Global superconvergence approximations of the mixed finite element method for the Stokes problem and the linear elasticity equation

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M2AN - M

ODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

A ^IHUI Z ^HOU

Global superconvergence approximations of the mixed finite element method for the Stokes problem and the linear elasticity equation

M2AN - Modélisation mathématique et analyse numérique, tome 30, n

^o

4 (1996), p. 401-411

<http://www.numdam.org/item?id=M2AN_1996__30_4_401_0>

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MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

(Vol. 30, n° 4, 1996, p. 401 à 411)

GLOBAL SUPERCONVERGENCE APPROXIMATIONS OF THE MIXED FINITE ELEMENT METHOD FOR THE STOKES PROBLEM AND THE

LINEAR ELASTICITY EQUATION (*)

Aihui ZHOU C1)

Abstract. — A three fields formulation for solving the Stokes problem and the équation of linear elasticity proposed by Baranger and Sandri is developed in this paper, using finite element sub spaces based upon a modified pair of « Qx - Qo » element. It is proved that global superconvergence approximations can be obtained for all three fields, not only extra stress tensor, velocity but also pressure.

Key words : Superconvergence, mixed finite element, Stokes and linear elasticity problem.

Résumé. — On développe une formulation à trois champs pour résoudre le problème de Stokes et l'équation d'élasticité linéaire proposée par Baranger et Sandri. On utilise les sous-espaces éléments finis basés sur Vêlement modifié Ql — Q(y On prouve que les approximations de superconvergence peuvent être obtenues pour tous les champs, non seulement pour le tenseur des extracontraintes et la vitesse, mais aussi la pression.

1. INTRODUCTION

Let Q be a domain consisting of connectée! rectangles and

: *u = V *„ * L2(Q), i,j = 1 , 2 }

) , i = 1, 2 } ,

It is known that both the Stokes problem and the linear elasticity équation can be written in Oldroyd's formulation (see Baranger and Sandri [1992]) :

(*) Manuscript received December 10, 1992, revised October 12, 1995.

C1) Institute of Systems Science, Academia Sinica, Beijing 100080, P.R. China Subject Classification. AMS(MOS) : 65N30.

M2 AN Modélisation mathématique et Analyse numérique 0764-583X/96/04/S 4.00

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Find ( <7, u, p) G S x V x L such that

(

(<7,T)-2a(<*(u),T) = 0, V T Ë L ,

( a , ^ ( u ) ) + 2 ( l - a ) ( J ( u )

1

£ f ( v ) ) - ( / ;

1

V - v ) = (f,v), V v e V , where a G [0, 1] and ƒ? ^ 0 are constants.

In the context of the Stokes problem, a dénotes extra stress tensor, u velocity and p pressure; d(u) = (Vu + Vu )/2 rate of strain tensor and* Vu gradient velocity tensor. While for the linear elasticity problem, u dénotes displacement, a — 2 a e ( u ) a scaled «non Newtonian » extra stress tensor and e(u) the strain tensor and / ? = ( l - 2 v ) / 2 v with the Poisson ratio v.

Mixed finite element schemes for this version have been recently proposed by Fortin and Pierre [1989] and Franca and Stenberg [1991] on the Maxwell model corresponding to a = 1 and by Baranger and Sandri [1992] for a e [0,1). For the approximation accuracy, only have the usual estimâtes been obtained in literature. Regarding the version of Oldroyd's model, whether superconvergence approximations can be obtained is an interesting problem and to solve it is certainly valuful for both the theory and practice,

In this paper, we shall pay attention to superconvergence aspect in this version. It is proved that global superconvergence approximations can be obtained for all three variables, not only extra stress tensor, velocity but also pressure if finite element spaces based upon a modified pair of

« Q\ — QQ» éléments are chosen. Our theoretical bases are the identity techniques and interpolation arguments developed by Lin, Yan and Zhou [19], see a survey by Zhou, Li and Yan [26]. It has been shown that these techniques and arguments are powerful for convergence analysis especially for improving accuracy analysis in mixed finite element methods for solving the Stokes équations provided certain preprocesses on the finite element mesh and imposed a kind of postprocess on the finite element approximation. We mention here related work [18, 25, 26] in this direction.

