Mixed Finite Element Methods for Incompressible Flow: Stationary Stokes Equations

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1 Mixed Finite Element Metods for Incompressible Flow: Stationary Stoes Equations Ziqiang Cai, Carles Tong, 2 Panayot S. Vassilevsi, 2 Cunbo Wang Department of Matematics, Purdue University, West Lafayette, Indiana Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, California 9455 Received 2 October 2008; accepted 27 January 2009 Publised online in Wiley InterScience ( DOI 0.002/num In tis article, we develop and analyze a mixed finite element metod for te Stoes equations. Our mixed metod is based on te pseudostress-velocity formulation. Te pseudostress is approximated by te Raviart- Tomas (RT) element of index 0 and te velocity by piecewise discontinuous polynomials of degree. It is sown tat tis pair of finite elements is stable and yields quasi-optimal accuracy. Te indefinite system of linear equations resulting from te discretization is decoupled by te penalty metod. Te penalized pseudostress system is solved by te H(div) type of multigrid metod and te velocity is ten calculated explicitly. Alternative preconditioning approaces tat do not involve penalizing te system are also discussed. Finally, numerical experiments are presented Wiley Periodicals, Inc. Numer Metods Partial Differential Eq 000: , 2009 Keywords: incompressible Newtonian flow; mixed finite element; multigrid; Stoes equations I. INTRODUCTION Let be a bounded, open, connected subset of R d (d = 2 or 3) wit a Lipscitz continuous boundary. Let f = (f,..., f d ) and ν>0be te given external body force and inematic viscosity, respectively. Denote u = (u,..., u d ), σ = ( σ ij ) d d, and p to be te velocity vector, stress tensor, and pressure, respectively. Wen te density of te fluid is practically constant, te basic equations for incompressible Newtonian flows consist of σ + pδ 2νɛ(u) = 0, (consitutive law) u + u u σ = f, (balance of linear momentum), (.) t u = 0, (conservation of mass) Correspondence to: Ziqiang Cai, Department of Matematics, Purdue University, 50 N. University Street, West Lafayette, IN ( zcai@mat.purdue.edu) Contract grant sponsor: U.S. Department of Energy by University of California Lawrence Livermore National Laboratory; Contract grant number: W-7405-Eng Wiley Periodicals, Inc.

2 2 CAI ET AL. were,, and δ denote te gradient operator, divergence operator, and identity tensor, respectively; and ɛ(u) = ( u + ( u) t )/2 is te deformation rate tensor. Here σ, p, and ν are scaled wit te density. To close te system, bot initial and boundary conditions are needed. Te initial condition sould be given as u t = 0 = u 0, were u 0 is te initial velocity. Tere are different inds of boundary conditions. Many applications in incompressible Newtonian flow are posed under te Diriclet boundary condition for te velocity u = g on, (.2) were g = (g,..., g d ) is prescribed velocity on te boundary satisfying te compatibility condition n g ds = 0. (.3) In tis case, te pressure is only unique up to an additive constant. System (.) is nown as te stress-velocity-pressure formulation of incompressible Navier- Stoes equations. Eliminating te stress from (.) gives te velocity-pressure formulation of Navier-Stoes equations u + u u (νɛ(u)) + p = f, t (.4) u = 0. Witout te nonlinear term u u, Eq. (.4) becomes te Stoes problem. Te Stoes problem is linear and plays a critical role in numerical metods for solving Navier-Stoes equations. Te velocity-pressure formulation in (.4) as long been te mainstream in computational incompressible Newtonian flows. However, researc on te stress-velocity-pressure formulation is gaining consistent attention recently because of te arising interest in non-newtonian flows. For non-newtonian flows in wic te constitutive law is nonlinear, te stress cannot be eliminated. Terefore, a formulation containing te stress as a fundamental unnown is unavoidable. Notice tat te main advantage of te stress-velocity-pressure formulation is tat it provides a unified framewor for bot te Newtonian and te non-newtonian flows. It as also been pointed out [] tat an accurate and efficient numerical sceme for Newtonian flows under formulation (.) is necessary for te successful computation of non-newtonian flows. Anoter advantage of te stress-velocity-pressure formulation is tat, pysical quantity lie te stress is computed directly instead of by taing derivatives of te velocity. Tis avoids degrading of accuracy wic is inevitable in te process of numerical differentiation. Accurate calculation of te stress is paramountly important for any flow problems involving obstacle bodies since it is crucial for, e.g., te design of solid structure and te reduction of drag. However, te stress-velocity-pressure formulation as some obvious disadvantages. Te most significant ones are te increase in te number of unnowns and te symmetry requirement for te stress tensor [2]. Bot of tem pose extra difficulty in te numerical computation. To avoid tese disadvantages, tis article studies mixed finite element metods based on te pseudostressvelocity formulation [3, 4]. Raviart-Tomas (RT) elements of index 0 [5] are used for approximating eac row of te pseudostress, and discontinuous piecewise polynomials of degree 0 for approximating eac component of te velocity. It is sown tat tis pair of mixed finite elements is stable and yields quasi-optimal accuracy O( + ) for sufficiently smoot solutions. Tis discretization as two obvious advantages: (i) accurate approximation to pysical quantities Numerical Metods for Partial Differential Equations DOI 0.002/num

