A posteriori energy-norm error estimates for advection-diffusion equations approximated by weighted interior penalty methods

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1 A posteriori energy-norm error estimates for advection-diffusion equations approximated by weighted interior penalty methods Alexandre Ern, Annette Stephansen o cite this version: Alexandre Ern, Annette Stephansen. A posteriori energy-norm error estimates for advection-diffusion equations approximated by weighted interior penalty methods. pages <hal > HAL Id: hal Submitted on 3 Sep 007 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. he documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 A posteriori error estimates for weighted interior penalty methods A POSERIORI ENERGY-NORM ERROR ESIMAES FOR ADVECION-DIFFUSION EQUAIONS APPROXIMAED BY WEIGHED INERIOR PENALY MEHODS Alexandre Ern (Cermics, Ecole des Ponts, Parisech, 6 et 8 avenue Blaise Pascal, Champs sur Marne, Marne la Vallée Cedex, France. ern@cermics.enpc.fr) Annette F. Stephansen (Cermics, Ecole des Ponts, Parisech, 6 et 8 avenue Blaise Pascal, Champs sur Marne, Marne la Vallée Cedex, France and Andra, Parc de la Croix-Blanche, -7 rue Jean Monnet, 998 Châtenay-Malabry cedex, France. stephansen@cermics.enpc.fr) Abstract We propose and analyze a posteriori energy-norm error estimates for weighted interior penalty discontinuous Galerkin approximations to advection-diffusion-reaction equations with heterogeneous and anisotropic diffusion. he weights, which play a key role in the analysis, depend on the diffusion tensor and are used to formulate the consistency terms in the discontinuous Galerkin method. he error upper bounds, in which all the constants are specified, consist of three terms: a residual estimator which depends only on the elementwise fluctuation of the discrete solution residual, a diffusive flux estimator where the weights used in the method enter explicitly, and a non-conforming estimator which is nonzero because of the use of discontinuous finite element spaces. he three estimators can be bounded locally by the approximation error. A particular attention is given to the dependency on problem parameters of the constants in the local lower error bounds. For moderate advection, it is shown that full robustness with respect to diffusion heterogeneities is achieved owing to the specific design of the weights in the discontinuous Galerkin method, while diffusion anisotropies remain purely local and impact the constants through the square root of the condition number of the diffusion tensor. For dominant advection, it is shown, in the spirit of previous work by Verfürth on continuous finite elements, that the constants are bounded by the square root of the local Péclet number. Mathematics subject classification: 65N30, 65N5, 76Rxx Key words: Discontinuous Galerkin, weighted interior penalty, a posteriori error estimate, heterogeneous diffusion, advection-diffusion. Introduction In this work, we are interested in a posteriori energy-norm error estimates for a particular class of discontinuous Galerkin (dg) approximations of the advection-diffusion-reaction equation { (K u) + β u + µu = f in Ω, (.) u = 0 on Ω, where for simplicity homogeneous Dirichlet boundary conditions are considered. Here, Ω is a polygonal domain in R d with boundary Ω, µ L (Ω), β [L (Ω)] d with β L (Ω),

