Yongdeok Kim and Seki Kim

Transcription

1 J. Korean Math. Soc. 39 (00), No. 3, pp STABLE LOW ORDER NONCONFORMING QUADRILATERAL FINITE ELEMENTS FOR THE STOKES PROBLEM Yongdeok Kim and Seki Kim Abstract. Stability result is obtained for the approximation of the stationary Stokes problem with nonconforming elements proposed by Douglas et al [1] for the velocity and discontinuous piecewise constants for the pressure on quadrilateral elements. Optimal order H 1 and L error estimates are derived. 1. Introduction In this paper we obtain the stability result for the finite elements approximation of the stationary Stokes problem with nonconforming elements proposed by Douglas et al [1] for the velocity and discontinuous piecewise constants for the pressure on quadrilateral elements. The stability result was obtained by the macroelement technique. Also, optimal order error estimates on quadrilateral elements will be given. The finite element space Q 1 P 0 combination for the velocity and pressure for the Stokes problem does not satisfy inf-sup condition: it strongly depends on the mesh. For a regular mesh the kernel of (div v, p) is two dimensional. More precisely, the well known checkerboard pressure is in the kernel. This instability can be cured by using nonconforming velocity as will be shown in Theorem. The Q P 1 element is probably the most popular quadrilateral two dimensional element for the Stokes problem at the present time. It appeared as a cure for the instability of the Q Q 1 element which appears quite naturally in the use of reduced integration penalty methods. Conforming Q 1 Q 1 combination for the velocity and pressure with three internal degrees of freedom added to the velocity space for Stokes problem was given by [7]. Received April 6, Mathematics Subect Classification: 65N15, 65N30. Key words and phrases: stability, quadrilateral element, Stokes problem.

2 364 Yongdeok Kim and Seki Kim Nonconforming finite elements [3,8] are attractive for discretizing the Stokes problem since they possess favorable stability properties. For example, it removes the checkerboard pressures. Rannacher and Turek [4], in the setting of the stokes problem, analyzed two forms of nonconforming elements based on simply rotating the usual bilinear element to employ Span{1, x, y, x y } as the local basis. On rectangles, they construct a very clever argument that uses a cancellation property on each rectangle, plus a series of an inverse property, to show optimal order approximation of the solution of the Stokes problem; however, if the usual definition of the global nonconforming space by requiring continuity at interfacial midpoints is adopted, there is a loss of optimality for truly quadrilateral partitions of the domain. Douglas et al [1], modified the rotated bilinear local basis Span{1, x, y, x y } as Span{1, x, y, (3x 5x 4 ) (3y 5y 4 )} which has the following orthogonalities 1, w w k Γk = 0 and 1, w Γ = 0 on quadrilateral elements where Γ k = Ω Ω k and Γ = Ω Ω k. By imposing these orthogonalities to local basis they can avoid that loss of optimality for quadrilateral elements. A stability result for the stationary Stokes problem with the following velocity-pressure finite element spaces for quadrilateral elements was given in []: V b = Span{1, x, y, (3x 5x 4 ) (3y 5y 4 ), (1 x )(1 y )}, P 1 = Span{1, x, y}. The conforming bubble function (1 x )(1 y ) was necessary for the stability, because they use linear function for pressures. The bubble function can be removed by using constant function for pressures. This is the obective of this paper. Now, let V = Span{1, x, y, (3x 5x 4 ) (3y 5y 4 )}, P = Span{1}. In this paper, we use V as an approximation space for velocity and P for pressure of the Stokes problem. Then, applying a standard macroelement technique [, 4, 5, 6], we obtain the inf-sup or Babuška and Brezzi stability condition which is essential to finite elements approximation of Stokes problem. Optimal order error estimates derived for H 1 -norm and L -norm, respectively. An outline of this paper is as follows. In Section, we briefly review Stokes problem. In Section 3, finite elements approximation space for the Stokes problem on quadrilateral elements was given. Stability results

