Mixed finite element technologies

CISM COURSES AND LECTURES Series Editors: The Rectors Giulio Maier - Milan Jean Salençon - Palaiseau Wilhelm Schneider...

Author: Peter Wriggers | Carsten Carstensen

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CISM COURSES AND LECTURES

Series Editors: The Rectors Giulio Maier - Milan Jean Salençon - Palaiseau Wilhelm Schneider - Wien

The Secretary General Bernhard Schreﬂer - Padua

Executive Editor Paolo Seraﬁni - Udine

The series presents lecture notes, monographs, edited works and proceedings in the ﬁeld of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series is to make known in the international scientiﬁc and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences.

INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No. 509

MIXED FINITE ELEMENT TECHNOLOGIES

EDITED BY CARSTEN CARSTENSEN HU BERLIN, GERMANY PETER WRIGGERS UNIVERSITY OF HANNOVER

This volume contains 55 illustrations

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned speciﬁcally those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 2009 by CISM, Udine Printed in Italy SPIN 12641321

All contributions have been typeset by the authors.

ISBN 978-3-211-99092-6 SpringerWienNewYork

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C. Carstensen 1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

−0.8 −1 −1

−0.8 −0.5

0

0.5

1

−1 −1

−0.5

1

1 0.8

0.6

0.6

0.4

0.4

0.2

0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

−0.8

0.5

1

0.5

1

0.5

1

−0.8 −0.5

0

0.5

1

−1 −1

−0.5

0

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

−0.8

−0.8 −0.5

0

0.5

1

−1 −1

−0.5

0

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

−0.8 −1 −1

1

0.2

0

−1 −1

0.5

0.8

−1 −1

0

−0.8 −0.5

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0.5

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)'30% :SJBOHVMBUJPOT % % % HFOFSBUFE CZ '3,+3 XJUI WJTJCMF CBMBODF PG MPDBM NFTISFOFNFOU UPXBSET UIF PSJHJO UIF SFFOUFSJOH DPSOFS PG UIF 2TIBQF EPNBJO BOE HMPCBM SFOFNFOU FWFSZXIFSF FMTF

Lectures on Adaptive Mixed Finite Element Methods

27

)'30% 6PTTJCMF SFOFNFOUT PG POF USJBOHMF * JO UIF TUFQ 8+,/4+ VQ UP SPUBUJPO PG UIF CBTFMJOF XIJDI JT DIPTFO GPS NBJOUBJOJOH TIBQF SFHVMBSJUZ

'!"$

#&(

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*' (6/%5+0/4 #/& %#/ $' %*04'/ 46%* 5*#5 . &

#/&

4F &

&'/05'& $:

300( )IPPTF

#/&

TVDI UIBU GPS BMM D

1 > 1D 4F &

UIFSF IPMET

28

C. Carstensen

0

10

adaptive refinement uniform refinement

−1

10

−2

10

2

3

10

4

10

5

10

6

10

10

)'30% )POWFSHFODF IJTUPSZ$ +TUJNBUFE FSSPS />> / PG '3,+3 EBTIFE MJOF BOE 3,+3 TPMJE MJOF WT OVNCFS PG EFHSFFT PG GSFFEPN JO MPHBSJUINJD TDBMF

:IFO ? $& 1 > JT EJWFSHFODFGSFF EJW ? & JO :IF -BVTT EJWFSHFODF UIFPSFN QSPWFT ? 4A &

'DDPSEJOH UP :IFPSFN XJUI ! & XF DBO OE ? & )VSM 4PUJDF UIBU GPS BMM OBMJUZ

BOE BMM

TVDI UIBU

UIFSF IPMET UIF PSUIPHP

1 )VSM 4F &

JO QBSUJDVMBS

1 )VSM 4F &

:IF TFDPOE TFU PG BOBMZUJDBM UPPMT DPODFSOT B QSJNBM GPSNVMBUJPO PG UIF NJYFE UZQF

Lectures on Adaptive Mixed Finite Element Methods

29

6!,/+% 6PJTTPO 6SPCMFN :IF VY > $& 1C # BOE C PG UIF 6PJTTPO QSPCMFN TPMWF UIF QSPCMFN 7D 4F #> C? D $& 1> ? 2> D 2? C & " GPS BMM ? D . & # PG UIF GPSN DPOTJEFSFE JO 9VCTFDUJPO XJUI > ? 4F BOE 2> D $& > 1D 4F 1> ? $&

:IF PQFSBUPS # $ . . EFOFE JO " JT CPVOEFE MJOFBS BOE CJKFDUJWF ' EJSFDU QSPPG PG BO JOGTVQ DPOEJUJPO GPS # GPMMPXT GPS BOZ > C # GSPN ? D $& > 1C% C # BOE /> C/ /? D/ />/ /C/ />/ /C/ />/ /C/ & #> C? D

:IF HFOFSBMJTFE 2BY3JMHSBN MFNNB UIFO ZJFMET CJKFDUJWJUZ PG # 6!,/+% 9UPLFT 6SPCMFN 2FU EFW $& US BOE US $&

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$ D 4F GPS D # % !

$& * # % $ US 4F & + BOE . $& 2FU # % ! # % ! :IF PQFSBUPS # $ . . EFOFE GPS C D . CZ # C D $& 1 2 D 2 C

JT MJOFBS CPVOEFE BOE CJKFDUJWF :IJT SFTVMU IPMET GPS * BOE * 6!,/+% 2JOFBS +MBTUJDJUZ 6SPCMFN -JWFO QPTJUJWF NBUFSJBM 2BNF QBSBNFUFST BOE MFU 1 $& $ 4F GPS # % !

30

C. Carstensen

SFQMBDF # BOE BEPQU 2 % D GSPN # :IFO UIF PQFSBUPS # $ . . GSPN JT MJOFBS CPVOEFE BOE CJKFDUJWF BOE UIF PQFSBUPS OPSNT PG # BOE # BSF JOEFQFOEFOU ' QSPPG JT JODMVEFE JO (SBFTT FU BM :IF PQFSBUPS # SFQSFTFOUT UIF XFBL GPSN PG UIF .FMMJOHFS8FJTTOFS QSJODJQMF JO NFDIBOJDT

( *"( ''%' %$)'%"

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$&

8 /?> A /

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EFOF

PTD6 % $&

8 /6

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#

#

Lectures on Adaptive Mixed Finite Element Methods

31

XJUI TPNF QSPKFDUJPO PQFSBUPST BOE :IF TBNF DPOTUSVDUJPO JT QPTTJCMF GPS (*,3 BOE 3,+3T BT XFMM )BSTUFOTFO ### BT EFQJDUFE JO :BCMF +MFNFOU 8: (*3 (*,3

*? &

$ * & F & & $ ? =. # * +

* & & &

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BOE />

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(Z PSUIPHPOBMJUZ PG 1 BOE )VSM BOE BO JOUFHSBUJPO CZ QBSUT UIFSF IPMET 1 > > 4F /1/ & & EJW> > 4F

32

C. Carstensen

4PUF UIBU UIFSF BSF OP CPVOEBSZ UFSNT CFDBVTF WBOJTIFT PO 9JODF EJW> > & 7 7 BOE XJUI UIF QJFDFXJTF DPOTUBOU JOUFHSBM NFBO PG UIFSF IPMET EJW> > 4F & 7 7 4F PTD7 % 8 ' 6PJODBSF JOFRVBMJUZ QSPWFT 8 /1/

:IF DPNCJOBUJPO PG QSPWFT *FOF BT TPNF JOUFSQPMBUJPO PG TVDI UIBU & % BOE )VSM & % BOE )VSM 1 JF 1D )VSM 4F & GPS BMM D

XIFODF EJW )VSM & :IF TFDPOE LFZ PCTFSWBUJPO SFBET )VSM )VSM 4F &

(Z PSUIPHPOBMJUZ PG 1 BOE )VSM UIFSF IPMET )VSM )VSM 4F & > > )VSM 4F

'O JOUFHSBUJPO CZ QBSUT GPS > & 1C BOE UIF EJTDSFUF FRVBUJPOT GPS ? & )VSM (* % XJUI EJW ? & TIPX UIBU UIF MBTU FYQSFTTJPO FRVBMT > > )VSM 4F & C C EJW )VSM 4F &

'MUPHFUIFS POF EFEVDFT :IF UIJSE LFZ BTTFSUJPO SFBET /)VSM / ,SPN BOE UIF PSUIPHPOBMJUZ PG 1 BOE )VSM POF IBT /)VSM / & > > )VSM 4F

Lectures on Adaptive Mixed Finite Element Methods

33

:IJT IPMET GPS BMM $ % XIJDI XF NBZ DIPPTF BT TPNF )MFNFOUUZQF BQQSPYJNBUJPO PG XIJDI DBO CF QSPWFO )MFNFOU #!% )BSTUFOTFO BOE ,VOLFO UP TBUJTGZ / / 8 /1/

BOE

/ / 8 /1/

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> > )VSM 4F & DVSM> > 4F

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8 /?> A / /)VSM /

!

