A posteriori error estimator based on gradient recovery by averaging for discontinuous Galerkin methods

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1 A posteriori error estimator based on gradient recovery by averaging for discontinuous Galerkin methods Emmanuel Creusé, Serge Nicaise October 7, 29 Abstract We consider some (anisotropic and piecewise constant) diffusion problems in domains of R 2, approximated by a discontinuous Galerkin method with polynomials of any degree. We propose an a posteriori error estimator based on gradient recovery by averaging. It is shown that this estimator gives rise to an upper bound where the constant is one up to some additional terms that guarantee reliability. The lower bound is also established. Moreover these additional terms are negligible when the recovered gradient is superconvergent. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests. Key Words A posteriori estimator, Discontinuous Galerkin finite elements. AMS (MOS) subject classification 65N3; 65N5, 65N5, Introduction Among other methods, the finite element method is one of the more popular that is commonly used in the numerical realization of different problems appearing in engineering applications, like the Laplace equation, the Lamé system, the Stokes system, the Maxwell system, etc... (see [7, 8, 23]). More recently discontinuous Galerkin finite element methods become very attractive since they present some advantages, like flexibility, adaptivity, etc... In our days a vast literature exists on the subject, we refer to [3, ] and the references cited there. Adaptive techniques based on a posteriori error estimators have become indispensable tools for such methods. For continuous Galerkin finite element methods, there now Université des Sciences et Technologies de Lille, Laboratoire Paul Painlevé UMR 8524, EPI SIMPAF - INRIA Lille Nord Europe, UFR de Mathématiques Pures et Appliquées, Cité Scientifique, Villeneuve d Ascq Cedex creuse@math.univ-lille.fr Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes, F Valenciennes Cedex 9 France, Serge.Nicaise@univvalenciennes.fr

2 exists a vast amount of literature on a posteriori error estimation for problems in mechanics or electromagnetism and obtaining locally defined a posteriori error estimates. We refer to the monographs [2, 4, 24, 3] for a good overview on this topic. On the other hand a similar theory for discontinuous methods is less developed, let us quote [5, 2, 7, 8, 9, 27, 3]. Usually upper and lower bounds are proved in order to guarantee the reliability and the efficiency of the proposed estimator. Most of the existing approaches involve constants depending on the shape regularity of the elements and/or of the jumps in the coefficients; but these dependences are often not given. Only a few number of approaches gives rise to estimates with explicit constants, let us quote [2, 6, 2, 22, 25, 26, 6] for continuous methods. For discontinuous methods, we may cite the recent papers [, 2, 9, 4, 5]. Our goal is therefore to consider second order elliptic operators with discontinuous diffusion coefficients in two-dimensional domains with mixed boundary conditions and a discontinuous Galerkin method with polynomials of any degree. Inspired from the paper [6], which treats the case of continuous diffusion coefficients approximated by a continuous Galerkin method, we further derive an a posteriori estimator with an explicit constant in the upper bound (more precisely ) up to some additional terms that are usually superconvergent and some oscillating terms. The approach, called gradient recovery by averaging [6] is based on the construction of a Zienkiewicz/Zhu estimator, namely the difference in an appropriate norm of a h u h Gu h, where h u h is the broken gradient of u h and Gu h is a H(div )-conforming approximation of this variable. Here special attention has to be paid due to the assumption that a may be discontinuous. Moreover the non conforming part of the error is managed using a Helmholtz decomposition of the error and a standard Oswald interpolation operator [9, ]. Furthermore using standard inverse inequalities, we show that our estimator is locally efficient. Two interests of this approach are first the simplicity of the construction of Gu h, and secondly its superconvergence property (validated by numerical tests). The schedule of the paper is as follows: We recall in section 2 the diffusion problem, its numerical approximation and an appropriate Helmholtz decomposition of the error. Section 3 is devoted to the introduction of the estimator based on gradient averaging and the proofs of the upper and lower bounds. The upper bound directly follows from the construction of the estimator and some results from [6], while the lower bound requires the use of some inverse inequalities and a special construction of Gu h. Finally in section 4 some numerical tests are presented that confirm the reliability and efficiency of our estimator and the superconvergence of Gu h to a u. Let us finish this introduction with some notation used in the remainder of the paper: On D, the L 2 (D)-norm will be denoted by D. In the case D =, we will drop the index. The usual norm and semi-norm of H s (D) (s ) are denoted by s,d and s,d, respectively. Finally, the notation a b and a b means the existence of positive constants C and C 2, which are independent of the mesh size, of the quantities a and b under consideration and of the coefficients of the operators such that a C 2 b and C b a C 2 b, respectively. In other words, the constants may depend on the aspect ratio of the mesh as well as the polynomial degree l (see below). 2

