UNIFIED A POSTERIORI ERROR ESTIMATOR FOR FINITE ELEMENT METHODS FOR THE STOKES EQUATIONS

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1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 10, Number 3, Pages c 013 Institute for Scientific Computing and Information UNIFIED A POSTERIORI ERROR ESTIMATOR FOR FINITE ELEMENT METHODS FOR THE STOKES EQUATIONS JUNPING WANG, YANQIU WANG, AND XIU YE Abstract. This paper is concerned with residual type a posteriori error estimators for finite element methods for the Stokes equations. In particular, the authors established a unified approach for deriving and analyzing a posteriori error estimators for velocity-pressure based finite element formulations for the Stokes equations. A general a posteriori error estimator was presented with a unified mathematical analysis for the general finite element formulation that covers conforming, non-conforming, and discontinuous Galerkin methods as examples. The key behind the mathematical analysis is the use of a lifting operator from discontinuous finite element spaces to continuous ones for which all the terms involving jumps at interior edges disappear. Key words. A posteriori error estimate, finite element methods, Stokes equations 1. Introduction A posteriori error estimator refers to a computable formula that offers a measure for judging the reliability and efficiency of a particular numerical scheme employed for approximating the solution of partial differential equations or alike. With a mathematically justified a posteriori error estimator, one would be able to generate a mesh that is tailored at reducing computational errors at places of great need. This process is commonly known as adaptive mesh refinement which has become a useful and important tool in today s scientific and engineering computing. The goal of this paper is to offer a systematic framework for developing and analyzing a posteriori error estimators for finite element methods for model Stokes equations. This paper is concerned with residual type a posteriori error estimators. In other words, the computable formula for judging the efficiency and reliability of numerical schemes shall be given by functions of residuals. Along this avenue, several fine results have been developed for finite element methods for the Stokes equations. For conforming finite element methods, some a posteriori error estimators have been derived for mini-elements by Verfurth [1] and Bank-Welfert [4]. Ainsworth- Oden [3] and Nobile [18] have considered more general conforming finite elements in their study. For nonconforming finite elements, a posteriori error estimation for the Crouzeix-Raviart element [8] has been developed by several researchers such as Verfurth [], Dari-Durán-Padra [9] and Doerfler-Ainsworth [10]. Carstensen, Gudi, and Jensen [5] proposed and analyzed an a posteriori error estimator for discontinuous Galerkin methods by using a stress-velocity-pressure formulation for the Stokes equations. Kay and Silvester[16] established a posteriori error estimation Received by the editors August 1, Mathematics Subject Classification. 65N15, 65N30, 76D07, 35B45, 35J50. The research of Junping Wang was supported by the NSF IR/D program, while working at the Foundation. However, any opinion, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. The research of Xiu Ye was supported in part by National Science Foundation Grant DMS

2 55 J. WANG, Y. WANG, AND X. YE for the stabilized finite element formulation. The recovery based a posteriori error estimate for the Stokes equations is investigated in [1]. In both [9] and [10], the analysis for their a posteriori error estimators was based on a Helmholtz decomposition for decomposing the Crouzeix-Raviart element into two parts: an exactly divergence-free part and the second as its orthogonal complement. While the Helmholtz decomposition offers an applaudable approach for analyzing the efficiency and reliability of a posteriori error estimators for the Stokes equations, the method has difficulty in being extended to finite element approximations arising from discontinuous Galerkin methods. The main difficulty comes from the fact that the approximate velocity field from the discontinuous finite element methods is not divergence-free in the classical sense. Therefore, other analytical techniques have been developed for discontinuous finite elements; but most of them requires special and unnecessary properties about the finite element mesh. For example, Houston, Schötzau and Wihler [14] have developed an a posteriori error analysis for the discontinuous Q k Q k 1 element on partitions consisting of parallelograms only. In this paper, we establish a unified approach for deriving and analyzing a posteriori error estimators of residual type for velocity-pressure based formulations of the Stokes equations. In particular, we shall develop a general finite element formulation that covers conforming, non-conforming, and discontinuous Galerkin methods as examples. Then, a general a posteriori error estimator shall be presented with a unified mathematical analysis. The key behind the analysis is the use of a lifting operator from discontinuous finite element spaces to continuous ones for which all the terms involving jumps at interior edges disappear. A similar lifting operator was employed by Karakashian and Pascal [15] for analyzing a posteriori error estimates for a discontinuous Galerkin approximation to second order elliptic equations. The paper is organized as follows. In Section, a model Stokes problem and some notations are introduced. In Section 3, we shall first present a general finite element formulation for the Stokes equations, and then illustrate how most existing conforming, nonconforming, and discontinuous Galerkin methods be represented by the general framework. In Section 4, we establish an analytical tool for analyzing the general a posteriori error estimator of residual type. Finally in Section 5, we present some numerical results to confirm the theory developed in previous sections.. Preliminaries and notations Let Ω be an open bounded domain in R d,d =,3. Denote by Ω the boundary of Ω. The model problem seeks a velocity function u and a pressure function p satisfying (1) () (3) u+ p = f in Ω, u = 0 in Ω, u = 0 on Ω, where,, and denote the Laplacian, gradient, and divergence operators, respectively, and f is the external volumetric force acting on the fluid. For simplicity, the algorithm and its analysis will be presented for the model Stokes problem (1)-(3) only in two-dimensional spaces (i.e.; d = ) with polygonal domains. An extension to the Stokes problem in three dimensions can be made formally for general polyhedral domains.

