ANALYSIS OF RECOVERY TYPE A POSTERIORI ERROR ESTIMATORS FOR MILDLY STRUCTURED GRIDS

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1 MATHEMATICS OF COMPUTATION Volue 73, Nuber 247, Pages S (03) Article electronically published on August 19, 2003 ANALYSIS OF RECOVERY TYPE A POSTERIORI ERROR ESTIMATORS FOR MILDLY STRUCTURED GRIDS JINCHAO XU AND ZHIMIN ZHANG Abstract. Soe recovery type error estiators for linear finite eleents are analyzed under O(h 1+α )(α>0) regular grids. Superconvergence of order O(h 1+ρ )(0<ρ α) is established for recovered gradients by three different ethods. As a consequence, aposteriorierror estiators based on those recovery ethods are asyptotically exact. 1. Introduction A posteriori error estiates have becoe standard in odern engineering and scientific coputation. There are two types of popular error estiators: the residual type (see, e.g., [2], [4]) and the recovery type (see, e.g., [21]). The ost representative recovery type error estiator is the Zienkiewicz-Zhu error estiator, especially the estiator based on gradient patch recovery by local discrete least-squares fitting [22], [23]. The ethod is now widely used in engineering practice for its robustness in a posteriori error estiates and its efficiency in coputer ipleentation. It is a coon belief that the robustness of the ZZ estiator is rooted in the superconvergence property of the associated gradient recovery under structured eshes. Superconvergence properties of the ZZ recovery based on local least-squares fitting are proven by Zhang [17] for all popular eleents under rectangular eshes, by Li-Zhang [11] for linear eleents under strongly regular triangular eshes, and by Zhang-Victory [18] for tensor product eleents under strongly regular quadrilateral eshes. While there is a sizable literature on theoretical investents for residual type error estiators (see, e.g., [1], [3], [10], [14] and references therein), there have not been any theoretical results on recovery type error estiators. Nevertheless, the recovery type error estiators perfor astonishingly well even for unstructured grids. The current paper intends to explain this phenoenon. We observe that for Received by the editor June 26, 2002 and, in revised for, Deceber 15, Matheatics Subect Classification. Priary 65N30; Secondary 65N50, 65N15, 65N12, 65D10, 74S05, 41A10, 41A25. Key words and phrases. Gradient recovery, ZZ patch recovery, superconvergence, a posteriori error estiates. The work of the first author was supported in part by the National Science Foundation grant DMS and the Center for Coputational Matheatics and Applications, Penn State University. The work of the second author was supported in part by the National Science Foundation grants DMS , DMS , and INT c 2003 Aerican Matheatical Society

2 1140 JINCHAO XU AND ZHIMIN ZHANG an unstructured esh, when adaptive procedure is used, a esh refineent will usually bring in soe kind of local structure. It is then reasonable to assue that for ost of the doain, every two adacent triangles for an O(h 1+α ) approxiate parallelogra. Under this assuption, we are able to establish superconvergence of the gradient recovery operator for three popular ethods: weighted averaging, local L 2 -proection, and the ZZ patch recovery. Furtherore, by utilizing an integral identity for linear eleents on one triangular eleent developed by Bank and Xu [5], we are able to generalize their superconvergence result between the finite eleent solution and the linear interpolation fro an O(h 2 ) regular grid to an O(h 1+α ) regular grid. Finally, we are able to prove asyptotic exactness of the three recovery error estiators. The topic of a posteriori error estiates has recently attracted ore and ore attention in the scientific counity (see, e.g., [5], [6], [7], [9], [16], [20]; also see recent books [1], [3] for soe general discussions). The literature regarding finite eleent superconvergence theory can be found in the books [8], [10], [12], [15], [19]. 