3 MIXED METHOD FOR STOKES PROBLEM 3 suc as te stress and vorticity and (ii) no essential boundary condition posed in approximation space. Moreover, te metod can be easily extended to applications wit variable viscosity and/or variable density. One possible disadvantage on using te pseudostress in incompressible Newtonian flows is tat it increases te number of variables. Indeed, at te continuous level, te pseudostress-velocity formulation as d times more independent variables tan te velocity-pressure formulation. However, at te discrete level, for lower order elements te number of degrees of freedom for te pseudostress-velocity using Raviart-Tomas elements is comparable wit tat for te velocity-pressure using Crouzeix-Raviart elements [6 8] (nonconforming velocity and discontinuous pressure) and bot approaces ave te same accuracy for te H seminorm of te velocity and te L 2 norm of te pressure. More specifically, for te lowest order elements, te pseudostressvelocity as dn f + dn t unnowns and te velocity-pressure as dn f + N t unnowns, were N f and N t are te number of edges/faces and elements, respectively. Te velocity wit dn t unnowns in te pseudostress-velocity form is furter troug eiter te penalty metod for stationary problems or natural time discretization for unsteady-state problems so tat we only need to solve numerically te symmetric and positive definite pseudostress system wit dn f unnowns. Similarly, for stationary problems one can use te penalty metod to eliminate te pressure in te velocity-pressure form to get te Lamé system wit dn f unnowns. Te large Lamé constant is te reciprocal of te penalty parameter. To solve te indefinite system of linear equations resulting from te discretization efficiently, we eliminate te velocity by using te penalty metod for stationary problems to obtain a smaller system involving only te pseudostress. To avoid accuracy loss in te penalty metod, te penalty parameter is cosen to be proportional to te discretization accuracy. Tis means tat = O( + ) for RT elements of index 0. Wit tis coice of, te condition number of te pseudostress system is O( 2 ) = O( 2 (+) ) and, ence, very ill-conditioned. Tis is an apparently very difficult problem to solve by any conventional iterative metods wose convergence factor depends also on te penalty parameter. In tis article, we numerically solve te reduced pseudostress system by te H(div) type of multigrid metod introduced in [9 ]. Preconditioning te pseudostress system by a V(, )-cycle multigrid metod for a weigted H(div) problem, it is sown tat te corresponding preconditioned conjugate gradient (PCG) metod converges uniformly wit respect to te mes size, te number of levels, and te penalty parameter provided tat is bounded above by a constant. Tis is confirmed by our numerical results on uniform rectangular RT elements of te lowest order ( = 0). Wit computed pseudostress, te velocity can ten be calculated eiter explicitly for = 0 or locally for. Te penalty approac is not te only one possible. We suggest a bloc-diagonal preconditioner for te (unpenalized) saddle-point problem wic utilizes te same tools as te preconditioner for te penalized matrix, namely, one needs an optimal preconditioner for a similar H(div) problem. Te article is organized as follows. Te pseudostress-velocity formulation is derived in Section II. Sections III and IV describe and analyze mixed finite element metod and te penalty metod, respectively. Our preconditioning tecnique is discussed in Section V. Finally, numerical experiments on te accuracy of mixed finite element metod and te condition number of te preconditioned system are presented in Section VI. We end wit some concluding remars in Section VII. A. Notation We use te standard notations and definitions for te Sobolev spaces H s ( ) d and H s ( ) d for s 0. Te standard associated inner products are denoted by (, ) s, and (, ) s,, and teir Numerical Metods for Partial Differential Equations DOI 0.002/num

4 4 CAI ET AL. respective norms are denoted by s, and s,. (We suppress te superscript d because te dependence on dimension will be clear by context. We also omit te subscript from te inner product and norm designation wen tere is no ris of confusion.) For s = 0, H s ( ) d coincides wit L 2 ( ) d. In tis case, te inner product and norm will be denoted by and (, ), respectively. Set H0 ( ) := {q H ( ) : q = 0on }. We use H ( ) to denote te dual of H0 ( ) wit norm defined by φ = (φ, ψ) sup. 0 =ψ H 0 ψ ( ) Denote te product space H ( ) d = d i= H ( ) wit te standard product norm. Finally, set wic is a Hilbert space under te norm H(div; ) ={v L 2 ( ) d : v L 2 ( )}, v H(div) = ( v 2 + v 2 ) 2. II. PSEUDOSTRESS-VELOCITY FORMULATION For a vector function v = (v,..., v d ), define its gradient as a d d tensor v v x x d ( ) v = v d v = vi. x j d d d x x d For a tensor function τ = (τ ij ) d d, let τ i = (τ i,..., τ id ) denote its it-row for i =,..., d and define its divergence, normal, and trace by τ = ( τ,..., τ d ), n τ = (n τ,..., n τ d ), and trτ = d τ ii, i= respectively. Let A : R d d R d d be a linear map, wic is singular, defined by Aτ = τ d (trτ)δ, it is easy to see tat Aτ is trace free and tat τ R d d as te following ortogonal decomposition wit respect to te product of tensors for σ, τ R d d. τ = Aτ + (trτ)δ, (2.) d σ : τ d σ ij τ ij, i,j= Numerical Metods for Partial Differential Equations DOI 0.002/num

5 MIXED METHOD FOR STOKES PROBLEM 5 Introducing a new independent, nonsymmetric tensor variable, te pseudostress, as follows σ = ν u pδ, (2.2) taing trace of (2.2) and using te divergence free condition in te tird equation of (.) give Ten (2.2) may be rewritten as p = trσ. (2.3) d κaσ u = 0, were κ = /ν. For incompressible fluids, because te divergence of ( u) t vanises, te stress and pseudostress ave te same divergence; i.e., σ = σ. Hence, we ave te following pseudostress-velocity formulation of te Navier-Stoes equation κaσ u = 0, u (2.4) t + u u σ = f. Te incompressibility condition is implicitly contained in te constitutive equation [te first equation in (2.4)]. Tere are two reasons for eliminating te pressure. An obvious one is to reduce one variable and, ence, many degrees of freedom in te discrete system. A more important reason is tat we are able to use economic and accurate stable elements and able to develop fast solvers for te resulting discrete system so tat computational cost will be greatly reduced. In tis article, we concentrate on te pseudostress-velocity formulation of te stationary Stoes problem { κaσ u = 0 in, (2.5) σ = f in, wit boundary condition (.2) and compatibility condition (.3). Mixed finite element metods based on te pseudostress-velocity formulation for te stationary Navier-Stoes equation will be presented in [2]. Wen te viscosity parameter is constant, problem (2.5) is independent of ν (or κ) by scaling te σ and f wit te viscosity. Oterwise, assume tat tere exist positive constants κ 0 and κ suc tat 0 <κ 0 κ(x) κ, (2.6) for almost all x. It is well-nown tat te stationary Stoes equation as a unique solution provided tat pdx = 0, Numerical Metods for Partial Differential Equations DOI 0.002/num