3 UNIFIED A POSTERIORI ERROR ESTIMATE FOR STOKES EQUATIONS 553 For any given polygon D Ω, we use the standard definition of Sobolev spaces H s (D) with s 0 (e.g., see [1, 6] for details). The associated inner product, norm, and seminorms in H s (D) are denoted by (, ) s,d, s,d, and r,d,0 r s, respectively. When s = 0, H 0 (D) coincides with the space of square integrable functions L (D). In this case, the subscript s is suppressed from the notation of norm, semi-norm, and inner products. Furthermore, the subscript D is also suppressed when D = Ω. Denote by L 0 (D) the subspace of L (D) consisting of functions with mean value zero. The above definition/notation can easily be extended to vector-valued and matrixvalued functions. The norm, semi-norms, and inner-product for such functions shall follow the same naming convention. In addition, all these definitions can be transferred from a polygonal domain D to an edge e, a domain with lower dimension. Similar notation system will be employed. For example, s,e and e would denote the norm in H s (e) and L (e) etc. Throughout the paper, we follow the convention that a bold Latin letter denotes a vector. Let u = [u i ] 1 i, v = [v i ] 1 i be two vectors, and σ = [σ ij ] 1 i,j, τ = [τ ij ] 1 i,j be two matrices, define ) v = u v = v σ = ( v1 x v x It is not hard to see that v 1 y v y, v = v 1 x + v y, ( ) u1 v 1 u 1 v, σ : τ = σ u v 1 u v ij τ ij, ( ) σ11 v 1 +σ 1 v, v σ u = σ 1 v 1 +σ v σ : (u v) = v σ u. i,j=1 σ ij u i v j. i,j=1 Let T h be a geometrically conformal triangulation of the domain Ω; i.e., the intersection of any two triangles in T h is either empty, a common vertex, or a common edge. Denote by h T the diameter of triangle T T h, and h the maximum of all h T. We assume that T h is shape regular in the sense that for each T T h, the ratio between h T and the diameter of the inscribed circle is bounded from above. The shape regularity of T h ensures a validity of the inverse inequality for finite element functions. In addition, shape regularity allows one to apply the routine scaling arguments in finite element analysis. Let us introduce two finite dimensional spaces V s and P t as follows: V s = {v V : v T [P s (T)], for all T T h }, P t = {q L 0 (Ω) : q T P t (T), for all T T h }, wherev = [L (Ω)] or [H 1 0(Ω)], s and tarenon-negativeintegers, and P k (T)is the set of polynomials of degree no more than k on T. A finite element method usually seeks discrete velocity and pressure approximations in some subspaces V h V s and Q h Q t. Certain continuity condition may be imposed on V h and Q h, depending on the type of finite elements. For a posteriori error estimates only, V h and Q h are not required to satisfy the discrete inf-sup condition. However, we do need to make the following assumption (4) {v [H 1 0 (Ω)] : v T [P 1 (T)], for all T T h } V h,

4 554 J. WANG, Y. WANG, AND X. YE in the analysis of the a posteriori error estimates. For a finite element partition T h, with triangles only, (4) is satisfied as long as k, order of the polynomials, is no less than 1. Therefore, the assumption (4) is reasonable and examples that satisfy it will be given later. Denote by E h the set of all edges in T h, and denote by E 0 h := E h\ Ω the collection of all interior edges. Next, we define the average and jump on edges for scalar-valued function q, vector-valued function w, and matrix-valued function τ, respectively. For any interior edge e E 0 h, let T 1, T be two triangles sharing e and n 1 and n be the unit outward normal vectors on e, associated with T 1 and T, respectively. Define the average { } and jump [ ] on e by {q} = 1 (q T 1 +q T ), [q] = q T1 n 1 +q T n, {w} = 1 (w T 1 +w T ), [w] = w T1 n 1 +w T n, {τ} = 1 (τ T 1 +τ T ), [τ] = n 1 τ T1 +n τ T. We also define a matrix-valued jump [ ] for w on e by [w] = w T1 n 1 +w T n. If e is a boundary edge, the above definitions need to be adjusted accordingly so that both the average and the jump are equal to the one-sided values on e. That is, {q} = q e, {w} = w e, {τ} = τ e, [q] = q e n, [w] = w e n, [τ] = n τ e, [w] = w e n, where n is the unit outward normal of Ω. Let q, v and τ be scalar-, vector-, and matrix-valued functions that are regular enough to make all involving terms well-defined, then the following identities are standard []: qv nds = [q] {v}ds+ (5) {q}[v]ds, T T T h e Eh 0 e e E e h n τ vds = [τ] {v}ds+ (6) {τ} : [v]ds. T T T h e Eh 0 e e E e h 3. Finite element formulation We first derive a general weak formulation for the Stokes equations that covers a set of existing numerical methods including discontinuous Galerkin, conforming finite elements, and some nonconforming finite element schemes. To this end, let v and w be vector-valued functions, and we introduce the following notation (v, w) Th := v w dx, K T h K (v, w) Eh := e E h e v w ds. Testing the momentum equation (1) by v V h yields (7) ( u, v)+( p, v) = (f, v).