3 2 The boundary value problem and its discretization Let be a bounded open domain of R 2 with a Lipschitz boundary Γ that we suppose to be polygonal. We further assume that is simply connected and that Γ is connected. We consider the following elliptic second order boundary value problem with non homogeneous mixed boundary conditions: div (a u) = f in, u = g D on Γ D, a u n = g N on Γ N, where Γ = Γ D Γ N and Γ D Γ N =. For convenience we suppose that meas Γ D >. In the sequel, we suppose that a is a symmetric positive definite matrix which is piecewise constant, namely we assume that there exists a partition P of into a finite set of Lipschitz polygonal domains,, J such that, on each j, a = a j where a j is a symmetric positive definite matrix. The variational formulation of () involves the bilinear form B(u, v) = a u v and the Hilbert space H D() = {u H () : u = on Γ D }. Given f L 2 (), g D H 2 (Γ D ) and g N L 2 (Γ N ), the weak formulation consists in finding u w + HD () such that B(u, v) = fv + g N v, v HD(), Γ N (2) where w H () is a lifting for g D, i.e., w = g D on Γ D. Invoking the positiveness of a, ( ) /2 the bilinear form B is coercive on HD () with respect to the norm a /2 u 2 and this coerciveness guarantees that problem (2) has a unique solution by the Lax-Milgram lemma. 2. Discontinuous Galerkin approximated problem Following [9, 3], we consider the following discontinuous Galerkin approximation of our continuous problem: We consider a triangulation T made of triangles T whose edges are denoted by e. We assume that this triangulation is regular, i.e., for any element T, the ratio h T ρ T is bounded by a constant σ > independent of T and of the mesh size h = max T T h T, where h T is the diameter of T and ρ T the diameter of its largest inscribed ball. We further assume that T is conforming with the partition P of, i.e., the matrix a being constant 3 ()

4 on each T T, we then denote by a T the value of a restricted to an element T. With each edge e of an element T, we associate a unit normal vector n e, and n T stands for the outer unit normal vector of T. E (resp. N ) represents the set of edges (resp. vertices) of the triangulation. In the sequel, we need to distinguish between edges included into, Γ D or Γ N, in other words, we set E int = {e E : e }, E D = {e E : e Γ D }, E N = {e E : e Γ N }. For shortness, we also write E ID = E int E D. Problem (2) is approximated by the (discontinuous) finite element space: X h = { v h L 2 () v h T P l (T ), T T }, where l is a fixed positive integer. The space X h is equipped with the norm ( q DG,h := a /2 q 2 T + γ T T e E ID h e [q ] 2 e) /2, where γ is a positive parameter fixed below. Later on we also need the continuous counterpart of X h, namely we introduce as well as We further need S h = { v h C() v h T P l (T ), T T }, S h, = { v h C() v h T P (T ), T T }. X h, = { v h L 2 () v h T P (T ), T T }. For our further analysis we need to define some jumps and means through any e E of the triangulation. For e E such that e, we denote by T + and T the two elements of T containing e. Let q X h, we denote by q ±, the traces of q taken from T ±, respectively. Then we define the mean of q on e by For v [X h ] d, we denote similarly {{q}} = q+ + q. 2 {{v}} = v+ + v. 2 The jump of q on e is now defined as follows: [q ] = q + n T + + q n T. 4