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34

C. Carstensen

2FU $ . . CF B MJOFBS BOE CPVOEFE PQFSBUPS CFUXFFO UIF SFBM .JMCFSU TQBDFT . BOE JUT EVBM . 9VQQPTF UIBU JT TVSKFDUJWF BOE JOKFDUJWF TVDI UIBU JT CPVOEFE BT XFMM /O QBSUJDVMBS HJWFO G . UIFSF FYJTUT B VOJRVF F . TVDI UIBU F & G " 9VQQPTF XF BSF HJWFO TPNF OJUF FMFNFOU BQQSPYJNBUJPO F . QPTTJCMZ XJUI TPNF BEEJUJPOBM QSPQFSUJFT HFOFSBUFE CZ UIF DPNQVUBUJPOBM TDIFNF UIBU QSPWJEFE F :IFO XF BEESFTT UIF JTTVF PG BQQSPYJNBUJOH UIF FSSPS /F F / 4PUJDF UIBU UIF MJOFBS GVODUJPOBM #5A $& G F & F F .

#

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Lectures on Adaptive Mixed Finite Element Methods

35

:IJT JT UIF TUBSUJOH QPJOU PG UIF VOJGZJOH UIFPSZ :IF GBDU UIBU C] IBT UP CFMPOH UP XIJMF C EPFT OPU JT TPNF DSVDJBM QPJOU JO UIF TFRVFM /U TIPVME CF OPUJFE DMFBSMZ UIBU JT B 13+.#- .+9'& GPSNVMBUJPO BMTP DBMMFE IZCSJE JO (SF[[J BOE ,PSUJO ## XIFSF # JT /05 B TVCTQBDF PG EJW% :IJT JT CFDBVTF UIF EFSJWBUJWFT BDU PO C & ,PS UIF 6PJTTPO FRVBUJPO UIF CJMJOFBS GPSN 2 SFBET ? 1D 4F 2? D &

XJUIPVU BO JOUFHSBUJPO CZ QBSUT BT GPS UIF EVBM NJYFE GPSNVMBUJPO BOE TJNJMBS FYQSFTTJPOT IPME GPS 9UPLFT BOE 4BWJFS2BN F FRVBUJPOT EFTDSJCFE JO +YBNQMF \ :IJT MFDUVSF FNQIBTJTFT UIBU UIFSF JT FTTFOUJBMMZ POMZ POF UZQF PG SFTJEVBM UIBU BSJTF JO B QPTUFSJPSJ FSSPS DPOUSPM$ -JWFO TPNF 7 # BOE TPNF 7 # ( POF FODPVOUFST UIF MJOFBS GVODUJPOBM #5A $ XSJUUFO #5A EFOFE CZ #5AD $& 7D 4F 7 D 4A

XJUI UIF -BMFSLJO QSPQFSUZ $ % $& & % )

LFS #5A

:IF OPSN PG #5A FOUFST BT BO FYQMJDJU CPVOE JO UIF GPSN PG /#5A/ $&

TVQ $

#5AD/1D/

:P Y JEFBT TVQQPTF OPUBUJPO GSPN 9FDUJPO ,PS BOZ > # $& # BOE C ] $& UIFSF IPMET GPS #5A # BOE #5A EFOFE GPS BMM ? # BOE D CZ ? 1] C > 4F BOE #5A ? $& #5A D $& 7D 4F > 1D 4F

9JODF UIF EJTDSFUF TPMVUJPO C PG UIF 3,+3 JT JO HFOFSBM OPU JO UIF ] / GPS TPNF GPSNVMBUJPO BCPWF JT BQQMJFE UP FTUJNBUF /> > / /C C C ] EJFSFOU GSPN C :IF FWBMVBUJPO PG #5A GPMMPXT XJUI B 6PJODBSF JOFRVBMJUZ

36

C. Carstensen

2FU 7 & % EFOPUF UIF QJFDFXJTF JOUFHSBM NFBO PG 7

:IFO

7 7 & EJW> > :IJT BOE BO JOUFHSBUJPO CZ QBSUT GPMMPXFE CZ UIF 6PJODBSF JOFRVBMJUZ TIPX > 1D 4F 7D 4F #5A D & & 7 EJW > D 4F & 7 7 D D 4F 8 17 /1D/ .FODF IPU BCCSFWJBUFT IJHIFS PSEFS UFSNT /#5A / 8 17 & IPU JT PG RVBESBUJD XIFODF PG IJHIFS PSEFS XIFO DPNQBSFE UP /> > / GPS (* ] EFOPUFT BO BSCJUSBSZ OPU ,PS UIF FWBMVBUJPO PG /#5A / SFDBMM UIBU C OFDFTTBSJMZ EJTDSFUF FMFNFOU JO %-)2).- -JWFO > # EFOF UIF MJOFBS GVODUJPOBM #5A $ CZ #5A D $& > )VSM D 4F

GPS D

9FU /#5A / $&

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> )VSM D 4F/ )VSM D/

XIFSF % SFGFST UP 4PO)POGPSNJOH TFF 9VCTFDUJPO " GPS NPSF EFUBJMT (%.0%, 03 #/: > )'/'3#5'& $: #/ 0( #$-' 5*'3' *0-&4 NJO /> 1] C / & /#5A /

# "

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$ % LFS #5A EJW% .FODF

EJW #

Lectures on Adaptive Mixed Finite Element Methods

37

:IF MBTU JODMVTJPO IPMET GPS UIF 3,+3 TQBDFT PG :BCMF :IVT ? NBZ CF DPOTJEFSFE JO EJTDSFUF 3,+3 BOE TIPXT > )VSM D 4F & C EJW )VSM D 4F & #5A D &

:IF NBJO PCTFSWBUJPO JT UIBU TJODF > )VSM D & >] 1D GPS TPNF SPUBUFE >] BOE XJUI JOUFHSBUJPO CZ QBSUT #5A IBT UIF GPSN :IVT UIF BOBM ZTJT PG FH 'JOTXPSUI BOE 5EFO % (BCV TLB BOE 9USPVCPVMJT % ( $& /8 17/ > $&

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$& C > # $& # % ! BOE GPS C

% BOE > #

$& C > #

,PS MJOFBS FMBTUJDJUZ )BSTUFOTFO BOBMZTFT SPCVTU BQQSPYJNBUJPOT XJUI B NJYFE OJUF FMFNFOU NFUIPE BOE FTUBCMJTIFT GPS BOZ C ] UIBU ] / /] C TZN / /#5A / / TZN / /C C

38

C. Carstensen

:IJT FTUJNBUF MFBET UP / / /#5A / /#5A / / TLFX / BOE IFODF UP TJNJMBS FTUJNBUF BT JO UIF MJUFSBUVSF

''%' *) %$ $

:IJT MFDUVSF JT EFWPUFE UP UIF FSSPS SFEVDUJPO QSPQFSUZ PG BEBQUJWF NJYFE OJUF FMFNFOU NFUIPET '3,+3 GPS UIF 6PJTTPO 6SPCMFN ;OMJLF GPS DPOGPSNJOH ',+3 UIFSF JT OP GVSUIFS SFTUSJDUJPO MJLF JOOFSOPEF QSPQFSUJFT FUD JO 8+,/4+ :IF SFTVMUT PG UIJT MFDUVSF BSF UBLFO GSPN )BSTUFOTFO BOE .PQQF :ISPVHIPVU UIJT MFDUVSF % BOE % EFOPUF UXP TIBQFSFHVMBS USJBOHVMBUJPOT PG JOUP USJBOHMFT XIFSF % JT TPNF SFOFNFOU PG % TVDI UIBU UIF SFOF NFOU * . $& *" % $ " * + PG FBDI FMFNFOU * JO % JT EFQJDUFE JO ,JHVSF " 3PSFPWFS MFU > (* % EFOPUF UIF EJTDSFUF 3,+3 TPMVUJPO PO UIF DPBSTF USJBOHVMBUJPO % :IF TVCJOEJDFT BOE 8 SFGFS UP UIF DPBSTF BOE OF USJBOHVMBUJPO % BOE % SFTQFDUJWFMZ FH > > > EFOPUF UIF FYBDU BOE EJTDSFUF VYFT PG (* ',+3 PG 9VCTFDUJPO XJUI SFTQFDU UP UXP DPOTFDVUJWF NFTIFT % BOE % (%.0%, +SSPS 8FEVDUJPO 6SPQFSUZ

*'3' *0-&4

/> > / /> > / PTD / 6 / (03 40.' 104+5+7' %0/45#/54 #/& 8*+%* '9%-64+7'-: &'1'/& 0/ 5*' .+/+.#- #/)-' +/ % #/& % #/& 8+5* 6 & % &'/'& $: 6 $& 6 #/& . $& EJBN* (03 * % #/& PTD $&

8 /6 6 /

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Lectures on Adaptive Mixed Finite Element Methods

39

:IFSFGPSF UIF KVNQ ?> A BMPOH & * ) * JO UIF UBOHFOUJBM EJSFDUJPO BCCSFWJBUFE ?> A JO UIF TFRVFM TBUJTFT UIBU /?> A/ & /?> A / 9JODF ?> A JT B TDBMBS B OF GVODUJPO BMPOH UIF FEHF JU SFBET ?> A F & F NJE GPS BMM F XJUI YFE & ?> A 4A8 BOE 8 & EJBN 0./.1)2).- *JTDSFUF 2PDBM + DJFODZ

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140

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146

P. Wriggers

HH AMQ=PEKJO OP=PA@ =>KRA ?=J >A PN=JOBKNIA@ PK PDA ?QNNAJP ?KJCQN= PEKJ SDANA PDA ?KJOPEPQPERA AMQ=PEKJO =NA BKNIQH=PA@ EJ PANIO KB PDA "=Q?DU OPNAOO PAJOKN =J@ PDA *EN?DDK OPNAOO PAJOKN *EJAI=PE?O