5 UNIFIED A POSTERIORI ERROR ESTIMATE FOR STOKES EQUATIONS 555 Next, using integration by parts, Equation (6) and the fact that [ u] = 0 on all e Eh 0, we have ( u, v) = ( u, v) Th n u vds K T h K = ( u, v) Th ({ u}, [v]) Eh. Analogously, since [p] = 0 on all e Eh 0, then by using Equation (5) we arrive at ( p, v) = ( v, p) Th + v np ds Thus, (7) can be rewritten as K T h K = ( v, p) Th +({p}, [v]) Eh. (8) ( u, v) Th ({ u}, [v]) Eh ( v, p) Th +({p}, [v]) Eh = (f, v). The mass conservation equation () can be tested by using any q Q h, yielding (9) ( u, q) Th = 0. In order to derive a general weak formulation, we introduce the following bilinear forms: a(u,v) := ( u, v) Th ( (10) γ ({ u}, [v]) Eh +δ({ v}, [u]) Eh α(h 1 e [u], [v]) Eh ), (11) b(v,p) := ( v, p) Th +γ([v],{p}) Eh, where γ = 0 or 1, δ = 1, 1 or 0, and α 0 are parameters with various values. The general weak formulation for the Stokes equations is then given as follows: (1) (13) Algorithm G: Find (u h ;p h ) V h Q h such that a(u h,v)+b(v,p h ) = (f,v), b(u h, q) = 0 for all (v;q) V h Q h. When γ = 1, it is not hard to see that system (1)-(13) is consistent with the system (8)-(9), as the exact solution of the Stokes problem satisfies [u] = 0 and [u] = 0 on all e E h. When γ = 0, these two systems are in general not consistent. It should be pointed out that by choosing different values of γ and the spaces V h and Q h, Algorithm G represents different types of finite element formulations. For illustrative purpose, we shall list three most important examples; all of which satisfy the assumption (4) on V h. Example 1: Discontinuous Galerkin. Set γ = 1 and V h {v [L (Ω)] : v T [P s (T)], for all T T h }, Q h {q L 0(Ω) : q T P t (T), for all T T h }. When δ = 1, the corresponding formulation is symmetric. One example of such a discontinuous Galerkin formulation was discussed in [5], where V h, Q h are chosen to be H(div) conforming elements and the well-known inf-sup condition was inherited to be valid.

6 556 J. WANG, Y. WANG, AND X. YE Example : Conforming finite element method. Set γ = 1 and V h {v [H 1 0 (Ω)] : v T [P s (T)], for all T T h }, Q h {q L 0(Ω) : q T P t (T), for all T T h }. By the continuity requirement on V h, we easily see that the two bilinear forms are simplified to a(v,w) = ( v, w), b(v,q) = ( v, q). The conforming finite element scheme is one of the most well-studied formulations for the Stokes problems. Later in the section for numerical experiments, we shall consider one such element, namely, the Taylor-Hood element [13]. Example 3: Nonconforming finite element method. Set γ = 0 and chose a finite element space V h such that V h [H 1 0 (Ω)]. This gives some nonconforming finite element methods, depending on the selection of V h and Q h. In this paper, we only consider nonconforming finite elements which satisfies the following condition: Assumption (H): ({ u pi}, [v]) Eh = 0 for all u, v V h, p Q h, where I is the identity matrix. It is not hard to see that the Crouzeix- Raviart type nonconforming elements [8, 11, 7] satisfy this assumption, because by its definition, { u pi} is a polynomial one degree lower than [v] and [v] vanishes at all Gaussian points on each edge e E h. The goal of this manuscript is to provide a tool that can be employed to analyze a posteriori error estimators for the unified formulation (1)-(13). To this end, we first derive an orthogonality property of the error between the exact solution and its finite element approximation: e = u u h, ǫ = p p h. By subtracting (1) from (8) and using ({p h }, [v]) Eh = ({p h I}, [v]) Eh, we have ( ) ( e, v) Th ({ e}, [v]) Eh +γ δ({ v}, [u h ]) Eh α(h 1 e [u h], [v]) Eh ( v,ǫ) Th +({ǫ}, [v]) Eh (1 γ)({ u h p h I}, [v]) Eh = 0. Note that for all of the above-mentioned discontinuous Galerkin, conforming and nonconforming cases, we must have (14) (1 γ)({ u h p h I}, [v]) Eh = 0. In fact, (14) follows from γ = 1 for the discontinuous Galerkin and the conforming cases, and from Assumption (H) for the nonconforming case. Combining this and the fact that [u] = 0 on all edges, the above equation can be rewritten as (15) ( ) ( e, v) Th ({ e}, [v]) Eh γ δ({ v}, [e]) Eh α(h 1 e [e], [v]) Eh ( v,ǫ) Th +({ǫ}, [v]) Eh = 0. Moreover, if v V h [H 1 0 (Ω)], then (15) becomes (16) ( e, v) Th +γδ({ v}, [u h ]) Eh ( v,ǫ) Th = 0.

