Multi-parameter asymptotic error resolution of the mixed finite element method for the Stokes problem

ESAIM: M

A ^IHUI Z ^HOU

Multi-parameter asymptotic error resolution of the mixed finite element method for the Stokes problem

ESAIM: Modélisation mathématique et analyse numérique, tome 33, n^o1 (1999), p. 89-97

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Mathematical Modelling and Numerical Analysis M2AN, Vol. 33, N° 1, 1999, p. 89-97 Modélisation Mathématique et Analyse Numérique

MULTI-PARAMETER ASYMPTOTIC ERROR RESOLUTION OF THE MIXED FINITE ELEMENT METHOD FOR THE STOKES PROBLEM

AIHUI ZHOU1

Abstract. In this paper, a multi-parameter error resolution technique is applied into a mixed finite element method for the Stokes problem. By using this technique and establishing a multi-parameter asymptotic error expansion for the mixed finite element method, an approximation of higher accuracy is obtained by multi-processor computers in parallel.

Resumé. Dans cet article une méthode de résolution d'erreurs multi-paramétrique est appliquée à la méthode des éléments finis mixte pour le problème de Stokes. À l'aide de cette technique, et en utilisant un développement asymptotique multi-paramétrique de l'erreur, on obtient une plus grande précision par calcul parallèle sur des machines multi-processeurs.

AMS Subject Classification. 65N15, 65N30.

Received: June 26, 1998. Revised October 21, 1997 and Mardi 3, 1998.

1. INTRODUCTION

The Stokes équations have become an important model problem in computational fiuid dynamics for de- signing and analyzing finite element algorithms. In the context of the Galerkin variational formulation, the computational accuracy for the velocity components can be compared to that known for the Poisson équation.

For the pressure, however, one frequently observes a drastic réduction of the accuracy along the boundary.

Therefore, one has anyway to employ some techniques or arguments to recover and increase the accuracy.

The extrapolation approach established by Richardson in 1926, is an important technique to obtain approxi- mations of higher accuracy. The application of this approach in the finite différence method can be found in [13].

In 1983, Lin, Lü and Shen [10] have applied the technique to the finite element method. Development in this direction can be found in [3,12,14]. However, this approach has a limitation since it requires a global refinement and hence wastes computing time and memory. Recently, a so-called multi-parameter error resolution technique has been introduced to obtain approximations of higher accuracy (see [19,20]). This new approach is based on a multi-parameter asymptotic error expansion and only uses partial refinements. It is shown that a combination of discrete solutions related to such partial refinements can produce an approximation of higher accuracy and the procedure can be done by multi-processor computers in parallel. In this paper, we apply this technique to a mixed finite element method for the Stokes problem.

Consider the homogenous Stokes problem

- A u + Vp = f, in fi,

divu = 0, infi, (1.1) u = 0, on <9Q,

Key words and phrases. Multi-parameter asymptotic error resolution, multi-parameter extrapolation, mixed finite element, Stoke problem, parallel computing.

1 Institute of Systems Science, Academia Sinica, Beijing 100080, China, e-mail: azhou@bamboo.iss.ac.cn.

© EDP Sciences, SMAI 1999

90 A. ZKOU

where ft — [0, l]2, u = (1*1,^2) is the velocity, p is the pressure and f indicates the external force. We know that (1.1) is equivalent to the variational formulation( [8]): Find (u,p) € (HQ(£1))2 X LQ(Q) such that

B(u,p;v,<z) = (f,v), V(v,g) e (H^iï))2 x Lg(ïl), (1.2) where

i?(u,p; v,g) — (Vu, Vv) — (p, div v) + (g, divu) and other notations are identical to these of références [5,8].

Divide ft into subrectangles ftj(j = 1, • • • , m) whose edges are parallel to the sc-axis and the y-axis respec- tively. On each ftj(j = 1, • • • , m) a uniform mesh is imposed, and globally a quasi-uniform partition on ft is formed. Dénote the mesh sizes on ftj by hj^ in x-direction and hj^ in y-direction (j — 1,--- , m). Among these mesh parameters, some are independent, say /ii, • • • , hs. It can be proved that 2 < s < m + 1 and one can choose s > 2. Let /i — max{/ij, * • • , /is}, Th be the partition on ft and (V^, L^) be a pair of finite element spaces on T^, e,^. the Bernardi-Raugel finite element space or the filtered bilinear-constant finite element space, satisfying the Babuska-Brezzi conditon (see [2,8]). The interpolated finite element approximations correspond- ing to hi, • • - , h$ is denoted by (u(^i, • • • , hs),p(hiy • • • , hs)). Then there exist multi-parameter error résolution expressions on ft (see Sect. 4)

u(hij • • • , hs) ~ u + £ *= 1 dijh2 + O(h3), in H1 — norm,

u(Ai, • • • , hs) = u + £ J= 1 ajft| + O(h4), in L2 - norm, (1.3) p(/ii, • - • , h8) = P + £^=i ^i^j + O(h3), in L2 - norm

where a^, bj(j — 1,-'* ;^s) ^{a r e} independent of (fti,--- ,/i^s)- Based on (1.3), a multi-parameter extrapolation yields approximations of higher accuracy