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!U L=NPE=H EJPACN=PEKJ KB PDA NOP PANI =LLHE?=PEKJ KB PDA @ERANCAJ?A PDA KNAI =J@ EJPNK@Q?PEKJ KB PDA PN=?PEKJ >KQJ@=NU ?KJ@EPEKJ KJA K>P=EJO PDA SA=G BKNI KB HEJA=N IKIAJPQI ! 2 < 2+ &N=@ 2+ #

3DA CN=@EAJP KB PDA PAOP BQJ?PEKJ ?=J =HOK >A EJPANLNAPA@ =O R=NE=PEKJ KB PDA @ABKNI=PEKJ CN=@EAJP (J PDA SA=G BKNI PDA NOP /EKH= *EN?DDK OPNAOO PAJOKN ?=J >A NALH=?A@ PDNKQCD >U PDA OA?KJ@

Mixed Finite-Element-Methods

147

/EKH=*EN?DDK OPNAOO PAJOKN HA=@EJC PK &N=@ &N=@ &N=@ &N=@ SDANA PDA B=?P D=O >AAJ QOA@ PD=P PDA O?=H=N LNK@Q?P KB = OUIIAPNE?=H PAJOKN DANA SEPD = =JPEOUIIAPNE?=H L=NP KB = PAJOKN EO VANK @AJKPAO PDA R=NE=PEKJ KB PDA &NAAJ+=CN=JCA OPN=EJ PAJOKN 4OEJC AMQ=PEKJ ?=J >A NASNEPPAJ =O # ! 2 < 2+ 2+

3DA NOP PANI EJ @AJKPAO PDA EJPANJ=H RENPQ=H SKNG =HOK ?=HHA@ OPNAOO @ERANCAJ?A PANI 3DA H=OP PSK PANIO @AO?NE>A PDA RENPQ=H SKNG KB PDA =LLHEA@ HK=@EJC =J@ PDA EJANPE= PANI !& "*-( *" '%)!- (*(!)/0( 0--!)/ *)#0-/%*) 3DA PN=JO BKNI=PEKJ KB PDA SA=G BKNI PK PDA ?QNNAJP KN OL=PE=H ?KJCQN=PEKJ EO LAN BKNIA@ >U GEJAI=PE?=H KLAN=PEKJO EJ SDE?D PDA >=OA RA?PKNO LQOD BKNS=N@ PK PDA ?KJCQN=PEKJ 6EPD PDA PN=JOBKNI=PEKJ KB PDA NOP /EKH=*EN?DDK OPNAOO PAJOKN PK PDA "=Q?DU OPNAOO PAJOKN OAA 3=>HA PDA RENPQ=H SKNG ?=J >A BKNIQH=PA@ =O &N=@ ! &N=@ ! &N=@ ! CN=@ %QNPDANIKNA NAH=PEKJO 2> ! 2+ ! AJ=>HA PK SNEPA PDA SA=G BKNI EJ PANIO KB PDA ?QNNAJP ?KJCQN=PEKJ # ;! 2/ < 2> 3 CN=@ 2>

(J PDEO NAH=PEKJO AMQ=PEKJ D=O >AAJ QOA@ PK PN=JOBKNI PDA PN=?PEKJ RA? PKN ! PK 3DA OUIIAPNU KB PDA "=Q?DU OPNAOO PAJOKN B=?EHEP=PAO PDA NALH=?AIAJP KB PDA OL=PE=H CN=@EAJP KB PDA PAOP BQJ?PEKJ >U EPO OUIIAPNE? L=NP 'AJ?A SEPD PDA @AJEPEKJ ,

CN=@ CN=@

PDA SA=G BKNI SEPD NAOLA?P PK PDA OL=PE=H ?KJCQN=PEKJ BKHHKSO # ;! 2/ < 2> 3 , 2>

148

P. Wriggers

3DEO NAH=PEKJ EO EJ = BKNI=H OAJOA AMQER=HAJP PK PDA LNEJ?ELHA KB RENPQ=H SKNG KB PDA CAKIAPNE?=HHU HEJA=N PDAKNU !QP DANA PDA EJPACN=H PDA OPNAOO =J@ RENPQ=H OPN=EJ IA=OQNAO D=RA PK >A AR=HQ=PA@ SEPD NAOLA?P PK PDA ?QNNAJP ?KJCQN=PEKJ #QA PK PDEO PDA JKJHEJA=NEPEAO @K =LLA=N DKSARAN DE@@AJ (J PDA BQNPDAN R=NE=PEKJ=H BKNIQH=PEKJO LNAOAJPA@ EJ PDEO OA?PEKJ SA SEHH JACHA?P PDA EJANPE= PANIO #< =J@ SEHH ?KJ?AJPN=PA KJ OP=PE? AMQEHE>NEQI AMQ= PEKJO !& "*-( %) /$! #!*(!/-%''4 '%)!- .! %KN PDA CAKIAPNE?=H HEJ A=N PDAKNU PDA SA=G BKNI EO BKNI=HHU AMQER=HAJP PK .JHU PDA KLAN=PKN @AJA@ EJ D=O PK >A AR=HQ=PA@ SEPD NAOLA?P PK PDA ?KKN@EJ=PAO KB PDA EJEPE=H ?KJCQN=PEKJ =O SAHH =O PDA EJPACN=HO 2> ;! 2/ 3 " $ " 2>

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.B =HH LKOOE>HA @ABKNI=PEKJO PDA KJAO SDE?D I=GA OP=PEKJ=NU BQHH PDA AMQEHE>NEQI AMQ=PEKJ 3DA OP=PEKJ=NU R=HQA KB ?=J >A ?KILQPA@ >U PDA R=NE=PEKJ KB SEPD NAOLA?P PK PDA @ABKNI=PEKJ %KN PDEO LQNLKOA SA EJPNK@Q?A PDA @ENA?PEKJ=H @ANER=PERA 2 2

Mixed Finite-Element-Methods

149

SDE?D EO =HOK ?=HHA@ NOP R=NE=PEKJ KB 3DA =LLHE?=PEKJ KB PDEO I=PDAI=PE?=H KLAN=PEKJ UEAH@O , : 2+ ! 2 " 9

3DA @ENA?PEKJ=H @ANER=PERA KB PDA NECDP "=Q?DU&NAAJ OPN=EJ PAJOKN ?=J A=OEHU >A SNEPPAJ EJ PANIO KB PDA &NAAJ+=CN=JCA OPN=EJ PAJOKN 2 2 2 9&N=@ : &N=@ 2

&N=@ &N=@

3DA H=OP NAOQHP ?KNNAOLKJ@O PK PDA R=NE=PEKJ @AJA@ =HNA=@U 3DA L=NPE=H @ANER=PERA KB , SEPD NAOLA?P PK HA=@O PK PDA OA?KJ@ /EKH=*EN?DDK OPNAOO PAJOKN OAA , 'AJ?A AMQ=PEKJ EO AMQER=HAJP PK PDA SA=G BKNI BKN = DULANAH=OPE? I=PANE=H 3DA ?KJOPNQ?PEKJ KB OQ?D LNEJ?ELHA D=O OARAN=H =@R=JP=CAO EP ?=J >A PDA >=OEO BKN I=PDAI=PE?=H EJRAOPEC=PEKJO NAC=N@EJC ATEOPAJ?A =J@ QJEMQAJAOO KB OKHQPEKJO EP HA=@O PK PDA @ARAHKLIAJP KB A?EAJP =HCKNEPDIO BKN PDA OKHQPEKJ KB PDA JKJHEJA=N AMQ=PEKJO KJ PDA >=OEO KB KLPEIEV=PEKJ OPN=PACEAO 0.$%50 +-%)%+'! "*- )%/! !"*-(/%*). JKPDAN R=NE=PEKJ=H LNEJ?ELHA EO PDA 'Q6=ODEVQ LNEJ?ELHA OAA 6=ODEVQ (P D=O C=EJA@ OECJE?=J?A @QNEJC PDA H=OP UA=NO BKN PDA ?KJOPNQ?PEKJ KB JEPA AHAIAJPO 3DEO LNEJ?ELHAO ?=J >A @ANERA@ >U SNEPEJC PDA SA=G BKNIQH=PEKJ SEPD =@@EPEKJ=H ?KJOPN=EJP AMQ=PEKJO SDE?D ?KJP=EJ GEJAI=PE?O =J@ ?KJOPEPQPERA AMQ=PEKJO #QA PK PDEO @ABKNI=PEKJO OPN=EJO =J@ OPNAOOAO K??QN =O EJ@ALAJ@AJP R=NE =>HAO "KJPN=NU KJ?A PDA LNEJ?ELHA EO ?KJOPNQ?PA@ EPO R=NE=PEKJ SEPD NAOLA?P PK =HH R=NE=>HAO UEAH@O PDA OP=PE? AMQEHE>NEQI AMQ=PEKJO PDA GEJAI=PE?=H NA H=PEKJO =J@ PDA ?KJOPEPQPERA AMQ=PEKJ 3DA BKNIQH=PEKJ KB PDA JKJHEJA=N RANOEKJ KB PDA 'Q6=ODEVQ LNEJ?ELHA ?=J >A CERAJ >U =JU OAP KB SKNG ?KJFQ C=PA@ R=NE=>HAO 'ANA SA SEHH OP=PA EP EJ PANIO KB PDA @ABKNI=PEKJ CN=@EAJP PDA NOP /EKH=*EN?DDK OPNAOO PAJOKN =J@ PDA #ABKNI=PEKJ 9 , &N=@ : 2+