7 UNIFIED A POSTERIORI ERROR ESTIMATE FOR STOKES EQUATIONS 557 Next, we introduce a norm for the space [H0 1(Ω)] +V h as follows (17) v = v 1,T + h 1 e [v] e. e E h For a well-crafted numerical scheme, we usually expect to have an a priori error estimate like the following (18) e + ǫ Ch k+1 ( u k+1 + p k ), wherec isapositiveconstantandk isdeterminedbythe orderofthecorresponding finite elements and the regularity of the exact solution (u; p). For some elements, the a priori error estimate (18) may have variations on the right-hand side, but this does not affect our analysis to be presented. A priori error estimation like (18) is well known for conforming finite element methods, for which v = v 1. Such an estimate is also known for some discontinuous Galerkin formulations; e.g., see [4, 5] for an approach with H(div)-elements. For nonconforming finite elements, such as the Crouzeix-Raviart type elements, similar error estimates hold true due to the fact that the jump term, [u h ], vanishes at all Gaussian points on the edge e. In fact, for any interior edge e shared by two elements T 1 and T, let ē be the average value of e on edge e. Since [e] = [u h ] vanishes at all Gaussian points on the edge e, then ē has the same value when the trace of e was taken from either T 1 or T. Hence, he 1 [e] e = h 1 e [e ē] e Consequently, we have e E h h 1 h 1 e e ē e T i T 1,T C e 1,T i. T 1,T e [e] e C e 1,T. Therefore, e is in the same order as e 1,T for the Crouzeix-Raviart type elements. The analysis here shows that the norm as defined in (17) is a reasonable one to use in the a priori and later the a posteriori error estimates for the unified formulation (1)-(13). 4. A posteriori error estimation The goal of this section is to derive an a posteriori error estimation for the Algorithm G by using a unified framework. For simplicity of notation, we shall use to denote less than or equal to up to a constant independent of the mesh size, variables, or other parameters appearing in the inequality. To this end, define { [ uh p J 1 ( u h p h I) = h I] on e Eh 0, 0 on boundary edges, and { [uh ] on e E J (u h ) = h 0, u h n on boundary edges. Our residual-based global error estimator is given by (19) η = η T,

8 558 J. WANG, Y. WANG, AND X. YE where ηt =h T f h + u h p h T + u h T + 1 ( he J 1 ( u h p h I) +h 1 e J (u h ) ) ds, e E h e with h e being the length of edge e, and f h the L projection of the load function f onto V h. It is also convenient to introduce an oscillation quantity for the load function f on T h as follows ( ) 1/ osc(f) = h T f f h T. Our ultimate goal is to establish the following result. Theorem 1. Let (u;p) be the solution of (1)-(), and (u h ;p h ) be its finite element approximation arising from (1)-(13). Then, one has ( ) 1/ (0) (u u h ) T + p p h η +osc(f) and (1) η ( (u u h ) T ) 1/ + p p h +osc(f). For convenience, the relation (0) is referred to as a reliability estimate and (1) as an efficiency estimate A lifting operator and some technical estimates. Define H 1 (T h ) = H 1 (T) and S k = P k (T). For any triangle T T h, denote by T (T) the set of all triangles in T h having a nonempty intersect with T, including T itself. Denote by E(T) the set of all edges in E h having a nonempty intersection with T, including all three edges of T. Similarly, for any point x in Ω, denote by E(x) the set of all edges that pass through x. Note that E(x) is nonempty only when x lies on E h. Let e be an edge of triangle T. For any v S k, let us define a lifting operator () L k : v L k (v) S k H 1 0 (Ω) that lifts a discontinuous piecewise polynomial to a continuous piecewise polynomial with vanishing boundary trace as follows. Let G k (T) be the set of all Lagrangian nodal points for P k (T). At all internal Lagrangian nodal points x j G k (T), we set L k (v)(x j ) = v(x j ). At boundary Lagrangian points x j G k (T) T, we let L k (v)(x j ) be either a traceof v from anyside oraprescribed weighted averageof all possible traces. At global Lagrangian points x j T Ω, we set L k (v)(x j ) = 0. Note that for any function g H 1 (T) the following estimate holds: (3) g e h 1 T g T +h T g T. Lemma 1. For any v S k, k 1, the lifting operator L k as defined in () satisfies the following estimate: (4) v L k (v) T +h T (v L k(v)) T h e [v] e T T h. e E(T)