2. Geoetry identities of a triangle In this section, we shall generalize the result in [5] for α =1toallα > 0. Following the arguent in [5], we consider in Figure 1, a triangle τ with vertices p t k =(x k,y k ), 1 k 3, oriented counterclockwise, and corresponding nodal basis functions (barycentric coordinates) {φ k } 3 k=1.let{e k} 3 k=1 denote the edges of eleent τ, {θ k } 3 k=1 the angles, {n k} 3 k=1 the unit outward noral vectors, {t k} 3 k=1 the unit tangent vectors with counterclockwise orientation, {l k } 3 k=1 theedgelengths, and {d k } 3 k=1 the perpendicular heights. Let p be the point of intersection for the perpendicular bisectors of the three sides of τ. Let s k denote the distance between p and side k. Ifτ has no obtuse angles, then the s k will be nonnegative. Otherwise, the distance to the side opposite the obtuse angle will be negative. Let D τ be a syetric 2 2 atrix with constant entries. We define ξ k = n k+1 D τ n k 1. The iportant special case D τ = I corresponds to, and in this case ξ k =cosθ k. Let q k = φ k+1 φ k 1 denote the quadratic bup function associated with edge e k and let ψ k = φ k (1 φ k ). p 3 p 3 θ 3 p d 3 s 3 p 1 p 2 p l 1 e 3 3 t 3 n 3 p 2 Figure 1. Paraeters associated with the triangle τ

3 ANALYSIS OF RECOVERY TYPE A POSTERIORI ERROR ESTIMATORS 1141 The following fundaental identity is proved in [5] for v h P 1 (τ): (2.1) 3 { ξ k q k (u u I ) D τ v h = (l 2 k+1 l 2 τ k=1 e k 2sinθ k 1) 2 u 2 } u vh k t 2 +4 τ k t k n k t k 3 { l k ξ k 3 u 3 } u vh τ 2sin 2 l k+1 ψ k 1 θ k 2 + l k 1 ψ k+1 t k+1 t k 1 2, t k 1 t k+1 t k k=1 where u I P 1 (τ) is the linear interpolation of u on τ. We say that two adacent triangles (sharing a coon edge) for an O(h 1+α ) (α >0) approxiate parallelogra if the lengths of any two opposite edges differ only by O(h 1+α ). Definition. The triangulation T h = T 1,h T 2,h is said to satisfy Condition (α, σ) if there exist positive constants α and σ such that every two adacent triangles inside T 1,h for an O(h 1+α ) parallelogra and Ω 1,h Ω 2,h = Ω, Ω 2,h = O(h σ ), Ωi,h τ T i,h τ, i =1, 2. Reark. There are two iportant ingredients in an autoatic esh generation code. One, called swap diagonal, changes the direction of soe diagonal edges in order to obtain near parallel directions for adacent eleent edges and to ake as any nodes as possible have six triangles attached. Another, known as Lagrange soothing, iteratively relocates nodes to place each node near a esh syetry center (see condition (3.1) in Section 3). Clearly, both swap diagonal and Lagrange soothing are intended to ake every two adacent triangles for an O(h 1+α ) parallelogra. Eventually, only a sall portion of eleents (including boundary eleents) do not satisfy this condition. These eleents then belong to Ω 2,h, which has a sall easure. Therefore, Condition (α, σ) is a reasonable condition in practice and can be satisfied by ost eshes produced by autoatic esh generation codes. Denote V h H 1 (Ω), the C 0 linear finite eleent space associated with T h. Lea 2.1. Assue that T h satisfy Condition (α, σ). Let D τ be a piecewise constant atrix function defined on T h, whose eleents D τi satisfy D τi 1, D τi D τ i h α, i =1, 2; =1, 2. Here τ and τ are a pair of triangles sharing a coon edge. Then for any v h V h (2.2) (u u I ) D τ v h h 1+ρ ( u 3,Ω + u 2,,Ω ) v 1,Ω, ρ =in(α, σ 2, 1 2 ), τ T h τ where u I V h is the interpolation of u. Proof. Applying (2.1), (2.3) τ T h τ (u u I ) D τ v h = I 1 + I 2

