "(!! " -"(!% #"# 1 ,%% eh ≤ c

h2 f . ε

># "8,"# ##%&' 8"&' #%"( 3 1 N(1 ! - &%- $ - "8&-% -' #"$"1 %"&' #%"( HIJS5T $ - "8&-% "&% -' #"$"1 %"&' #%"( U/"&%V % θ , SBT - 9 ,-% "% ε /%# #"(2 HIJS5T "% =%" ,-#"% #" " ,81 -%! PQ + ,

8 & &'

4& #"("# # %" #"%# -"" " "1 %'0 #%"(0 (%, -! "-1(22, ! =%"$" #"%# "("#7 ,(& −εu − u = 0, u(0) = 0, u(1) = 1, SBBCT #797 %"&" ,"8 " B 1\ u(x) =

1 − exp(− xε ) . 1 − exp(− 1ε )

SBBT

4 "%& "% "%" &% $ - =%"# $2 (! 1 "# (22"$" -! 8(% # #%"( "&'0 =#%"- ,"%' 0#' /"("8" %"1 ,"%'0 0# &%% #" % ","#%! - PQ %"( "&'0 =#%"- (! ! "-1(22, 8(% #%-%! - (79# $2 4'# (! SBBCT ,"%7 0# 1"$" "!( %"&"% −ε

ui+1 − 2ui + ui−1 ui+1 − ui−1 = 0, i = 1, . . . , M − 1, SB5T − 2 h 2h u0 = 0, uM = 1,

$( ui = uh (ih), h = 1/M 4 &%"# & 0#' SB5T %"&1 " % ""%"! ,&! - ,0 % "&7%! , #%"( "&'0 =#%"- (! SBBCT "&"1'# uh ? ,"%" 0#' #" % 8'% 8, %( S# # P QT ("* uh (ih) =

1 − ri , 1 − rM

−h . $( r = 2ε 2ε + h

SB T

? SB T S# " B 1]T #% "! h ≥ 2ε % - =%"# & ri #!% , "% , , 4 '-1 "# SBBT 18" "! "%%%-7% "=%"1 # 0 "!- !-!%! %'# - &"# ! %% - SB T -& "%"! 2εh "- h < 2ε #" % 8'% -# "8#%'# S""81 " - 0 01#"# "%%-T % -"(% (! #'0 ε ##" 8""# & ,-%'0 - -",79 %# $8&0 - /"% 0#" (79 &%-" ##' ,%% - & h ≥ 2ε !-!%! ( " # &'# ' ! ! # % # !! " # & # " )"") " %

$ & " & !

"

) F &! & 2 " % ' # ! " & !' " ' )"") # &$ ! & "& &!% " ' )"") " $ "

) ! H1 % $! , ! " 8 "

$ # # !

34# 3G4

5

1.5

1.5

1.5

a.

u(x)

b

c.

1

1

1

0.5

0.5

0.5

0 0

0.2

x

0.4

0.6

0 0

0.2

0.4

x

0.6

0 0

0.2

x

0.4

? B * \ ? -! "-1(22, SBBCT [0, 0.6] ε = 10−2 [ ] ? 0#' %'# ,"%!1 # SB5T h = 201 [ A ? 0#' ,"%!# "%- "%" SBT 0# -"$" "!( ,(& SBBCT =%" −ε

!

4 &

ui+1 − 2ui + ui−1 ui+1 − ui = 0, i = 1, . . . , M − 1, − 2 h h u0 = 0, uM = 1,

SBT

%"&'# # S# " B 1T uh (ih) =

1 − ri , 1 − rM

$( r = ε +ε h .

SBT

<0# SBT #" " % - -( −(ε +

h ui+1 − 2ui + ui−1 ui+1 − ui−1 ) = 0, i = 1, . . . , M − 1, − 2 2 h 2h u0 = 0, uM = 1,

- "%""# #' ,# 0# SB5T %'# ,"%!# (! -! % SBBCT - # ε ,#% ε+ h2 /"=%"# 0# ,"%!# "%- "%" #" " %%"-% #%"( ("8-!79 %-7 S&7T (22,7 .%9 3) \ /"-% &%" 0# −ε

ui+1 − 2ui + ui−1 ui − ui−1 = 0, i = 1, . . . , M −1, u0 = 0, uM = 1, − h2 h

0.6

"#79! SBBCT (% ("-%-"%" h ≥ 2ε ] /"%"% "$&" ("-% 0# - %'0 ,"%!0 "%- "%" (! -! −εu + u = 0,

.%9 3+ "

u(0) = 0, u(1) = 1,

% &%" #%"(' Kgh e"8 1 #%"# ω = 12 "8(7% $ -79# -"%-"# S T η(ν) = √cν (! %#' %01($"" #% ⎛

β ⎜ −γ 1 ⎜ ⎜ A= 2⎜ h ⎜ ⎝

−δ β

0 −δ

... 0

0 ...

−γ 0

β −γ

⎞

⎟ ⎟ ⎟ ⎟, ⎟ −δ ⎠ β

",-"'# "" %'# "%%# γ, β, δ %# &%" γ+δ ≤ β S(" %- ( (! #%"( S T B = 2(D+L) #" " % - PBQT &-(" (%-* #%"(' Kgh e"8 ω = 12 "8(7% $ -79# -"%-"# (! %#' - SBT 6 0#' ,"%!# "%- "%" /#% %"# B > - #%"( Kgh ω = 12 I − 2W −1 A =

−1

1 1 L+R . D+ L 2 2

+ # " ST (% , "" "#' #% -" &% -%- -(*

$(

γ .−1 1 γ (D + L)−1 = (β I − J)−1 ≤ h2 β − 2 2 2 . γ 1 T −2 γ +δ , L + R = J + δ J ≤ h 2 2 2 I 6 (&! #% J 6 #% -( ⎛

0 1

⎜ ⎜ 1 ⎜ # J = 2 ⎜ ## h ⎜ ⎝ ... ...

0 ## # 0

##

# 1 0

⎞

... ... 0 1

⎟ ⎟ ⎟ ⎟. ⎟ ⎠ 0

N#%% &%" J = 1 /#%" 8" "89# &7 -! −εΔu + a1

∂u ∂u + a2 =f ∂x ∂y

- Ω,

u|∂Ω = 0

SBBT

,"%' 0#' ("8-!79 %# '# "8,"# 1 %-7 (22,7 #'0 ε %"& ""%" h ε (a1 + a2 ) -" -# "!' - -'&%" $(1 "(# 4 &%"% =%" 0#' ",79 "#1 ∂u 7 &"- a1 ∂u ∂x a2 ∂y ,"%!# "%- "%" /"%""# "89# & !%" ,'-% -%" 27 a = (a1 , a2 ) 4 78" %"& "8% (x , y ) - &%"% - 78"# , %1 #" " "(% - "%" 6 - -%" (a1 (x , y ), a2 (x , y ))

! ,(& SBBT #"%# & "%"!'0 a1 > 0 a2 > 0 "## ,(& #!! (! "-%-'0 &"- ,"% "%- "%" 1"$" "!( %"&"% G8" "# #% -(* ⎡

0 ¯ h (ε) = 1 ⎣ −ε − a1 h L h2 0

⎤ −ε 0 4ε + a1 h + a2 h −ε ⎦ . 0 −ε − a2 h

SBT

N#%# &%" (! ! "&" %#' '0 $81 &0 - #%"( 3 1 N(! 8(% U%"&'#V ("# & ε → 0 <("-%" -'"" "80"(1 #" S0"%! #" % 8'% ("%%"&"T "- (! "%"! -"$" #"$"%"&"$" #%"( ! %"$" &%"8' =%" -"1 %-" "0" - & #'0 a1 a2 "80"(#" ,1 #"-% ,' % % &%" "# ,"- -",%7% -(" "%" @# (! , ""(%# (ihx , jhy ) -'1 "" a1 (ihx , jhy ) > 0,

a2 (ihx , jhy ) < 0,

%" ,' ((i+1)hx , jhy ) (ihx , (j−1)hy ) (" ' #% 8" "1 # " -7 "#"# , (ihx , jhy ) 4 =%"# & #%1 %#' ,"%'0 - "0!% %% 9 ' " ' & ! ε " ! ) & " &!