3 Stable low order quadrilateral elements for Stokes problem 365 will be given in Section 4 and optimal order error estimates will be derived in Section 5.. Stokes problem In this section we will consider nonconforming finite element method for the Stokes equations: Find the velocity u and the pressure p such that (1) ν u + p = f in Ω, div u = 0 in Ω, u = 0 on Ω, where Ω R is a bounded domain, f is the given body force, and ν > 0 is the viscosity. From now on we will assume ν = 1 for simplicity. The usual variational formulation of (1) is the following. Find u H0 1(Ω) and p L 0 (Ω) such that () a(u, v) + b(p, v) = (f, v) v H 1 0 (Ω), b(q, u) = 0 q L 0(Ω), where a(u, v) = ( u, v), b(p, v) = (p, div v) and (, ) denotes the inner product in L (Ω) or L (Ω), and L 0 (Ω) is the space { } L 0(Ω) = p L (Ω) p dx = 0. The finite element methods we consider may be described abstractly as follows. We let T h denote a partition of Ω by quasi-regular convex quadrilaterals K of diameter h. We then denote by V h a finitedimensional approximation of H0 1(Ω). Since we are considering nonconforming methods, V h H0 1(Ω), but v h H 1 (K) for all v h V h and K T h. We assume that ( ) 1/ v h 1,h = v h 1,K K T h is a norm on V h. Let P h denote a finite-dimensional subspace of L 0 (Ω). The approximation scheme is then: Find u h V h, p h P h satisfying (3) a h (u h, v h ) + b h (p h, v h ) = (f, v h ) v h V h, b h (q, u h ) = 0 q P h, Ω

4 366 Yongdeok Kim and Seki Kim where a h (u, v) = K T h ( u, v) K, b h (p, v) = K T h (p, div v) K = (p, div h v). And div h v is the L (Ω) function whose restriction to each quadrilateral K T h is given by (div v) K. Combining () and (3), we obtain the following consistency error identity (4) a h (u u h, v h ) + b h (p p h, v h ) + b h (q h, u u h ) = ( u pν K ) v h, v h V h, q h P h, ν K K T h K where ν K denotes the unit outward normal to K. The analysis of this type of method is well understood. If the space V h were conforming, the general theory of saddle-point problems developed by Babuška and Brezzi could be applied directly. In that case the only difficulty in the analysis is the verification of the inf-sup condition (5) inf 0 q h P h b(q h, v h ) sup γ > 0. 0 v h V h q h 0 v h 1 When (5) holds, one obtain the quasi-optimal error estimate u u h 1 + p p h 0 C inf ( u v h 1 + p q h 0 ), where the inf is taken over all v h V h and q h P h. For nonconforming methods, a straightforward modification of this result leads to the conclusion that if (6) inf 0 q h P h then one obtain the error estimate b h (q h, v h ) sup γ > 0, 0 v h V h q h 0 v h 1,h (7) u u h 1,h + p p h 0 ( K K C inf( u v h 1 + p q h 0 ) + sup ( u ) ν K pν K ) w h, w h 1,h where the inf is taken over all v h V h and the sup is taken over all w h V h.

5 Stable low order quadrilateral elements for Stokes problem Finite elements spaces on quadrilateral elements Let us partition the domain Ω to quasi-regular convex quadrilateral elements denoted by Ω. Let Ω = Ω, Γ = Ω, Γ = Γ Ω, Γ k = Γ k = Ω Ω k, and denote the centers of Γ and Γ k by c and c k, respectively. Let P l (E) denote the class of polynomials of degree l on the set E. For an arbitrary quadrilateral K with vertices a i, 1 i 4, let ˆR be the reference square [ 1, 1] [ 1, 1] in the (ξ, η) reference space, with vertices â i, 1 i 4, as in Fig. 1. Let F K be the bilinear mapping from ˆR onto K, that is, (x, y) = F K (ξ, η) = (FK 1, F K ) = ( 4 i=1 x i ˆφ i, 4 i=1 y i ˆφ i ), where ˆφ 1 = (1 ξ)(1 η)/4, ˆφ = (1 + ξ)(1 η)/4, ˆφ 3 = (1 + ξ)(1 + η)/4, ˆφ 4 = (1 ξ)(1 + η)/4. We will use nonconforming finite elements proposed by [1] as an approximation space for velocity of Stokes problem. The nodal basis on reference element is the following: ˆϕ 1 (ξ, η) = η {(3ξ 5ξ 4 ) (3η 5η 4 )}, (8) ˆϕ (ξ, η) = ξ 1 8 {(3ξ 5ξ 4 ) (3η 5η 4 )}, ˆϕ 3 (ξ, η) = η {(3ξ 5ξ 4 ) (3η 5η 4 )}, ˆϕ 4 (ξ, η) = ξ 1 8 {(3ξ 5ξ 4 ) (3η 5η 4 )}. Fig. 1. Illustration of quadrilateral K and reference square ˆR.