4 1142 JINCHAO XU AND ZHIMIN ZHANG where I 1 = 3 τ T h k=1 I 2 = τ T h e k ξ k q k 2sinθ k 3 l k ξ k { (l 2 k+1 l 2 k 1) 2 u t 2 k 2 u +4 τ t k n k } vh t k, τ 2sin 2 θ k=1 k { 3 u 3 } u vh l k+1 ψ k l k 1 ψ k+1 t k+1 t k 1 2. t k 1 t k+1 t k I 2 is easily estiated by (2.4) I 2 h 2 u 3,Ω v h 1,Ω. To estiate I 1, we separate all interior edges into two different groups. E 1 is the set of edges e such that the two adacent triangles sharing e for an O(h 1+α ) approxiate parallelogra and E 2 is the set of the reaining interior edges. The set of all interior edges is given by E = E 1 + E 2. For each e E, there are two triangles, say τ and τ, that share e as a coon edge. Denote, with respect to τ, α e = ξ k (l 2 k+1 2sinθ l2 k 1 ), β e = ξ k 4 τ, k 2sinθ k and with respect to τ, α e = ξ k (l 2 k 2sinθ +1 l2 k 1 ), β e = ξ k 4 τ. k 2sinθ k Taking n and t to correspond to τ, wecanwrite I 1 = I 11 + I 12 + I 13, where I 1 = { } q e (α e α u e ) 2 e E e t 2 +(β e β e ) 2 u vh t n t for =1, 2, and I 13 = { 2 u q e α e e Ω e t 2 + β 2 } u vh e t n t. It is easy to see that, if v h =0on Ω, then I 13 =0. Otherwise,wehavethe following estiate: (2.5) I 13 h 3/2 u 2,, Ω v h 1,Ω. Setting z = t and z = n, weestiate (2.6) 2 u v h q e e t z t h 1 u 2,,Ω v h. τ By definition, for e E 1, α e = α e (1 + O(h α )) and β e = β e (1 + O(h α )). Therefore α e α e h2+α, β e β e h2+α. Cobining this with (2.6), we have (2.7) I 11 h 1+α u 2,,Ω v h h 1+α u 2,,Ω v h 1,Ω. Ω

5 ANALYSIS OF RECOVERY TYPE A POSTERIORI ERROR ESTIMATORS 1143 Now we turn to the estiate for I 12. Since adacent eleents in Ω 2,h do not for an O(h 1+α ) approxiate parallelogra, we siply estiate α e α e α e + α e h 2, β e β e β e + β e h 2. Siilarly to (2.7), this leads to I 12 h u 2,,Ω τ T 2,h Cobining this with (2.5) and (2.7) leads to τ v h h u 2,,Ω v h 0,Ω2,h h σ/2. (2.8) I 1 h 1+ρ u 2,,Ω v h 1,Ω. Finally, applying (2.4) and (2.8) to (2.3), we obtain (2.2). 3. Gradient recovery operators We define N h as the nodal set of a quasi-unifor triangulation T h.givenz N h, we consider an eleent patch ω around z, which we choose as the origin of a local coordinates. Let (x,y ) be the barycenter of a triangle τ ω, =1, 2,...,.We require that one of the following two geoetric conditions be satisfied for α 0: 1 (3.1) (x,y )=O(h 1+α )(1, 1). (3.2) =1 =1 τ ω (x,y )=O(h 1+α )(1, 1). Here we use (x,y ) to represent a vector in conditions (3.1) and (3.2). Reark. Condition (α, σ) iplies both conditions (3.1) and (3.2) for z N h Ω 1,h. Indeed, conditions (3.1) and (3.2) are trivially (with α = ) satisfied by unifor eshes of the regular pattern, the Union Jack pattern, and the criss-cross pattern, and allow an O(h 1+α ) deviation fro those eshes. For exaple, a strongly regular esh is an O(h 2 ) deviation fro a unifor esh of the regular pattern. Note that the condition (3.1) depends only on relative positions of the barycenters of the triangles and is independent of the shapes, sizes, and nubers of those triangles. A boundary node z usually leads to α = 0. However, if z is an interior node with α = 0, then there are no restrictions and we have a copletely unstructured esh around z. Let u I V h be the linear interpolation of a given function u. We shall discuss a gradient recovery operator G h and prove the superconvergent property between u and G h u I. The value of G h u I is first deterined at a vertex and then linearly interpolated over the whole doain. There are three popular ways to generate G h u I at a vertex z. (a) Weighted averaging. (3.3) G h u I (z) = =1 τ ω u I(x,y ).