"%"" -01%$"' 8" "8!%! ε → 0 $"1 %#' %" # #" " % - %% 1 # - PCQ PQ " 8"%7% "%"'0 ("" !0 a1 a2 -! ,&'0 ""8"- # ,-%'0 - "-%-'# &# #" " "&% - P Q PBQ <%# ,"%'0 - 8""# SBT -",1 % ","- #%"( "&'0 =#%"- - $" &1 "# -( % " " - ,( "=%"# -'8" "1 %" $8" % - #"$"%"&"# #%"( (! %" %1 #' "(",& > "%" ,(& $8" % " #1 %"( 3 A2h = r Ah p %8% ""8"$" "(0"( ""1 "&"# -'8" p r %% 8" ("8! (! "%"! =22%-'0 $ - % 2h %!%! %%-" #" % 8'% 8" -'8" p r ,-!90 "% a1 a2 8" "(%-! (%,! SBBT $8" % ","-# ,"% "%- "%" 4 "(# & "1 %" (a1 , a2 ) "%%! $87 % /- "%"! p r - ,-#"% "% (a1 , a2 ) (79 -' #"$"1 %"&' #%"( (! "%"'0 -%"12 (a1 , a2 ) -(' - PQ <8" %""" -("$" - =%"# $2 "(0"( &"# 7 -! "-1(22, !-!%! , S 1'T "!(" %"&"% ,"%" 0#' $# ("1 %%"# !-!%! %("% "%"# =22%-'0 $"1 %#"- # ,"- % (! ,(& " " '# "1 %"# 4 (79# $2 #' "("8 ,&# 0# 8"1 -'""$" "!( %"&"% "%"! (% %"&-"% 0#' ,"%!# "%- "%" "%"%" -","-! &'0 "! - (" "%#%# &%" 1 ","- ,"%'0 0# 8" -'""$" "!( -"%-" $ -790 % % -'" (7 ,(& %"&" %1 !%! ># # #"$"%"&' #%"( "%%! =22%-1 '# %#%"# (! ! -",79 %#' '0 $8&0 - 6 ! ' " ! $" ! &%

) >GG# " " " " ' "

) Ah "& )" ! " # & a# # # 3HG4

B

+ 9

:7;< * () * *

4 =%"# $2 8(% #"% ""8 "%"! 0# -'1 ""$" "!( 0"(#"% (! - "-1(22, 0# %"&-'0 "!-7 &'0 "! %"( ,1 -% "( ,-# KJLM1#%"( 6 88-% "% Kolj\smcqj J`cqbcqd Ljolpk M\mjltcq 8' (" "# W7$1 "# - $"( ", , -# #"$0 ("-%1 %"( KJLM 0""" "(0"(% (! (%, - "-%-'# &# #%"("# "&'0 =#%"- /("" # &%" ,( %$!! Th "8% ! 1 ("$" =#% %$! ,( "%"' #% δτ ,1 -!9 "% ε a(x) = {a1 (x), a2 (x)} 4 #%"( KJLM "&"1 =#%" uh ∈ Uh ("-%-"!% (79# ""%"1 7 (! 78" 2 vh , Uh +

$

ε(∇uh , ∇vh ) + (a·∇uh , vh ) τ ∈Th δτ (−εΔuh + a·∇uh − f, a·∇vh )τ = (f, vh ).

SBT

?( 8" "#%" , #' ","- ,( "891 !%" "8",& (! "-%-'0 &"a·∇u = a1

∂u ∂u + a2 . ∂x ∂y

@"80"(#' (79 "!! /-' -%"" & - -" &% SBT % 1 -! &% -",7% , 8" "%"- ,(& SBBT "%-!7% %(%' #%"( "&'0 =#%"- ""1 %' & " "%""$" 8(% -( ", #" " #%-% -%-" SBBT !" #" " a·∇vh ("# =#% τ ! ! (221 " ,(& SBBT =%"% & "89%! - " >% & - SBT -'&!%! "=#%" @"## "8",& (φ, ψ)τ :=

φ(x)ψ(x) dx τ

> ("# "%("# =#% %$! 21 ! uh !-!%! $(" S""# "&" %T %" %% & - SBT "( "%" " %"$" (! '0 8'0 "&'0 =#%"- -(1 -"* Δuh = 0 O δτ = 0 (! 78"$" τ %" #%"( SBT -9%! - %(%' #%"( 3 "&'0 =#%"- (! -! SBBT <%8,79 =22% (""%1 "$" & #% !7 %%7 (! '0 8'0 "&'0 =#%"- "(""("$" #% δτ = δ 4 & Δuh = 0 (%-'# U"-'#V &"# ("8-!#'# - SBT ,-!9# "% uh !-!%! δ(a·∇uh , a·∇vh ) = δ([a ⊕ a]∇uh , ∇vh ).

/"( -' #" " %%"-% (%,1 7 #%"("# "&'0 =#%"- ,"%"" (22,1 S(! $(" uT* −δ div ([a ⊕ a]∇u).

SBT

3-! " %," [a⊕a] "-(% -# "%" a /"=%"# SBT ("8-!% %-7 (22,7 %"1

" -(" "%" ,"%"" #%"(' %-" -!,"% &%" (% #%"( KJLM 8" %"&'# 4 %" -#! δ > 0 8$"(! (""%" %-1 " (22, #' #" # " (% & %"&-"1 % 0#' "!-7 &'0 "! B 4 "%& "% ,"%" 0#' "%- "%" 1"$" "!( , ('(9$" $2 #%"("- %-" -!,"1 % - #%"( KJLM (""%' &

δτ (f, a·∇vh )

τ

("8-!%! % - -7 &% -! "%"%" uh # ""%" SBT > "8&-%! "-#%"% "&"1=#%" "%"- ,(& SBT* (! 78"

'-" - SBBT ("-%-"1 !% SBT $" "(%-% -#%" uh X%" ",-"!% "&% ,%%' " 0"(#"% -'""$" "!( (! ! SBT %#' -'8" %8,""$" #% δτ !-1 !%! U%"#V -"""# 3"-"! 0#%&" 0""" 8' ("8-!% %-7 (22,7 %"" - %0 "("81 %!0 Ω $( %"&" &" /% Peh -" 6 Peh "(!%! "" (! ("$" =1 #% τ ∈ Th vh ∈ Uh

Peh =

hτ aτ , 2ε

$( aτ 6 "! "# a =#% τ S- (# (! a 8(# ","-% "# , L∞ T hτ 6 (#% =1 #% τ 89 "#( !-!%! -'8" #% δτ % &%"8' -'"!! U," (-"" #%"%V* h2 ε hτ δτ ≈ aτ δτ ≈

(!

Peh → 0,

SBCT

(!

Peh → ∞.

SBT

%%- SBCT "$( "#(7% -'8% δτ = 0 (! Peh ≤ 1 -",-9! - =%"# & %(%"# #%"( 3 /-(# "%"' "%' #' 2"# (! -'1 8" δτ , %0 &%" &%" -%&7%! - %%* 1 , δτ = 12 Apoz(Peh ) − Pe h Peh hτ ¯ ¯ δτ = δ aτ (1+Peh ) , δ ∈ [0.2, 1], hτ 1 δτ = 2aτ 1 − Pe (! Peh ≥ 1, δτ = 0 h

SB5T SB T &. SBT

>"# (%-!% %&' ,%% S# 1 # PB Q T " 0"(#"% -'""$" "!( (! KJLM #%"( <1 & (# "%"' ("" ! " Uh

/("" # &%" Uh "%"% , "&"1""#'0 S"!( l ≥ 1T 2 ,( %$!! %! &%" 1 -(-' %(%' %"!"' ,%%' P Q* (! 71 8" ("%%"&" $(" 2 v : v|∂Ω = 0 #" " % 1 %"!% vˆ ∈ Uh %" &%" ("# =#% τ ∈ Th v − vˆH (τ ) ≤ chk−i+1 vH (τ ) , i = 0, 1, k = 0, . . . , l. SBT τ N#%# &%" SBT !-!%! %- (# " "" %1 "! 4 &%"% , "" %"! (% $"1 8! % -" - "8% Ω - %0 "#0 %# %!# h O%%-" "#" (! , %"&-"% 0"(#"% #%"( KJLM !-!%! (79! "# ,-!9! "% %$1 ! H1 (Ω)* i

k+1

2

|||u||| :=

ε∇u +

12

δτ a·∇u2τ

,

τ ∈Th

$( ("$%! &%" a #% "&7 L∞ 1"# ("# τ ! ($" ("8%- --(# 9 "8",& aτ := a∞,τ = j__ sup(|a1 (x)| + |a2 (x)|). x∈τ

@" # "("8%! "%% μh , "8%"$" -%- Δvh τ ≤ μh h−1 ∀ τ ∈ Th , vh ∈ Uh . SBBT τ ∇vh τ N#%# &%" μh = 0 (! '0 "&'0 =#%"-

% 3- ! SBT a ∈ L∞(Ω)2 div a = 0 u(x, y) ! " SBBT SBT δτ !!