6 6 CAI ET AL. wic, togeter wit (2.3), implies trσ dx = 0. Terefore, we introduce a subspace of H(div; ) d : H(div; ˆ ) d ={τ H(div; ) d : trτdx = 0}. To obtain te wea formulation of (2.5), we multiply te first equation in (2.5) by a test tensor function τ H(div; ˆ ) d, integrate it over te domain, and use integration by parts and boundary condition (.2) (κaσ, τ) + (u, τ) = g (n τ)ds g(τ). Multiplying a test vector function v L 2 ( ) d on bot sides of te second equation in (2.5) and integrating it over te domain give tat ( σ, v) = (f, v) f(v). Now, te variational problem of te pseudostress-velocity formulation is to find a pair (σ, u) H(div; ˆ ) d L 2 ( ) d suc tat { (κaσ, τ) + (u, τ) = g(τ) τ H(div; ˆ ) d, (2.7) ( σ, v) = f(v) v L 2 ( ) d. It follows from te fact tat A is singular and te ortogonal decomposition in (2.), tat trτ can not be controlled by Aτ alone for any τ H(div; ˆ ) d, but we do ave te following inequality (see [3, 4]). Lemma 2.. For any τ ˆ H(div; ) d, we ave tr τ C( Aτ + τ ). (2.8) (We use C wit or witout subscripts in tis article to denote a generic positive constant, possibly different at different occurrences, tat is independent of te mes size and te penalty parameter introduced in subsequent sections but may depend on te domain.) It is easy to see tat wic, togeter wit Lemma 2., implies τ 2 = Aτ 2 + d trτ 2, (2.9) τ C( Aτ + τ ) C( Aτ + τ ). (2.0) To prove te existence and uniqueness of problem (2.7), it is convenient to use te following lemma (see, e.g., [5]). Numerical Metods for Partial Differential Equations DOI 0.002/num

7 MIXED METHOD FOR STOKES PROBLEM 7 Lemma 2.2. For any q L 2 ( ), tere exists a v H ( ) d suc tat v = q in and v C q. (2.) Teorem 2.3. Te variational problem in (2.7) as a unique solution. Proof. By te Caucy-Scwarz inequality, definition of norms, and a trace teorem, it is easy to see tat linear forms f(v) and g(τ) are continuous in L 2 ( ) d and H(div; ˆ ) d, respectively; tat is and f(v) f v, (2.2) g(τ) g /2, n τ /2, g /2, τ H(div). (2.3) It follows from (2.0) and (2.6) tat te bilinear form (κaσ, τ) = (κaσ, Aτ) is coercive in te divergence free subspace of H(div; ˆ ) d Cκ 0 τ 2 H(div) κ 0 Aτ 2 (κaτ, τ), (2.4) for any τ in Hˆ 0 (div; ) d ={τ H(div; ˆ ) d τ = 0}. For any v L 2 ( ) d, Lemma 2.2 implies tat tere exists τ H ( ) d d suc tat τ = v in and τ C v. (2.5) Let a = tr τdx and τ = τ a δ were denotes te volume of te domain, ten it is d easy to cec tat Hence, τ ˆ H(div; ) d, τ = v in, and τ C v. (2.6) ( γ, v) sup v 2 β v. (2.7) γ H(div; ˆ ) d γ H(div) τ H(div) Now, te coercivity condition in (2.4) and te inf-sup condition in (2.7) imply [6] tat te variational problem in (2.7) as a unique solution. It is important to point out tat te pseudostress contains more information tan te stress σ = pδ + ν( u + ( u) t ) = σ + ν( u) t. Pysical quantities suc as te velocity gradient, stress, vorticity, and pressure can be algebraically expressed in terms of te pseudostress: u = Aσ, σ = σ + ν(aσ ) t, ω = 2 (Aσ (Aσ )t ), and p = trσ, (2.8) d Numerical Metods for Partial Differential Equations DOI 0.002/num

8 8 CAI ET AL. respectively. Terefore, tese pysical quantities (if needed) can be computed in a postprocessing procedure witout degrading accuracy of approximation. In (2.8), we conveniently represent te vorticity u as te sew symmetric part of te velocity gradient: ω = 2 ( u ( u)t ). Tis defines te vorticity (or te curl operator) in all dimensions by one formula. III. FINITE ELEMENT APPROXIMATION Assume tat is a polygonal domain, let T be a quasi-regular triangulation of wit (triangular/tetraedral or rectangular) elements of size O(). Denote spaces of polynomials on an element K R d P (K) is te space of polynomials of degree ; P, 2 (K) = p(x, x 2 ) : p(x, x 2 ) = a ij x i xj 2 d = 2; i,j 2 P, 2, 3 (K) = p(x, x 2, x 3 ) : p(x, x 2, x 3 ) = a ij x i xj 2 x 3 d = 3; i,j 2, 3 { P, (K) d = 2, Q (K) = P,, (K) d = 3. Denote te local Raviart-Tomas (RT) space of index 0 on an element K: { P (K) RT (K) = d + (x,..., x d )P (K) K = triangle/tetraedral, Q (K) d + (x,..., x d )Q (K) K = rectangle/cube. In two dimensions, degrees of freedom for RT 0 (K) = (a + bx, c + bx 2 ) on triangle or RT 0 (K) = (a + bx, c + dx 2 ) on rectangle are normal components of vector field on all edges. For te coice of degrees of freedom of te RT space of index, see [7]. Tey are cosen for ensuring continuity of te normal component of vector field at interfaces of elements. Ten one can define te H(div; ) conforming Raviart-Tomas space of index 0 [5] by Let RT ={v H(div; ) : v K RT (K) K T }. { P (K) K = triangle/tetraedral, D (K) = F(Q (K)) K = rectangle/cube, were F( ˆv) F and F are affine map from te reference element ˆK to te pysical element K [7]. Denote te space of piecewise polynomials of degree by P ={q L 2 ( ) : q K D (K) K T }. Numerical Metods for Partial Differential Equations DOI 0.002/num