9 UNIFIED A POSTERIORI ERROR ESTIMATE FOR STOKES EQUATIONS 559 Proof. The proof follows from a routine scaling argument and the fact that all norms on finite dimensional spaces are equivalent. To this end, we observe that v L k (v) T +h T (v L k (v)) T h T (v L k (v))(x j ), x j G k (T) where G k (T) is the set of all Lagrangian nodal points for P k (T). For simplicity, let L k (v) be defined by taking a random one-sided trace on the boundary Lagrangian points. It follows from the definition of L k that v L k (v) vanishes at all internal Lagrangian points in T. At Lagrangian points on T, there are two possibilities: (1) x j is in the interior of an edge e; () x j is a vertex of T. In the first case, we see that (v L k (v))(x j ) is either 0 or [v] e (x j ), where [v] e denotes the jump on e. In the second case, we can use the triangle inequality to traverse through all edges e E(x j ) and to obtain (v L k (v))(x j ) [v] e (x j ). e E(x j) The above analysis, together with the shape regularity of T h, implies that (v L k (v))(x j ) [v] e (x j ), x j G k (T) e E(T) x j G k (e) where G k (e) was used to denote the corresponding Lagrangian points on edge e. Then, using the routine scaling argument on edges, inequality (4) follows immediately. Thefollowingisanotherresultthatturnsouttobe veryuseful intheforthcoming analysis. Lemma. For any v H 1 (T h ), there exists a v I S k H0(Ω), 1 k 1, satisfying (5) v v I T +h T (v v I ) T h T v T + h e [v] e T T h. T T (T) e E(T) Proof. First of all, there exists a piecewise constant v 0 S k (e.g., the cell average of v) such that (6) v v 0 T +h T (v v 0 ) T h T v T T T h. Furthermore, by the approximation property, inequality (3) and the shape regularity of T h, we have for each e Eh 0 h e [v 0 ] e = h e v 0 T1 n 1 +v 0 T n ds e (7) h e (v 0 v) T1 n 1 ds+h e (v 0 v) T n ds+h e [v] e e h T 1 v T 1 +h T v T +h e [v] e, where T 1 and T are the two triangles sharing e as a common edge. On boundary edges, similar result can obviously be obtained without any difficulty. Since v 0 S k with k = 1, according to Lemma 1, one may lift v 0 to a continuous piecewise linear function L 1 (v 0 ) S 1 H0 1 (Ω) which satisfies (4). By taking e

10 560 J. WANG, Y. WANG, AND X. YE v I = L 1 (v 0 ), we obtain from the usual triangle inequality, (6), and (7) that v v I T +h T (v v I) T v v 0 T +h T (v v 0) T + v 0 L 1 (v 0 ) T +h T (v 0 L 1 (v 0 )) T h T v T + h e [v 0 ] e e E(T) h T v T + h e [v] e, T T(T) e E(T) which completes the proof. 4.. Reliability estimate. We first establish an estimate for the pressure error in terms of the a priori error estimator and the velocity error. The result can be stated as follows. Lemma 3. Let (u;p) be the solution of (1)-() and (u h ;p h ) be its finite element approximation arising from (1)-(13). Then, we have (8) p p h η +osc(f)+ ( (u u h ) T ) 1/. Proof. Let v [H 1 0(Ω)] and v I V h [H 1 0(Ω)] be an interpolation of v such that both components satisfy (5). Observe that such an interpolation v I is possible if V h satisfiesthe assumption(4). Notethat [v] = 0oneveryedgesincev [H 1 0(Ω)]. Let e = u u h and ǫ = p p h be the error for velocity and pressure approximations, respectively. Using integration by parts, (6), (16), (3), (5), and the fact that both v and v I are continuous across each interior, we arrive at ( v, ǫ) = ( (v v I ), ǫ)+( v I, ǫ) = ( (v v I ), ǫ)+( e, v I ) Th +γδ({ v I }, [e]) Eh = ( (v v I ), ǫi e) Th +( e, v) Th +γδ({ v I }, [e]) Eh = (f + u h p h, v v I ) Th +({v v I }, [ u h p h I]) E 0 h γδ({ v I }, [u h ]) Eh +( e, v) Th ( v 1 ( h T f + u h p h T) 1/ +( h e [ u h p h I] e) 1/ e Eh 0 +( e [u h ] e) 1/ +( ) e T) 1/, e E h h 1 which, together with the following inf-sup condition ( v, p p h ) (9) p p h sup v [H0 1 v (Ω)] 1 yields the required estimate (8). The next result is concerned with an estimate for the velocity approximation, which can be stated as follow.