'( ! '

0 ≤ δτ ≤

h2τ με

SBT

∀ τ ∈ Th ,

" μ = min{1, μh } F" SBT ! uh Uh 2

|||u − uh ||| ≤ c

τ ∈Th

C

ε+

a2τ δτ

!

+ min

a2τ 1 , ε δτ

/

h2τ

2 h2k τ uHk+1 (τ )

SBT

k = 1, . . . , l

' -"(# "" (",%%-" %"#' %#%# &%" "" 8,%! "- "" SBT SBCT (! 81 " 2"#' as (u, v) := ε(∇u, ∇v) + (a·∇u, v) +

δτ (−εΔu + a·∇u, a·∇v)τ ,

τ ∈Th

,(" H10 (Ω) × H10 (Ω) 4"1-'0 as (vh , vh ) ≥

1 |||vh |||2 2

∀vh ∈ Uh .

SBT

4"1-%"'0 (! 78'0 v ∈ H10 (Ω) %0 &%" Δv ∈ L2 (Ω) vh ∈ Uh -'"!%! " as (v, vh )

≤

|||vh ||| ε∇v2 +

min{

τ ∈Th

+

a2k 1 , }v2τ ε δτ

SBCT 12

(εh2τ Δv2τ

+

δτ ak ∇v2τ )

.

τ ∈Th

> ",%! "- "%"$""% KJLM #%"(* as (u − uh , vh ) = 0 ∀ vh ∈ Uh .

! %"$" &%"8' -'-% 2"# (! #% %8,1 δτ 8(# "%#,"-% " (! "8 SBT %# --! &"- ,-!90 "% δτ /"&# δτ a2τ h2τ ≈ min

!

a2τ 1 , ε δτ

/

h4τ ,

&%" -"(% ," (-"" #%"% SBCT 1 SBT N#1 %# &%" δτ ,(-#" 2"## SB5T SB T ("-%-"!% =%"# ," "$&7 SBT

"$& - 4'8# #% δ

τ " 2"# SB T /"(%-1 !! - SBT "&# " =#%'0 "8,"- 2

|||u − uh |||

≤ c

εh2τ + aτ

τ ∈Th

Peh h3 Peh + 1 τ

! / Peh + 1 +aτ min Peh , h3τ h2(l−1) u2Hl+1 (τ ) Peh εh2τ + aτ h3τ h2(l−1) u2Hl+1 (τ ) . ≤ c SBT τ ∈Th

' #" # "%#%8"-% #" ! "(0"(!97 "1 %% -! SBBT % &%"8' -'"!" aτ ≤ 1 >"$( , SBT (% 0""" ,-%' "!(" 0"(#"% 32 (! #%"1 ( KJLM #%" '# "&'# =#%# (! " ε Sε hT* |||u − uh ||| ≤ c

-√

. 3 εhτ + h 2 uH2 (Ω) ,

h = max hτ . τ

SB5T

@ #"# ( - & '0 %" #" % 8'% (", 8" -'" "!(" 0"(#"% # PBQ %#%# &%" (! #'0 ε "# ||| · ||| #" % -'" (%! @,-!9! "% ε " "&%! (! -%""$" & - "1 ( "#' |||u − uh |||*

δτ a·∇(u − uh )2τ .

SB T

τ

4' SB T "-% -(% L2 1"#' "8 (! "1 %""$" "$" 2 a # a 6 "("1 "(' "%" - "#18" - " ,(% "# "89# & ># # &' =#%' KJLM #%"("# (#"%7% "!(" 0"(#"% 32 S -'T - L2 1 "# <-!,'-! " SBT %"&-"%7 #%"( "%#%# ( !& " 3+>4 " ! !)% !)% &

−εΔu + ux + u = f.

B5

&%" ,-!9! "% ε " ",-"(7 -(" "%" (! "8 % a·∇(u − uh ) ",&% "(- &'0 "! - 8 "# #'0 ε <%# '0 $8&0 - SAh x = bT ""%1 -%%-79! "&"1=#%" ,(& SBT #" % 8'% =21 2%-" "#"97 #"$"%"&"$" #%"( '% &1 '0 =#%"- ",'-% &%" (! ("% ! -1 " 0"(#"% #% $8" % A2h (" 8'% "%" %# !#" (%, - SBBT $8" % "#"97 #%"( KJLM SBT #%"# %8, δτ ! !'( " >"$( " # (! ("%%"&" "%'0 %& #"$"%"&' #%"( "8&-% ,-#7 "% ε ""% 0"(#"% PBQ N"&! %"! 0"(#"% " 9%-% (! "%"$" &! # PBBQ @"%"' , (! &! 0&0 8,"- #" " 1 % - P Q

* . , ' &

4 =%"# ,( #' "8(# # #"$"%"&'0 #%"("(! &"$" ! %#' - @- 1 <%" "1 '-790 %"--! (- ##" -!," (1 "%* −νΔu + (u · ∇)u + ∇p = f - Ω, SBT −div u = 0 - Ω, SBT u|∂Ω = 0. SBBT 3( u(x, y, z) = (u1 (x, y, z), u2 (x, y, z), u3 (x, y, z)) 6 -%"12! #% 2,& #' ""% ("% S$,T - %"& "1 %%- (x, y) "%"%" %""$" 87(%![ ⎧ ⎨ Δu1 Δu2 ; Δu = ⎩ Δu3

⎧ ∂u1 ∂u1 ∂u1 ⎪ ⎨ u1 ∂x + u2 ∂y + u3 ∂x ∂u2 ∂u2 2 u1 ∂x + u2 ∂y + u3 ∂u (u · ∇)u = ; ∂x ⎪ ⎩ u ∂u3 + u ∂u3 + u ∂u3 1 ∂x 2 ∂y 3 ∂x

p(x, y, z) 6 !!2! "%"7 "(# (%-'# "8,"# "%8"-- Ω p d x = 0 =% 2! #% #' 1 B

#%&"$" (-! ("% - %"& (x, y, z) ν > 0 6 1 #% "=22% -!,"% ! (%"$" "#! 2, -'-"( - @- 1 <%" "#(# #""$2 PCQ P 5Q N( ( 8(# -'(!% '# 2%"# S#u f U, H10 T -%"' 2 "%%- -%"'0 21 "8'&'# 2%"# !' ? - 9%! - "%%-0 H10 (Ω) := H10 (Ω) 3 (! u L2 (Ω) (! p ("8" ","-% (79 "%1 %-" (! (-!* L02 :=

+

! / p ∈ L2 (Ω) : p dx = 0 . Ω

=# %

N(& SBT 1 SBT (%-!% "8" (221 '0 - @&# ,& #%"("- ! "("8'0 %# " ,(& <%"* −Δu + ∇p −div u u|∂Ω

= f = g = 0.

-

Ω, Ω,

SBT SBT SBT

" %"$" (! 8" !"% ," ! #"%# %"" 1 2 2 1 (-#' &* Ω ⊂ R u = (u1 , u2 ) H0 = H0 (Ω) <%# <%" 0""" 8 % %# @- 1 <%" & 8"0 ,& ν U#'0 ""%!0V u "1 # %"$" ,(& <%" -",% -"#"$%! 1 "'0 - @- 1 <%" - =%"# & 21 ! g - -" &% SBT #" % 8'% "%& "% ! +%1 2"# <%" $" "&% $! SBT " Ω #!! "80"(#" "- g* Ω g d x = 0 ?-%-" ν = 1 - SBT #% "89"% ,%%"-* - ,(& <%" -! SBT -$( #" " "%#%8"-% ν −1 ",-(! ,# pnew := ν −1 pold -" &% ,8-%! "% !-"$" -0" (! #% ν

! -'-"( 8" 2"#"- ,(& #" # (- -1 ! - SBT ",-"' v1 ∈ H10 (Ω) v2 ∈ H10 (Ω) +%$1 B

# "&-! -%- " Ω ",! 2"#' %$1 "-! " &%!# N%# ('-# (- -%- ?-%-" SBT !" #" # q ∈ L02 (Ω) @,"-# 8'# 1 ,(& 2 u ∈ H10 (Ω) p ∈ L02 (Ω) ("-%-"!79 ""%"!# (∇u, ∇v) − (p, div v) = (f , v) −(div u, q)

$( (∇u, ∇v) =

=

(g, q)

2 ∂u1 ∂v1 Ω

i=1

∀ v ∈ H10 (Ω), ∀q ∈

L02 (Ω),

∂u2 ∂v2 + ∂xi ∂xi ∂xi ∂xi

SBCT SBT

d x.