2. PRELIMINAIRES

The space £ of symmetrie tensors with L

2

( Q ) components is equiped with the scalar product (<r, T ) = a

i}

T

tj

with associated norm | T |

0

; V is equiped

M² AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis

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with the scale product (u, v )

v

= (d( u ), d( v ) ) with associated norm

|v|, = (d(u),d(\))

m

. Other notations see Baranger and Sandri [1992] and Girault and Raviart [1986].

The variational formulation of problem (1) can be written in the following form (see Baranger and Sandri [1992]) :

Find ( ö\ u, ƒ? ) e L x V x L such that

(2) £ ( < 7 , u , / ? ; T , v , 4 ) = - 2 a ( f , v ) ,

V ( T , V , ^ ) G L X V X L ,

where

With fi E [0,/?

0

).

For approximating (2), we need introducé three finite element spaces. We start with a subdivision T

h

of Q a R into rectangles whose edges respectively parallel to x - and y - axis with size h. Subsequently we divide each rectangle into four smaller rectangles by joining the midpoints, thus creating another subdivision T

hl2

of Q into rectangles. For a path e e T

h

consisting of éléments k

e t

e T

h/2

( 1 = 1, 2, 3, 4) numbered in the counterclockwise sense.

Defïne a pressure space L

h

through the foliowing choice of basis. For each rectangle e e T

h

we define three basis functions

l, o n ^ , u k

et2

, ), others ,

others , , o n t

é 3

u ^

4 l

, others .

Of course, outside the particular rectangle e e T

^h

, the basis functions vanish as well. This pressure space L

h

consists of three of the four possible piecewise constants associated with the four rectangles in T

h/2

contained within a single rectangle in T

h

and that have zero mean over Q, i.e.,

(3) L

h

= {qe L

20

(Q):q\

e

e span {fu

e

,. : l = 1, 2, 3}, e e T

h

} .

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The velocity space V

^h

and the extra stress tensor space £

^h

are defined as (4) V

h

= {v e V : T|

4

e Q,{k)\ VA: e T

m

},

(5) Sh = {a e £ : a..\e Ê 8 , ( 4 1 « ij « 2, Ve e T j .

It is known that the pair of spaces (V

h

, L

h

) satisfies the inf-sup condition (see Brezzi and Fortin [1991], Girault and Raviart [1986], or Gunzburger [1986]) : (6) sup

{W

,

yy

,'

9)

»c\g\

0

, VqeL

h

V € Vh ' I l

for some constant c > 0.

Dénote

Â P = ƒ * / > + ( - O

¹

' a

^e

( / > ) . on*

^eil

.

^f

/ = 1 , 2 , 3 , 4 where

e,\\ Jketl \Ke,2\ Jke2

L rf ^p -vri\

e.3\ J ^3 \Ke,4\ Jjfee>4

One sees that a

^e

(p) - 0 if p is linear on e, which yields, by Bramble-Hilbert lemma :

(7) | a

^e

( p ) | ^ch\\p\\

^2e

, (8) \\J

^H

P-J

^h

p\\

^o

^ch

²

\\p\\

²

.

LEMMA :

(H

²

(Q))

⁴

x (ffj(fl) n H

³

(f2))

²

x (L

^2Q

(Q) n then

(9) # O - j

h

< 7 , u - i

h

u , / ? - j

f t

/ 7 ; T , v,

AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis

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where i

h

is the usual bilinear interpolation operator with respect to T

h/2

and Jh

f f = =

(i*(

f f

),y)i

< / i

;«2

f

°

r ff

=((

ff

)(,-)i

</

,y«2-

Proof: For any k e T

hn

, let (x

k

, y

k

) be the center of /c, /^

x

, h

k

be its widths in the x- and y-direction, respectively, and dénote

By the identities

(10) [ d

x

( w - i

h

w) = [ F

k

d

2x

d

x

w , k e T

h/2 Jk Jk

(11) f d

^y

( w - i » = f £

⁴

d,V '