6 1144 JINCHAO XU AND ZHIMIN ZHANG (b) Local L 2 -proection. We seek linear functions p l P 1 (ω) (l =1, 2), such that (3.4) [p l (x, y) l u I (x, y)]q(x, y)dxdy =0, q P 1 (ω), l =1, 2. ω Then we define G h u I (z) =(p 1 (0, 0),p 2 (0, 0)). (c) Local discrete least-squares fitting proposed by Zienkiewicz-Zhu [22]. seek linear functions p l P 1 (ω) (l =1, 2), such that (3.5) [p l (x,y ) l u I (x,y )]q(x,y )=0, q P 1 (ω), l =1, 2. =1 Then we define G h u I (z) =(p 1 (0, 0),p 2 (0, 0)). Note that (c) is a discrete version of (b). The existence and uniqueness of the iniizers in (b) and (c) can be found in [11, Lea 1]. The following theore generalizes the result in [11] fro α =1toα>0. Theore 3.1. Let ω be an eleent patch around a node z N h,letu W 3 (ω), and let G h u I (z) be produced by either the local L 2 -proection or the weighted averaging under condition (3.2), or by the local discrete least-squares fitting under condition (3.1). Then G h u I (z) u(z) h 1+α u 3,,ω. Proof. (a) For the weighted averaging, we have τ ω lu I (x,y ) l u(0, 0) = = =1 =1 =1 τ ω l(u I u)(x,y )+ where, by the Taylor expansion, =1 τ ω l(u I u)(x,y )+ l u(0, 0) R 1 (u) h 2 u 3,,ω. τ ω [ lu(x,y ) l u(0, 0)] =1 τ ω (x,y )+R 1 (u), Since the barycenter is the derivative superconvergent point for the linear interpolation, then l (u I u)(x,y ) h 2 u 3,,ω, =1, 2,...,. Recall the condition (3.2), and we derive τ l u(0, 0) ω (x,y ) h 1+α u 2,,ω. Therefore, (3.6) =1 =1 τ ω lu I (x,y ) l u(0, 0) h 1+α u 3,,ω. We

7 ANALYSIS OF RECOVERY TYPE A POSTERIORI ERROR ESTIMATORS 1145 (b) For the local L 2 -proection, we set q = 1 in (3.4) to obtain τ p l (x,y )= τ l u I (x,y ). Therefore, p l (0, 0) =1 =1 Using (see [11, Lea 2]) =1 τ ω lu I (x,y )=p l (0, 0) =1 = p l (0, 0) (3.7) p l (0, 0) u 3,,ω =1 τ ω p l(x,y ) =1 τ ω (x,y ). and condition (3.2), we obtain τ (3.8) p l (0, 0) ω lu I (x,y ) h 1+α u 3,,ω. Cobining (3.6) and (3.8), we have proved (3.9) p l (0, 0) l u(0, 0) h 1+α u 3,,ω. (c) For the local discrete least-squares fitting, we set q = 1 in (3.5) to obtain p l (x,y )= l u I (x,y ). Therefore, =1 =1 =1 p l (0, 0) 1 l u I (x,y )=p l (0, 0) 1 p l (x,y ) =1 = 1 p l(0, 0) (x,y ). Using (3.7) and condition (3.1), we obtain (3.10) p l (0, 0) 1 l u I (x,y ) h 1+α u 3,,ω. Next, 1 = 1 =1 l u I (x,y ) l u(0, 0) =1 l (u I u)(x,y )+ 1 =1 =1 =1 [ l u(x,y ) l u(0, 0)] = 1 l (u I u)(x,y )+ 1 lu(0, 0) (x,y )+R 2 (u), =1 =1