6 368 Yongdeok Kim and Seki Kim With the above notations, we construct the following velocity-pressure finite element spaces: (9) Ṽ h = {v(x) = ˆv(F 1 K (x)), ˆv N CQ 1( ˆR), K T h }, V h = {v Ṽh v(c ) = 0, for all midpoints c on Γ}, P h = {p L 0(Ω) p P 0, K T h }, where N CQ 1 ( ˆR) = Span{ ˆϕ 1, ˆϕ, ˆϕ 3, ˆϕ 4 }. From the definition of the space V h, any v V h has midpoints continuity between adacent quadrilateral elements. Using the orthogonal properties for scalar function [1], that is, 1, w w k Γk = 0, 1, w Γ = 0, we can derive the following orthogonal properties for vector functions, (10) (11) 1, w w k Γk = 0, w V h, 1, w Γ = 0, w V h, which guarantees optimal order error estimates. For convenience in the analysis below, let Λ h = {λ λ k = tr Γk (λ Ω ) P 0 (Γ k ); Define proections Π, P 0 and I by λ k + λ k = 0; λ = tr Γ (λ Ω ) P 0 (Γ )}. (1) Π : H (Ω) V h : (v Πv)(c) = 0, c = c k or c ; P 0 : H (Ω) Λ h : v ν P 0 v, z Γ = 0, z P 0 (Γ), Γ = Γ k or Γ ; I : L (Ω) P h : (p Ip, q h ) Ω = 0 q h P h, i.e., Ip be the mean value of p in each K T h. Since Ip reproduces constant functions on elements we obtain the following: (13) p Ip 0,Ω C h p 1,Ω, p H 1 (Ω).

7 Stable low order quadrilateral elements for Stokes problem 369 Moreover, since Π reproduces linear functions on elements and P 0 reproduces constants on faces, it follows from standard polynomial approximation results that 1 1 (14) v Πv 0, + h v Πv 1, +h v Πv, 1 + h 1 v Πv 0, 1 +h 3 (v Πv) 0, 1 + h 3 v P 0 v 0, 1 C h v, v H (Ω), where z m, = z H m (Ω ), z m, = k z H m (Γ k ) with Γ replacing Γ k for boundary faces. (See [1, Section 6]) 4. Stability results using macroelement technique In this section we slightly modify the macroelement technique [, 4, 5, 6] and obtain the inf-sup stability condition. By a macroelement M we define a connected set of elements of which the intersection of any two is either empty, a vertex, or an edge. Two macroelements M and M are said to be equivalent if they can be mapped continuously onto each other [6]. For a macroelement M we define the spaces V 0,M = {v V h v(c ) = 0 for all midpoints c on M, v = 0 in Ω \ M}, P M = {p P h p = 0 in Ω \ M}, N M = {p P M (div h v, p) M = 0 v V 0,M }. Theorem 1. Suppose that there is a fixed set of equivalent classes ε i, i = 1,,..., l, of macroelements, a positive integer L, and a macroelement partitioning M h such that

8 370 Yongdeok Kim and Seki Kim (M1) For each M ε i, i = 1,,..., l, the space N M is one-dimensional, consisting of functions that are constant on M; (M) Each M M h belongs to one of the classes ε i, i = 1,,..., l; (M3) Each K T h is contained in at least one and not more than L macroelements of M h ; (M4) Each e Γ h is contained in the interior of at least one and not more than L macroelements of M h, where Γ h denotes the collection of edges, of the elements of T h, in the interior of Ω. Then the inf-sup condition (6) is valid. We now show that the finite elements approximation scheme (3) with V h and P h defined by (9) satisfy the inf-sup condition by using the macroelement technique. Fig.. Reference and arbitrary two elements macroelement. Theorem. For the Stokes problem (1), the velocity-pressure finite element formulation (3) satisfies the inf-sup stability condition (6). Proof. Consider a macroelement consisting of two elements: M = K 1 K as in Fig.. By virtue of the macroelement technique, it is sufficient to show that (15) N M = {w W M (div h v, w) M = 0, v V 0,M } = {constants on M}.