9 Let P be te L 2 projection onto P. It is well-nown tat MIXED METHOD FOR STOKES PROBLEM 9 q P q C r q r for 0 r +, (3.) for all q H r ( ). Also, it is well-nown tat tere exists an interpolation operator : H(div; ) L t ( ) d RT for t>2 satisfying te commutativity property and te following approximation properties ( v) = P v v H(div; ) L t ( ) d, (3.2) v v C r v r for r +, (3.3) (v v) C r v r for 0 r +. (3.4) Denote te product spaces by RT d = d i= RT and P d = d i= P and define { RTˆ d = τ RT d trτdx = 0 Ten our mixed finite element approximation is to find a pair (σ, u ) RT ˆ d P d suc tat { (κaσ, τ) + (u, τ) = g(τ) τ RT ˆ d, ( σ, v) = f(v) v P d. (3.5) To establis well-posedness of (3.5) and error bounds, define an interpolation operator : H(div; ˆ ) d L t ( ) d d RT ˆ d by τ = ( τ,..., τ d ) t bδ wit b = tr( τ,..., τ d ) t dx, d }. and te L 2 projection operator onto P d by P v = (P v,..., P v d ). By (3.2), (3.3), (3.4), and (3.), it is ten easy to cec te validity of te commutativity property ( τ) = P τ τ H(div; ) d L t ( ) d d, (3.6) and te approximation properties τ τ C r τ r for r +, (3.7) (τ τ) C r τ r for 0 r +, (3.8) v P v C r v r for 0 r +. (3.9) Let D = { τ RT ˆ d ( τ, v) = 0 v P } d, Numerical Metods for Partial Differential Equations DOI 0.002/num

10 0 CAI ET AL. and denote bilinear forms by a(σ, τ) = (κaσ, τ) = (κaσ, Aτ) and b(τ, v) = ( τ, v). Next two lemmas verify te coercivity of te bilinear form a(, ) in D and te inf-sup condition of te bilinear form b(, ) in RT ˆ P d. Lemma 3.. Tere exists a positive constant ˆα independent of te mes size suc tat Cκ 0 τ 2 H(div) a(τ, τ) τ D. (3.0) Proof. Te commutativity property (3.6) gives tat RTˆ d P d, wic, in turn, implies tat D is te divergence free subspace of ˆ. Hence, coercivity (3.0) follows from (2.4). RT d Lemma 3.2. Tere exists a positive constant ˆβ independent of te mes size suc tat ( τ, v) sup ˆβ v v P d τ RT ˆ d τ. (3.) H(div) Proof. operator By te triangle inequality and (3.7) wit r = we ave te stability of te interpolation τ C τ τ H ( ) d d. (3.2) For any v P d L2 ( ) d, tere exists a τ ˆ H(div; ) d satisfying (2.6): Taing γ = τ Hence, by (3.2) and (2.6) τ = v in and τ C v. RT ˆ d and using commutativity property (3.6), we ave γ = ( τ) = P τ = P v = v. γ H(div) = τ H(div) = ( τ 2 + ( τ) 2 ) 2 (C τ 2 + v 2 ) 2 C v. Now, for any v P d L2 ( ) d ( τ, v) ( γ, v) sup ˆβ v, τ RT ˆ d τ H(div) γ H(div) were ˆβ is independent of te mes size. Tis proves te lemma. Now, we are ready to establis te well-posedness and error bounds of mixed finite element approximation. Numerical Metods for Partial Differential Equations DOI 0.002/num

11 MIXED METHOD FOR STOKES PROBLEM Teorem 3.3. Te discrete problem in (3.5) as a unique solution (σ, u ) in RTˆ d P d. Let (σ, u) be te solution of (2.7), we ten ave σ σ H(div) C inf τ RT ˆ d σ τ H(div) (3.3) and ( u u C inf u v + v P d inf τ RT ˆ d ) σ τ H(div). (3.4) Moreover, for r +, assume tat f H r ( ) d and (σ, u) H r ( ) d d H r ( ) d. Ten we ave te following error bounds: and σ σ H(div) C r ( σ r + f r ) (3.5) u u C r ( u r + σ r + f r ). (3.6) Proof. Existence and uniqueness of problem (3.5) and error bounds in (3.3) and (3.4) follow from te abstract teory for te saddle-point problem (see, e.g., [6, 7]) and Lemmas 3. and 3.2. Error bounds in (3.5) and (3.6) follow from (3.3), (3.4), and te approximation properties in (3.7), (3.8), and (3.9). We end tis section by establising an a priori estimate for a sligtly more general system tat contains bot (3.5) and its perturbation. Tis estimate will be used for bounding te penalty error in next section. Lemma 3.4. For a constant parameter 0 <, let a pair (γ, w ) RT ˆ d P d unique solution of be te { (κaγ, τ) + (w, τ) = g (τ) τ RT ˆ d, ( γ, v) (w, v) = f (v) v P d. (3.7) Assume tat g and f are continuous linear functionals defined on H(div; ) d and L 2 ( ) d wit norms g and f, respectively. Ten te following a priori estimate olds γ H(div) + w C( f + g ), (3.8) were C is a positive constant independent of te mes size and te parameter. Proof. To bound (γ, w ) in H(div) L 2 norm, we first bound w above in terms of Aγ and g by using (3.), te first equation of (3.7), te Caucy-Scwarz inequality, and (2.6) ˆβ w ( τ, w ) sup τ RT ˆ d τ H(div) = sup τ ˆ RT d Numerical Metods for Partial Differential Equations DOI 0.002/num g (τ) (κaγ, τ) τ H(div) g +κ Aγ. (3.9)