2+

;! 2

150

P. Wriggers

3DA R=NE=PEKJ =??KN@EJC PK PDA @AJEPEKJ KB PDA @ENA?PEKJ=H @ANER=PERA CERAJ =>KRA UEAH@O JKS PDNAA EJ@ALAJ@AJP AMQ=PEKJO

2+ &N=@

;! 2

&N=@ 2+

, 2+

SDANA =J@ =NA PDA RENPQ=H @EOLH=?AIAJPO PDA RENPQ=H OPNAOOAO =J@ PDA RENPQ=H OPN=EJO NAOLA?PERAHU .>OANRA PD=P PDAU NALNAOAJP PDA SA=G BKNI PDA GEJAI=PE?=H NAH=PEKJ =J@ = DULANAH=OPE? ?KJOPEPQPERA AMQ=PEKJ BKN -KPA BQNPDAN PD=P PDA H=OP PSK PANIO ?=J >A EJPANLNAPA@ =O ?KJOPN=EJP PANIO SDE?D D=RA PK >A BQHHHA@ =@@EPEKJ=HHU PKCAPDAN SEPD PDA NOP AMQ=PEKJ (J PD=P ?=OA =J@ =NA +=CN=JCE=J IQHPELHEANO 3DA R=NE=PEKJ=H LNEJ?ELHA ?=J =HOK >A SNEPPAJ EJ PANIO KB PDA OA?KJ@ /EKH=*EN?DDK OPNAOO =J@ PDA &NAAJ+=CN=JCE=J OPN=EJO "

;

" : 2+ 9,

2+ "

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SDANA " SEPD &N=@ " OAA 3=>HA (PO R=NE=PEKJ BKHHKS =J=HKCKQO PK 0.$%50 +-%)%+'! "*- .('' !"*-(/%*). 3DA 'Q6=ODEVQ LNEJ ?ELHA S=O KNECEJ=HHU @ANERA@ EJ 6=ODEVQ BKN OI=HH @ABKNI=PEKJO (P PDAJ EO OP=PA@ EJ PANIO KB PDA OPN=EJ PAJOKN PDA NOP OPNAOO PAJOKN =J@ PDA @EOLH=?AIAJP "

"

PN " : 2+ 2+ ;! 2

9

Mixed Finite-Element-Methods

151

3DA R=NE=PEKJ =??KN@EJC PK PDA @AJEPEKJ KB PDA @ENA?PEKJ=H @ANER=PERA CERAJ =>KRA UEAH@O JKS PDNAA EJ@ALAJ@AJP AMQ=PEKJO

"

2+

"

;! 2

" 2+

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9 PN : 2+

SDANA =J@ =NA PDA RENPQ=H @EOLH=?AIAJPO PDA RENPQ=H OPNAOOAO =J@ PDA RENPQ=H OPN=EJO NAOLA?PERAHU .>OANRA PD=P PDAU NALNAOAJP PDA SA=G BKNI PDA GEJAI=PE?=H NAH=PEKJ =NA CERAJ EJ 3=>HA =J@ PDA HEJA=N AH=OPE? ?KJOPEPQPERA AMQ=PEKJ BKN BKHHKSO BNKI 3=>HA 0.$%50 +-%)%+'! "*- %)*(+-!..%%'%/4 OLA?E=H BKNI KB PDA 'Q6=ODEVQ R=NE=PEKJ=H LNEJ?ELHA ?=J >A =LLHEA@ BKN PDA ?KJOPNQ?PEKJ KB JEPA AHAIAJPO SDE?D D=RA PK NALNAOAJP JA=NHU EJ?KILNAOOE>HA I=PANE=H >AD=REKQN 2EJ?A EJ?KILNAOOE>EHEPU EO =OOK?E=PA@ SEPD = ?KJOPN=EJP BKN PDA RKHQIAPNE? @ABKNI=PEKJ ! KJA ?=J QOA PDA OLHEP PK @EOPEJCQEOD RKHQIAPNE? =J@ EOK?DKNE? L=NPO KB PDA @ABKNI=PEKJ !=OA@ KJ PDEO E@A= 2EIK AP =H BKNIQH=PA@ = PDNAAAH@ BQJ?PEKJ=H SDE?D EO KJHU @AJA@ BKN PDA RKHQIAPNE? L=NP KB PDA @ABKNI=PEKJ 'AJ?A PDA EJ@ALAJ@AJP R=NE=>HAO =NA JKS PDA @A BKNI=PEKJ PDA LNAOOQNA 8 =J@ = OPN=EJ R=NE=>HA SDE?D EO AMQER=HAJP PK ! 3DA H=OP R=NE=>HA D=O PK BQHH PDA ?KJOPN=EJP ?KJ@EPEKJ ! 6EPD PDA IQHPELHE?=PERA OLHEP KB PDA @ABKNI=PEKJ CN=@EAJP

KJA K>P=EJO EJ PDEO ?=OA PDA OLHEP EJPK RKHQIAPNE? =J@ @ARE=PKNE? L=NPO ! &N=@ %QNPDANIKNA -KPA PD=P SA D=RA EJ NAH=PEKJ DKH@O SEPD (J PDEO R=NE=PEKJ=H LNEJ?ELHA =HOK PDA ! OPN=EJ AJANCU BQJ?PEKJ , OAA D=O PK >A @AJA@ KJ PDA >=OEO 4OEJC PDA =@@EPERA OLHEP , KB PDA JAS R=NE=>HAO , , OAA PDA BKHHKSEJC PDNAAAH@ R=NE=PEKJ=H BQJ?PEKJ=H ?=J , ,

152

P. Wriggers

>A ?KJOPNQ?PA@ 8

, 8 ! : 2+ 9 ,

2+

;! 2

!U ?KJOE@ANEJC PDA NAH=PEKJ PDA $QHAN+=CN=JCA AMQ=PEKJO K>P=EJA@ BNKI PDEO R=NE=PEKJ=H LNEJ?ELHA =NA 8

9

;! 2

8 9

, 2+ : 8 !

9 ! 2+

8

, 8 2+

'ANA 9 =J@ =NA PDA RENPQ=H @ABKNI=PEKJO RENPQ=H LNAOOQNA =J@ RENPQ=H RKHQIAPNE? OPN=EJO NAOLA?PERAHU !U ?KIL=NEOKJ KB NAH=PEKJ SEPD KJA ?=J ?KJ?HQ@A PD=P KJA ?=J QOA EJ AMQ=PEKJ =HOK PDA ATLNAOOEKJ BKN PDA NOP PANI EJ PDA EJPACN=H 3DEO ATLH=EJO PDA OLHEP EJPK EOK?DKNE? =J@ RKHQIAPNE? L=NPO 3DA PANIO EJ AMQ=PEKJ =NA KBPAJ SNEPPAJ SEPD NAOLA?P PK PDA OL=PE=H ?KJCQN=PEKJ OEJ?A PDEO OEILHEAO PDA JQIANE?=H EILHAIAJP=PEKJ SEPDEJ PDA JEPA AHAIAJP IAPDK@ 6EPD PDA R=NE=PEKJ KB PDA "=Q?DU&NAAJ PAJOKN &N=@ &N=@ CN=@ CN=@

KJA K>P=EJO SEPD PDA @AJEPEKJ 8

9

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Mixed Finite-Element-Methods

153

3DA J=H NAOQHP BKHHKSO SEPD =J@ 2+

(% , +%" @ER 8

;! 2

9 ! 2+

8 9

8

, 8 2+

(J PDEO BKNIQH=PEKJO PDA EJPACN=J@O =NA CERAJ SEPD NAOLA?P PK PDA OL=PE=H KN ?QNNAJP ?KJCQN=PEKJO SDANA=O PDA EJPACN=PEKJ OPEHH EO LANBKNIA@ EJ PDA EJEPE=H ?KJCQN=PEKJ 3DA NOP AMQ=PEKJ @AJKPAO PDA SA=G BKNI KB PDA AMQEHE>NEQI SEPD PDA *EN?DDK OPNAOOAO EJOPA=@ KB PDA "=Q?DU OPNAOOAO 3DA OA?KJ@ AMQ=PEKJ NALNK@Q?AO PDA ?KJOPN=EJ ?KJ@EPEKJ ! =J@ PDA PDEN@ AMQ=PEKJ UEAH@O PDA ?KJOPEPQPERA AMQ=PEKJ BKN PDA LNAOOQNA 8 OAA =HOK !''%)#!-!%..)!- +-%)%+'! .('' ./-%). 3DA 'Q6=ODEVQ LNEJ?E LHA EO PDA IKOP CAJAN=H LNEJ?ELHA SDE?D @ALAJ@O KJ PDNAA EJ@ALAJ@AJP AH@O !=OA@ KJ PDA BQHHHIAJP KB PDA ?KJOPN=EJPO SA ?=J AHEIEJ=PA KJA KB PDA ?KJOPN=EJPO @ENA?PHU 3DEO UEAH@O AC = PSKAH@ BKNIQH=PEKJ SEPD @EOLH=?A IAJPO =J@ OPNAOOAO =O LNEI=NU R=NE=>HAO 3DA =@AMQ=PA 'AHHEJCAN1AEOOJAN BQJ?PEKJ=H ?=J >A SNEPPAJ BKN OI=HH OPN=EJO =O ; 2+ ;!" 2 " 2+ 9: 2+ " "