9 Stable low order quadrilateral elements for Stokes problem 371 By elementary calculation, we get div h v w dx dy = div h v w dx dy M K 1 K ( y u = ˆR η ξ y ) ( u α 1 dξ dη + x v ξ η ˆR η ξ + x ) v α dξ dη ξ η = {((y 3 + y 4 ) (y 1 + y ) + ξ((y 1 + y 3 ) (y + y 4 )) u ˆR ξ ((y + y 3 ) (y 1 + y 4 ) + η((y 1 + y 3 ) (y + y 4 )) u η }α { ((x 3 + x 4 ) (x 1 + x ) + ξ((x 1 + x 3 ) (x + x 4 )) v ˆR ξ + ((x + x 3 ) (x 1 + x 4 ) + η((x 1 + x 3 ) (x + x 4 )) v η }α 1 {((y 6 + y 3 ) (y + y 5 ) + ξ((y + y 6 ) (y 5 + y 3 )) u ˆR ξ ((y 5 + y 6 ) (y + y 3 ) + η((y + y 6 ) (y 5 + y 3 )) u η }α { ((x 6 + x 3 ) (x + x 5 ) + ξ((x + x 6 ) (x 5 + x 3 )) v ˆR ξ + ((x 5 + x 6 ) (x + x 3 ) + η((x + x 6 ) (x 5 + x 3 )) v η }α. Let ϕ be the function defined on M which has nodal value equal to one at the point denoted by and zeros at. (See Fig..) By taking v = (u, v) = (ϕ, 0) we have (16) α 1 (x x 3 ) + α (x 3 x ) = 0 and by taking v = (u, v) = (0, ϕ) we have (17) From these two equations we get ( ) x x α α y y 3 This leads (18) α 1 (y y 3 ) + α (y 3 y ) = 0. ( ) x3 x = (α y y 1 α ) 3 α 1 = α. ( ) x x 3 = 0. y y 3 From (18) we obtain (15). The theorem follows from Theorem 4.1

10 37 Yongdeok Kim and Seki Kim 5. Error estimates In this section we will estimate the energy and L errors. At first by estimating the consistency error, we obtain optimal energy norm error. Next using the duality argument, we obtain optimal L norm error. From now on we will use C denotes generic positive constants independent of the mesh parameter h and is different where it occurs. Now, let us estimate the consistency error given by the following: u pν, w Ω = u pν, w Γ ν ν + u pν, w ν Ω \Γ. From the definition of proection P 0 and the orthogonal properties (10), (11), the following orthogonalities hold: P 0 u, w Γk + P 0 u k, w k Γk = P 0 u, w w k Γk = 0, w V h, u P 0 u, 1 Γ = 0, Γ = Γ or Γ k, 1, w Γ = 0, w V h. We will first estimate the term: u pν, w ν Ω \Γ = u P 0 u, w m Ω \Γ (p p)ν, w w k Γk. By (14), a standard trace theorem, and certain approximation of w, p by properly chosen constants m, p (its average over Ω ), (19) u P 0 u, w m Ω \Γ C u h 1 w m (w m ) 1 C u h 1 1 w C h u w 1,h.

11 Stable low order quadrilateral elements for Stokes problem 373 Since w w k vanishes at the midpoints of Γ k, we get w w k, w w k Γk C h w 1,h, and (0) (p p)ν, w w k Γk C h p 1 w 1,h. Summing (19) and (0) we get (1) u pν, w ν Ω \Γ C h ( u + p 1 ) w 1,h. Since w(c ) = 0 at the boundary midpoints, we get the following: () w, w Γ C h ( w, w) Ω. Then, (3) u pν, w Γ = u P 0 u, w Γ ν C h ( u + p 1 ) w 1,h. (p p)ν, w Γ Therefore, from (1) and (3) we get the following consistency error estimates, (4) u pν, w Ω C h ( u + p 1 ) w 1,h. Theorem 3. Let (u, p) be the solution of () and (u h, p h ) be the solution of (3). Then the following optimal error estimates holds: (5) u u h 1,h + p p h 0 C h ( u + p 1 ). Proof. From the error estimates (7) and apply v h = Πu and q h = Ip and using approximation properties of (14), (13) and the consistency