12 2 CAI ET AL. Next, coosing v = γ P d inequality give in te second equation of (3.7) and using te Caucy-Scwarz γ 2 = (w, γ ) + f ( γ ) ( w + f ) γ. Dividing γ on bot sides and using (3.9) yield, for 0 <, γ w + f C( g + f +κ Aγ ), wic, togeter wit (2.0), implies γ H(div) C( Aγ + γ ) C( g + f + Aγ ). (3.20) Finally, we establis an upper bound for Aγ. To tis end, in (3.7), we tae τ = γ and v = w and subtract te second equation from te first equation to obtain κaγ 2 + w 2 = g (γ ) f (w ) g γ H(div) + f w. It ten follows from (2.6), (3.20), (3.9), and te δ-inequality (2ab δa 2 + b 2 /δ for all positive δ) tat Hence, κ 0 Aγ 2 κaγ 2 g γ H(div) + f w C g ( g + f + Aγ ) + C f ( g + Aγ ) C( g + f ) g +C( g + f ) Aγ C( g 2 + f 2 ) + 2 Aγ 2. Aγ C( g + f ). (3.2) Now, (3.8) is a direct consequence of (3.9), (3.20), and (3.2). Tis completes te proof of te lemma. Corollary 3.5. Let (σ, u ) RT ˆ d P d estimate olds be te solution of (3.5), ten te following a priori σ H(div) + u C( f + g /2, ). (3.22) Proof. Te a priori estimate in (3.22) follows from Lemma 3.4 wit = 0, g = g /2,, and f = f. IV. PENALTY METHOD To solve te saddle-point problem in (3.5) efficiently, we eliminate te velocity by using te penalty metod [7 9] to obtain a smaller system involving only te pseudostress wic will be solved by a fast multigrid metod. Te velocity can ten be calculated for piecewise polynomials Numerical Metods for Partial Differential Equations DOI 0.002/num

13 MIXED METHOD FOR STOKES PROBLEM 3 of degree eiter explicitly for = 0 or locally for. To tis end, let 0 <be a small parameter. We perturb (3.5) by finding (σ, u ) RT ˆ d P d suc tat { (κaσ, τ) + ( ( u, τ) = g(τ) τ RT ˆ d, σ, v ) ( u, v) = f(v) v P d. (4.) It is easy to cec tat te perturbed problem in (4.) as a unique solution (σ, u ) in RT ˆ d P d. By using Lemma 3.4, next lemma sows tat te perturbed solution of (4.) is close to te original solution of (3.5). Lemma 4.. Let (σ, u ) and (σ, u ) be te solutions of (3.5) and (4.), respectively. Ten, for all 0 <, tere exists a positive constant C independent of bot and suc tat u u + σ σ H(div) C u C( f + g /2, ). (4.2) Proof. Let γ = σ σ and w = u u. Ten difference of (3.5) and (4.) gives te following well-posed system { (κaγ, τ) + (w, τ) = 0 τ RT ˆ d, ( γ, v) (w, v) = (u, v) v P d. (4.3) Now, te first inequality in (4.2) follows from te a priori estimate in Lemma 3.4 wit g =0 and f = u, and te second inequality is a direct consequence of a priori estimate (3.22) for te discrete solution. Tis completes te proof of te lemma. Teorem 4.2. Let (σ, u) and (σ, u ) be te solutions of (2.7) and (4.), respectively. Ten, for all 0 <, tere exists a positive constant C independent of bot and suc tat u u + σ σ C ( inf v P d u v + H(div) inf τ RT ˆ d σ τ H(div) + ( f + g /2, ) ). (4.4) Moreover, coosing = O( r ), we ten ave te following error estimate: u u + σ σ H(div) C r ( u r + σ r + f r + g /2, ) (4.5) for all r +. Proof. Inequality (4.4) is an immediate consequence of te triangle inequality, Teorem 3.3, and Lemma 4.. Error bound in (4.5) follows from (4.4) and te approximation properties in (3.7), (3.8), and (3.9). Tis proves te teorem. Remar 4.3. Teorem 4.2 indicates tat te penalty metod does not deteriorate te accuracy of approximation provided tat = O( r ). Corollary 4.4. Let (σ, u ) be te solution of (4.). Let σ, ω, and p be te respective stress, vorticity, and pressure defined in (2.8) and define teir approximations as follows σ = σ + ν( ) Aσ t, ω = ( Aσ 2 (Aσ )t), and p = d trσ. (4.6) Numerical Metods for Partial Differential Equations DOI 0.002/num

14 4 CAI ET AL. Ten, for = O( r ), we ave te following error estimate: σ σ + ω ω + p p C σ σ C r ( u r + σ r + f r + g /2, ) (4.7) for all r +. Proof. Let γ = σ σ, ten it is an immediate consequence of (2.8) and (4.6) tat σ σ = γ + ν(aγ )t, ω ω = 2 (Aγ (Aγ )t ), and p p = d trγ. Now, te first inequality in (4.7) follows from te triangle inequality and te second inequality from (4.5). Te penalty system in (4.) can be efficiently solved by decoupling te velocity and pseudostress as follows. Coosing v = τ P d in te second equation of (4.) gives ( u, τ) = ( σ, τ) f( τ) τ RT ˆ d. (4.8) Substituting (4.8) into te first equation of (4.) yields te penalized system for only te pseudostress ( κaσ, τ) + ( σ, τ) = g(τ) + f( τ) τ RT ˆ d. (4.9) Tis system will be numerically solved by effective multigrid metods discussed in te next section. As RTˆ d = P d, wit nown pseudostress σ, te velocity u can ten be calculated by u = ( σ + P f ), (4.0) were P is te L 2 projection operator into P d defined in te previous section. For = 0, te calculation of P f is explicit because for every K T P f K = fdx. K K For, te calculation of P f requires numerical solutions of local problems on eac element K T (P f K, v) K = (f, v) K v P (K). We end tis section wit description of matrix forms of (4.9) and (4.0). To do so, let { i, i =, 2,..., N} and {ψ i, i =, 2,..., M} be basis functions for RTˆ d and P d, respectively. Te solutions σ and u of (4.9) and (4.0) may be represented in terms of tese basis functions N σ = M j j and u = U j ψ j, Numerical Metods for Partial Differential Equations DOI 0.002/num j= j=