(4 E j = i U3 - (4 s - 3)uo)/3 = u + O(h3), in Hl - norm,

(4 £5=i uj - (4s - 3)uo)/3 = u + O(h4), in L2 - norm, (1.4) (4 EUi Pi ~ (4s ~ 3)^o)/3 = P + ° ^3) ' i n ^2 " n o r m'

w h e r e ( u0, P o ) = ( u ( h i}- - - , / is) , p ( / i i , • • • , / is) ) a n d ( u j , p j ) = ( u ( h i , - - - , ftj_i, / i j / 2 , / i j+ 1 ) • - • , / is) ,

Therefore, a parallel algorithm for approximations of higher accuracy can be designed as follows.

Algorithm.

Step 1. Compute (\ij7pj)(0 < j < s) in parallel.

Step 2. Set

J1 Ui ~ (4s - 3)uo)/3, Pe = (4 E,5 = 1 Pj - (45 - 3)po)/3.

A MIXED FINITE ELEMENT FOR THE STOKES PROBLEM

2. PRÉLIMINAIRES

For any element e = [x^e — h^eiy^e — k^e]², we define the interpolation operators U = Qi,2 ^x $2,1 and jh : £²(e) —> QQ by (c/. [2,8])

(v - i^v)(o;e ± /l^e, Ve ± *e) = 0, Vv (

/ («^ - ih,i<p)dy = 0, V</? G C(e),

J(x,-h

I e | Je

91

: (C(e))²

(2.2)

where Q^r,n — span{x^ly³ : 0 < i < r, 0 < j < n} and Q^r = Q^fjr. Introducé two other postprocessing interpolation operators. Let Th be a rectangular partition on O with size h. We assume that Th has been obtained from T^h by dividing each element into nine congruent rectangles. Let r = uf⁼¹e^z G T$^h with e^z ÉT/J, and I³h = (13/1,1^3/1,2) and J3^ are defined as

J = 1,2,

, j = 1,2, ^(2.3)

l /C I( J 3 H P - P ) = O , Z = l , - - - , 9 ,

where pt (i = 1, • • • , 16) are all vertices of ei, • • • , eg. It can be proved that (cf. [11,17,18])

- w||0 - w]|a < , Vw e H^1+r(n), 1 < r < 3,

(2.4)

(2.5)

< c||g||o, Vq£L^h,

\\J3hW - w||o < c/i^r||i(;||^r, Vw € H^r(Ü), 1 < r < 3.

(2.6)

In order to solve the problem (1.1), many pairs of finite element spaces were introduced (see [2,8]). One of them is the so-called Bernardi-Raugel element, which is defined by

J V f c ^ v É (C(Ü))², v |^ee Qi^l2(e) x Q².i(e), v |en= 0, e e T^h}, { L^h = {q € L§(fï), ç|e € (30(6), /ⁿ q = 0, e G T^h}.

We mention here that the Bernardi-Raugel element was presented but not enalysed in [7].

92 A ZHOU

3. ERROR RESOLUTION FOR INTERPOLANTS We have the following multt-parameter error resolution for interpolants.

T h e o r e m 3 . 1 . For (u,p) G (H4(Ü))2 x H3{Ü), there exist î3 <E (L2(Ü))2,g3 G that

B(u-i^hu,p- j^hp] v, q)

Jlï

w h e r e T = i 3 8 ^ Proof. Since

(g,div(u-i^hu)) it is only necessary to prove two expansions

( V ( U - Ï M I ) , V V ) = \

for v = , and

(p - jfcp,diw) = ^

1 r r

°s,su s , i c r „ J s e-3 Js

j = 1,. • • ,s) such

(3.1)

:yPVl

(3.2)

where TXiTy is the part of F parallel to the z-direction and the y-direction respectively, and se,i = {{x, y) : x = xe - he,ye - ke < y < ye + fce},

<se,2 = {(x, y) : xe - h€ < x < xe + he) y = ye - A;e}, 5e,3 = {(x,y) '• x = Xe + he,ye - ke < y < ye + ke}, S&A = {(x, y) : xe - he < x < xe + he, y — ye + ke}.