'ANA EO PDA ?KJOPEPQPERA PAJOKN SDE?D NAH=PAO OPN=EJO =J@ OPNAOOAO EJ ?=OA KB HEJA=N AH=OPE?EPU OAA3=>HA 9 " : 9 " :

; =J@ ;! =NA PDA =LLHEA@ >K@U BKN?AO =J@ >KQJ@=NU PN=?PEKJO NAOLA?PERAHU 3DA R=NE=PEKJ KB PDEO R=NE=PEKJ=H BQJ?PEKJ=H UEAH@O ; 2+ ;! 2 2+

" 2+

: 2 9

154

P. Wriggers

SDANA =J@ =NA PDA R=NE=PEKJO KB PDA OPNAOOAO =J@ PDA @EOLH=?AIAJPO NAOLA?PERAHU 3DA NOP AMQ=PEKJ NALNAOAJPO PDA SA=G BKNI KB AMQEHE>NEQI 3DA OA?KJ@ AMQ=PEKJ EO PDA SA=G OP=PAIAJP KB PDA ?KJOPEPQPERA AMQ=PEKJ (P ?KQH@ =HOK >A SNEPPAJ =O " 9 : 2+

OEJ?A PDA H=OP PANI EO JKPDEJC AHOA PDAJ PDA EJRANOEKJ KB 'AJ?A PDA PSKAH@ 'AHHEJCAN1AEOOJAN LNEJ?ELHA D=O >AOE@AO PDA SA=G BKNI KB AMQEHE> NEQI KJA =@@EPEKJ=H ?KJOPN=EJP AMQ=PEKJ SDE?D EO PDA EJRANOA KB ?KJOPEPQPERA AMQ=PEKJ ,KNA CAJAN=H PDA NAH=PEKJ OPAIIEJC BNKI PDA +ACAJ@NA PN=JOBKNI=PEKJ , * EO R=HE@ (J HEJA=N AH=OPE?EPU PDA OPN=EJ AJANCU D=O PDA BKNI , SDE?D UEAH@O SEPD *

9:

9 :

'AJ?A EP ?=J >A OAAJ PD=P PDA NOP PSK PANIO EJ =NA AMQER=HAJP PK PDA EJPACN=H KB PDA OPN=EJ AJANCU , 0' (%3! "*-(0'/%*) JKPDAN RANOEKJ D=O =O KJHU QJGJKSJO PDA OPNAOOAO 3DEO EO >=OA@ KJ PDA @Q=H BKNIQH=PEKJ 9: 2+ *

EJ SDE?D PDA ?KILHAIAJP=NU AJANCU * D=O PK >A IEJEIEVA@ >U = OPNAOO AH@ SDE?D BQHHHO PDA OPNKJC BKNI KB HEJA=N =J@ =JCQH=N IKIAJPQI =J@ EJ BKN PDA ?=OA KB OP=PE?O =J@ PDA PDA OPNAOO >KQJ@=NU ?KJ@EPEKJO -AQI=JJ >KQJ@=NU ?KJ@EPEKJO KJ (P EO KB ?KQNOA ?KILHE?=PA@ PK J@ = OPNAOO AH@ SDE?D BQHHHO PDAOA HK?=H ?KJ@EPEKJO 'AJ?A KJA ?=J =@@ PDA HEJA=N IKIAJPQI PK >U QOEJC PDA +=CN=JCE=J IQHPELHEAN IAPDK@ 9: 2+ @ER " 2+ -) '

-KS PDA OPNAOOAO D=RA PK >A ?KILQPA@ BKN AH@O SDE?D BQHHH PDA @ERANCAJ?A KLAN=PKN =J@ PDA =JCQH=N IKIAJPQI EJ =J@ PDA -AQI=JJ >KQJ@=NU

Mixed Finite-Element-Methods

155

?KJ@EPEKJO KJ 3DA +=CN=JCA IQHPELHEAN " ?=J >A EJPANLNAPA@ =O = @EO LH=?AIAJP 2EJ?A EP EO JKP OEILHA PK J@ OP=>HA JEPA AHAIAJP @EO?NAPEV=PEKJO BKN OAA !NAVVE =J@ %KNPEJ KJA ?=J NAH=T PDA OPNKJC OUIIAPNU ?KJ@EPEKJ KB PDA OPNAOO AH@ OAA AC %N=AEFO @A 5AQ>AGA NJKH@ AP =H =J@ 2PAJ>ANC 3DEO UEAH@O PDA IK@EA@ BQJ?PEKJ=H 9: 2+ @ER " 2+ 2+ -) '

'ANA PDA OPNAOO AH@ D=O PK BQHHH PDA PDA @ERANCAJ?A KLAN=PKN =J@ PDA -AQ I=JJ >KQJ@=NU ?KJ@EPEKJO 3DA =@@EPEKJ=H +=CN=JCA IQHPELHEAN EO =O OK?E=PA@ SEPD PDA HEJA=N NKP=PEKJ AH@ P PDA ATPNAIQI EP D=O PDA R=HQA " " 5=NE=PEKJ KB HA=@O PK PDA SA=G BKNI BKN PDA PDNAA EJ@ALAJ@AJP AH@O " =J@ 9: 2+ @ER " 2+

@ER 2+

2+

'ANA =J@ =NA PDA PAOP BQJ?PEKJO 3DEO OAP KB AMQ=PEKJO ?=J =HOK >A K>P=EJA@ BNKI PDA OPNKJC BKNIO 3DAJ KJA D=O PK IQHPELHU HEJA=N =J@ =JCQH=N IKIAJPQI >U =J@ NAOLA?PERAHU SDE?D UEAH@O $MQ=PEKJ BKHHKSO >U IQHPELHUEJC PDA HEJA=N ?KJOPEPQPERA AMQ=PEKJ 9: " >U = PAOP BQJ?PEKJ SDE?D EO BNKI PDA OPNAOO OL=?A BQHHHEJC PDA @ERANCAJ?A KLAN=PKN @@EPEKJ=H L=NPE=H EJPACN=PEKJ UEAH@O PDA J=H NAOQHP !''%)#!-!%..)!- +-%)%+'! )%/! ./-%). %KN JEPA AH=OPE? @ABKN I=PEKJO SA ?KJOE@AN PDA -AK 'KKGA ?KJOPEPQPERA AMQ=PEKJ

!

=J@ =OOQIA PD=P EP EO LKOOE>HA PK EJRANP PDEO AMQ=PEKJ

2EJ?A PDA +ACAJ@NA PN=JOBKNI=PEKJ ?=JJKP >A =LLHEA@ EJ PDA JEPA @ABKN I=PEKJ ?=OA SA BKNIQH=PA PDA LNEJ?ELHA =J=HKCKQO PK AMQ=PEKJ 3DEO

156

P. Wriggers

UEAH@O

2+

; 2+

;! 2

9 " : 2

SDANA EO PDA RENPQ=H OPNAOO &N=@ &N=@ EO PDA RENPQ=H OPN=EJ =J@ " EO PDA NECDP "=Q?DU&NAAJ PAJOKN PDA H=PPAN @ALAJ@EJC KJHU KJ PDA @ABKNI=PEKJ -KPA PD=P PDA EJRANOA EO JKP QJEMQAHU @AJA@ AEPDAN HK?=HHU KN CHK> =HHU OAA .C@AJ 'KSARAN SA ?=J EJRANP >U HKKGEJC =P @EANAJP OKHQPEKJ >N=J?DAO %KN PDEO SA NASNEPA =O ; ; SEPD ; !

;

; BNKI PDA HABP OE@A HA=@O PK 3DA IQHPELHE?=PEKJ KB SEPD ; ;

; ; 3DA ?KILQP=PEKJ KB -KS EP NAI=EJO PK ?KILQPA EJ @ALAJ@AJ?A KB PDA @APANIEJ=JP KB UEAH@O ; ; @AP !

; =J@ PDEO HA=@O PK OEJ?A ! @AP 6EPD PDA =>>NARE=PEKJ / ; @AP ; = ?Q>E? AMQ=PEKJ BKN ; ; ; / ; ; ; 3K K>P=EJ = OEILHAN OKHQPEKJ KJA ?=J @AJA 1 1 =J@ / @AP 3KCAPDAN SEPD 1 =J@ 1 1 1 SDE?D KJHU @ALAJ@ KJ PDA +=IA ?KJOP=JPO PDEO LNKRE@AO PDA ?Q>E? AMQ=PEKJ BKN / / #ALAJ@EJC KJ PDA @EO?NEIEJ=JP / / SA D=RA PK @EOPEJCQEOD PDNAA @EANAJP OKHQPEKJO

Mixed Finite-Element-Methods

157

%KN PDEO ?=OA PDANA D=O KJHU KJA NA=H OKHQPEKJ :

SEPD : /

/ :

/ /

3DEO ?=OA EO AMQER=HAJP PK / =J@ UEAH@O PDNAA OKHQPEKJO BKN

/ ?KO 9 =N??KO 5 : /

5

(J PDA LDUOE?=H LNK>HAI SA D=RA PK BQHH ! %NKI =J@ EP PDAJ BKHHKSO PD=P 'AJ?A KJHU PDA OKHQPEKJ SEPD 5 NAI=EJO QJ@AN PDAOA ?EN?QIOP=J?AO

/ ?KO =N??KO /

(J PDEO ?=OA PDA @APANIEJ=JP / AEPDAN EO / KN / %KN / PDA KJHU OKHQPEKJ

EO SDE?D UEAH@O SEPD =J@ BKN PDA )=?K>E=J ! %KN / ?=J >A K>P=EJA@ BNKI 3DEO UEAH@O EJ PDA HEIEP 6EPD PDEO OKHQPEKJ SA ?=J JKS >=OA@ KJ =J@ @ANERA PDA ATLNAO OEKJ BKN PDA EJRANOA KB PDA -AK'KKGA I=PANE=H 6EPD

SA K>P=EJ PDA J=H NAOQHP

%)!-%5/%*) *" 1-%/%*)' "*-(0'/%*).