15 respectively. Denote te unnown vectors by MIXED METHOD FOR STOKES PROBLEM 5 = (, 2,..., N) t and U = ( U, U 2,..., U M) t, te coefficient matrices by A = A 0 + A wit A 0 = (( κa j, )) i and A N N = (( j, )) i D = ( ) Dij wit M M D ij = ( ψ j, ψ ) i, and te rigt-and side vectors by G = ( ) G i wit G N i = g ( ) i + f ( ) i, F = ( F i wit F )M i = f ( ) ψ i, ten te matrix forms of (4.9) and (4.0) are and N N, A = G (4.) D U = (B + F ), (4.2) respectively, were B = (Bij ) M N wit Bij = ( j, ψ i ). Note tat support of te basis function ψ i for te velocity is one element. Hence, te coefficient matrix D is a bloc-diagonal mass matrix wit eac bloc of size ( + ) ( + ), were is te degree of piecewise discontinuous polynomials approximating te velocity. Tis indicates tat computational cost of solving (4.2) is negligible and, ence, te main cost of te new metod for solving te Stoes equation is te solution of (4.). V. MULTIGRID PRECONDITIONERS In tis section, we study efficient multigrid preconditioners for bot te penalty system (4.9) and te global (unpenalized) saddle-point problem (3.5). Consider first te penalized problem (4.9). Denote te corresponding bilinear form of (4.9) by A (σ, τ) = (κaσ, τ) + ( σ, τ) and introduce a weigted H(div) inner product by B (σ, τ) = (σ, τ) + ( σ, τ). Teorem 5.. Assume tat te penalty parameter is bounded above by a constant. Ten bilinear forms A (, ) and B (, ) are spectrally equivalent and uniform in ; i.e., tere exist two positive constants C and C 2 independent of suc tat C B (τ, τ) A (τ, τ) C 2 B (τ, τ) Numerical Metods for Partial Differential Equations DOI 0.002/num τ ˆ H(div; ) d. (5.)

16 6 CAI ET AL. Proof. Equality (2.9) gives (κaτ, τ) = κaτ 2 C τ 2, wic, in turn, implies te upper bound in (5.). Te lower bound in (5.) is a direct consequence of (2.0) and te assumption tat C. Te above teorem sows tat te form B (, ) can be used to precondition te form A (, ) effectively. It is well-nown (see, e.g., [20]) tat te form B (, ) can be efficiently preconditioned by te multigrid (V-cycle) preconditioner wit appropriate additive or multiplicative Scwarz smooters. Tis, in turn, implies tat te multigrid V-cycle for te form B (, ) is an efficient preconditioner for te form A (, ). As an alternative, we discuss a spectrally equivalent preconditioner for te saddle-point problem in (3.5) witout using te penalty metod. Tis discrete problem taes a saddle-point two-by-two bloc matrix form: M = ( A 0 B T B 0 ). (5.2) Denote by H = (B ( j, i )) N N te matrix representation of te H(div) bilinear form B (σ, τ), and denote by, te Euclidean inner product. In wat follows will study te spectral relations between te symmetric and indefinite matrix M and te symmetric, positive definite, and bloc-diagonal matrix D = ( ) H 0. 0 D Lemma 5.2. Tere exist positive constants α and β independent of suc tat D M F, M F β D F, F (5.3) for any F = (g, f) t and tat M X, X α D X, X (5.4) for any X = (x, y) t. Proof. For any given F = (g, f) t, let X = (x, y) t be te unique solution of te following saddle-point problem M X = F. (5.5) Denote by g and f te corresponding linear functionals of g and f, respectively, and by (τ, v) te corresponding function representation of X in RT ˆ d P d. Te a priori estimate in Corollary 3.5 implies τ 2 H(div) + v 2 β( g + f ), (5.6) Numerical Metods for Partial Differential Equations DOI 0.002/num

17 MIXED METHOD FOR STOKES PROBLEM 7 wic translates to D X, X β ( H g, g + D f, f ) = β D F, F in terms of matrices and coefficient vectors. Now, (5.3) follows from te fact tat X = M F due to (5.5). For any X = (x, y) t, let (τ, v) be te corresponding function representation of X in RT ˆ d P d. Ten (5.4) may be rewritten as (Aτ, τ) + 2(v, τ) α ( τ 2 H(div) + v 2), wic is an immediate consequence of te definition of A and te Caucy-Scwarz inequality. We note tat te same result as in Lemma 5.2 olds if D is replaced wit any spectrally equivalent matrix. As D is a simple mass-matrix coming from discontinuous elements, ence easily invertible, we only need a spectrally equivalent preconditioner for H wic was already discussed previously in te case of te penalty matrix. We comment at te end tat, in oter words, Lemma 5.2 sows tat te absolute value of eigenvalues of te symmetric matrix D 2 M D 2 are bounded above and away from te origin and tat tese bounds are independent of. As it is well nown (see, e.g., te original reference [2]), tese facts are sufficient to prove mes-independent convergence bounds for te preconditioned minimum residual metod applied to te system ( ) ( ) x g M =, (5.7) y f using D as a preconditioner. A mes-independent convergence bound is also valid, if one simply uses te preconditioned conjugate gradient metod applied to te (weigted) normal system ( ) ( ) x g M D M = M y D, (5.8) f using D as a preconditioner. In conclusion, te saddle-point matrix M can be optimally preconditioned by appropriate bloc-diagonal matrix D in a preconditioned minimum residual algoritm, or in te preconditioned conjugate gradient metod applied to te weigted normal form (5.8). Suc tecniques were explored previously, as early as in [22]. VI. NUMERICAL RESULTS In tis section, we present numerical results on accuracy of mixed finite element approximation and on te condition number of te preconditioned pseudostress system in (4.9). Test problems are defined on te unit square = (0, ) 2 wit te viscosity parameter being one (ν = ). To measure te discretization error, we consider a model problem wit a nown nonzero solution. Let ( ) ( ) sin(2πx) (2π) 2 sin(2πx)cos(2πy)+ x, if y = 0,, 0 f = 2 (2π) 2 cos(2πx)sin(2πy)+ y and g = ( Numerical Metods for Partial Differential Equations DOI 0.002/num 0 sin(2πy) ), if x = 0,