We employ the technique in [11,17,18] and introducé the two error functions

F = Fe = \{{y - yef - - xe)2 - h2e), then, it is obvious that F (y) = 0 on se^ and se ) 4, £?(a;) = 0 on se>i and 56}3.

== g \-^ Jxxxi

A MIXED FINITE ELEMENT FOR THE STOKES PROBLEM 9 3

F = - U ? +

Let u = (ui, ^2), we only need to go to prove

~~ IHUIIvlli. (3-3)

A Taylor expansion yields

dxvx = dxvi(x,ye) + -(F^yyydxyV^x^ye) + {-^{F3)yyyy + — k2e)dxyyvi.

The définition (2.1), intégration by parts and inverse estimâtes lead to ƒ öa;(ui -ih,iui)dxvi(x,ye) = 0,

t/e

g / (F2)yyydx(ui ~ ih,iUi)dxyvi(x,ye) = - - J F2dxyyyuidxyvl{x,ye

J dx(u! - »fc,l«l)(gö(F3)vwwi/ + jQkî)dxyyvl = ~^ J ( ^ y

Thus, we obtain

(öx(ui-iMWi)^xVi)=O(ft3)||u||4||v||i (3.4)

For (dy(ui — ih,iv>i),dyVi)j using the définition (2.1),

ôyUi = Exxdyv1{xe,y) + -(^2)xa:xöa;yi;i, and intégration by parts,

/ 9y(îxi -2fc1iui)(^)xxoyî;i(a:e,2/) Je

= ( / - / )Exdy{u1-ihliu1)dyV1(xe,y)dy- ƒ Exdxy(ui - ihtiui)dyvi(xe, y)

J Se,3 Jse,l Je

= - ( / - ƒ )Sa;(ui - 4 , 1 ^ 1 ) ^ ^ 1 (xe,y)dy+ Ed^yindyViixe.y)

Jse,3 Jse,i Je

= ~hl f ÔxxyU^dyV! - ExdxyVl) + i J(E2)xxdxxyUldyv(xe,y)

= ~\h2e J d^yUidyVi + f dXXXyUl(~hlEdxyVl - ^(E2)XdyVl + £ ( ^ ) XExdXyVl )

= -ô^e( f - f )d

u

dx + -hl l d

u

+ O(/i

)t|u||

||v||i,

6 Jse4. J Se,2 ó Je

94 A. ZHOU

- / {E2)xxxdy{ux — ihUi)dxyvi

b Je

= l(f - f ){E

)

d

{

- i

u

)d

v

- \ [(

O JSe.3 J^e.! b ie

= i f(E2)xdxxyu1dxyv1

/le(( ƒ - ƒ )OœxyWiVidx + / Therefore, we have

||u||4||v||i. (3.5)

By the quasi-uniformity of T^, one sees that line intégrais in (3.5) disappear during the summation, thus (3.3) folïows from (3.4, 3.5).

We turn to prove (3.2). Similar to the proof of (3.1), we only need to prove ( P - JhP^dxVx) = - Y^ k\ / dxyypv! - - Yl fce( / - ƒ

eeTh J e ó sË,4U5ei2crx J s ^ J s

+ O(/»3)lb|||3||v||i. (3.6)

By the Taylor expansion, we have

J{P ~ JhP)OxV1 - ƒ (p ~ jhpXdvV^X, ye) + FydvyV^X, Ve

(3.7)

The définition (2.2) shows that / = 0 and II = - Fdypdxyvi(x,ye)

Je

= 3^e / dy/

*2(

= \kl( [ - [ )dyPdyv1(x,ye)dy - \k2e \ dxyVdvvl + OC/*3)|i

= ^ e ( / - / )dypdyvl(x,ye)dy-\k2e([ - [ )dxypv1dx + Ik^2e^d^xyypv¹+O(h³)\\p\\^\3\\v\\u

A MIXED FINITE ELEMENT FOR THE STOKES PROBLEM 95

III = ^ Jj2dyyPdxyyV1

= fiW ~ / ) F 2 d

0 J$e>3 J5e,1

Similarly, line intégrais along y-direction in the above terms will disappear during the summation of e G T^.

Thus we complete the proof,

Fr om the proof of Theorem 3.1, we can also obtain the following

Theorem 3.2. For (u,p) G [H^(p)f x HS(Ü), there exist ïó G (L2(^))2 ) g j- G (H1/2^))2^ = 1, • • • , s) such that

B(u-ihu,p-jhp;v,q) = ^h){ i fj-v + / gj-v

||w||2), V(v, q)eVhx Lh, (3.8) where w G (ftf(£l) n H⁴{Ü))².