%KN PDA OKHQPEKJ KB PDA OAP KB JKJHEJA=N AMQ=PEKJO I=JU @EANAJP =HCK NEPDIO =NA GJKSJ .BPAJ -ASPKJO IAPDK@ EO =LLHEA@ OEJ?A EP LKOOAOOAO PDA =@R=JP=CA KB = MQ=@N=PE? ?KJRANCAJ?A ?HKOA PK PDA OKHQPEKJ LKEJP (J ?=OA KB -ASPKJO IAPDK@ =J EILNKRA@ OKHQPEKJ EO K>P=EJA@ BNKI PDA 3=UHKN OANEAO ATL=JOEKJ KB PDA JKJHEJA=N AMQ=PEKJ =P PDA =HNA=@U ?KILQPA@ =LLNKTEI=PA OKHQPEKJ 3DEO 3=UHKN ATL=JOEKJ ?KNNAOLKJ@O EJ JEPA AHAIAJP =LLHE?=PEKJO PK PDA HEJA=NEV=PEKJ KB PDA SA=G BKNI KN EJ OKHE@ IA?D=JE?O PK PDA HEJA=NEV= PEKJ KB PDA RENPQ=H SKNG KN R=NE=PEKJ=H LNEJ?ELHAO =J@ ?=J >A K>P=EJA@ >U PDA @ENA?PEKJ=H @ANER=PERA

158

P. Wriggers

3DA HEJA=NEV=PEKJ SEHH >A OP=PA@ NOP BKN PDA SA=G BKNI SEPD NABANAJ?A PK PDA EJEPE=H ?KJCQN=PEKJ (J CAJAN=H SA SEHH ?KILQPA PDA HEJA=NEV=PEKJ =P = @ABKNI=PEKJ OP=PA KB PDA OKHE@ SDE?D EO EJ AMQEHE>NEQI 3DEO OP=PA SEHH >A @AJKPA@ >U 3DA HEJA=N L=NP KB PDA SA=G BKNI EO CERAJ >U " # 9 :

?KNNAOLKJ@O PK KJHU PDA OP=PA EO EJOANPA@ 3DA KLAN=PKN =P SDE?D PDA HEJA=NEV=PEKJ SEHH >A ?KILQPA@ !U =OOQIEJC PD=P PDA HK=@ EO ?KJOANR=PERA KJA ?=J ?KILQPA PDA @ENA?PEKJ=H @ANER=PERA KB EJ PDA @ENA?PEKJ KB " BNKI PDA NOP PANI EJ " 9 ": &N=@ 2+

DAJ?A =HH KPDAN PANIO =NA EJ@ALAJ@AJP KB PDA @ABKNI=PEKJ 3DA HEJA=NEV=PEKJ KB PDA NOP /EKH=*EN?DDK OPNAOO PAJOKN UEAH@O SEPD 9 ": &N=@ 2+ " &N=@ "

3DA 3ANIO SEPD = >=N D=RA PK >A AR=HQ=PA@ =P PDA @ABKNI=PEKJ OP=PA HEJA=NEV=PEKJ KB PDA OA?KJ@ /EKH=*EN?DDK OPNAOO PAJOKN BKHHKSO =O 9

: "

SDANA PDA H=OP PANI EO PDA HEJA=NEV=PEKJ KB PDA &NAAJ+=CN=JCA OPN=EJ PAJ 3DA AH=OPE?EPU PAJOKN SDE?D EO =HOK NABANNA@ PK PDA EJEPE=H OKN =P ?KJCQN=PEKJ EO CERAJ >U , =P PDA OP=PA 3DA QOA KB EJ ?KILHAPAO PDA HEJA=NEV=PEKJ 9

: &N=@ 2+ " &N=@ "

D=O PK >A AR=HQ=PA@ =P OP=PA !U =LLHUEJC PDA PN=?A -KPA PD=P =HOK KJA ?=J J@ KLAN=PEKJ PK PDA OA?KJ@ PANI =J@ >U QOEJC PDA OUIIAPNU KB = IKNA ?KIL=?P BKNI KB 9

&N=@

: 2+ " &N=@ "

Mixed Finite-Element-Methods

159

'ANA PDA OUIIAPNU KB PDA HEJA=NEV=PEKJ KLAN=PEKJ EO =J@ " ?=J >A K> OANRA@ 3DA NOP PANI EJ AMQ=PEKJ EO KBPAJ J=IA@ EJEPE=H OPNAOO PANI OEJ?A PDA OPNAOOAO =P PDA CERAJ OP=PA =LLA=N @ENA?PHU 3DA OA?KJ@ PANI ?KJ =HOK PDA R=NE=PEKJ KB P=EJO >AOE@AO PDA EJ?NAIAJP=H ?KJOPEPQPERA PAJOKN =J@ PDA PDA &NAAJ+=CN=JCA OPN=EJ PAJOKN &N=@ &N=@ &N=@ " EJ?NAIAJP KB PDA &NAAJ+=CN=JCA OPN=EJ PAJOKN

&N=@ " 3DA HEJA=NEV=PEKJ KB PDA LNEJ?ELHA KB RENPQ=H SKNG EJ PANIO KB PDA ?QNNAJP ?KJCQN=PEKJ EO K>P=EJA@ >U = LQOD BKNS=N@ KB PDA HEJA=NEV=PEKJ PK 4OEJC PDA PN=JOBKNI=PEKJO BKN PDA PDA =HNA=@U ?KILQPA@ ?KJCQN=PEKJ HEJA=NEV=PEKJ KB PDA &NAAJ+=CN=JCA OPN=EJ PAJOKN SDE?D NAOQHPA@ =O LQOD BKNS=N@ EJ , " EP BKHHKSO BKN PDA OA?KJ@ PANI EJ 9 ": 2+ , ,

BNKI OAA (J PDEO AMQ=PEKJO KJA ?=J ?KILQPA PDA BKQNPD KN@AN PAJOKN AC 6NECCANO =O 3DA NOP PANI EJ ?=J >A @ENA?PHU NA?=OP SEPD &N=@ &N=@ " &N=@ , " , &N=@ " 3DEO NAOQHPO EJ PDA HEJA=NEV=PEKJ SEPD NAOLA?P PK PDA GJKSJ ?QNNAJP ?KJCQ SDANA =HH MQ=JPEPEAO D=RA PK >A AR=HQ=PA@ =P N=PEKJ 9 " : 2+ " , " , , 3 ,

3DA EJPACN=H ?=J JKS >A NABANNA@ PK PDA ?QNNAJP ?KJCQN=PEKJ SEPD %KN PDEO LQNLKOA SA EJPNK@Q?A "=Q?DU OPNAOO PAJOKN PDA NAH=PEKJ 2 > !2+ =J@ @AJA = BQNPDAN EJ?NAIAJP=H ?KJOPEPQPERA PAJOKN ! OQ?D PD=P SA K>P=EJ

" 3

, " , , 9, " : 2>

3DEO NAH=PEKJ EO GJKSJ EJ PDA HEPAN=PQNA =HOK =O QL@=PA@ +=CN=JCA BKNIQH= PEKJ OAA AC !=PDA AP =H KN !=PDA DAJ?A PDA @ABKNI=PEKJ

160

P. Wriggers

PK SDE?D PDA BKNIQH=PEKJ EO NABANNA@ EO KJHU K>P=EJA@ SEPDEJ PDA OP=PA EO ?KILQPA@ SEPDEJ =J EJ?NAIAJP=H JKJHEJA=N OKHQPEKJ LNK?AOO 3DA OP=PA OKHQPEKJ LNK?A@QNA >U =J QL@=PEJC KB =HH @ABKNI=PEKJ OP=PAO EJ = OQ??AOOERA I=JJAN 6EPD PDA =>KRA CERAJ HEJA=NEV=PEKJO =HH NAH=PEKJO JAA@A@ SEPDEJ PDA -AS PKJ IAPDK@ =NA =R=EH=>HA =O SAHH =O BKN BKNIQH=PEKJO SEPD NAOLA?P PK PDA EJEPE=H ?KJCQN=PEKJ =O SEPD NAOLA?P PK PDA ?QNNAJP ?KJCQN=PEKJ 3DAOA NA H=PEKJO ?=J >A QOA@ EJ PDA O=IA S=U BKN PDA HEJA=NEV=PEKJO KB KPDAN R=NE=PEKJ=H LNEJ?ELHAO @EO?QOOA@ EJ PDA LNAREKQO OA?PEKJO

!,0%-!(!)/. "*- *)/%)00( )%/! !'!(!)/.