5 Remark that the jump [q ] of q is vector-valued. For a boundary edge e, i. e., e, there exists a unique element T + T such that e T +. Therefore the mean and jump of q are defined by {{q}} = q + and [q ] = q + n T +. For q X h, we define its broken gradient h q in by : ( h q) T = q T, T T. With these notations, we define the bilinear form B h (.,.) as follows: B h (u h, v h ) := a u h v h ({{a h v h }} [u h ] + {{a h u h }} [v h ]) T T T e E e ID + γ [u h ] [v h ], u h, v h X h, h e e E ID e where the positive parameter γ is chosen large enough to ensure coerciveness of the bilinear form B h on X h (see, e.g., Lemma 2. of [9]), namely according to the results from [29], the choice γ > (l + )(l + 2) 2 max (a T T T e T T ) h e T yields the coerciveness of B h. The discontinuous Galerkin approximation of problem (2) reads now: Find u h X h, such that B h (u h, v h ) = F (v h ), (4) where F (v h ) = fv h + e E D e g D (γh e v h a v h n T ) + g N v h, v h X h. Γ N As our approximated scheme is a non conforming one (i.e. the solution does not belong to HD ()), as usual we need to use an appropriate Helmholtz decomposition of the error (see Lemma 3.2 of [3] or Theorem of [] in 2D). Lemma 2. (Helmholtz decomposition of the error) We have the following error decomposition a h (u u h ) = a ϕ + curl χ, (5) with χ H () such that and ϕ HD (). Moreover the next identity holds: (3) curl χ n = on Γ N, (6) a /2 h (u u h ) 2 = a /2 h ϕ 2 + a /2 curl χ 2. (7) For the detailed proof of this Lemma we refer to Lemma 2. of [9]. 5

6 3 The a posteriori error analysis based on gradient recovery by averaging Error estimators can be constructed in many different ways as, for example, using residual type error estimators which measure locally the jump of the discrete flux [9]. A different method, based on equilibrated fluxes, consists in solving local Neumann boundary value problems [2] or in using Raviart-Thomas interpolant [, 9, 4, 5]. Here, as an alternative we introduce a gradient recovery by averaging and define an error estimator based on a H(div )-conforming approximation of this variable. In comparison with [6], we here allow the case of discontinuous diffusion coefficient and use a discontinuous Galerkin method. Inspired from [6] the conforming part of the estimator η CF involves the difference between the broken gradient a h u h and its smoothed version Gu h, where Gu h is for the moment any element in Xh, 2 satisfying Gu h H(div, ) = {v L 2 () 2 : div v L 2 ()}, (8) (Gu h ) j H ( j ), j =,, J. (9) Hence conforming part of the estimator η CF is defined by η 2 CF = T T η 2 CF,T, () where the indicator η CF,T is defined by η CF,T = a /2 (a u h Gu h ) T. For the nonconforming part of the error, we associate with u h, its Oswald interpolation operator, namely the unique element w h S h defined in the following natural way (see Theorem 2.2 of [9]): to each node n of the mesh corresponding to the Lagrangian-type degrees of freedom of S h, the value of wp h is the average of the values of u h at this node n T n if it belongs to Γ N (i.e., w h (n) = T u h T (n) ) and the value of g Pn T T D at this node if it belongs to Γ D (here we assume that g D C( Γ D )). Then the non conforming indicator η NC,T is simply η NC,T = a /2 (w h u h ) T. The non conforming part of the estimator is then η 2 NC = T T η 2 NC,T. () Similarly we introduce the estimator corresponding to jumps of u h : η 2 J = e E ID η 2 J,e, 6

7 with { γ ηj,e 2 h = e [u h ] 2 e if e E int, γ h e u h g D 2 e if e E D. As in [6], we introduce some additional superconvergent security parts. In order to define them properly we recall that for a node x N, we denote by λ x the standard hat function (defined as the unique element in S h, such that λ x (y) = δ x,y for all y N ), let ω x be the patch associated with x, which is simply the support of λ x and let h x be the diameter of ω x (which is equivalent to the diameter h K of any triangle K included into ω x ). We now denote by r the element residual and for all x N, we set r = f + div (Gu h ) r x = ( ω x λ x ) ω x rλ x if x N \ N D, r x = if x N D. We further use a multilevel decomposition of S h,, namely we suppose that we start from a coarse grid T and that the successive triangulations are obtained by using the bisection method, see [28, 6]. This means that we obtain a finite sequence of nested triangulations T l, l =,, L such that T L = T. Denoting by S l the space then we have S l = { v C() v T P (T ), T T l }, S l S l+ and S h, = L l=s l = S L. Furthermore if we denote by N l the nodes of the triangulation T l, we have N l N l+. As usual for all z N l we denote by λ lz the hat function associated with z, namely the unique element in S l such that For all l we finally set λ lz (z ) = δ zz z N l. Ñ l = (N l \ N l ) {z N l : λ lz λ l z }, and Ñ = N. It should be noticed (see for instance [6]) that to each z Ñl, the corresponding hat function λ lz does not belong to S l. Now we define ρ and γ by ρ 2 = x N ρ 2 x, γ 2 = L γlz, 2 l= z Ñl\Γ D 7