'-% ("8 ","-% =--%" "( 8"$" ! ' {u, p} ∈ H10 (Ω) × L02 (Ω) ("-%-"!79 a(u, p ; v, q) = (f , v) − (g, q)

∀ {v, q} ∈ H10 (Ω) × L02 (Ω),

A "" %" " ##%&! 8" 2"#" a(· ; ·)* a(u, p ; v, q) := (∇u, ∇v) − (p, div v) + (div u, q).

' % "(" # ","-% -' 1

u1 := (∇u, ∇u) 2 ,

"%"" "(!% "# H10 (Ω) =--%7 H1 1"# 4 &%"# & g = 0 "! " (! u "(%-" (% , SBCT -,!% v = u ","-% SBT q = p* u21 = (f , u) ≤ f u ≤ CF f u1 .

<("-%" u1 ≤ CF f (! (-! -",#" " "&% 8$"(! (71 9# -%- ,-%"# -%-" @& Iww S88-% "0"(% "% 2# (' ! w\]i_zt\ wljcT* c0 q ≤ ∇qH−1

B

(! 78" q ∈ L02 "" %" "%%" c0 H−1 6 "1 %%-" "! " H10 (Ω) "%"%" !"$" ",1 -(! , L2 /" "(7 2" ∇q H10 (Ω) 1 -%-" @& #" % 8'% "* c0 q ≤

sup

v∈H10 (Ω)

(q, div v) v1

∀ q ∈ L02 .

SBC5T

> " (! (-! (% , SBCT* c0 p

(p, div v) (∇u, ∇v) − (f , v) = sup ≤ u1 + CF f v v1 v v 1 ≤ 2CF f .

≤ sup

4 8" "89# & g = 0 " (! ! "&%! #"$" " #% -(* u1 + p ≤ c (f + g).

! 9%-"-! (%-"% ! %#' SBCT 1 SBT ("%%"&" "%8"-% f ∈ L2 (Ω)2 g ∈ L02 (Ω) /% ,(' "%%- "&'0 =#%"- Uh Ph (! "# u p ""%-%%-" /"%%-" Uh "%"1 % , !#" ##' (-0 "&"1=#%'0 "%%- S%0#"# & , %0T (" , "%"'0 "#% H10 (Ω) &%" ""%-%%-% (-# "#"%# -%"12 u "&"1=#%" {uh , ph } "(!%! , ""%"1 * (∇uh , ∇vh ) − (ph , div vh )

=

(f , vh ) ∀ vh ∈ Uh ,

−(div uh , qh )

=

(g, qh ) ∀ qh ∈ Ph .

SBC T SBCT

! 9%-"-! (%-"% ! {uh , ph } "80"1 (#" -'" "-! "$"-"% "%%- Uh Ph X%" "- "% ,- Iww ! * 9%-% "1 %% μ(Ω) > 0, ,-!9! "% h %! &%" μ(Ω) ≤ inf

sup

qh ∈Ph vh ∈Uh

BB

(qh , div vh ) . qh vh 1

SBCT

@-%-" SBCT #" % #%-%! (%' 1 "$ -%- @& "- SBCT (! #%"( "&'0 =1 #%"- SBC T 1 SBCT --' "!-" - 8"%0 P Q PCQ / -%- -' ! - -" &% SBCT 7 #" % 9%-"-% uh ph ("-%-"!790 SBC T 1 SBCT 8"1 "- =%"$" 2% "( %! - BB O cq_i`1 -' , SBCT "" %" " %#%! 0 h → 0 %" (%-" SBC T 1 SBCT 0"%! 9%-% " #"1 % 0"(%! '-"# 7 (22" ,(& SBT 1 SBT 4 "# "-7 SBCT ("-%-"!% # (79 -'8"* - &%- Uh -",## "&"17 "1 2"#7 "#7 H10 (Ω) - # C , x &%- Ph "&"1' S "&"1"%"!'T 21 ph <""%-%%-79 ' #7% "8",&! P1 − P1 P1 − P0 4'8" ("-%-"!79 Iww "-7 8(% "&"1 " S"%"!"T (- ph "%"%" "%"" %1 $! ""% uh "&"1! "%"%" %1 $! -(-" &9 # " B <%(%" "8",& =%0 * P1 isoP2 − P1 P1 isoP2 − P0 > SBCT -'"" (! ' P2 − P1 S"&"1-(%&! ""% "&"1 " (-T $" # ' "%%- ("-1 %-"!79 SBCT 6 =%" "2"#' =#%' ,1?- 6 P˜1 −P0 S"&"1"%"!" (- ph "&"1! "1 "% uh %" %$! =%"# uh 6 '- %"1 " - (0 %"" =#%"- %$!T > "1 2"#' =#%' uh #"$% 8'% ,'-' 80 %$"1 "- - %$! %" 8' 2"#' - SBC T 1 SBCT (" ' 8'% "(' "=#%" %' , (1 $ ' Uh , Ph #" " % - PQ 4-(# "8",& (! "%%- (%"1""(1 '0 2 U0h := {vh ∈ Uh : (div vh , qh ) = 0

∀ qh ∈ Ph }.

O "%%- Uh U0h "-(7% %" 2"#"- #%"( "&'0 =#%"- (! ,(& <%" #% "& "%" -(* % uh ∈ U0h ("-%-"!797 -%- (∇uh , ∇vh ) = (f , vh ) ∀ vh ∈ U0h .

B

uh

-

ph uh ph

? B* /# Iww1%"&-" ' SP1 isoP2 − P0 T R! (-! #" % 8'% "( ( " 1 0" (! uh /"%" ","- %0 "%%- Uh #% "(" !( "8# # "8 ( - PQ PCQ .%9 33 /% #%! "%%- Uh Ph &# U0h = Uh ⊂ H10 (Ω) " % (79 %- (!* \ cqqis - -" &% SBCT ("%$%! "%"" qˆh ∈ Ph qˆh = 0[ ] μ(Ω) = 0 - SBCT %" 9%-% %-" {ˆ uh , pˆh } %#' SBC T 1 SBCT f = 0 g = 0[ A ("" !0 , % ] 9%-% f ∈ L2 (Ω)2 g ∈ L02 (Ω) % &%" ,(& SBC T 1 SBCT #% ! "- \ -"",% "&"#"%7 ˆ h = 0 Ph ! "- ] "" % pˆh -'# qˆh , % \ u %- ( % A (% , ] %%-' R($"# ##%&"% #%' %"% ,(& # x B > - & -'0 ,(& ,"8'0 1 , #"$"%"&"$" #%"( (! ,(& <%" "%8% "" 0"(#"% "&"1=#%'0 (221 '# / -'" SBCT %(%'0 ("" !0 "8 (" & " " !" Rn I ) A

$ & JKL(A)⊥ = CM(AT )

B

"#"'0 -"%-0 Uh Ph * inf v − vh 1

vh ∈Uh

≤ c h vH 2

inf q − qh ≤ c h qH 1

qh ∈Ph

∀ v ∈ H2 (Ω) ∩ H10 (Ω), SBCBT ∀ q ∈ H1 (Ω) ∩ L01 (Ω),

SBCT

-(- " 0"(#"% {uh , ph } 7 (221 " ,(&* u − uh 1 + p − ph ≤ C

-

. inf v − vh 1 + inf q − qh SBCT

vh ∈Uh

qh ∈Ph

u − uh 1 + p − ph ≤ c h(uH 2 + pH 1 ).SBCT

/ (""%'0 ("" !0 Ω g #7% #%" "1 * u − uh ≤ c h u − uh 1 , SBCCT uH + pH ≤ c (f + gH ). SBCT SBCT 1 SBCT ("%%"&" %(%' ",%%- #" " % -" #"$0 %"&0 (" , &0 $ " #%"( "&'0 =#%"- (! - @- 1 <%" !-!1 %! PQ -( SBCT - & "&"1$(" $' % - PQ @-%- SBCT 1 SBCT % 8,"# (! "- -"1 %- "# - "# ,-!9 "% h* 2

1

1

1

|||uh , ph ||| := (uh 2 + h2 ph 2 ) 2 .