8 1146 JINCHAO XU AND ZHIMIN ZHANG with R 2 (u) h 2 u 3,,ω. Therefore, (3.11) 1 l u I (x,y ) l u(0, 0) h 1+α u 3,,ω. =1 Cobining (3.10) and (3.11), we obtain (3.9) for the current case. Theore 3.2. The recovery operator G h satisfies G h v(z) = c v(x,y ), c =1, =1 in all three cases unconditionally. Furtherore, c > 0 for (a) the weighted averaging unconditionally; (b) the local L 2 -proection under the condition (3.2); (c) the local discrete least-squares fitting under the condition (3.1). Proof. The assertion is obvious for the weighted averaging case. Choose v = x + y. Then the iniizer p 1 =1andp 2 =1inbothcases(b)and (c). Therefore, G h v(z) =(1, 1) = c (x + y) = c (1, 1). =1 Now we let p l (x, y) =a 0 + a 1 x + a 2 y. Then for the local discrete least-squares fitting, a i s are given by x y (3.12) x x2 x y a 0 y x a 1 = lu h (x,y ) y y2 a x l u h (x,y ). 2 y l u h (x,y ) Note that x 2 = O(h2 ), x y = O(h 2 ), y 2 = O(h2 ); and under condition (3.1), x = O(h 1+α ), =1 =1 y = O(h 1+α ). By scaling arguent we see that a 1 = O(h α 1 ), a 2 = O(h α 1 ). Therefore, a 0 = 1 l u h (x,y ) a 1 x a 2 = c l u h (x,y ) with c = 1 + O(h2α ) > 0. A siilar arguent shows that c = τ ω + O(h2α ) > 0 y

9 ANALYSIS OF RECOVERY TYPE A POSTERIORI ERROR ESTIMATORS 1147 for the local L 2 -proection when condition (3.2) is satisfied. Under the given condition, the recovered gradient at a vertex z is a convex cobination of gradient values on the eleent patch surrounding z. 4. Superconvergence of the recovery operators We consider the non-self-adoint proble: find u H 1 (Ω) such that (4.1) B(u, v) = [(D u + bu) v + cuv] =f(v), v H 1 (Ω). Ω Here D is a 2 2 syetric, positive definite atrix, and f( ) is a linear functional. We assue that all the coefficient functions are sooth, and the bilinear for B(, ) is continuous and satisfies the inf-sup condition on H 1 (Ω). These conditions insure that (4.1) has a unique solution. The finite eleent solution u h V h satisfies (4.2) B(u h,v h )=f(v h ) v h V h. To insure a unique solution for (4.2), we further assue the inf-sup condition of B to be satisfied on V h. We define the piecewise constant atrix function D τ in ters of the diffusion atrix D as follows: D τi = 1 D i dx. τ τ Note that D τ is syetric and positive definite. Theore 4.1. Let the solution of (4.1) satisfy u H 3 (Ω) W 2 (Ω), letu h be the solution of (4.2) and let u I V h be the linear interpolation of u. Assue that the triangulation T h satisfies Condition (α, σ). Then u h u I 1,Ω h 1+ρ ( u 3,Ω + u 2,,Ω ), ρ =in(α, 1 2, σ 2 ). Proof. We begin with the identity B(u u I,v h )= (u u I ) D τ v h dx+ τ T h τ τ T h τ (u u I ) (D D τ ) v h dx + (u u I )(b v h + cv) dx = I 1 + I 2 + I 3. Ω The first ter I 1 is estiated using Lea 2.1 and I 2 and I 3 can be easily estiated by I 2 + I 3 h 2 u 2,Ω v 1,Ω. Thus B(u u I,v h ) h 1+ρ ( u 3,Ω + u 2,,Ω ) v h 1,Ω. We coplete the proof using the inf-sup condition in B(u h u I,v h ) B(u u I,v h ) u h u I 1,Ω sup = sup v h V h v h 1,Ω v h V h v h 1,Ω h 1+ρ ( u 3,Ω + u 2,,Ω ).