8 where ρ 2 x = h 2 x r r x 2 λ x + h x ω x Gu h n g N 2 λ x, ω x Γ N γ lz = R, λ lz, R being the residual defined by R, ϕ = Gu h ϕ fϕ g N ϕ, ϕ H (). Γ N 3. Upper bound Theorem 3. Assume that there exists w h X h H () such that g D = w h ΓD. Let u w + HD () be a solution of problem (2) and let u h be its discontinuous Galerkin approximation, i.e. u h X h solution of (4). Then there exists C > such that a /2 h (u u h ) (ηcf 2 + ηnc) 2 /2 + C( ρ + γ), (2) and consequently u u h DG,h (( + ɛ)(ηcf 2 + ηnc) 2 + ηj) 2 /2 + C( + 4ɛ )/2 ( ρ + γ), ɛ >. (3) Proof: From the Helmholtz decomposition of the error we have a /2 h (u u h ) 2 = a /2 ϕ 2 + a /2 curl χ 2. (4) We are then reduced to estimate each term of this right-hand side. For the non conforming part, we proceed as in [], namely by Green s formula we have a /2 curl χ 2 = h (u u h ) curl χ = h u h curl χ + g D curl χ n Γ D = h (w h u h ) curl χ, since w h curl χ = Γ D g D curl χ n. By Cauchy-Schwarz s inequality we directly obtain a /2 curl χ 2 η NC a /2 curl χ. (5) For the conforming part, we write a /2 ϕ 2 = a h (u u h ) ϕ = (a u Gu h ) ϕ + (Gu h a h u h ) ϕ. 8

9 By Cauchy-Schwarz s inequality we obtain a /2 ϕ 2 a /2 (a h u h Gu h ) a /2 ϕ + (a u Gu h ) ϕ. (6) Using problem (2), the second term of this right-hand side is bounded by the residual, indeed (a u Gu h ) ϕ = Gu h ϕ fϕ g N ϕ = R, ϕ. Γ N Using the arguments from Theorem 4. of [6], we have R, ϕ C( ρ + γ) a /2 ϕ. (7) Coming back to the identity (4), and using the estimates (5), (6) and (7) we conclude by discrete Cauchy-Schwarz s inequality and again using (4): a /2 h (u u h ) 2 η NC a /2 curl χ + (η CF + C( ρ + γ)) a /2 ϕ (η 2 NC + η 2 CF ) /2 ( a /2 curl χ 2 + a /2 ϕ 2 ) /2 + C( ρ + γ) a /2 ϕ [(η 2 NC + η 2 CF ) /2 + C( ρ + γ)] a /2 h (u u h ). 3.2 Lower bound Our lower bound is based on the equivalence of the local L 2 -norm of any element in X h with a local L 2 -norm in the interfaces. First of all for any vertex x of one j and belonging to more than one sub-domain, we introduce the following local notation: let i, i =,, n, n 2, the sub-domains that have x as vertex. We further denote by n i the unit normal vector along the interface I i between i and i+ (modulo n if x is inside the domain ) and oriented from i and i+. Now we are able to prove the following lemma: Lemma 3.2 Assume that x is a vertex of one j and belonging to more than one subdomain, and use the notations introduced above. Then there exists a positive constant C that depends only on the geometrical situation of the i s near x such that for all v (i) R 2, i =,, n, there exist vectors g(v) (i) R 2, i =,, n satisfying and such that the following estimate holds (g(v) (i+) g(v) (i) ) n i =, i =,, n, (8) n v (i) g(v) (i) C i= n [v n] i, (9) i= 9