3 A. ERN and A. F. SEPHANSEN µ := µ β is assumed to be uniformly positive, the diffusion tensor K is a symmetric, uniformly positive definite field in [L (Ω)] d,d and f L (Ω). Owing to the above assumptions, (.) is well-posed. DG methods received extensive interest in the past decade, in particular because of the flexibility they offer in the construction of approximation spaces using non-matching meshes and variable polynomial degrees. For diffusion problems, various DG methods have been analyzed, including the Symmetric Interior Penalty method [5, 6], the Nonsymmetric method with [34] or without [30] penalty, and the Local Discontinuous Galerkin method [6]; see [4] for a unified analysis. For linear hyperbolic problems (e.g., advection reaction), one of the most common approaches is to use upwind fluxes to formulate the DG method [6, 9]. A unified theory of DG approximations encompassing elliptic and hyperbolic PDE s can be found in [9, 0]. he approximation of the advection-diffusion-reaction problem (.) using DG methods has been analyzed in [5] and more recently in [] with a focus on the high Péclet regime with isotropic and uniform diffusion. he case of high contrasts in the diffusivity poses additional difficulties. Recently, a (Symmetric) Weighted Interior Penalty method has been proposed and analyzed to approximate satisfactorily (.) in this situation [3]. he key idea is to use weighted averages (depending on the normal diffusivities at the two mesh elements sharing a given interface) to formulate the consistency terms and to penalize the jumps of the discrete solution by a factor proportional to the harmonic mean of the neighboring normal diffusivities; the idea of using weighted interior penalties in this context can be traced back to []. he present paper addresses the a posteriori error analysis of the weighted interior penalty method. Many significant advances in the a posteriori error analysis of dg methods have been accomplished in the past few years. For energy-norm estimates, we refer to the pioneering work of Becker, Hansbo and Larson [8] and that of Karakashian and Pascal [7], while further developments can be found in the work of Ainsworth [, 3] regarding robustness with respect to diffusivity and that of Houston, Schötzau and Wihler [4] regarding the hp-analysis; see also [3, 35]. Furthermore, for L -norm estimates, we mention the work of Becker, Hansbo and Stenberg [9], that of Rivière and Wheeler [3], and that of Castillo [5]. Broadly speaking, two approaches can be undertaken to derive a posteriori energy-norm error estimates; in [, 8, 3], a Helmholtz decomposition of the error is used, following a technique introduced in [7, 4], while the analysis in [4, 7] relies more directly on identifying a conforming part in the discrete solution. he analysis presented herein will be closer to the latter approach. We also mention recent work relying on the reconstruction of a diffusive flux; see [, 8]. his paper is organized as follows. presents the discrete setting, including the weighted interior penalty bilinear form used to formulate the discrete problem. 3 contains the main results of this work. he starting point is the abstract framework for a posteriori error estimates presented in 3. and which is closely inspired from the work of Vohralík for mixed finite element discretizations [4]. hen, 3. addresses the case of pure diffusion with heterogeneous and possibly anisotropic diffusivity. We derive an upper bound for the error consisting of three error indicators, i.e. a residual, a diffusive flux and a non-conforming one. his form is similar to that obtained in previous work. he key point however is that the diffusive flux error indicators also provide local lower error bounds that are fully robust with respect to diffusivity heterogeneities and that depend on the local (elementwise) degree of anisotropy; see Propositions 3. and 3.. A key ingredient to obtain this result is the use of weighted averages in writing the consistency term. 3.3 extends the previous analysis to the advection-diffusion-reaction problem. Here, the