10 1148 JINCHAO XU AND ZHIMIN ZHANG Theore 4.2. Let the solution of (4.1) satisfy u W 3 (Ω), letu h be the solution of (4.2), andletg h be a recovery operator defined by one of the three: (a) the weighted averaging, (b) the local L 2 -proection, and (c) the local discrete least-squares fitting. Assue that the triangulation T h satisfies Condition (α, σ). Then u G h u h 0,Ω h 1+ρ u 3,,Ω. Proof. We decopose (4.3) u G h u h =( u ( u) I )+(( u) I G h u I )+G h (u I u h ), where ( u) I V h V h is the linear interpolation of u. By the standard approxiation theory, (4.4) u ( u) I 0,Ω h 2 u 3,Ω. We observe that when we pick an eleent patch on T 1,h, both conditions (3.1) and (3.2) are satisfied. Therefore, using Theore 3.1, we have ( u) I G h u I 0,Ω1,h 1/2 τ G h u I (z) u(z) 2 τ Ω 1,h z N h τ (4.5) On the other hand, h 1+α u 3,,Ω Ω 1,h 1/2 h 1+α u 3,,Ω. (4.6) ( u) I G h u I 0,Ω2,h h u 3,,Ω Ω 2,h 1/2 h 1+σ/2 u 3,,Ω by Condition (α, σ). Cobining (4.5) with (4.6), we have (4.7) ( u) I G h u I 0,Ω h 1+in(α,σ/2) u 3,,Ω. Siilarly as in (4.5), we have, by using the fact proved in Theore 3.2, that G h v(z) is a convex cobination of v τz s; G h (u I u h ) 0,Ω1,h 1/2 τ G h (u I u h )(z) 2 τ T 1,h z N h τ 1/2 = (u I u h ) 0,Ω1,h τ T 1,h τ (u I u h ) τ 2 (4.8) h 1+ρ u 3,,Ω, by Theore 4.1. In addition, G h (u I u h ) 0,Ω2,h τ τ T 2,h h u 3,,Ω z N h τ τ τ T 2,h (4.9) h 1+σ/2 u 3,,Ω. Cobining (4.8) and (4.9) yields (4.10) G h (u I u h ) 0,Ω h 1+ρ u 3,,Ω. G h (u I u h )(z) 2 1/2 1/2

11 ANALYSIS OF RECOVERY TYPE A POSTERIORI ERROR ESTIMATORS 1149 The conclusion follows by applying (4.4), (4.7), and (4.10) to the right-hand side of (4.3). Theore 4.2 requires the global regularity u W 3 (Ω) which is too restrictive in practice. The next theore turns to interior axiu nor estiates and relaxes the global regularity assuption on the solution. Theore 4.3. Consider an interior patch ω z Ω d Ω 1,h with d =dist(ω z, Ω d ) Kh for soe constant K>0. Letu W 2 (Ω) W 3 (Ω d) be the solution of (4.1), let u h be the solution of (4.2), andletg h be a recovery operator defined by one of the three: (a) the weighted averaging, (b) the local L 2 -proection, and (c) the local discrete least-squares fitting. Then we have ( u G h u h )(z) h 1+in(1,α) u 3,,ωz +d 1 h 2 ln 1 h u 2,,Ω+h 1+α ln d h u 2,,Ω d. Proof. We denote Vh 0(Ω d) as the finite eleent subspace that has a copact support on Ω d and start fro B(u h u I,χ)=B(u u I,χ)=F(χ), χ Vh 0 (Ω d), with F (χ) = { } q e (α e α u e ) 2 e E e t 2 +(β e β e ) 2 u χ t n t, d where E d is the edge set of Ω d. By the sae arguent as in (2.7), we have F (χ) h 1+α u 2,,Ωd Ω d χ. Therefore, (4.11) F 1,,Ωd = sup F (χ) h 1+α u 2,,Ωd. χ W1 1(Ω d), χ W 1 1 (Ω d ) =1 Recall Theore 1.2 of Schatz-Wahlbin [13] (it is straightforward to verify that all conditions of that theore are satisfied under the current situation): e W 1 (Ω 0) + d 1 e L (Ω 0) C in ( w χ W 1 χ S h (Ω d ) + d 1 w χ L (Ω d )) + Cd 1 s N/q e W s q (Ω d ) + C ln d h F 1,,Ω d, where e = w w h satisfies B(e, χ) =F (χ). Now, setting q =,s =0,w =0, Ω 0 = ω z,andw h = u I u h,weobtain u h u I 1,,ωz d 1 u h u I L (Ω d ) +ln d h F 1,,Ω d. Applying (4.11) and u h u I L (Ω d ) h 2 ln 1 h u 2,,Ω results in (4.12) u h u I 1,,ωz d 1 h 2 ln 1 h u 2,,Ω + h 1+α ln d h u 2,,Ω d. Now we decopose ( u G h u h )(z) =( u G h u I )(z)+g h (u I u h )(z). By Theore 3.2, G h v(z) is a convex cobination of values of v on τ ω z. Consequently, G h is a bounded operator in the sense G h v h (z) v h 1,,ωz, v h V h.