11 UNIFIED A POSTERIORI ERROR ESTIMATE FOR STOKES EQUATIONS 561 Theorem. Let (u;p) and (u h ;p h ) be the solutions of (1)-() and (1)-(13). Then one has the following global reliability estimate: (30) ( (u u h ) T ) 1/ η +osc(f). Substituting (30) into (8) yields the following estimate for the pressure approximation: (31) p p h η +osc(f). Thus, it follows from the definition of that e + ǫ η +osc(f). Proof. Let e I V h [H 1 0 (Ω)] be an interpolation of e satisfying (5). Again, such a choice of e I is possible because V h was assumed to satisfy (4). Observe that [e] = [u h ]. Thus, it follows from (5) and the mesh regularity that (3) e e I T+h T (e e I ) T h T T T (T) e T + e E(T) e J (u h ) e. Using(16),integrationbyparts,(6), andthefactthate I, u, u, parecontinuous across all interior edges, we obtain h 1 ( e, e) Th = ( e, (e e I )) Th +( e, e I ) Th = ( e, (e e I )) Th +( e I, ǫ) γδ({ e I }, [e]) Eh = ( (e e I ), e ǫi) Th +( e, ǫ) Th γδ({ e I }, [e]) Eh = (f + u h p h, e e I ) Th +( e, ǫ) Th γδ({ e I }, [e]) Eh ({e e I }, [ u h p h I]) E 0 h ({ e ǫi}, [u h ]) Eh = (f + u h p h, e e I ) Th +( e, ǫ) Th γδ({ e I }, [e]) Eh (33) ({e e I }, [ u h p h I]) E 0 h ({ e ǫi}, [u h χ]) Eh = I 1 +I +I 3 +I 4 +I 5 ; whereχ = L k (u h ) V h [H 1 0(Ω)] isacontinuousinterpolationofu h suchthat (4) is satisfied, and I j represents the corresponding term from the previous equation. The first term I 1 can be estimated by using the inequality (3) as follows I 1 (34) ( )1 ( h T e e I T h T f + u h p h T ( )1 ( h T f + u h p h T e T + ( ) 1/ (η +osc(f)) η + e T. )1 e E h h 1 e J (u h ) e )1

12 56 J. WANG, Y. WANG, AND X. YE The term I can be handled by using (8) and the fact that e = u h as follows ( ) 1/ I u h T p p h ( ) 1/ (35) η η +osc(f)+ e T. Using [e] = [u h ] and the estimate (3), the term I 3 can be bounded as follows ( )1 ( )1 I 3 γδ h e e I e h 1 e [u h ] e e E h e E h ( )1 ( )1 e I T h 1 e [u h] e e E h ( )1 e T + ( )1 h 1 e [u h] e h 1 e [u h] e e E h e E h ( )1 (36) e T +η η. The same argument can be applied to provide an estimate for the term I 4 : ( )1 (37) I 4 e T +η η. As to I 5, we first use (15) to obtain I 5 =({ e ǫi}, [u h χ]) Eh =({ e}, [u h χ]) Eh ({ǫ}, [u h χ]) Eh =( e, (u h χ)) Th γδ({ (u h χ)}, [e]) Eh +γα(h 1 e [u h χ], [e]) Eh ( (u h χ), ǫ) Th =( e, (u h χ)) Th γδ({ (u h χ)}, [e]) Eh γα(h 1 e [u h], [u h ]) Eh ( (u h χ), ǫ) Th =J 1 +J +J 3 +J 4. Recall that χ = L k (u h ) is a lift of u h so that the estimate (4) holds true. It follows from (4) that ( )1 J 1 η e T Next, we use [e] = [u h ] to obtain J γδ e E h (u h L k (u h )) e [u h ] e, which, with the help of (3) and (4), can be bounded by J η.

13 UNIFIED A POSTERIORI ERROR ESTIMATE FOR STOKES EQUATIONS 563 It is obvious that J 3 η, and it follows from (8) and (4) that ) J 4 η (η +osc(f)+( e, e) 1. Th Collecting all the estimates for J s yields ) (38) I 5 η (η +osc(f)+( e, e) 1. Th Now we substitute all the estimates for I s into (33) to obtain ( ) 1/ ( e, e) Th (η +osc(f)) η +osc(f)+ e T This completes the proof of (30). 3 (η +osc(f)) + 1 ( e, e) T h Efficiency. We first define two bubble functions, which are widely used in a posteriori error estimations [3]. For each triangle T T h, denote by φ T the following bubble function { 7λ 1 λ λ 3 in T, φ T = 0 in Ω\T, where λ i, i = 1,,3 are barycentric coordinates on T. It is clear that φ T H 1 0 (Ω) and satisfies the following properties [3]: (39) (40) For any polynomial q with degree at most m, there exist positive constants c m and C m, depending only on m, such that c m q T q φ T dx q T, (qφ T ) T C m h 1 T q T. T For each e Eh 0, we can analogously define an edge bubble function φ e. Let T 1 and T be two triangles sharing the edge e. To this end, denote by ω e = T 1 T the union of the elements T 1 and T. Assume that in T i, i = 1,, the barycentric coordinates associated with the two ends of e are λ Ti 1 and λ Ti, respectively. The edge bubble function can be defined as follows 4λ T1 1 λt1 in T 1, φ e = 4λ T 1 λt in T, 0 in Ω\ω e. Then φ e H0 1 (Ω) and satisfies the following properties [3]: For any polynomial q with degree at most m, there exist positive constants d m, D m and E m, depending only on m, such that (41) d m q e q φ e ds q e, (4) (43) (qφ e ) ωe D m h 1/ e q e, qφ e ωe E m h 1/ e q e. Then we have the following efficiency bound. e