4 A posteriori error estimates for weighted interior penalty methods 3 focus is set on achieving a certain degree of robustness in the high Péclet regime, namely that achieved by Verfürth [38] for a posteriori energy-norm error estimates with conforming finite elements and SUPG stabilization. Although these estimates are not independent of the Péclet number (see, e.g., [39] for fully robust estimates with suitable norm modification), their present extension to dg methods constitutes the first results of this type. Finally, numerical results are presented in 4.. he discrete setting Let { h } h>0 be a shape-regular family of affine triangulations covering exactly the polygonal domain Ω. he meshes h may possess hanging nodes, as long as the number of hanging nodes per mesh element is uniformly bounded. A generic element in h is denoted by, h denotes the diameter of and n its outward unit normal. Let an integer p. We consider the usual dg approximation space V h = {v h L (Ω); h,v h P p }, (.) where P p is the set of polynomials of degree less than or equal to p. he L -scalar product and its associated norm on a region R Ω are indicated by the subscript 0,R. For s, a norm (seminorm) with the subscript s,r designates the usual norm (seminorm) in H s (R). For s, H s ( h ) denotes the usual broken Sobolev space on h and for v H ( h ), h v denotes the piecewise gradient of v, that is, h v [L (Ω)] d and for all h, ( h v) = (v ). We say that F is an interior face of the mesh if there are (F) and + (F) in h such that F = (F) + (F). We set (F) = { (F), + (F)} and let n F be the unit normal vector to F pointing from (F) towards + (F). he analysis hereafter does not depend on the arbitrariness of this choice. Similarly, we say that F is a boundary face of the mesh if there is (F) h such that F = (F) Ω. We set (F) = { (F)} and let n F coincide with the outward normal to Ω. All the interior (resp., boundary) faces of the meshes are collected into the set Fh i (resp., F Ω h ) and we let F h = Fh i F Ω h. Henceforth, we shall often deal with functions that are double-valued on Fh i and single-valued on F Ω h. his is the case, for instance, of functions in V h. On interior faces, when the two branches of the function in question, say v, are associated with restrictions to the neighboring elements (F), these branches are denoted by v and the jump of v across F is defined as [[v]] F = v v +. (.) We set [[v]] F = v F on boundary faces. On an interior face F Fh i, we also define the standard (arithmetic) average as {v} F = (v + v + ). he subscript F in the above jumps and averages is omitted if there is no ambiguity. We define the weighted average of a two-valued function v on an interior face F Fh i as where the weights are defined as ω = {v} ω = ω v + ω + v +, (.3) δ K+, ω + δ K =, (.4) δ K+ + δ K δ K+ + δ K with δ K = n F (K )n F. We extend the above definitions to boundary faces by formally letting δ K+ = + so that ω = and ω + = 0. For the standard average, it is instead more

5 4 A. ERN and A. F. SEPHANSEN convenient to set {v} F = v F on boundary faces. On interior faces F Fh i, we will also need the conjugate weighted average defined such that {v} ω = ω + v + ω v +, (.5) and make use of the identity [[vw]] = {v} ω [[w]] + {w} ω [[v]]. he weak formulation of (.) consists of finding u V := H 0(Ω) such that with the bilinear form B(u,v) = (f,v) 0,Ω, v V, (.6) B(v,w) = (K h v, h w) 0,Ω + (β h v,w) 0,Ω + (µv,w) 0,Ω. (.7) Piecewise gradients are used so as to extend the domain of B to functions in V + V h. he energy norm is v B = v B,, v B, = (K h v, h v) 0, + ( µv,v) 0,. (.8) he discrete problem consists of finding u h V h such that with the bilinear form B h (u h,v h ) = (f,v h ) 0,Ω, v h V h, (.9) B h (v,w) = (K h v, h w) 0,Ω + ((µ β)v,w) 0,Ω (v,β h w) 0,Ω + F F h [(γ F [[v]],[[w]]) 0,F (n t F {K h v} ω,[[w]]) 0,F θ(n t F {K h w} ω,[[v]]) 0,F ] + F F h (β n F {v},[[w]]) 0,F. (.0) he penalty parameter γ F is defined for all F F h as γ F = γ K,F + γ β,f with γ K,F = h F δ F, γ β,f = β n F, (.) where δ F = δ K+δ K δ K+ + δ K, (.) and is a positive parameter ( can also vary from face to face). Note that by the above convention, γ K,F = h F δ K on boundary faces. Finally, the parameter θ can take values in {,0,+}. he particular value taken by θ plays no role in the subsequent analysis. o avoid technicalities, the diffusion tensor K is assumed to be piecewise constant on h and its restriction to an element h is denoted by K. We will indicate by λ m, and λ M, respectively the minimum and the maximum eigenvalue of K on. he minimum value of µ on is indicated by µ m,. he degree of diffusion anisotropy on an element is evaluated by the condition number of K, namely = λm, λ m,.