18 8 CAI ET AL. TABLE I. L 2 errors for te pseudostress σ σ. d.o.f. = 2 = /4 = /2 = = 2 = e 7.878e 7.895e 7.928e 7.997e e e e e e e e 6, e.7833e.7834e.7837e.7842e.7922e 66, e e e e e e 2 be te rigt-and side function and te prescribed velocity on te boundary, respectively. Ten ( ) sin(2πx)cos(2πy) u = and p = x 2 + y 2 cos(2πx)sin(2πy) are te exact solution of te stationary Stoes equation. By te definition of te pseudostress in (2.2), we ave σ = pδ + u cos(2πx)cos(2πy) = 2π 2π (x2 + y 2 ) sin(2πx)sin(2πy) sin(2πx)sin(2πy) cos(2πx)cos(2πy) 2π (x2 + y 2 ) Obviously, (σ, u) is ten te exact solution of variational problem (2.7) in te pseudostressvelocity formulation. Partition te domain = (0, ) 2 by uniform rectangular elements K ij = (i, j) for i, j = 0,,..., N wit = /N. Finite element approximation σ RT 0 d to te pseudostress is computed troug solving system (4.9) wit te lowest order RT element ( = 0) by a direct metod. Finite element approximation u P 0 d to te velocity is calculated explicitly by using (4.0). Discretization errors for = c and m wit different values of constant c and exponent m are reported in Tables I III. Te pseudostress and velocity are O() accurate in te L 2 norm for m as predicted teoretically in Section IV and teir dependence on te constant c is wea. Te second equations in (4.) and (2.5) imply u = σ + P f = σ P σ.. As u is bounded (see Lemma 3.4), P σ σ =O()wic is confirmed numerically in Table II. Next we study te multigrid convergence rates using different values of. Random rigt and sides are used wit te zero energy mode eliminated. We apply a classical V(,)-cycle multigrid algoritm wit multiplicative Scwarz smooters were te overlapped blocs are formed Numerical Metods for Partial Differential Equations DOI 0.002/num

19 TABLE II. MIXED METHOD FOR STOKES PROBLEM 9 Discrete L 2 errors for te divergence of pseudostress P σ σ. d.o.f. = 4 = 3 = 2 = /4 = = e e e e 2.242e e e 3.048e e e e e e e 3.088e e e e e e e e e 6, e e 6.724e e 3.033e 2.023e 66, e e e e e e 2 by collecting te edge variables incident on eac node (tat is, eac bloc is 8 8, because tere are 4 edges and eac edge as degree of freedom 2); and te coarsest problem is solved by te conjugate gradient metod. Te prolongation (or coarse-to-fine) operators, wic are widely used for nested rectangular meses, are defined by (from te coarser level l + to te finer level l): P l l+ (e j) l+ = (e ) l if (e j ) l+ (e ) l 2 (e ) l if (e j ) l+, (e ) l (E n ) l+, (e j ) l+ (e ) l, (e j ) l+ (e ) l were (e j ) l denotes te j-t edge on level l, (E j ) l denotes te j-t element on level l, and denotes tat te two edges are parallel. Te second clause of te above formula basically states tat te fine edges tat are not part of te coarse mes are interpolated by teir neigboring edges tat are parallel to tem. Te restriction (or fine-to-coarse) operators are defined as te transpose of te corresponding prolongation operators. Finally, we form te Galerin coarse operators for all coarse levels via A l+ = (Pl+ l )T A l Pl+ l. TABLE III. L 2 errors for te velocity u u. d.o.f. = 2 = /4 = /2 = = 2 = e 4.25e 4.28e 4.225e 4.240e e e e e e e e e.288e.288e.289e.29e.307e e e e e e e 2 6, e e e e e e 2 66, e 2.469e 2.469e 2.470e 2.470e 2.477e 2 Numerical Metods for Partial Differential Equations DOI 0.002/num

20 20 CAI ET AL. TABLE IV. MG convergence rate for different s. ρ(iter) N = 2 = 0. = 0.5 = = 5 = (2) 0.20(2) 0.20(2) 0.20(2) 0.20(2) 0.20(2) (2) 0.2(2) 0.2(2) 0.2(2) 0.2(2) 0.2(2) (2) 0.2(2) 0.2(2) 0.2(2) 0.2(2) 0.2(2) 6, (2) 0.2(2) 0.2(2) 0.2(2) 0.2(2) 0.2(2) 66,048 NC 0.2(2) 0.2(2) 0.2(2) 0.2(2) 0.2(2) 263,68 NC NC 0.24(3) 0.2(2) 0.2(2) 0.2(2) N, total degree of freedom; ρ, average convergence rates; (iter), number of MG iterations (V(,) using Scwarz smooter); NC, does not converge. Te results for a few different values of are given in Table IV. We observe ere tat multigrid is not robust wen is small. A remedy is to use te generalized minimal residual (GMRES) (or conjugate gradient (CG)) metod wit one V(,)-cycle multigrid as te preconditioner, te results of wic are given in Table V. In te next set of numerical experiments we use te bilinear form B (, ) as a preconditioner of te bilinear form A (, ). We again apply GMRES wit one V(,)-cycle multigrid as te preconditioner. Again te iteration counts for a few different values of are given in Table VI. In all our experiments we observe essential spectral equivalent convergence rates for small, i.e., = O(). Wen te metod converges, its convergence rate is independent of. Finally, Tables V and VI sow tat te preconditioned GMRES using te bilinear form A (, ) is twice faster tan tat using te bilinear form B (, ). TABLE V. MG convergence rate for different s. ρ(iter) N = 2 = 0. = 0.5 = = 5 = (9) 0.24(9) 0.24(9) 0.24(9) 0.30(9) 0.30(9) (0) 0.46(0) 0.46(2) 0.46(0) 0.46(0) 0.46(0) (0) 0.54(0) 0.54(0) 0.54(0) 0.54(0) 0.54(0) 6, (0) 0.55(0) 0.55(0) 0.55(0) 0.55(0) 0.55(0) 66, (0) 0.57(2) 0.57(0) 0.57(0) 0.57(0) 0.57(0) 263, (0) 0.58(0) 0.58(0) 0.58(0) 0.58(0) 0.58(0) N, total degree of freedom; ρ, average convergence rates; (iter), number of GMRES iterations using MG V(,) as preconditioner. Numerical Metods for Partial Differential Equations DOI 0.002/num