3DA OA=N?D BKN SAHH LANBKNIEJC JEPA AHAIAJPO SDE?D ?=J >A =LLHEA@ PK =N>EPN=NU LNK>HAI ?H=OOAO SEPDEJ OKHE@ IA?D=JE?O D=O = HKJC DEOPKNU =O KJA ?=J OAA BNKI PDA JQIANKQO O?EAJPE? L=LANO @ARKPA@ PK PDEO PKLE? ,=EJ P=NCAPO KB = @ARAHKLIAJP KB JEPA AHAIAJPO =NA +K?GEJC BNAA >AD=REKQN BKN EJ?KILNAOOE>HA I=PANE=H CKK@ >AJ@EJC LANBKNI=J?A JK HK?GEJC EJ PDEJ AHAIAJPO JK OAJOEPEREPU =C=EJOP IAOD @EOPKNPEKJO CKK@ ?K=NOA IAOD =??QN=?U OEILHA EILHAIAJP=PEKJ KB JKJHEJA=N ?KJOPEPQPERA AMQ=PEKJO =J@ A?EAJ?U AC BAS JA?AOO=NU EJPACN=PEKJ LKEJPO 3DA NOP LKEJP EO =OOK?E=PA@ SEPD PDA JQIANE?=H OEIQH=PEKJ KB OLA?E=H LNK>HAI ?H=OOAO SDE?D EJ?HQ@A EJ OKHE@ IA?D=JE?O NQ>>AN HEGA I=PANE=HO =J@ AH=OPKLH=OPE? I=PANE=H AMQ=PEKJO EJ PDA BN=IASKNG KB ! LH=OPE?EPU 3DA ?KJOPN=EJP NAH=PA@ PK PDA EJ?KILNAOOE>HA >AD=REKQN HA=@O ARAJ BKN CAKIAPNE?=H HEJA=N AHAIAJPO PK HK?GEJC OAA AC !N=AOO =J@ 'QCDAO

= <18>5- +. .691*<2C.- =1*= =1. 527.*;2C*=287 8/ =1. ,87=27>8>< 9;8+5.6 *< <=*=.- 27 -8.< 78= ,827,2-. 27 *55 ,*<.< @2=1 =1. 527.*;2C*=287 8/ =1. -2<,;.=. 9;8+5.6 @12,1 ;.<>5=< /;86 * 72=. .5.6.7= -2<,;.=2C*=287 7 ,*<. 8/ * 72=. .5.6.7= /8;6>5*=287 8/ * <=*=2, 9;8+5.6 @2=1 ,87=27>8>< 27=.;985*=287< />7,=287< /8; =1. -2<95*,.6.7=< 87. ,*7 <18@ =1. .:>2?*5.7,. 8/ =1. -2<,;.=2C*=287 8/ =1. 527.*;2C*=287< 02?.7 *+8?. *7=1. -2;.,= 527.*;2C*=287 8/ =1. -2<,;.=2C*=287 8/ =1. @.*4 /8;6 ><270 72=. .5.6.7=<

7 ,*<. =1*= =1. 72=. .5.6.7= 6.=18- 2< *9952.- =8 9;8+5.6< 8/ .5*<=8 95*<=2,2=B 8; 8=1.; 27.5*<=2, ,87<=2=>=2?. .:>*=287< =1.7 =1. 527.*;2C*=287 8/ =1. -2<,;.=. /8;6 2< 78 5870.; .:>2?*5.7= =8 =1. -2<,;.=2C*=287 8/ =1. <8 /*; 02?.7 527.*;2C*=287< <27,. =1. 27=.0;*=287 *508;2=16 /8; =1. .?85>=287 .:>*=287< 8/ =1. 27.5*<=2, 6*=.;2*5 ;.<987<. 95*B< * 9;8627.7= ;85. *7- 1.7,. =1. 527.*;2C*=287 -.9.7-< 87 =12< *508;2=16 <.. . 0

$268 .= *5 8; /8; * 68;. 0.7.;*5 8?.;?2.@ $268 *7- >01.<

Mixed Finite-Element-Methods

161

3DA OA?KJ@ =J@ PDEN@ LKEJPO =NA KB OECJE?=J?A SDAJ PDNAA@EIAJOEKJ=H OKHE@ AHAIAJPO OD=HH >A AILHKUA@ PK OKHRA >A=I KN ODAHH LNK>HAIO OEJ?A >A=I =J@ ODAHH OPNQ?PQNAO =NA KBPAJ @KIEJ=PA@ >U >AJ@EJC >AD=REKQN =J@ =NA >U ?KJOPNQ?PEKJ PDEJ EJ KJA KN PSK OL=PE=H ?KKN@EJ=PAO 3DA BKQNPD LKEJP EO AOOAJPE=H SDAJ IK@ANJ IAPDK@O BKN IAOD CAJAN=PEKJ =NA AILHKUA@ 3DAOA IAPDK@O HA=@ BKN =N>EPN=NU CAKIAPNEAO PK OK ?=HHA@ QJOPNQ?PQNA@ IAODAO SDE?D ?KJOEOP KB JEPA AHAIAJPO OD=LAO SEPD =N>EPN=NU CAKIAPNU 3DA BPD LKEJP EO NAH=PA@ PK PDA B=?P PD=P KJA D=OKBPAJ PK =J=HUVA PDNAA @EIAJOEKJ=H ?KILKJAJPO EJ NA=H AJCEJAANEJC =LLHE?=PEKJO SDE?D OEVA =J@ ?KILHATEPU ?=JJKP >A IK@AHHA@ QOEJC = ?KJRANCA@ IAOD AOLA?E=HHU SDAJ PDA OEIQH=PEKJ EO JKJHEJA=N 'AJ?A PDANA EO OPEHH JAA@ BKN AHAIAJPO SDE?D @ALE?P = CKK@ =??QN=?U ARAJ SDAJ QOA@ SEPDEJ = ?K=NOA IAOD $JCEJAANEJC =LLHE?=PEKJO @AI=J@ KBPAJ PDA QOA KB JAS JKJHEJA=N ?KJ OPEPQPERA AMQ=PEKJO 'ANA = OEILHA EJPANB=?A PK PDA JEPA AHAIAJP ODKQH@ OQLLKNP PDA QOAN EJ KN@AN PK A?EAJPHU ?D=JCA ATEOPEJC JEPA AHAIAJPO =J@ PK >A =>HA PK EILHAIAJP ?KILHAT ?KJOPEPQPERA AMQ=PEKJO %EJ=HHU SA HEGA PK IAJPEKJ PD=P A?EAJ?U EO JKP KJHU NAH=PA@ PK OLAA@ KB PDA AHAIAJP BKNIQH=PEKJ >QP =HOK PK PDA IAIKNU NAMQENAIAJP 3DA H=P PAN @AI=J@ EO AOOAJPE=H SDAJ AC EJAH=OPE? LNK>HAIO SEPD OARAN=H DQJ@NA@ PDKQO=J@ KN IEHHEKJO KB JEPA AHAIAJP D=RA PK OKHRA@ SEPDEJ = CERAJ PEIA BN=IA 2LAA@ KB PDA AHAIAJP BKNIQH=PEKJ EO AOOAJPE=H SDAJ EPAN=PERA OKHRANO =NA =LLHEA@ OEJ?A EJ PD=P ?=OA PDA PEIA BKN PDA ?KILQP=PEKJ KB NAOE@Q=HO =J@ P=JCAJP I=PNE?AO EO KB PDA O=IA KN@AN =O PDA PEIA QOA@ >U PDA OKHRAN SEPDEJ KJA EPAN=PEKJ (P EO SAHH GJKSJ PD=P = LQNA @EOLH=?AIAJP AHAIAJP SEPD >E KN PNEHEJA=N =JO=PV BQJ?PEKJO D=O >=@ ?KJRANCAJ?A >AD=REKQN EJ >AJ@EJC LNK>HAIO =J@ SEPDEJ =LLHE?=PEKJO SEPD EJ?KILNAOOE>HA I=PANE=HO 3DQO @EANAJP R=NE= PEKJ=H BKNIQH=PEKJO D=RA >AAJ ATLHKNA@ EJ KN@AN PK ?KJOPNQ?P JEPA AHAIAJPO SDE?D BQHHH =O I=JU NAMQENAIAJPO OP=PA@ =>KRA =O LKOOE>HA IKJC PDAOA =NA NA@Q?A@ EJPACN=PEKJ OP=>EHEV=PEKJ IAPDK@O DU>NE@ KN IETA@ R=NE=PEKJ=H LNEJ?ELHAO SDE?D >=OA KJ ?KILHEIAJP=NU AJANCU SNEPPAJ EJ PANIO KB PDA OPNAOO AH@ IETA@ R=NE=PEKJ=H LNEJ?ELHA KB 'Q6=ODEVQ PULA =J@ IETA@ R=NE=PEKJ=H LNEJ?ELHA BKN OLA?E=H MQ=JPEPEAO (J PDA BKHHKSEJC SA SEHH OQII=NEVA @EANAJP LKOOE>EHEPEAO =J@ @EO?QOO PDAEN @EANAJ?AO 3DAJ OKIA KB PDA PA?DJEMQAO =NA LNAOAJPA@ EJ @AP=EH ! 0! %)/!#-/%*) ) ./%'%5/%*) 3DA IKOP OEILHA IAPDK@ EO PDA QJ@ANEJPACN=PEKJ KB PDA EJPACN=HO HA=@EJC PK PDA JEPA AHAIAJP