%-%" SBCT 1 SBCT -% (! ,"% #1 " $8" % " S 27 g #"$% "("8%! (""%' "-! # % T |||uh − u2h , ph − p2h ||| ≤ c h2 (f + g1 ). SB5T 4 (79# $2 #' #"%# %# '0 1 $8&0 - ""%-%%-797 ,(& SBCT 1 SBT + ", -"",#! SB5T (! , -"%- "#1 - -'&" #%&" 2"# / "%" #"$"%"&"$" #%"( (! ,(& <%" "%"' "%"-! "(" ! %"!%! %(%1 '# "8,"# - "%%-0 Uh Ph ,-#" ($ "% ($ B

+

* % 3

! %"$" &%"8' % "% SBC T 1 SBCT %# '0 1 $8&0 - ## - 8,' 2 , Uh * ψi , i = 1, . . . , n -" 8'-% ("8" & 1 #"-% 8,' 2 (! -" "#"%' -%" ""% "%"# (! -%"" ,## - 8,' 2 , Ph * φi , i = 1, . . . , m ?-%- SBC T 1 SBCT "" 1 (7% %# -(

A B

BT 0

u p

=

fh gh

,

SB T

$( A > 0 u, p 6 -%" "=22%"- 2 , Uh Ph "=22%' #% A B -'&!7%! , ""%" A¯ ei , e¯j = B¯ ei , e¯j =

(∇ψi , ∇ψj ) i, j = 1, . . . , n, (div ψi , φj ) i = 1, . . . , n, j = 1, . . . , m,

$( e¯i 6 (&' -%" 1 i1"# #% ·, · 6 !" ",-( - Rn 8" - Rm #%8"-" #" % O(h2 ) #"% S BT ,#&! " +, 9%-"-! %1 -'0 u ("-%-"!790 "-7 Bu = 0 (% &%" #% B -""89 $"-"! "8%# 4 "89# & B !-!%! ( -(%" #% 8",&# &, M #%1 # - "%%- Ph S(! (-!T* M e¯i , e¯j =

(φi , φj ) i, j = 1, . . . , m.

SBT

N# %# SB T - -( Au

+ BT p Bu

= =

fh , gh .

SBT

#" # -" -%-" - SBT A−1 X%" #" " (% % A > 0 ("-%" "8%# /-" -%-" 1 #% -( u + A−1 B T p = A−1 fh .

BC

#" # =%" -%-" B $"(! -%""# ""%"1 7 - %# SBT "&# B A−1 B T p = B A−1 fh − gh .

SBBT

4%" ""% u &, , ""%"! ' "& %1 # '0 $8&0 - "%"%" ,-%1 "$" -%" p 6 (-! % S = B A−1 B T - -" &% #% ,- S %#' SBT X% #%1 S"%"T $% - 7 " , %#' N#1 %# &%" -"1-'0 #% S - "%& "% #% A B !-!%! , " 6 S%&T - =#%' S "%&' "% ! % "%&' "% ! - =#%' A−1 4"1-%"'0 =% -' =#%' ("-"" U(""$"V -'&% 6 %8%1 ! -'& A−1 - !-"# -( /"=%"# % #% S -'&!7% !-" 4 (79 ## (",'-7%! "%"1 ' ",' -"%- S "%"' #"$% 8'% %"-' 8, -'&! =#%"- S

3+ T== ! #RU:& !

- ! Ph μ(Ω) A = AT > 0 M ,

! ! ! #R8%& "

S = B A−1 B T !

Uh

μ2 (Ω) M ≤ S ≤ M.

SBT

! ",-"'0 -%""- q ∈ Rm v ∈ Rn #"%# ""%-%%-79 "&"1=#%' 2* $m $n qh = i=1 qi φi vh = i=1 vi ψi >"$( ) !

qh 2 = M q, q,

vh 1 = A v, v,

(qh , div vh ) = q, B v,

(qh , div vh ) q, B v = sup sup 1 . vh 1 vh ∈Uh v∈Rn Av, v 2

supx #%-%! %"" (! x = 0 x "1 B

!-!%! - ,#% <-(-' (79 ""%"!* sup

v∈Rn

q, B v2 B T q, v2 A−1 B T q, Av2 = sup = sup Av, v Av, v v∈Rn Av, v v∈Rn = sup

A−1 B T q, v2A = A−1 B T q 2A v2A

= A

B T q, A−1 B T qA = B T q, A−1 B T q

v∈Rn −1

= S q, q.

N( #' ","- &%" (! ",-""$" !"$" "1 b) ,-(! ""%-%%-79 "#' -'"" b = sup (a, a a∈R (! 78"$" b ∈ Rn S# BT $" "-% &%" -%-" ∇v2 = div v2 + rot v2 -'"!%! (! ",-""$" v ∈ H10 (Ω) S# BT /"( -&% div v ≤ ∇v ("-%" (! 78" 21 qh ∈ Ph -(-" n

sup

vh ∈Uh

(qh , div vh ) qh div vh ≤ sup ≤ qh . vh 1 vh 1 vh ∈Uh

> (! ",-"" q ∈ Rm #"%% qh = %"$( -0!! !! " SBCT SBT -%

SBT $m

i=1 qi φi

μ2 (Ω)M q, q ≤ S q, q ≤ M q, q,

&%" ",&% SBT " "(7 @" ##%&"% A -&% ##%&"% A−1 , ##%&"% A−1 (% ##%&"% #%' S

.%9 31 /"-% &%" (! ",-""$" !1 "$" ",-(! ""%-%%-79 "#' -'"" (a, b) sup (! 78"$" b ∈ Rn a

b =

a∈Rn

"

%- " (" % 5

(a, b) "#"97 -1 a∈Rn a (a, b) b ≤ sup a∈Rn a

% - b ≥

sup

.%9 35 /"-% &%" (! ",-"" v ∈ H (Ω) 1 0

-'"!%! -%-"

∇v2 = div v2 + rot v2 ,

$(

/"-% -%"" %" (%-" Δ = ∇div + rot T rot /#% $" ",-"" ("%%"&" $(" 2 v ∈ H10 (Ω) #" % v "%$% " &%!# 4 "1 &"# %" (%- (% ,#'7 " -# 2!# , H10 (Ω) 1 rot v = − ∂v ∂y +

+

∂v2 ∂x

%*

?#"%# (-0%"&' #%"( ("" # &%" "%1 %- Uh Ph U2h P2h ""%-%%-7% ,87 "8% &%" $8" %" <2"## -"%-" "# /% %# SB T ""%-%%-% "%%-# Uh Ph 4-(# "8"1 ,& (! #%' %#'*

Ah =

A B

BT 0

SBT

.

"$&" "(!%! A2h (! ,(& $8" % /-(# "(" "%" -%-" /% - Rn ,(' (- "#' · α · β >"$( (! ",-"'0 #% A B -(-" A Bα ≤ Aβ→α Bα→β , SBCT $( Aβ→α = sup

x∈Rn

A xα , xβ

Bα→β = sup

x∈Rn

B xβ . xα

?#"%# #% % (-0%"&"$" #%"( #%1 $ -790 % Sh S# x BT ν M = (I − p A−1 2h r Ah )Sh .

! (",%%- 0"(#"% - "# · α ("%%"&" ("1 ,% - -%- SBCT −1 ν Mα ≤ A−1 h − p A2h rβ→α Ah Sh α→β < 1.

"18" "#" · β O%%-'# "8,"# -"%-" 1 "# (" "%8"-% - "# · β→α -"%-" $1 -! - "# ·α→β @"## &%" -'"! =%0 -"%S"&%T ("%%"&" (! (",%%- 0"(#"% W 1 #"1 $"%"&"$" #%"( ? "#' · α · β "(,#- "("-'# " "$( 0 ("8" -'8% ,&'# /#%" %# SB T #"%# (79 "1 #'* 1 u, pα = u2 + h2 p2 2 ,

1 u, pβ = u2 + h−2 p2 2 .