10 where here means the Euclidean norm and [v n] i derivative of v along the interface I i : means the jump of the normal [v n] i = (v (i+) v (i) ) n i, i =,, n. Proof: First introduce the following subspace of R 2n : W = {v = (v (i) ) n i= : v (i) R 2 and satisfying [v n] i =, i =,, n}. We take g(v) = Π W v, the orthogonal projection of v = (v (i) ) n i= into W. By construction g(v) trivially satisfies (8). On the other hand the estimate (9) is equivalent to v Π W v C n [v n] i, i= which is easily proved by a contradiction argument and the fact that we are in a finite dimensional space. Using the above lemma, we are now able to prove the asymptotic nondeterioration of the smoothed gradient if the following choice for Gu h is made: We distinguish the following different possibilities for x N. ) First for all vertex x of the mesh (i.e. vertex of at least one triangle) such that x is inside one j, we set (Gu h ) j (x) = T a T u h T (x). (2) ω x 2) Second if x belongs to the boundary of and to the boundary of only one j (hence it does not belong to the boundary of another k ), we define (Gu h ) j (x) as before. 3) If x belongs to an interface between two different sub-domain j and k but is not a vertex of these sub-domains, then we denote by n j,k the unit normal vector pointing from j to k and set t j,k the unit orthogonal vector of n j,k so that (n j,k, t j,k ) is a direct basis of R 2 ; in that case we set x T (Gu h ) j (x) n j,k = (Gu h ) k (x) n j,k = T a T u h T (x) n j,k, (2) ω x x T (Gu h ) j (x) t j,k = T a T u h T (x) t j,k, (22) ω x j (Gu h ) k (x) t j,k = ω x k T j :x T T k :x T T a T u h T (x) t j,k. (23) 4) Finally if x is a vertex of at least two sub-domains j, for the sake of simplicity we suppose that each triangle T having x as vertex is included into one j, and we take (Gu h ) j (x) = g(v) (j) j J x, (24)

11 where J x = {j {,, J} : x j }, g(v) (j) were defined in the previous Lemma 3.2 with here v given by v = (a j u h T (x)) j Jx. With these choices, we take (Gu h ) j = x N j (Gu h ) j (x)λ x, j =,, J, (25) where (Gu h ) j (x) was defined before. The main point is that by construction Gu h satisfies the requirements (8) and (9) but moreover we have the next asymptotic nondeterioration result: Theorem 3.3 If l 2, then for each element T T the following estimate holds a /2 T (Gu h a T u) T u u h DG,ωT + osc(f, ω T ), (26) where ω T denotes the patch consisting of all the triangles of T having a nonempty intersection with T and v 2 DG,ω T = a /2 h v 2 ω T + γ e [v ] 2 e, e E ID :e ω T h and osc(f, ω T ) 2 = T ω T h 2 T f Π T f 2 T, where Π T f is the L 2 (T )-orthogonal projection of f onto P (T ). Proof: By the triangle inequality we may write a /2 T (Gu h a T u) T a /2 T (Gu h a T u h ) T + a /2 T (a T u h a T u) T a /2 T (Gu h a T u h ) T + u u h DG,T. Therefore it remains to estimate the first term of this right-hand side. For that purpose, since T j for a unique j {,, J}, we may write (Gu h a T u h ) T = x T {(Gu h ) j (x) a j u h T (x)}λ x. As λ x, and since the triangulation is regular, we get a /2 T (Gu h a T u h ) T (Gu h ) j (x) a j u h T (x) h T. (27) x T We are then reduced to estimate the factor (Gu h ) j (x) a j u h T (x) for all nodes x of T. For that purpose, we distinguish four different cases:

12 ) If x j, then we use an argument similar to the one from Proposition 4.2 of [6] adapted to the DG situation. By the definition of Gu h, we have (Gu h ) j (x) = ω x T ω x T a j u h T (x), because in this case all T ω x are included into j. As a consequence, we obtain and therefore (Gu h ) j (x) a j u h T (x) = ω x (Gu h ) j (x) a j u h T (x) T ω x T a j ( u h T (x) u h T (x)), T ω x a j ( u h T (x) u h T (x)). For each T ω x, there exists a path of triangles of ω x, written T i, i =,, n such that T = T, T n = T, T i T j, i j, T i T i+ is an common edge i =,, n. Hence by the triangle inequality we can estimate n a j ( u h T (x) u h T (x)) a j ( u h Ti+ (x) u h Ti (x)). Now for each term, since a j is symmetric and positive definite, we have i= a j ( u h Ti+ (x) u h Ti (x)) {a j ( u h Ti+ (x) u h Ti (x))} n i + ( u h Ti+ (x) u h Ti (x)) t i, where n i is one fixed unit normal vector along the edge T i T i+ and t i is one fixed unit tangent vector along this edge. All together we have shown that (Gu h ) j (x) a j u h T (x) h T h T e E int :e ω x { [a j u h (x) n] e + [ hu h (x) t] e }. Using a norm equivalence and an inverse inequality we obtain (Gu h ) j (x) a j u h T (x) h T {h /2 e [a j h u h n] e e + h /2 e [u h ] e }. (28) e E int :e ω x 2) If the node x belongs to the boundary of and to the boundary of a unique j, since (Gu h ) j (x) is defined as in the first case, the above arguments lead to (28). 3) If x is a vertex of different sub-domains j, then by Lemma 3.2, we have (Gu h ) j (x) a j u h T (x) h T h T [a h u h (x) n] e, (29) e E int :e ω x 2