12 1150 JINCHAO XU AND ZHIMIN ZHANG Therefore, (4.13) ( u G h u h )(z) ( u G h u I )(z) + u I u h 1,,ωz. The conclusion follows by applying Theore 3.1 and (4.12) to the right-hand side of (4.13). Reark. When α<1, we choose d = h 1 α and obtain ( u G h u h )(z) h 1+α ln 1 h. When α 1, we choose d = h 1 β with β (0, 1] and obtain ( u G h u h )(z) h 1+β ln 1 h. We see that when α 1, the recovery is ore accurate as z leaves the boundary. 5. Asyptotic exactness of the recovery type error estiators With preparation in the previous sections, it is now straightforward to prove the asyptotic exactness of error estiators based on the recovery operator G h. The global error estiator is naturally defined by (5.1) η h = G h u h u h 0,Ω. Theore 5.1. Assue the hypotheses of Theore 4.2. Furtherore, assue that there exists a constant c(u) > 0 such that (5.2) (u u h ) c(u)h. Then η h 1 (u u h ) 0,Ω hρ, ρ =in( 1 2, σ 2,α). Proof. By Theore 4.2 and hypothesis (5.2), we have η h 1 (u u h ) 0,Ω G hu h u 0,Ω h1+ρ u 3,,Ω h ρ. (u u h ) 0,Ω c(u)h The pointwise error estiator at a vertex z τ Ω 1,h is naturally defined by (5.3) ηh z = G hu h (z) u h (τ). The next theore shows that the pointwise error estiator is asyptotically exact. Theore 5.2. Assue the hypotheses of Theore 4.3. Letz be a vertex of eleents τ Ω 1,h and assue that there exists a constant c(u) > 0 such that (5.4) u(z) u h (τ) c(u)h. Then we have (a) when α (0, 1), ηh z u(z) u h (τ) 1 hα, with dist(z, Ω 1,h ) Kh 1 α ;and(b)whenα 1, ηh z u(z) u h (τ) 1 hβ, β (0, 1], with dist(z, Ω 1,h ) Kh 1 β.

13 ANALYSIS OF RECOVERY TYPE A POSTERIORI ERROR ESTIMATORS 1151 Proof. We only prove the case when α (0, 1). By Theore 4.3 and hypothesis (5.4), we have ηh z u(z) u h (τ) 1 G hu h (z) u(z) h1+α = h α. u(z) u h (τ) h We see that the error estiators (5.1) and (5.3) based on the gradient recovery operator are asyptotically exact under Condition (α, σ). As we entioned above, this condition is not a very restrictive condition in practice. An autoatic esh generator usually produces soe grids which are ildly structured. In practice, a copletely unstructured esh is seldo seen. Our analysis explains in part the good perforance of the ZZ error estiator based on the local discrete least-squares fitting for general grids. Acknowledgents The authors would like to thank Professor Wahlbin for the intriguing discussion which led to the proof of Theore 4.3. References [1] M. Ainsworth and J.T. Oden, A Posteriori Estiation in Finite Eleent Analysis, Wiley Interscience, New York, MR 2003b:65001 [2] I. Babuška and W.C. Rheinboldt, A Posteriori Error Estiates for the Finite Eleent Method, Internat. J. Nuer. Methods Engrg., 12 (1978), pp [3] I. Babuška and T. Strouboulis, The Finite Eleent Method and its Reliability, OxfordUniversity Press, London, MR 2002k:65001 [4] R.E. Bank and A. Weiser, Soe a posteriori error estiators for elliptic partial differential equations, Math. Cop., 44 (1985), pp MR 86g:65207 [5] R.E. Bank and J. Xu, Asyptotically exact aposteriorierror estiators, Part I: Grid with superconvergence, preprint, to appear in SIAM J. Nuer. Anal. [6] R.E. Bank and J. Xu, Asyptotically exact a posteriori error estiators, Part II: General Unstructured Grids, preprint, to