14 564 J. WANG, Y. WANG, AND X. YE Theorem 3. Let (u;p) and (u h ;p h ) be the solutions of (1)-() and (1)-(13), respectively. Then for all T T h and e Eh 0, we have (44) h T f h + u h p h T e T + ǫ T +h T f f h T, ( ) 1/ (45) h 1/ e J 1 ( u h p h I) e h e f f h ωe + e T + ǫ ωe, T ω e (46) u h T e T. By summing the above estimates over all element T T h and edges e E h, we obtain the following efficiency estimate: η e + ǫ +osc(f). Proof. Let w T = (f h + u h p h )φ T. Since f = u+ p and w T vanishes on T, we clearly have (f f h, w T ) T +(f h + u h p h, w T ) T = ( e, w T ) T ( w T, ǫ ) T. Then, by inequalities (39)-(40), f h + u h p h T (f h + u h p h, w T ) T = ( e, w T ) T ( w T, ǫ ) T (f f h, w T ) T ) ( e T + ǫ T w T T + f f h T w T T ( h 1 T e T +h 1 T ǫ T + f f h T ) f h + u h p h T. This completes the proof of (44). Similarly, for any e Eh 0, let w e = ([ u h p h I])φ e. Using integration by parts and the fact that w e = 0 on ω e, we have (47) ( u h, w e ) ωe = ( u h,w e ) T + [ u h ] w e ds, T ω e e and (48) ( w e, p h ) ωe = ( p h,w e ) T + T ω e e [p h I] w e ds. Testing (1) by using w e over ω e and then using integration by parts give (49) (f, w e ) ωe = ( u, w e ) ωe ( w e, p) ωe. Using the properties of φ e and equations (47)-(49), we have [ u h p h I] e [u h p h I] w e ds e = ) ((f f h, w e ) T +(f h + u h p h, w e ) T ( e, w e ) T +(ǫ, w e ) T T ω e [ u h p h I] e ( h 1/ e f f h ωe +h 1/ e f h + u h p h ωe Combining the above with (44) gives (45). +he 1/ ( ) e T )1/ +he 1/ ǫ ωe T ω e

15 UNIFIED A POSTERIORI ERROR ESTIMATE FOR STOKES EQUATIONS 565 Finally, the estimate (46) holds true as u = 0 and clearly u h T = u h u T = e T e T. This completes the proof of the theorem. 5. Numerical results Some numerical results for such a posteriori error estimators have been reported in [10] for the Crouzeix-Raviart element, and in [4] for an H(div) based discontinuous Galerkin formulation. In this section, we would like to present some computational results for the conforming finite element method in order to verify the theory established in previous Sections. To this end, we consider the Taylor-Hood element [13] when applied to the Stokes problem. We use the BiConjugate Gradient (BICG) iterative solver for solving the resulting linear algebraic equations, with a relative residual = 10 8 being set as a stopping criteria Test problems. Three problems are considered in the numerical test; all are defined on the domain of unit square Ω = (0,1) (0,1). Two of them have exact solutions given by: Test Problem 1: A first Stokes problem with exact solution ( ) x u = y(x 1) (y 1)(y 1) xy (x 1)(x 1)(y 1), π(y x) p = sin, Test Problem : A second Stokes problem with exact solution in the polar coordinate system ( 3 ( )) r cos θ 3θ cos u = 3 ( r 3sin θ sin ), 3θ p = 6r 1/ cos θ. Observe that the solution has a corner singularity of order 0.5 at the origin (0,0). Test Problem 3: D lid driven cavity. This is a Stokes problem which describes the flow of fluid in a rectangular container driven by a uniform motion of the top lid [0]. Note that the boundary condition features a discontinuity at two top corners. The exact solution (u; p), if it exists, should not be sufficiently smooth so that (u; p) [H 1 ] L. However, the discrete problem is still well-posed and should provide an approximation to the actual solution. It must be pointed out that the finite element formulation was only presented for the homogeneous Dirichlet boundary condition in the previous Sections. But the formulation can be easily extended to problems with non-homogeneous Dirichlet boundary conditions such as u = g on Ω. In this case, the term J (u h ) in the a posteriori error estimator needs to be modified as follows { [uh ] on interior edge e E J (u h ) = h 0, (u h g) n on boundary edges. Also, in the computational implementation, we had replaced h T f h+ u h p h T by T f h + u h p h T, where T stands for the area of triangle T.