6 A posteriori error estimates for weighted interior penalty methods Abstract setting 3. A posteriori error analysis In this section we present the basic abstract framework for our a posteriori error estimates. he following result is directly inspired from the abstract framework introduced by Vohralík [4]. Lemma 3.. Let Z and Z h be two vector spaces. Let A be a bounded bilinear form defined on Z Z with Z := Z + Z h. Assume that A can be decomposed into the form A = A S + A SS where A S is symmetric and nonnegative on Z and where A SS is skew-symmetric on Z (but not necessarily on Z ). hen, defining the semi-norm := A S (, ) /, the following holds for all u,s Z and u h Z h, where φ = u s u s. u u h s u h + A(u u h,φ) + A SS (u h s,φ), (3.) Proof. Suppose first that u s u u h. hen, u u h = A(u u h,u u h ) A SS (u u h,u u h ) = A(u u h,u s) + A(u u h,s u h ) A SS (u u h,u u h ) = A(u u h,u s) + A S (u u h,s u h ) + A SS (u u h,s u h ) A SS (u u h,u u h ) = A(u u h,u s) + A S (u u h,s u h ) + A SS (u u h,s u) = A(u u h,u s) + A S (u u h,s u h ) + A SS (u h s,u s), where we have used A SS (u s,u s) = 0 since (u s) Z. Introducing φ yields u u h u s A(u u h,φ) + u u h s u h + u s A SS (u h s,φ). (3.) Having hypothesized that u s u u h, we infer u u h s u h + A(u u h,φ) + A SS (u h s,φ). (3.3) Consider now the case u u h u s. Since A SS (u s,u s) = 0, hus u s = A(u s,u s) = A(u u h,u s) + A S (u h s,u s) + A SS (u h s,u s) u s A(u u h,φ) + u h s u s + u s A SS (u h s,φ). u u h u s A(u u h,φ) + s u h + A SS (u h s,φ). (3.4) Combining the results we obtain (3.). 3.. Pure diffusion Let β = 0 and µ = 0 in (.), i.e., we consider a diffusion problem with anisotropic and heterogeneous diffusivity: { (K u) = f in Ω, (3.5) u = 0 on Ω.

7 6 A. ERN and A. F. SEPHANSEN he bilinear form B defined by (.7) becomes B(v,w) = (K h v, h w) 0,Ω, (3.6) while the definition of the (semi-)norm B involves only the diffusive contribution, i.e., v B, = (K hv, h v) 0,. he discrete problem is still (.9) with bilinear form B h defined by B h (v,w) = (K h v, h w) 0,Ω + F F h [(γ K,F [[v]],[[w]]) 0,F (n t F {K h v} ω,[[w]]) 0,F (n t F {K h w} ω,[[v]]) 0,F ]. (3.7) Lemma 3. can be applied by letting Z := V, Z h := V h, A = A S := B and A SS := 0. he semi-norm coincides with B. his yields u u h B inf u h s B + s V sup φ V, φ B= B(u u h,φ). (3.8) We now proceed to estimate the second term in the right-hand side of (3.8). Let Π h : L (Ω) V h denote the L -orthogonal projection onto V h. It is well-known that for v L (Ω), Π h v coincides on each mesh element with the mean value of v on the corresponding element. he projector Π h satisfies the following approximation properties: For all h and for all φ H (), φ Π h φ 0, C p h φ, (3.9) φ Π h φ 0, C h φ 0,. (3.0) he constant C p in the Poincaré-type inequality (3.9) can be bounded for each convex by d/π, see [7, 3], while it follows from [40] that the constant C in the trace inequality (3.0) is given by C = 3dρ with ρ = h / where denotes the (d )-measure of and the d-measure of ; note that ρ is uniformly bounded owing to the shape-regularity of the mesh family. For all h, define on the volumetric residual and on the boundary residual such that for F, where R(u h ) = f + h (K h u h ), (3.) J K (u h ) F = ω,f n t [[K h u h ]] + γ K,F [[u h ]], (3.) n t F ω,f = K n F n t F K n F + n t F K n, (3.3) F with F =. Note that the convention regarding δ K+ yields ω,f = 0 on boundary faces. Lemma 3.. he following holds: sup B(u u h,φ) φ V, φ B= ( (η + ζ ) ), (3.4)