appear in SIAM J. Nuer. Anal. [7] C. Carstensen and S. Bartels, Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforing, nonconforing, and ixed FEM, Math. Cop. 71 (2002), [8] C.M. Chen and Y.Q. Huang, High Accuracy Theory of Finite Eleent Methods. Hunan Science Press, Hunan, China, 1995 (in Chinese). [9] W. Hoffann, A.H. Schatz, L.B. Wahlbin, and G. Wittu, Asyptotically exact aposteriori estiators for the pointwise gradient error on each eleent in irregular eshes I: A sooth proble and globally quasi-unifor eshes, Math. Cop., 70 (2001), pp MR 2002a:65178 [10] M. Křížek, P. Neittaanäki, and R. Stenberg (Eds.), Finite Eleent Methods: Superconvergence, Post-processing, and A Posteriori Estiates, Lecture Notes in Pure and Applied Matheatics Series, Vol.196, Marcel Dekker, New York, MR 98i:65003 [11] B. Li and Z. Zhang, Analysis of a class of superconvergence patch recovery techniques for linear and bilinear finite eleents, Nuerical Methods for Partial Differential Equations, 15 (1999), pp MR 99:65201 [12] Q. Lin and N. Yan, Construction and Analysis of High Efficient Finite Eleents (in Chinese), Hebei University Press, P.R. China, [13] A.H. Schatz and L.B. Wahlbin, Interior axiu nor estiates for finite eleent ethods. Part II, Math.Cop., 64 (1995), pp MR 95:65143 [14] R. Verfürth, A Posteriori Error Estiation and Adaptive Mesh Refineent Techniques, Teubner Skripten zur Nuerik, B.G. Teubner, Stuttgart, [15] L.B. Wahlbin, Superconvergence in Galerkin Finite Eleent Methods, LectureNotesinMatheatics, Vol.1605, Springer, Berlin, MR 98:65083

14 1152 JINCHAO XU AND ZHIMIN ZHANG [16] N. Yan and A. Zhou, Gradient recovery type a posteriori error estiates for finite eleent approxiations on irregular eshes, Coput. Methods Appl. Mech. Engrg., 190 (2001), pp MR 2002c:65189 [17] Z. Zhang, Ultraconvergence of the patch recovery technique II, Math. Cop., 69 (2000), pp MR 2000:65107 [18] Z. Zhang and H.D. Victory Jr., Matheatical Analysis of Zienkiewicz-Zhu s derivative patch recovery techniques, Nuerical Methods for Partial Differential Equations, 12 (1996), pp MR 98c:65191 [19] Q.D. Zhu and Q. Lin, Superconvergence Theory of the Finite Eleent Method, Hunan Science Press, China, 1989 (in Chinese). [20] J.Z. Zhu and Z. Zhang, The relationship of soe a posteriori error estiators, Coput. Methods Appl. Mech. Engrg., 176 (1999), pp MR 2000f:65126 [21] O.C. Zienkiewicz and J.Z. Zhu, A siple error estiator and adaptive procedure for practical engineering analysis, Internat. J. Nuer. Methods Engrg., 24 (1987), pp MR 87:73055 [22] O.C. Zienkiewicz and J.Z. Zhu, The superconvergence patch recovery and a posteriori error estiates, Part 1: The recovery technique, Internat. J. Nuer. Methods Engrg., 33 (1992), pp MR 93c:73098 [23] O.C. Zienkiewicz and J.Z. Zhu, The superconvergence patch recovery and a posteriori error estiates. Part 2: Error estiates and adaptivity, Internat. J. Nuer. Methods Engrg., 33 (1992), pp MR 93c:73099 Center for Coputational Matheatics and Applications, Departent of Matheatics, The Pennsylvania State University, University Park, Pennsylvania E-ail address: xu@ath.psu.edu Departent of Matheatics, Wayne State University, Detroit, Michigan E-ail address: zzhang@ath.wayne.edu