/("" # -#! &%" "%%-" Ph -'8" % &%" (- ph '-" Ω (! 2 , Ph -'"" "8%" -%-" SBT %"$( -%! -'-"( , SB5T 1

|||uh − u2h , ph − p2h ||| ≤ c h2 (f 2 + h−2 g2 ) 2 ,

SBT

f ∈ Uh g ∈ Ph /%# %(%'0 ( S# x T , SBT ("-" 8' -"%-" "# - -( SB 55T

−1 2 A−1 h − p A2h rβ→α ≤ c h .

(" -",% ,%( 6 8,"-! " SB5T - %1 (%"# & -'"$" #"$"$" -'"!%! "81 #%"# "- g(x, y) = 0 - -0 #"$"$" X%" "- "8!," -'"!%! (! 2 , Ph SBT (! f ∈ Uh g ∈ Ph 8' -1% (", - PQ "1 #"97 (79 ## (",%%-" "%"" $"#",(" 8(% "9"

33

) ! Uh Ph U2h P2h ! ! ! #RU:& #RUR& #RUK& " ! ! - Uh U2h fh ∈ Uh gh ∈ Ph (fh , v2h ) + (gh , q2h ) = 0

∀ v2h ∈ U2h , q2h ∈ P2h ,

" #RU$&

#RU%& ! 1

|||uh , ph ||| ≤ c h2 (fh 2 + h−2 gh 2 ) 2 .

SB 5 T SB 5T

/" # "#"97 ##' "&% #" -"%-" "# >8%! (",% SBT > - SBCT "# ,"% # ( '-'# # "&"1=1 #%'# # "-%! &, "#"' -"1 %- "%%- Uh Ph " "%"7 H10 L20 "$&" % "#' ,"% # ( "&"1=#%'# !# #" $8" % "-7%! &, -"%-" "%%U2h P2h "#"-% Uh Ph %7( ,7&# |||uh − u2h , ph − p2h ||| ≤ C

inf

inf

v2h ∈U2h q2h ∈P2h

|||uh − v2h , ph − q2h |||.

4 &%- v2h q2h -",## "%"$"7 "7 uh ph U2h P2h N-("#" -'"!%! -%-" |||uh − u2h , ph − p2h ||| ≤ C |||uh − v2h , ph − q2h |||. SB 5T # -' - -" &% ! =%"$" #"%# wh rh ! ,(& <%"* (∇wh , ∇vh ) − (rh , div vh ) −(div uh , qh )

= =

(uh − v2h , vh ) ∀ vh ∈ Uh , (ph − q2h , qh ) ∀ qh ∈ Ph .

/-! &% =%" %#' ("-%-"!% "-!# ##' "1 =%"# -(- " |||wh , rh ||| ≤ c h2 (uh − v2h 2 + h−2 ph − q2h 2 ) . SB 5BT > ",! "%"$""%7 uh − v2h U2h ph − q2h P2h "(# uh , ph wh , rh ""%-%%-790 -"'0 ,(& N# 1 2

|||uh − v2h , ph − q2h |||2

(uh , uh − v2h ) + h2 (ph , ph − q2h ) (f , wh ) + h2 (g, rh ) 1 1 ≤ f 2 + h−2 g2 2 wh 2 + h6 rh 2 2 1 ≤ c h2 f 2 + h−2 g2 2 |||uh − v2h , ph − q2h |||. = =

4#% SB 5T =%" (",'-% SBT SB 55T (% (""% -"%-"# $ -! Ah Shν α→β ≤ η(ν)h−2 , η(ν) → 0 ν → ∞. SB 5T

/(# -"" "%"! $ -790 % 4 %1 % $ -! (! ,(& % <%" ,(!7%! (- %* 8"&' (' ?#"%# 8" ,-%1 ' #%"(' , (" %$" /((# &%%! &%" %"! %"'0 #%"("- (! #% -( SBT 9%-1 " " &# (! "" %" "('0 ##%&'0 #% -%&-0! - !0 $ -79 -"1 %-" ("," (! -0 S" ",#'0 - =%"# &1 %-T %"'0 #%"("- ++

%&' > %

X%"% #%"( "%"%! 8"&'# $ -!# 8' ("1 - " C510 - P Q %"( ,-% % "( 88-%" KEMK SKssjolcA Epi`mjb M\i__1KjcbjmT /"" #%"( 31 1 N(! #" % 8'% # SB T ,1, &! #%' -"$" $" -"$" 8" %" (! #%"( KEMK "%"% - # ,-%'0 % &%" #" " -'(% 8" $' ,-%'0 #% 8"&' #%"( 31 1 N(! "%"' 8(% "%" "( "%"%" -'('0 $ O%%-' ""8 %" # $"- -!, ,87 %$! Th #"1 =#%' ?#"%# KEMK $ -! # "&"1=#%1 " ' P1 isoP2 −P0 ! =%"$" ## - =#%' %1 $! " (-7 S8" %$"T* τi , i = 1, . . . , I. 8",&# &, ui , pi 8"' "=22%"- S,&T ""1 % (-! ,('0 =#% τi 4$" "=22%"- ("# 8" (! ("$" #* " ,& "% (" "#"%' u "(" ,& p N#%# &%" "( %"% "=22% , u #" % -0"(% - "" 8""- ui "("1 -#" /% Ai , Bi 6 #%' ,#"% S n T S n T "%-1 ' , =#%"- #% A B ""%-%%-790 i1"# 1 8" "=22%"- 4 Ai Bi -"(% =#%' %"!9 & %" %"8"- #%' Ah "## ,-%1 '0 , i1"$" 8" "$&" -%"# ui , pi "(# ri , qi B

⎫ ⎪ ⎪ ⎬

⎪ ⎪ ⎭ #

5

? B* @-" ," #%' "89#" - SB 5T (! -!,

r q

=

A B

BT 0

u p

−

fh 0

.

KEMK $ -! ,(7%! (79# "8,"# ! -'"%*

i =

1, . . . , I

ui pi

=

ui pi

−w

A˜i Bi

BiT 0

−1

ri qi

SB 5T

4'&! - SB 5T "0"(!% "("-%" (! ,1 &'0 ,& i / =%"# "-' ,&! ui , pi ",7%! -'& -!, ri+1 , qi+1 %( w 6 %"' 1 #% " " -'8% A˜i = Ai $" "!' -'8" 6 A˜i = bc\d (Ai ) 4 "(# & "89#! #% - SB 5T #% -( -(' B ! "89! "1 ,%! #%"( 3

"$"%"&' #%"( " $ -!# % KEMK ,"1 #("- 8! "( , &0 (! ! %# % @1 - 1 <%" 0"%! - %"!9 -#! ,-%" (",%%- $" 0"(#"% ( (! "%$" &! 6 %#' <%" ("%%# #" " "%% 0( 0"(#"% " ,"%"'0 %0 ! -! =%"$" ("%% "80"1 (#" ( #"(2"- #%"( # PQ

>8 B * /",% 0"(#"% #"$"%"&"$" #%"( KEMK $ -!# F" $ - %"&"# "- h B C 5 B | 5C 5 5 5 C 5 5 B | C 55 5 5 5 C 5 5 B /"##" KEMK $ - #" " #"%% $ -! % e"8 SKET $( "( $ - "-(% SB 5T %" , &%" % SB 5T % -'"!7%! (! -0 ("- i = 1, . . . , I S% (! -0 8""- ,-%'0T ,-#"*

u p

= new

u p

−w old

i∈I

Pi ·

A˜i Bi

BiT 0

−1

ri qi

,

$( Pi "%" "(" ! Rn×m @(-" - PBQ 8'" ("," $ -79 -"%-" KE %1 $ (! ,(& <%" - ("" &%" i∈I Pi PiT = 1 (" KE #%"( !-!%! # =22%-'# &# KEMK #1 %"( " # "#7%0 "("-%" 1 0%%" 4 !0 -! - %80 B B -(' &' ,%%' , PBQ (! ,(& <%" "#"1 -" "&'# =#%# ,1?- - (&"# -(1 % F - %80 ",'-7% ( # "8 , "( #%"( O &" $ -790 % ("# %"&"# "-1 # B %" #"$"%"&' #%"( KE $ -!# 0"(%! 4 "# KE #%"( %8% - B1 , 8" $ -1 790 % (! "&! %"$" ",%! 0"(#"% &%" (#"%% KEKM #%"( /" "$ #%"(# e"8 3 1 N(! (! - "-1(22, #" " " (% &%" -7& "-%-'0 &"- - -! <%" , # ( (-#! #%"(# 8(% 9 ,&% /#' &%"- "#"97 #"$"%"&"$" #%"( " $ 1 -!# KEMK #" " % ( #"$0 8"% - PB5Q PQ

>8 B* /",% 0"(#"% #"$"%"&"$" #%"( KE $ -!# F" $ - %"&"# "- h B C 5 B | 5 5C 5B5 5 5 55 | C 5 5C 5B 5C 5 55 +,

?