162

P. Wriggers

RA?PKNO =J@ I=PNE?AO (P EO =HOK RANU A?EAJP =J@ O=BAO IAIKNU BKN DEOPKNU @=P= OPKN=CA OEJ?A HAOO EJPACN=PEKJ LKEJPO =NA QOA@ 4J@AN EJPACN=PEKJ KN NA@Q?A@ EJPACN=PEKJ IA=JO PD=P HAOO &=QOO LKEJPO =NA QOA@ BKN PDA EJPACN=PEKJ KB P=JCAJP I=PNE?AO =J@ NAOE@Q=H RA?PKNO PD=J JA?AOO=NU BKN PDA ?DKOAJ LKHUJKIE=H @ACNAA KB PDA OD=LA BQJ?PEKJO BKN NOP =LLHE?=PEKJO OAA AC 8EAJGEASE?V AP =H 1A@Q?A@ EJPACN= PEKJ S=O @ARAHKLA@ PK =RKE@ HK?GEJC EJ ?=OA KB EJ?KILNAOOE>EHEPU (J PD=P ?=OA EP EO KBPAJ KJHU =LLHEA@ PK PDA LNAOOQNA L=NP KB PDA ?KJOPEPQ PERA AMQ=PEKJ OAA AC ,=HGQO =J@ 'QCDAO =J@ 'QCDAO 1A@Q?A@ EJPACN=PEKJ EO =HS=UO =OOK?E=PA@ SEPD = N=JG @A?EAJ?U KB PDA P=JCAJP I=PNE?AO SDE?D ?=J >A ?QNA@ >U @EANAJP IAPDK@O CAJ AN=HHU GJKSJ =O OP=>EHEV=PEKJ PA?DJEMQAO OAA AC !AHUPO?DGK AP =H =J@ 1AAOA =J@ 6NECCANO 4OEJC PDA NA@Q?A@ EJPACN=PEKJ PKCAPDAN SEPD OP=>EHEV=PEKJ HA=@O PK JEPA AHAIAJPO SDE?D BQHHH ?KJ @EPEKJO =J@ 'AJ?A PDAOA AHAIAJPO =NA HK?GEJC BNAA EJ ?=OA KB EJ?KILNAOOE>EHEPU PDAU D=RA = CKK@ ?K=NOA IAOD =??QN=?U =NA JKP OAJOEPERA =C=EJOP IAOD @EOPKNPEKJO =J@ ?=J >A QOA@ BKN =N>EPN=NU ?KJOPEPQPERA AMQ=PEKJO 3DA NA@Q?A@ EJPACN=PEKJ LNKRE@AO PDA IKOP AB ?EAJP LKOOE>EHEPU PK ?KILQPA@ PDA AHAIAJP NAOE@Q=H =J@ PDA AHAIAJP P=JCAJP OPEJAOO AC BKN =J AECDPJK@A >NE?G AHAIAJP KJHU KJA &=QOO LKEJP EO JAA@A@ 'KSARAN PDAOA AHAIAJPO JAA@ KBPAJ PDA ?DKE?A KB =NPE?E=H OP=>EHEV=PEKJ L=N=IAPANO (J PDA SKNOP ?=OA AC BKN OKIA >AJ@EJC LNK>HAIO PDA JEPA AHAIAJP OKHQPEKJ ?=J @ENA?PHU @ALAJ@ KJ PDA OP=>EHEV=PEKJ L=N=IAPAN 4-% *- (%3! 1-%/%*)' +-%)%+'!. 6DAJ IETA@ R=NE=PEKJ=H LNEJ?ELHAO =NA QOA@ =O >=OEO BKN JEPA AHAIAJP @EA?ONAPEV=PEKJ PDAJ @EB BANAJP LKOOE>EHEPEAO ATEOP BKN PDA ?KJOPNQ?PEKJ KB PDA JEPA AHAIAJP I= PNE?AO 3DEO EO @QA PK PDA B=?P PD=P KJA ?=J OP=PA I=JU @EANAJP IETA@ BKNIO OAA H=OP OA?PEKJ 2KIA KB PDAI JAA@ ?KJBKNIEJC @EOLH=?AIAJP AH@O PKCAPDAN SEPD JKJ?KJBKNIEJC OPNAOO KN OPN=EJ AH@O KPDANO NAHU KJ ?KJBKNIEJC OPNAOO AH@O >QP =HHKS JKJ?KJBKNIEJC @EOLH=?AIAJP AH@O %KN PDA ?=OA KB HEJA=N AH=OPE?EPU DU>NE@ AHAIAJPO SDANA NOP @A O?NE>A@ EJ /E=J SDE?D D=O HA=@ PK I=JU @EANAJP JEPA AHAIAJP BKNIQH=PEKJO QL PK JKS 6EPDEJ PDEO =LLNK=?D /E=J =J@ 2QIED=N= @ARAHKLA@ = JEPA AHAIAJP SDE?D EO A?EAJP =J@ =??QN=PA 'KS ARAN @QA PK PDA JAA@ PK EJRANP PDA ?KJOPEPQPERA AMQ=PEKJO SEPDEJ PDA BKNIQH=PEKJ EJ KN@AN PK K>P=EJ PDA ?KJOPEPQPERA AMQ=PEKJO EJ PANIO KB PDA OPNAOO AH@ PDANA =NA KJHU BAS AHAIAJPO BKN 2P 5AJ=JP I=PANE=HO GJKSJ SDE?D SKNG BKN H=NCA @ABKNI=PEKJO OLA?E=H BKNIQH=PEKJ BKN -AK'KKGA I=PANE=HO SEHH >A LNAOAJPA@ EJ OA?PEKJ )$)! ./-%) !'!(!)/. .! *) /$! 0.$%50 +-%)

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Mixed Finite-Element-Methods

173

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174

P. Wriggers 7 6.5 6 5.5

Tip displacement

5 4.5 4 3.5 3 2.5 2

Q1 Q1 / E4 CG4 PS8

1.5 1 0.5 0

0

50

100

Number of elements per side

%#0-! "KJRANCAJ?A KB PDA @EANAJP AHAIAJP BKNIQH=PEKJO

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Mixed Finite-Element-Methods

175

'! "KJRANCAJ?A KB PDA @EANAJP AHAIAJP BKNIQH=PEKJO

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176

P. Wriggers

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Mixed Finite-Element-Methods

177

) " 2EIK % NIANK =J@ 1 + 3=UHKN (ILNKRA@ 5ANOEKJO KB OOQIA@ $JD=J?A@ 2PN=EJ 3NE+EJA=N $HAIAJPO BKN # %EJEPA #ABKNI=PEKJ /NK> HAIO )'*.- + -#), $( **&$ #($, ( ("$( +$(" W 1 2PAJ>ANC B=IEHU KB IETA@ JEPA AHAIAJPO BKN AH=OPE?EPU LNK>HAIO .' +$,# -# '-$% W 3 2QOOI=J =J@ *) !=PDA %EJEPA $HAIAJP %KNIQH=PEKJ BKN -KJHEJA=N (J?KILNAOOE>HA $H=OPE? =J@ (JAH=OPE? J=HUOEO )'*.- +, ( -+.-.+ , W 1 + 3=UHKN / ) !ANAOBKN@ =J@ $ + 6EHOKJ -KJ"KJBKNIEJC $HAIAJP BKN 2PNAOO J=HUOEO (- +(-$)(& ).+(& !)+ .' +$& -#), $( ("$( +$(" W " 3NQAO@AHH =J@ 6 -KHH 3DA JKJHEJA=N AH@ PDAKNEAO KB IA?D=JE?O (J 2 %HX QCCA A@EPKN (.# + #1,$% 2LNEJCAN !ANHEJ 'AE@AH >ANC 6EAJ " 3NQAO@AHH =J@ 1 3KQLEJ 3DA ?H=OOE?=H AH@ PDAKNEAO (J (.# + #1,$% 2LNEJCAN !ANHEJ 'AE@AH>ANC 6EAJ * 6=ODEVQ +$-$)(& -#), $( &,-$$-1 ( &,-$$-1 /ANC=IKJ /NAOO .TBKN@ OA?KJ@ A@EPEKJ $ + 6EHOKJ 1 + 3=UHKN 6 / #KDANPU =J@ ) &D=>KQOOE (J?KIL=PE>HA #EOLH=?AIAJPO ,K@AHO (J .' +$& ( )'*.- + ) &, $( -+.-.+& #($, %AJRAO 2 ) /ANNKJA - 1K>EJOKJ 1 =J@ 2?DJK>NE?D 6 " $@O -AS 7KNG ?=@AIE? /NAOO W / 6NECCANO $#-&$( + $($- & ' (- 2LNEJCAN !ANHEJ / 6NECCANO )(&$( + $($- & ' (-, 2LNEJCAN !ANHEJ 'AE@AH>ANC -AS 7KNG / 6NECCANO =J@ 2 1AAOA JKPA KJ AJD=J?A@ OPN=EJ IAPDK@O BKN H=NCA @ABKNI=PEKJO )'*.- + -#), $( **&$ #($, ( ("$( +$(" W . " 8EAJGEASE?V =J@ 1 + 3=UHKN # $($- & ' (- -#) -# RKHQIA ,?&N=S 'EHH +KJ@KJ . " 8EAJGEASE?V 1 + 3=UHKN =J@ ) , 3KK 1A@Q?A@ (JPACN=PEKJ 3A?D JEMQA EJ &AJAN=H J=HUOEO KB /H=PAO =J@ 2DAHHO (- +(-$)(& ).+(& !)+ .' +$& -#), $( ("$( +$(" W

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180

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181

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184

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Stability of Some Finite Element Methods…

185

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Stability of Some Finite Element Methods…

187

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197

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Stability of Some Finite Element Methods…

199

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201

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Stability of Some Finite Element Methods…

203

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204

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Stability of Some Finite Element Methods…

205

2 * 0, 0 ,1 &: 2 1& "+ ,& S@ &4 7 5 A 4 & 54 & 5 !

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206

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