21 TABLE VI. MIXED METHOD FOR STOKES PROBLEM 2 GMRES-MG convergence rates for different s. ρ(iter) N = 2 = 0. = 0.5 = = 5 = (9) 0.38(9) 0.38(9) 0.38(9) 0.38(9) 0.38(9) (20) 0.40(20) 0.40(20) 0.40(20) 0.40(2) 0.40(2) (22) 0.42(22) 0.42(22) 0.42(22) 0.42(22) 0.42(22) 6, (22) 0.42(22) 0.42(22) 0.42(22) 0.42(22) 0.42(22) 66, (2) 0.4(2) 0.4(2) 0.4(2) 0.4(2) 0.4(2) 263,68 0.4(2) 0.4(2) 0.4(2) 0.4(2) 0.4(2) 0.4(2) N, total degree of freedom; ρ, average convergence rates; (iter), number of GMRES iterations using MG V(,) as preconditioner. VII. CONCLUSION REMARKS In tis article, we studied a new numerical metod for solving te stationary Stoes equation, wic may be easily extended to Navier-Stoes equations in principle. Te metod is more accurate tan existing metods for applications in wic te sear stress are important. Te main cost of te metod is te computation of te solution of te pseudostress system in (4.9). Even toug te pseudostress as more variables tan te velocity and pressure, te numbers of degrees of freedom for te pseudostress using Raviart-Tomas elements of index = 0, and BDM elements of index =, 2 [7] are comparable to tose for te velocity-pressure using Crouzeix-Raviart (nonconforming) elements of order =, 2 and = 2, 3, respectively. Calculations of te oter pysical quantities suc as te velocity, pressure, stress, and vorticity are straigtforward and ave negligible cost. Our numerical results ave sown tat te positive, definite pseudostress system can be solved by a igly efficient PCG wit a spectrally equivalent multigrid preconditioner. Uniform convergence analysis, wit respect to te mes size, te number of levels, and te large penalty parameter, on multigrid metod for te pseudostress system will be presented in [23]. If one wants to avoid te penalty formulation, ten one as to wor wit te indefinite saddle-point system in te way described in te second part of Section V. Te latter approac is somewat more expensive because one as to wor wit bigger size matrices and vectors, but neverteless te discussed preconditioned metods (te minimum residual and conjugate gradient applied to te weigted normal system) exibit proven convergence rates bounded independently of te mes size. References. J. Marcal and M. Crocet, Hermitian finite elements for calculating viscoelastic flow, J Non-Newtonian Fluid Mec 20 (986), D. N. Arnold and R. Winter, Mixed finite elements for elasticity, Numer Mat 42 (2002), Z. Cai, B. Lee, and P. Wang, Least-squares metods for incompressible Newtonian fluid flow: linear stationary problems, SIAM J Numer Anal 42 (2004), Numerical Metods for Partial Differential Equations DOI 0.002/num

22 22 CAI ET AL. 4. Z. Cai and G. Stare, Least-squares metods for linear elasticity, SIAM J Numer Anal 42 (2004), P. A. Raviart and I. M. Tomas, A mixed finite element metod for second order elliptic problems, Lecture Notes in Matematics 606, Springer-Verlag, Berlin, 977, pp M. Crouzeix and P. A. Raviart, Conforming and nonconforming finite element metods for solving te stationary Stoes equations, RAIRO Anal Numer 7 (973), R. Rannacer and S. Ture, Simple nonconforming quadrilateral Stoes element, Numer Metods PDE 8 (992), Z. Cai, J. Douglas Jr., and X. Ye, A stable nonconforming rectangular finite element metod for te Stoes and Navier-Stoes equations, Calcolo 36 (999), D. N. Arnold, R. S. Fal, and R. Winter, Preconditioning in H(div) and applications, Mat Comp 66 (997), R. Hiptmair, Multigrid metod for H(div) in tree dimensions, ETNA 6 (997), P. S. Vassilevsi and J. Wang, Multilevel iterative metods for mixed finite element discretizations of elliptic problems, Numer Mat 63 (992), Z. Cai, C. Wang, and S. Zang, Mixed finite element metods for incompressible flows: stationary Navier-Stoes equations, SIAM J Numer Anal, submitted. 3. D. N. Arnold, J. Douglas Jr., and C. P. Gupta, A family of iger order mixed finite element metods for plane elasticity, Numer Mat 45 (984), Z. Cai and G. Stare, First-order system least squares for te stress-displacement formulation: linear elasticity, SIAM J Numer Anal 4 (2003), S. C. Brenner and L. R. Scott, Te matematical teory of finite element metods, Springer-Verlag, New Yor, F. Brezzi, On existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO Anal Numér. 2 (974), F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Metods, Springer-Verlag, New Yor, M. Bercovier, Perturbation of mixed variational problems application to mixed finite element metods, Numer Mat RAIRO 2 (978), V. Girault and P. A. Raviart, Finite element metods for navier-stoes equations: teory and algoritms, Springer-Verlag, New Yor, D. N. Arnold, R. S. Fal, and R. Winter, Multigrid in H(div) and H(curl), Numer Mat 85 (2000), R. Candra, Conjugate Gradient Metods for Partial Differential Equations, Yale University, New Haven, CT, T. Rusten and R. Winter, A preconditioned iterative metod for saddle point problems, SIAM J Matrix Analysis Appl 3 (992), Z. Cai and Y. Wang, A multigrid metod for te pseudostress formulation of Stoes problems, SIAM J Sci Comput 29 (2007), Numerical Metods for Partial Differential Equations DOI 0.002/num