13 and therefore as before we conclude that (28) holds. 4) Finally if x belongs to an interface between two subdomains and is not a vertex of them, then it is not difficult to show that (29) holds (due to the regularity of the mesh), and consequently (28) is still valid. Summarizing the different cases, by (27) and (28), we have a /2 T (Gu h a T u h ) T {h /2 e [a h u h n] e e + h /2 e [u h ] e }. (3) e E int :e ω x The first term of this right hand side is a part of the standard residual error estimator and it is by now standard that (using appropriate bubble functions and Green s formula) h /2 e [a u h n] e e a h (u u h ) ωe + osc(f, ω e ), e E int. The second term is part of the DG-norm. Therefore the above estimate in (3) leads to (26). Now using the same arguments than in Proposition 4. of [6], we have Theorem 3.4 For all T T, x N and l, z N l, we have η CF,T a /2 T (u h u) T + a /2 (Gu h a u) T, ρ x a /2 (Gu h a u) ωx + osc(f, ω x ) + osc(g N, ω x ), γ lz a /2 (Gu h a u) ωlz, where osc(g N, ω x ) 2 = e ω x Γ N h e g N Π e g N 2 T, Π e g N being the L 2 (e)-orthogonal projection of g N onto P 2 (e). For the non conforming part of the estimator, we make use of Theorem 2.2 of [9] to directly obtain the Theorem 3.5 Let the assumptions of Theorem 3. be satisfied. For each element T T the following estimate holds η NC,T a /2 T u u h DG,ωT. (3) A direct consequence of these three Theorems is the next local lower bound: Theorem 3.6 Let the assumptions of Theorems 3. and 3.3 be satisfied. For each element T T the following estimate holds η CF,T + η NC;T + η J,T + x T ( ρ x + γ x ) u u h DG,ωT + osc(f, ω T ) + osc(g N, ω T ), where γ x = γ Lx recalling that L is such that N L = N. 3

14 Remark 3.7 Note that the lower bound on the non conforming estimator (see (3)) involves a constant that depends on the aspect ratio of the mesh and of the penalization parameter γ, and is specific to the discontinuous Galerkin method. Consequently it prevents the estimator to be asymptotically exact, as in the continuous Galerkin method [6]. Nevertheless the numerical tests show quite satisfactory effectivity indices (see below). As in Proposition 4.3 of [6], one has γ a /2 (Gu h a u), and therefore a global lower bound can be obtained: Theorem 3.8 Let the assumptions of Theorems 3. and 3.3 be satisfied. Then the following global lower bound holds η CF + η NC + η J + ρ + γ u u h DG,h + osc(f, ) + osc(g N, ). 4 Numerical results 4. The polynomial solution In order to illustrate our theoretical predictions, this first numerical test consists in validating our computations on a simple case, using an uniform refinement process. Let be the square (, ) 2, Γ D =, A = Id and f defined such that the exact solution u is given by : u(x, y) = (x + )(x )(y + )(y ). Let us recall that u h is the finite element solution, e(u h ) = u u h DG,h the error, η(u h ) = ηcf 2 + η2 NC + η2 J the estimator and Gu h the approximated value of a u given by (25). Computations are performed using a global mesh refinement process from a initial cartesian grid. First, it can be seen from Table that the convergence rate of the numerical method is equal to one, as theoretically expected. Then, the superconvergence property of the term Gu h u is actually observed, since the ratio Gu h u /e(u h ) goes towards zero when DoF goes towards infinity. The reliability of the estimator is ensured since the ratio in the last column, the so-called the effectivity index, converges fastly towards the constant The interior and boundary layer solution The following numerical test consists in solving the interior and boundary layer example given in [6]. Let the square (, ) 2, Γ D =, A = Id and f defined such that : u(x, y) = arctan(6(x 2 + y 2 )) 4