•/ A: J k

we obtain firstly

(12) \(V-(u-i

^h

u),q)\^ch

²

\\u\\

³

\\q\\

⁰

, V

(13) | ( r f ( u - i

h

u ) , T ) |

^ C / Z2| | U | |3| T |0,

Vr

Next we turn to prove

(14) | ( d ( u - i

h

u ) , d ( v ) ) |

^ C A ' H U I I J I V I , , VV

e V ,

In f act, combining the identities (10), (11) and

- h

^w

) (y - y

^k

**) = | {^(y - y*)**

3

X

( w - i

h

w ) ( x -

ik

= - Ek Jk

(7)

we obtain {cf. Lin, Yan and Zhou [19], Lin and Zhu [20] or Zhou and Li [25]), for bilinear function q> on k,

d

x

(w-i

h

w)d

x

(p= d

x

(w-i

h

w) {d

x

<p{x,y

k

) + {y - y

k

) d

x

d <p{x

t

y))

Jk Jk

(15) = f F, a

^2y

a

^xW

a

^x

<p-\\ F

^k

(

^y

- y

^k

) a

^2y

a

^x

w a^ix.y),

d

x

(w~ i

h

w ) d

y

<p =\d

x

{w-i

h

w){ d

y

(p(x

k

, y) + {x-x

k

) d

x

d tp{x

9

y))

J k J Jt

= f F k( dy< p - ( x - xk) dxdv<p(x,y)) d2vdxw

J k

=

" ( f " f )

^E

k

^a

xVà

²

w

(16) +\F^k(d^v<p -(x-x^k) d^xd^v<p(.x, y ) ) a²a^xw ,

J k '

where s

k

^i — 3,4) are the down and up edges of L If T

h/2

is almost uniformly, i.e., for any pair of neighboured éléments k and k\ there holds

then by Abel's summation, (14) is obtained from (15), (16) and similar identities to

(17) d

y

{ w — i

h

w) d

y

<p and d

y

( w - i

h

w ) d

x

q>.

Jk ' Jk

Finally, by the similar arguments, for bilinear function <p on k, we have

( p - JhP ) dx< P = ( P - J H P ) (dx<p(x, yk) + ( y - yk) dxdy( p ( x , y ) )

Jk Jk

= (p-j

h

p)(y-y

k

)à

x

à

Y

<p(x

9

y)

Jk

= - FkdyPdxdy<p(x>y)

v k

= - ( f " f r * â r ^p ^ ⁺ f ^F * â * â vP^•

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modeiling and Numerical Analysis

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where s

ki

(i= 1,2) are the left and right edges of k. If T

h/2

is almost uniformly, we obtain, by Abel's summation

(18)

(19) \(d(yXa-j

h

a)\^ch

2

\a\

2

\y\^ V v e V

h

. And (8) and (18) implies

(20) | ( V - v , p - 7 , p ) | S<

2

||/>|l

2

|v|

l f

V v € V

A >

which together with (12)-(14), (19), (20) and

(21) (0-J

^h

(T,t) + 2aP(p-j

^h

p,q) = O, V(r, q) e £

^A

x L

^h

leads to

(22) ^

c / 2 2

( |

T

|

o +

|

v

|

i +

| | ^ | |

o ) ï

V ( T , v , < ? ) e L

h

x V

h

x L

h

, where J

h

(J = U

h

{o)

ij

)

x

^

u

.^

2

for a = ( (cr)^)

1

^

/ y

^

2

. From (8), we complete the proof.

To obtain superconvergence approximations, we need to introducé two operators. One is the usual biquadratic nodal interpolation operator I

h

on T

h

, another is an interpolation operator J

h

with respect to the partition T

h

, which is defined by the follows : J

h

p\

e

e Q

x

(e), for any p e L

2

(Q) and e G T

h

and satisfying (cf. Lin, Yan and Zhou [19] and Zhou and Li [25]) :

Jh q = Q »

(23) I J

h

q=\ q, \/q^L

2

(Q), onk

ei

cze.