&'

N# %# SB T - "89# -( A x = b 89! (! #%"1 ( ('0 % "%"% - "(8" #%' B %" &%" #% A B ("% "%" =22%-'0 $ -1 790 % -(* ym+1 = ym − C −1 (A B ym − b), x = B y. SB 5T (# , #'0 "%'0 ""8"- ,(! B !-!%! -'8" S(! ,(& <%" A = AT T >"$( AB = AAT 6 ##%1 &! "" %"1"(! #% %"( % e"8 - &%- $ -790 % (! &! B = AT 8' 1 #"% - PQ P Q 4 PQ 8'" 2%& ("," 0"%! 2"#"-" - -( SB 5CT %79 $ -79 -"%-"* B = AT

c h−2 Ah (I − wh2 Dh−1 Ah Dh−1 Ah )ν α→β ≤ √ 2ν + 1

wh = O(h2 ),

SB 5CT

$( ($"! #% Dh #% -(

Dh =

I 0

0 h2 I

.

! "- SB 5CT # =% " 0"(! 1 %" "# %# ,#' x = Dh y A¯ = Dh− Ah Dh− * 1 2

−2 ¯ − wh2 A¯2 )ν ≤ √c h A(I . 2ν + 1

1 2

1 2

SB 5T

¯ = A¯ 6 ##%&! #% $" "-% &%" wh−1 := ρ(A) −2 ¯ c h <("-%" _` (wh A) ∈ [−1, 1] SB 5T (%

,

−2 ¯ − w2 A¯2 )ν ≤ w−1 max |x(1 − x2 )ν | ≤ √c h A(I . h h x∈[−1,1] 2ν + 1

$# #"# !-!%! #%"( ('0 $ -1 3 1 N (! (" ' (%"# "# S T 2"#"-' - 2"# SB 5T W8# - PQ 4 &%- "%1"8"--79$" "%" B 8%!

B =

&%" (%

AB =

I 0

BT BB T

,

AB T + B T (BB T ) BB T

A B

.

SB 5T

> ($"' 8" #%' AB !-!7%! ,""(1 '# " BB T "#% #% (%, !1 "$" -! /" (! (-! /("" # &%" Aˆ R 6 "%"' "8"--% A BB T ""%-%%-" #"$% 8'% "&' # "#"97 HIJ ," 1 ! >"$( "8"--% C (! - %#' , SB 5T "(!%!

C=

Aˆ B

0 R

.

<(79# #"# !-!%! "( , 8" "!'0 ( "- $"%#"- 6 #%"( KHrLIv PBQ* ,('0 -%"0 uold pold (79 8 unew pnew 0"1 (%! , "" $"-* @% "# %"&7 27 u˜ (- "" $"%""$" #%"( (! ! %#' A˜ u = fh − B T pold .

@% "# %"&7 27 p˜ , %#' S%# 1 "#% "#7 -! /"T* BD−1 B T p ˜ = αB˜ u, SB T C

$( D ≈ A # D = bc\d (A) α 6 %"' #% <%# SB T % #" % %! 81 " "!* unew = u ˜ − α−1 D−1 B T p ˜,

pnew = pold + p ˜.

7% ,& # ( -%# #%"( "%"' "%&7%1 ! ($ "% ($ ""8# 8 "$" ! %#' 1"# $ -'8""# #%' D - SB T 4%' KHrLIv -%&7%! - %% "( ,'# ,-!#* KHrLIvh ($# (" =#%' ",'-7% &%" # KHrLIv - &%- $ - -$( -"(% %&" =221 %-"% #"$"%"&"$" #%"( , $ -790 -"%('0 % #" " % - PCQ PQ .%9 3> /",% &%" #%"( KHrLIv #" % 8'% ,1 - -( ('0 % SB 5T ?#"%%

B =

I 0

D−1 B T I

.

.%9 32 :E%6C%D= @ # ,(& S"("1 #! ,(& <%"T*

γu − u + p = f −u = 0

(0, 2π), γ ≥ 0 (0, 2π),

u(0) = u(2π) = ur , 2π p dx = 0 0

",% &%" KHrLIv #%"( D = bc\d (A) !-!%! 0""# $ -# γ h−2 :" $ D = A# > !" ' &

#"%% & "$( 8 pold "%&1 %! "% %"&"$" ! p "( $#"* p = pold + cos(kx) k = 1, 2, . . . ! "9! -'& #" " -,!% -" %"&" 3

+9

%&' $ =

%"( 8' (" ="# N"# - PQ /% D ≈ A # D = bc\d (A) %"$( $ -79 % - #7% -(*

ui+1 pi+1

=

ui pi

−1 !

i

/

u fh αD B T A BT − − B 0 B 0 pi 0

SB T O D = bc\d (A) %" SB T #" " %%"-% $ -! e"8 "%%- 2 u* Bu = 0. %-%" 1 (! %! SB T %8% ! -"#"$%" %#'

αD B

BT 0

v q

=

ri B ui

.

SB T

+, SB T -(" &%" B v = B ui ("-%" Bui+1 = B(ui − v) = 0 (! -0 i ≥ 0 "$&" %# <%" S# SBT SBBTT ,(& SB T #" % 8'% -( ,(& (! -"#"$%" #" q* α−1 B D−1 B T q = α−1 B D−1 ri − Bui .

SB BT

?#"%# "% & D = I 4 =%"# & SB BT -"(%! 7 ,(& (! "- (-! α−1 BB T q = g "%"! !-!%! (%'# "$"# -! /" +, SB T #" " "&% %" ""%" (! "8 - ""%* u − ui+1 = (I − B T (BB T )−1 B)(I − α−1 A)(u − ui ).

SB T

<""%" SB T $!(" ",'-% &%" "8 - ""% ,-% "% "8 - (- ('(9 %a %" P := I−B T (BB T )−1 B !-!%! "%"$"'# "1 %""# Ker(B) "=%"# P = 1 α ≥ λmax (A) , 5

SB T "(%-" (% SB T

u − ui+1 ≤ u − ui .

"-" (! "- $ -79$" -"%- !-!%! ##

31 α ≥ λ

" ( ! ! q

max (A)

A(u − uν ) + B T q ≤

α u − u0 , eν

ν = 1, 2, . . . .

) ! ?%" ""%" "8 - -%"

""% eν = u − uν #% -(*

eν = P (I − α−1 A)eν−1 .

SB T

4-(# "8",& M = P (I − α−1 A) P &%'-! -%-" P e1 = e1 , SB T "&# eν = M ν−1 e1 .

SB CT

4 %" -#! #! -" -# P eν = eν ## α(I − M )eν = αeν − P (αI − A)eν = P Aeν = Aeν + B T q,

SB T

$( q = −(B B T )−1 B Aeν +, SB CT SB T (% Aeν + B T q = α(I − M )M ν−1 e1 ≤ α(I − M )M ν−1 e1 .

SB 5T

% M ##%& % ##%&' #%' A P _` (M ) ∈ [0, 1] "=%"# " (I −M )M ν−1 ≤ (e ν)−1 (% , ##' 4 SB T #' ## e1 ≤ e0 &%" ,-% (",%%-" ##' N#%# &%" - %- ( ##' "!-!%! ,-%' -%" q ∈ Ph /"=%"# (! (",%%- $ -79$" -"1 %- ,-! (",%%- 0"(#"% #"$"%"&"$" #1 %"( "80"(# 9 "( "# %"&' $ $"%# (1 9 " $ - " ( " $8" %

$" # "# # u0 8 $" ! u " ! !) $ ! e0

X%"% $ "%"% - 0" ( -%" p˜ν ##,79$" "# Auν + B T p˜ν − fh = A(u − uν ) + B T (p − p˜ν ) = A(u − uν ) + B T q.

SB T

"$&" "(" $ -79 % =%"% $ %8% 1 ! ,(& % /" (! p˜ν * B B T p˜ν = B(fh − A uν ).