15 k DoF e(u h ) Gu h u e(u h ) η(u h ) e(u h ) E- 7.9E E- 5.6E E E E E E-2.83E E-2.29E-.7 Table : The polynomial solution (uniform refinement) Figure : Interior and boundary layer solution. is the exact solution (see Figure ). Let us note that the boundary layer crosses the boundary and that the loading term oscillates across it, what constitutes the difficulty of the computation. This time, an adaptive mesh refinement strategy is used based on the estimator η T = η CF,T + η NC,T + η J,T and the marking procedure η T >.75 max T η T and a standard refinement procedure with a limitation on the minimal angle. Several mesh levels are displayed of Figure 2, to show the capability of the algorithm to track the high gradients regions. Furthermore, quantitative results are displayed on Table 2. Once again, the superconvergence property of Gu h u is observed as well as the reliability of the estimator. 5

16 Figure 2: Mesh levels, and 2, interior and boundary layer solution. k DoF e(u h ) Gu h u e(u h ) η(u h ) e(u h ) E+ 4.57E E+ 7.9E E+.E E+ 8.63E E+ 5.3E E+ 3.98E-.6 Table 2: The interior and boundary layer solution (adaptive refinement). 4.3 The discontinuous case This section is devoted to the case of the discontinuous coefficient a. Namely, the domain = (, ) 2 with Γ D = Γ is decomposed into 4 sub-domains i, i =,..., 4, with = (, ) (, ), 2 = (, ) (, ), 3 = (, ) (, ) and 4 = (, ) (, ). In that case we take discontinuous coefficient a, namely we take a = a i on i, with a = a 3 = and a 2 = a 4 = C to be specified. For this second test, and using usual polar coordinates centered at (, ), the exact solution is equal to the singular function u(x, y) = r α φ(θ), where α (, ) and φ are chosen such that u is harmonic on each sub-domain i, i =,..4, and satisfies the jump conditions : [u] = and [a u.n] = on the interfaces. Non-homogeneous Dirichlet boundary conditions on Γ are fixed accordingly. It is easy to see (see for instance []) that α is the root of the transcendental equation tan α π 4 = /C. Since α <, this solution has a singular behavior around the point (, ). For this test, we also compute the standard ZZ smoothed gradient G u h belonging to S (2) h, = 6

17 { vh C() 2 v h T P 2 (T ), T T } and characterized by its value at each node of the mesh given by : (G u h )(x) = ω x T a T u h T (x). (32) x T Figures 3 and 4 show some of the meshes obtained during the local refinement process. We first can see that the case C = provides a more locally refined mesh around the singularity than the case C = 5. Moreover, tables 3 and 4 display the corresponding quantitative results. These tables show that, even in the case of a singular solution and discontinuous coefficient a, the smoothed gradient is superconvergent, while the effectivity index is around.7 (resp. 2.8) which is quite satisfactory and comparable with results from [9, 5]. Note that the superconvergence of the smoothed gradient and the convergence of the effectivity index require more iterations when C becomes larger. Nevertheless our estimator is robust with respect to C. In contrast we may notice that the standard ZZ smoothed gradient G u h is no more superconvergent to a u, as in the case a =, which is not surprising since the statement of Theorem 3.3 is not valid for G u h if a. k DoF e(u h ) Gu h u e(u h ) G u h u e(u h ) η(u h ) e(u h ) E- 9.8E-.58E E-2 7.2E- 3.7E E E- 3.99E E E- 5.35E E E- 9.35E+.75 Table 3: Discontinuous coefficient a: C = 5, γ = 2, singular solution (local refinement). k DoF e(u h ) Gu h u e(u h ) G u h u e(u h ) η(u h ) e(u h ) E- 9.99E- 2.79E E- 9.93E- 2.88E E- 9.68E- 3.59E E-2 9.4E- 5.3E E E- 9.4E Table 4: Discontinuous coefficient a: C =, γ = 5, singular solution (local refinement). 7

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