It is easy to see that the operators I

h

and J

h

possess the following properties : (24) ' < W * , JJH = J„'*

(25) H ^ w - w l l ^ c ^ l l w l ^ , | | y

f c

« - 9 l l

0

« c

2

where I

h

w = (I

h

w

v

l

h

w

2

) for w = (w

v

w

2

).

Following Baranger and Sandri [1992], we present hère the mixed finite element approximation to the problem (2) :

Find ( a

h

, u

h

, p

h

) G L

h

x V

h

x L

h

such that

(26) B(a

^h

, u

^h1

p

^h

) = - 2 <(f, v ) ,V(r, y*

⁹

q) e L

^h

x V

^h

x L , .

(9)

For a e [ 0 , 1 ) and jSe [0,/?

0

) with some constant ji

0

> 0, the fini te element solution ( er

^h

, u

^h

, p

^h

) exists, and the following optimal error estimate is valid (see Baranger and Sandri [1992]) :

( 2 7 ) K - f f | I 0 + | | uh- u | |1 + \ \ ph- p \ \0 ^ c h ( \ \ o \ \x + \ u \2+ \ p \ , ) .

3. SUPERCONVERGENCE

We shall prove that all the three approximations are superconvergent globally. First of all, we have an error équation

(28) fl(<7-er

^h

, u - u

^h

, p - p

^A

;

T,

v, $ ) = 0 ,

V ( T , V , ? )

e L

^A

x \

^h

x L

^h

. By Preliminaries in the previous Section, we obtain the following theorem :

THEOREM : Let

be the exact solution of (2) and (<7

h

, u

h

,p

h

) e L

h

x V

h

x L

h

be the finite element solution of (26). Then

(29) l i Jh^ h - ^ l i o + I l I f c U h - u l l ^ \\JhPh-p\\Q^ch\

where J

h

0

h

= (J

h

(0

h

)

iJ

)

l

*

iJ

*

2

for a

h

= ( (Oi.y)i ^ /, j v* Proof : One sees that (28) implies

(30) =

Note

lemma in the previous section and the theorems 4.1 and 4.2 in [2],

Note that j

h

maps LQ( ^2 ) into L

h

, we have, by the BB stability (6), the

l

^u

h or

(31) K - /

h

* l l o

+

H

u

h "

i

h

u

l l i + l l ^ - J i P l I o ^ ^

2

-

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis

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On the other hand, one sees that

(32) I U

h

T | |

0

^ c | | T | |

0 >

V i e Z

h

, (33) IH.vll, ^ c | | v | |

I f

V v e \

k

, (34) \\J

^h

q\\oc\\<ih>*

^{V * E L}^A^.

Thus, combining (8), (24), (25), (31)-(34) and the identities (35) J

h

<J

h

- a = J

h

(a

h

- J

h

a

h

) + J J

h

a - a , (36) I

h

u

h

- u = I

h

(u

h

- i

h

u ) + I

h

i

h

u - u , (3V)

we finish the proof.

In this paper, only are superconvergence approximations for a e [0, 1 ) and

« Q\ - Q

o

» element presented. The forthcoming papers will deal with the case a = 1 and other mixed éléments.

Acknowledgements. The author would like to thank Professor Q. Lin and other members of our group for constant discussions and the référée for careful comments.

REFERENCES

[1] I. BABUSKA, 1973, The finite element method with Lagrangian multiplies, Numer.

Math., 20, pp. 179-192.

[2] J. BARANGER and D. SANDRI, 1992, A formulation of Stokes's problem and the linear elasticity équations suggested by the Oldroyd model for viscoelastic flow, M2AN, 26, pp. 331-345.

[3] F. BREZZI, 1974, On the existence,uniquence and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO Anal Numer., R-2, pp. 129-

15L

[4] F. BREZZI, 1986, A survey of mixed finite element methods, in : Finite Elements- Theory and Application (éd. Dwoyer et al), Springer-Verlag, pp. 34-49.

[5] F. BREZZI and M. FORTIN, 1991, Mixed and Hybrid Finite Element Methods, Springer-Verlag.