12 374 Yongdeok Kim and Seki Kim estimates (4) we obtain: u u h 1,h + p p h 0 ( C u Πu 1 + p Ip 0 + sup C h ( u + p 1 ). Ω ( u ) pν ) w h w h 1,h From now on we will estimate the L error estimate using the duality arguments. Let η = u u h. Let ϕ H0 1(Ω) and χ L 0 (Ω) be the unique solution of the auxiliary Stokes problem (6) a(ϕ, v) + b(χ, v) = (η, v) v H 1 0 (Ω), b(q, ϕ) = 0 q L 0(Ω), which satisfies the following a priori estimate (7) ϕ + χ 1 C u u h. Using the above notation, there holds η = (η, ϕ + χ) = a h (η, ϕ) + b h (χ, η) ϕ χν, η Ω. Using the error identity (4) and by taking v h = Πϕ, q h = χ h, we get (8) a h (η, Πϕ) + b h (p p h, Πϕ) + b h (χ h, η) = ( u pν ) Πϕ. Ω Since η is in L (Ω) we have ϕ H (Ω), χ H 1 (Ω). From this and by (6), we get (9) By (8) and (9), b h (p p h, ϕ) = 0, u pν, ϕ Ω = 0. η = a h (η, ϕ Πϕ) + b h (p p h, ϕ Πϕ) + b h (χ χ h, η) ϕ χν, η Ω u pν, (ϕ Πϕ) Ω. ν ν

13 Stable low order quadrilateral elements for Stokes problem 375 Expand the fourth and fifth terms as following and using (), (7) and (14), we get = ϕ χν, η Ω + u pν, (ϕ Πϕ) Ω ( ϕ P 0 ϕ) (χ χ)ν, η η Ω \Γ ( u P 0 u) (p p)ν, (ϕ Πϕ) Ω \Γ ( ϕ P 0 ϕ) (χ χ)ν, η Γ ( u P 0 u) (p p)ν, (ϕ Πϕ) Γ C h ( u + p 1 ) η. Therefore, using the above result and (5), (7) and (14), we obtain (30) u u h C h ( u + p 1 ). Theorem 4. Let (u, p) be the solution of () and (u h, p h ) be the solution of (3). Then the following optimal L error estimates holds: (31) u u h C h ( u + p 1 ). References [1] J. Douglas, Jr., J. E. Santos, D. Sheen, and X. Ye, Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems, MAN Math. Model. Numer. Anal. 33 (1999), no. 4, [] Y. Kim and S. Lee, Stable nonconforming quadrilateral finite elements for the Stokes problem, Appl. Math. Comput. 115 (000), no. -3, [3] N. Kechkar and D. Silvester, Analysis of locally stabilized mixed finite element methods for the Stokes problem, Math. Comp. 58 (199), no. 197, [4] R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numer. Methods Partial Differential Equations 8 (199), [5] R. Verfürth, The Stability of finite element methods, Numer. Methods Partial Differential Equations 11 ( 1995), [6] R. Stenberg, Error analysis of mixed finite element methods for the Stokes problem, Math. Comp. 54 (1990),

14 376 Yongdeok Kim and Seki Kim [7], Analysis of mixed finite element methods for the Stokes problem: A unified approach, Math. Comp. 4 (1984), no. 165, 9 3. [8] W. Bai, The quadrilateral Mini finite element for the Stokes problem, Comput. Methods Appl. Mech. Engrg. 143 (1997), [9] Z. Zhang, Analysis of some quadrilateral nonconforming elements for incompressible elasticity, SIAM J. Numer. Anal. 34 (1997), no., Yongdeok Kim Department of Mathematics KAIST Daeeon , Korea ydkimia@hananet.net Seki Kim Department of Mathematics SungKyunKwan University Suwon , Korea skim@yurim.skku.ac.kr