SB T

@(! %# "8,"# {uν , p˜ν } !-!%! ,%%"# $ - " #%"( =1N % -'"! ν %1 SB T 0" (! p˜ν , SB T /#! -" -# B(u − uν ) = 0 (! ν ≥ 1 α = O(h−2 ) "&# &%" (! "8 eν = u − uν qν = p − p˜ν ## B SB T -% A{eν , qν }β

1 = Aeν + B T qν 2 + h−2 Beν 2 2 = Aeν + B T qν 1 −2 −2 e0 2 + h2 q0 2 2 ≤ c hν e0 ≤ c hν =

c h−2 0 0 ν {e , q }α ,

&%" !-!%! "80"(#'# $ -79# -"%-"# SB 5T >1 $ -79 -"%-" (""!% -"%-" "# SB 55T 4 %8 B --7%! ",% 0"(#"% #"$"1 %"&"$" #%"( (-#! -(# $ - q0 ",'-% "1 =22% #! "8 ""%-%%-79 % X% ,%%' -,!%' , PQ $( " 8' "&' (! ,1 (& <%" "#"-" - (&"# -(% "1 #"97 "&'0 =#%"- SL c_pL1L T -"#" % +","-! ;1 1#! (1 1#! "%$ -!# +","- #$" & $ - =1N -"(% ,-!9 "% & %"&'0 "- 0"(#"% #%"( % -! %! (%-% " "%" $ - (! "8 F' ,%%' #1 %"("# KHrLIv "&' (! -'8" D = I [ - SB T % 1 ","-" D = I 4 "8"0 &!0 α 6 "%"! "%#! "%%

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?#"%# - (! (-! SBBT #% S Sp := B A−1 B T p = B A−1 fh .

SB T

# ,-%" , ##' B -'" Iww "-! #% S = B A−1 B T !-!%! "" %" "(" ##%&" %"1=--%" #% # M "%%# =--%"% ,-!9# "% h ! "&"1 "%"!'0 "&'0 =#%"- #% # M !-!%! (1 $"" ! "&'0 =#%"- 8" -'""$" "!( ,1 -%" S# PQT &%" ($"! #% Mˆ "&! , M ##"-# =#%"- " %"# %" =--% ˆ −1 M ) ≤ c "%%" c ,-!9 "% (#% M * Apqb (M ,8! h <("-%" Apqb (Mˆ −1 S) ≤ c μ(Ω)−2 /("" # &%" -",#" " % %# '0 $81 &0 - #% A -'"" %"&"%7 > #% A % ,%% (%, (-0 - /"1 S" "("# (! (" "#"%' ""%T %" #"$"%"&1 ' #%"( -" "(0"(% (! -'"! =%"$" ,(! #! % %# Av = r #' #" # -'&!% ",-( #%1 ' S ",-""$" -%" , Rm ,%% - (! (-! SBBT #" % 8'% 8 " " "#"97 "8"-"$" S M −1 Mˆ −1 - &%- "8"-1 -%!T %""$" #%"( @# #%"( "! '0 $(%"- %"" (- (" -%" ""% 1 0"(%! , -"$" -! %#' <%" %"( "%" % (! ! %#' SB T ,-% "( ,-# $"%# ,-' 8(% 0"(%! ("%%"&" #"# 1 #% w S# BT " 7 % %# Av = r ! ' ("# $ %""$" #%"( #" % 8'% ("-"" U("1 "$"V <9%-% "" #%"("- ",-"!790 ,#% -'1 & A−1 r ## "%""$" "8"--%! Aˆ−1 ( , %%$ 8,%! (79 ## ("," - PQ

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ν1 , ν2 , μ1 , μ2 F" ! Au

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SB T

V ' !

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SB T

<& ("" # &%" λ > ν2 >"$( "1 %" λAˆ − A #% -" !(" <("-%" " "8%# /## (λAˆ − A)−1 -"# -7 - SB T "&# (λAˆ − A)−1 B T p = u /#!! B "&# ) !

B(λAˆ − A)−1 B T p = λSˆ p.

# !" ",-( "80 &% -%- p +## (λAˆ − A)−1 B T p, B T p = λSˆ p, p.

"- SB T -&% λ λ S p, p ≤ (λAˆ − A)−1 B T p, B T p ≤ S p, p. μ2 μ1

8",&! v = B T p -"#! &%" S = BA−1 B T "&# λ λ (λAˆ − A)−1 v, v ≤ ≤ . μ2 A−1 v, v μ1

SB CT

4 %" -#! σmin ≤

(λAˆ − A)−1 v, v ≤ σmax , A−1 v, v

SB T

$( σmin σmax 6 !! -0!! $ (! "8%-'0 ,1 & ,(& (λAˆ − A)−1 v = σA−1 v.

X% ,(& #% % "8%-' ,&! &%" ,(& Av = σ(λAˆ − A)v.

! "( !! -0!! $ "(%-" "&1 7%! , "-! SB BT* σmin =

ν1 , λ − ν1

σmax =

ν2 . λ − ν2

> "8,"# " SB CT SB T -% &%" (! 78"$" "81 %-"$" ,&! ("-%-"!79$" λ > ν2 -'"" 8" 8"

ν1 λ ν2 ≤ ≤ , λ − ν1 μ2 λ − ν2

SB 5T

ν1 λ ν2 ≤ ≤ . λ − ν1 μ1 λ − ν2

SB T

?! SB 5T "&# (! λ > ν2 8

λ∈

ν1 +

ν12 + 4ν1 μ2 ν2 + , 2

"$&" " SB T -"(!% 8

λ∈

ν1 +

ν12 + 4ν1 μ1 ν2 + , 2

9 ν22 + 4ν2 μ2 . 2

9 ν22 + 4ν2 μ1 . 2

8Z( =%0 %-"- (% # -%"" #" %-" - SB T >% #" %-" "&%! "$&'# (!# (! &! λ < ν1 @" -" #" %-" "'-% "%-1 ! - #"%! & ν1 ≤ λ ≤ ν2 4",## (79 "8"--% (! SBT

B =

Aˆ−1 0

0 Sˆ−1

.

/",% 0"(#"% #%"( % "! '0 $(%"- S#1 %"( " # P Q "(0"(% (! ! %# ##%1 &'# " ,"1"('# %## #%" 1 #% % wcEM_o\]T - # 7 SBT #" " "% ,! "%%' ν1 , ν2 , μ1 , μ2 4 &%"% "1 ,%! "" "- #"$"%"&"$" #%"( (! "(! Aˆ−1 S# $2 T "(0"(!9 "8"--% (! S S# #% # (! "%%- (-! - & Iww %"&-'0 "&'0 =#%"-T %" =% "%%' ,-1 !% "% (#% ,8! h <("-%" ## $%% " 0"(#"% "8"-"$" #%"( ,-#7 "% $ % S(#% ,8!T >("#"% $"%# (! ! %#' ( ,(" %"&"%7 %% " -&# & ,-%'0 &%" !-!%! &#" ,-#"%7 <( 8"% "-!9'0 "8"-'# %"'# #%"(# (! ,(& <%" -'(# PQ $( 2"#"- "8"-- (% %# "" %" #""!1 " #% (! -",#" '# # #%"( "!1 '0 $(%"- - 8"%- #%"("- "80"(#! "%-!79! 6 =%" & "8"--% Aˆ Sˆ

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P Q 0-"- @< ("- @/ "8"- 3 * F' #%"(' "-* 8"%"! ,"-'0 N 555 PQ <! R* %"( "&'0 =#%"- (! =%&0 ,(& "-* C5 PQ \At]i_Az ;* rimoc1dlcb rjozpb_ \qb ``mcA\ocpq_ wjlmcq jcbjm]jld* K`lcqdjl C PBQ _jljqo\qo * gmb \qb qj ApqkjldjqAj `lpp_ pl simocdlcb sjozpb_ Ao\ isjlcA\ C6

# !!' ! 1-2 3 ! 4#5# 6 ! 78 99: .! %"%%*- &%;..'# 1.2 < =#># 8 )# 9 4! %;?"# 1@2 A 6#=# < ! 9 4! %;@*# 1?2 A ! A#B#! A! C#9# D ! , # 9 4! %;??# 1;2 5! =#=# E ! 9 4! %;?;# 1%"2 5 ! A#9# 9 ! - # ( ! 9 4! %;;,#

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