16 566 J. WANG, Y. WANG, AND X. YE 5.. Results with uniform meshes. In this experiment, we solve test problems 1and on uniform meshesand compute the asymptoticorderofthe errorestimator η. The coarsest mesh is generated by dividing Ω into 4 4 sub-rectangles, and then dividing each sub-rectangle into four triangles by connecting its two diagonal lines. We then use the routine uniform refinement procedure, which divides each triangle into four sub-triangles by connecting the center of its three edges, to generate several levels of fine meshes. For test problem 1, we know theoretically that η = O(h ). For test problem, the order of η is expected to be O(h 0.5 ). Numerical results for these two test problems are reported in tables 1 and. For test problem, the error of pressure is not calculated since the pressure goes to infinity at the point (0,0). Table 1. Error profiles for test problem 1 on uniform triangular meshes. h dofs η (u u h ) u u h p p h e e e e e e e e e-03.71e e e e e-05.13e e-05 Asym. Order O(h k ), k = Table. Error profiles for test problem on uniform triangular meshes. h ndofs η (u u h ) u u h e e e e e e e+00.33e-01.17e e e e-04 Asym. Order O(h k ), k = Results with adaptive refinement. We perform adaptive refinements for test problems and 3, which have corner singularities. Two different refinement strategies are employed in this study. The first one is based on a comparison of each error η T with the maximum value of all the error estimators. The strategy can be described as follows: Local Refinement by Maximum Strategy : (1) Given a current triangular mesh, error indicators η T on each triangle, and a threshold θ (0,1) (e.g., θ = 0.5). One computes the maximum error η max = maxη T. () For each triangle T, if η T θη max, mark this triangle for refinement. (3) The actual refinement is done by the newest node bisection method [17, 19]. It has been proved that this method will not cause mesh degeneration. The only requirement is that the newest nodes for the coarsest mesh must be assigned carefully such that every triangle is compatibly divisible. This can be easily checked.

17 UNIFIED A POSTERIORI ERROR ESTIMATE FOR STOKES EQUATIONS 567 Table 3. Error profiles for test problem using adaptive mesh refinements. Strategy Refinement times dofs η (u u h ) u u h Maximum Local The second refinement strategy is based on a comparison of η T with those for its neighbors. To explain the main idea, let ρ > 0 be a prescribed distance parameter and set T ρ,t = { T : 0 < T T ρ}, where T T stands for the distance of the centers of T and T. With a given threshold θ > 1, we mark a triangle T for refinement if η T θη N(T), where η N(T) is the average of the local error indicator on all the neighboring triangles T T ρ,t. The following is such a refinement strategy that was numerically investigated in this study. Local Refinement by Local Strategy : (1) Given a current triangular mesh, error estimators η T on each triangle, and a threshold θ > 1.0 (e.g., θ = 1.5). One computes an error indicator η N(T) as the average of the local error indicator on neighboring triangles that share a vertex or an edge with T, not including T itself. () For each triangle T, if η T θη N(T), mark this triangle for refinement. (3) The actual refinement is again done by the newest node bisection method [17, 19]. The residual estimator η and errors between the true solution and its finite element approximations for test problem are reported in Table 3. By comparing tables and 3, one clearly sees the power of adaptive refinements in numerical methods for PDEs. For example, the error in L and H 1 norms are given by and with only 1117 degree of freedoms when the local adaptive refinement strategy was used, while the uniform partition requires more than degree of freedoms in order to reach a comparable accuracy. The refined meshes after 0, 8, and 16 refinements are illustrated in Figures 1 and, which show that our residual estimator really captures the corresponding corner singularity correctly, under both refinement strategies. Furthermore, in Figures 3 and 4, we examine the relation of η, (u u h ), and u u h with the degree of freedoms N during the process of the adaptive refinement. The plots start from the coarsest mesh and ends after 16 refinements. For the driven cavity problem, since the exact solution may not be sufficiently as smooth as in [H 1 ] L, the residual error η is not expected to decrease when the mesh is refined. Our numerical experiments show that the error indicator η T is able to locate both corner singularities for this problem. The meshes after 0, 8, and 16 refinements are plotted in Figures 5 and 6. Readers are invited to make their own conclusions for the numerical results illustrated in this Section.

18 568 J. WANG, Y. WANG, AND X. YE Figure 1. Test problem, maximum strategy adaptive refinement. Meshes after 0, 8, and 16 refinements. Figure. Test problem, local strategy adaptive refinement. Meshes after 0, 8, and 16 refinements O(1/N) η H1 semi norm L norm N: degrees of freedom Figure 3. Test problem, maximum strategy adaptive refinement. Plot of η, (u u h ) and u u h, versus the degrees of freedom N. References [1] R. Adams and J. Fournier, Sobolev Spaces, Academic press, 003. [] D. Arnold, F. Brezzi, B. Cockburn and D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (00),

19 UNIFIED A POSTERIORI ERROR ESTIMATE FOR STOKES EQUATIONS O(1/N) η H1 semi norm L norm 10 3 N: degrees of freedom 10 3 Figure 4. Test problem, local strategy adaptive refinement. Plot of η, (u u h ) and u u h, versus the degrees of freedom N. Figure 5. Test problem 3, maximum strategy adaptive refinement. Meshes after 0, 8, and 16 refinements. Figure 6. Test problem 3, local strategy adaptive refinement. Meshes after 0, 8, and 16 refinements. [3] M. Ainsworth and J.T. Oden, A Posteriori Error Estimators for Stokes and Oseen s E- quations. SIAM J. Numer. Anal., 17 (1997), [4] R. Bank and B. Welfert A Posteriori Error Estimates for the Stokes Problem, SIAM J. Numer. Anal. 8, (1991),

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