4. E R R O R R E S O L U T I O N F O R F I N I T E E L E M E N T S O L U T I O N S

A mixed finite element method based on (1.1) reads as follows: Find (uh,,Ph) ^ ^h x Lh such that

B(uhyPh-v,q) - (f,v), V(v,g) G V„ x Lh. (4.1) or

B(uh -u,ph- p; v, g) = 0, V(v, q) e Vh x Lh. (4.2) One has the following error estimate (see [2,8])

h^Wu - UfcHo + ||u - UfcHi + ||p - phllo < cMI|u||2 + HPIII). (4.3) Now, we shall present the multi-parameter error resolution for the numerical solution.

Theorem 4.1. If (u,p) G (H^(ü) D H4(Ü))2 x (L§(îî) n ^3( ^ ) ) is the exact solution of (1.1), then there exist

^ 2 2 lx that

Proof. Let (a^, 6j) E (H^(Q))2 x L0(fi) (j = 1, • • • , s) satisfy

Bi^b^q) - (/,,v) + / "g iv , V(v,g) e ( ^ ( î î ) )2 x ig(fi) (4.4) and (ajihl bjih) G Vh x Lh (j = 1, • • • , s)

,-, v) + / gj-v, V(v,g) G Vh x Lh, (4.5) Jv

96 A ZHOU then we have

Taking v = u^, — i^hn — J2*=i a.j,hh² and q — Ph — JhP — YL^S3=i bj,hh² in the above équation, we obtain by (4.2) and the Babuska-Brezzi condition (see e.g. [1,2,4])

(4.6) Ph~ 3hP = E * = i 6 J , ^ J + Ö(/i3), in L2(tt).

Regularity results imply that (a^fcj) G ( ^ ( n ) fl i72(f2 \ T))2 x (L§(fi) x i fx( ^ \ r ) ) ( j = 1,--- , s) (see [9]) and a similar estimate to (4.3) yields

lia, - a^Hx + \\b3 - b3ih\\0 < ch(\\*3\\'2 + H^Hi), j = 1, • • • ,s, (4.7) where || • \\ft means piecewise H%— norm(i=l,2). Thus, (4.7) together with (2.5, 2.6, and 4.6) complete the proof.

T h e o r e m 4 . 2 . If (u,p) G ( # o ( ^ ) H #4( H ) )2 x (Zg(fi) H i73(Q)) Z5 iAe exac^ solution of (1.1), then there exist SL3 G ( ^ ( f i ) fi iï"2(O \ T))2(j = 1, • • • , s) such that

s

- u = 5 3 a,^2 + O(h4), m L2(Q \ T). (4.8)

3=1

Proof. Let a,, 6 ^ ^ ^ ^ and b³ⁱh satisfy (4.4, 4.5) and set r] = u^ - i^u - Y^^Sp=i ^a3,hh²³ as well as ^ = ph - j ^ p - E " = i Khh], then there exists (w, ip) € (fi3(n) n F2( O ) )2 x (Z,2(O) n f f1^ ) ) satisfying

,<^; v,g) = (77, v), V(v,q) € ( ^ ( ) ) (),

(4.9) Thus

= B(w-ihMr,<p-jh<p;r),Ç)+B(T),£;ih-w,-jh<p). (4.10) Prom Theorem 3.1, Theorem 3.2, (4.6, 4.9), we have

< ch4\\r,\\o (4.11)

and

\B(r},£;i^hvr,-j^h<p)\ < ch⁴(h'¹ \\w - i^hw\\i + \\w\\²)

< e/i4||w||2

< ch'\\r)\\0. (4.12)

Combining (2.5, 4.10, 4.11) and (4.12), we achieve the proof.

A MIXED FINITE ELEMENT FOR THE STOKES PROBLEM 97 5. REMARKS

Only the multi-paramemter error resolution for the Bernardi-Raugel finite élément spaces has been discussed above. In fact, the error résolution arguments présentée in this papcr can bc applied to other pairs of finite élément spaces with similar results, for instance, the filtered bilinear-constant finite élément spaces. Moreover, if the Green-function technique as in [6] is adapted, then maximum norm estimâtes can also be obtained.

The author would like to express his appréciation to the référées for careful corrections and suggesting several im- provements. This work was supported in part by the Chinese National Science Foundation and the Educational Center for Mathematical Research sponsored by the Ministry of Education of China.

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