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[6] F. BREZZI and J. PITKÀRANTA, 1984, On the stabilization of finite element approximations of the Stokes équations, in : Efficient Solutions of Elliptic Sys- tems, Notes on Nurnen Fluid Mech., 10 (ed. Hackbush), Vieweg, Wiesbaden, pp. 11-19.

[7] R G. ClARLET, 1976, The Finite Element Methods for Elliptic Problems, North- Holland.

[8] P G. CIARLET and J. L. LIONS, 1991, Handbook of Numerical Analysis, Vol II, Finite Element Methods (Part I), North-Holland, Amsterdam.

[9] M. FORTIN and R. PIERRE, 1989, On the convergence of the mixed method of Crochet and Marchai for viscoelastic flows, Comp. Meth. AppL Mech. Engrg., 73, pp. 341-350.

[10] L. P. FRANCA, 1989, Analysis and finite element approximation of compressible and incompressible linear isotropic elasticity based upon a variational principle, Comp. Meth. AppL Mech. Engrg., 76, pp. 259-273.

[11] L. P. FRANCA and T. J. R. HUGHES, 1988, Two classes of mixed finite element methods, Comp. Meth. AppL Mech. Engrg., 69, pp. 89-129.

[12] L. P. FRANCA and R. STENBERG, 1991, Error analysis of some Galerkin-least- sequares methods for the elasticity équations, SIAM J. Num. Anal, 78, pp. 1680-

1697.

[13] V. GlRAULT and P. A. RAVIART, 1986, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer-Verlag.

[14] R. GLOWINSKI and O. PIRONNEAU, 1979, On a mixed finite element approxima- tion of the Stokes problem I, Convergence of the approximate solution, Numer.

Math., 33, pp. 397-424.

[15] M. D. GUNZBURGER, 1986, Mathematical aspects of finite element methods for incompressible viscous flows, in : Finite Eléments-Theory and Application (ed.

Dwoyer et al), Springer-Verlag, pp. 124-150.

[16] M. D. GUNZBURGER, 1989, Finite Element Methods for Incompressible Viscous Flows : A Guide to Theory, Pratice and Algorithms, Academie, Boston.

[17] J. Li and A. ZHOU, 1992, Notes on « On mixed mesh finite éléments for sol ving the stationary Stokes problem », Numer. AnaL J. Chinese Univ., 14, 3, pp. 287- 289 (Chinese).

[18] Q. LIN, 1 Li and A. ZHOU, 1991, A rectangle test for the Stokes équations, in : Proc. of Sys. Sci. Sc Sys. Eng., Great Wall (Hongkong), Culture Publish Co., pp. 240-241.

[19] Q. LIN, N. Y A N and A. ZHOU, 1991, A rectangle test for interpolated finite éléments, ibid., pp. 217-229.

[20] Q. LlN and Q. ZHU, 1994, The Proeprocessing and Postprocessing for the Finite Element Method, Shangai Scientific & Technical Publishers (Chinese).

[21] R. STENBERG, 1984, Analysis of mixed finite element method for the Stokes problem : a unified approach, Math. Comp., 42, pp. 9-23.

[22] R, STENBERG, 1991, Postprocess schemes for some mixed finite éléments, RAIRO Model. Math. Anal. Numer., 25, pp. 152-168.

[23] R. TEMAN, 1979, Navier-Stokes Equations, North-Holland, Amsterdam.

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[24] R. VERFURTH, 1984, Error estimâtes for a mixed finite element approximation of the stockes équations, RAIRO Numen Anal, 18, pp. 175-182.

[25] A. ZHOU and J. Li, 1994, The full approximation accuracy for the stream function-vorticity-pressure method, Numen Math., 68, pp. 427-435.

[26] A. ZHOU, J. LI and N. Y AN, 1992, On the full approximation accuracy in finite element methods, in : Proc. Symposium on Applied Math, for Young Chinese Scholars (éd. F. Wu), Inst of Applied Math., Academia Sinica, Beijing, July, pp. 544-553.

Global superconvergence approximations of the mixed finite element method for the Stokes problem and the linear elasticity equation

M2AN - M

A IHUI Z HOU