8 A posteriori error estimates for weighted interior penalty methods 7 where the residual error indicator η is and the diffusive flux error indicator is η = C p h λ m, (I Π h)r(u h ), (3.5) ζ = C h λ m, J K(u h ) 0,. (3.6) Proof. Let φ V such that φ B =. Using B(u,φ) = (f,φ) 0,Ω and integrating by parts we obtain B(u u h,φ) = (f + h (K h u h ),φ) 0, (n t F[[K h u h ]],φ) 0,F F F i h since φ V = H 0(Ω). esting the discrete equations with Π h φ yields F F h (γ K,F [[u h ]] n t F {K h u h } ω,[[π h φ]]) 0,F = (f,π h φ) 0,Ω. Observe that ( h (K h u h ),Π h φ) 0, = (n t F {K h u h } ω,[[π h φ]]) 0,F + (n t F[[K h u h ]], {Π h φ} ω ) 0,F. F F h F F i h Combining the above equations and using [[φ]] = 0 leads to B(u u h,φ) = (f + h (K h u h ),φ Π h φ) 0, (γ K,F [[u h ]],[[φ Π h φ]]) 0,F F F h (n t F[[K h u h ]], {φ Π h φ} ω ) 0,F = F F i h (R(u h ),φ Π h φ) 0, F n n F (J K (u h ),φ Π h φ ) 0,F. he conclusion is straightforward using (3.9) (3.0) and the fact that Π h (R(u h )) and (φ Π h φ) are L -orthogonal on each h. Remark 3.. aking off the mean value of R(u h ) in the residual error estimator is possible because the discrete space contains piecewise constant functions. his is a feature of dg approximations, but not, for instance, of continuous finite element approximations. heorem 3.. Pick any s h V and define the non-conforming error indicator ι as hen, the following holds u u h B ι = u h s h B,. (3.7) ( (η + ζ ) ) + ( ι ). (3.8)

9 8 A. ERN and A. F. SEPHANSEN Proof. Direct consequence of Lemma 3. and of (3.8). We now investigate the local efficiency of the above error indicators η, ζ and ι. Here, x y indicates the inequality x cy with positive c independent of the mesh and of the diffusion tensor. o simplify, the data f is assumed to be a polynomial; otherwise, the usual data oscillation term has to be added to the estimates. he following two propositions establish that the error indicators η and ζ are fully robust with respect to heterogeneities in the diffusion tensor, while the dependency on anisotropies remains local, i.e., only the square root of the condition numbers on and neighboring elements appears in the local lower bounds, but not the ratios of two diffusion tensor eigenvalues from different elements. Proposition 3.. For all h, η u u h B,. (3.9) Proof. Since (I Π h )R(u h ) 0, R(u h ) 0,, we simply bound R(u h ) 0,. o this purpose, we use the technique of element bubble functions introduced by Verfürth [36, 37]; the arguments, which are fairly standard, are only briefly sketched. Let h, let b be a suitable local bubble function in vanishing on and set ν = b R(u h ). hen, R(u h ) 0, (R(u h ),ν ) 0, = (K h (u u h ), ν ) 0, λ M, h u u h B, R(u h ) 0,, from which follows (3.9). Proposition 3.. For all h, ζ N where N is the set of elements sharing a face with the element. Proof. Let h. Observe that ζ λ m, F δ F h F [[u h]] F + λ u u h B,, (3.0) m, h F ω,f [[K h u h ]] F X + Y, and let us bound X and Y. (i) Bound on X. Let F. We use the result obtained by Achdou, Bernardi and Coquel []: to infer X λ m, = F (F) h F [[u h]] 0,F (λ M, λ F (F) (F) δ F h (u u h ) 0, M, δ F) h (u u h ) 0,, (3.) u u h B, N u u h B,,

10 A posteriori error estimates for weighted interior penalty methods 9 since δ F min(n F K n F,n F K n F ), n F K n F λ M, and n F K n F λ M,. (ii) Bound on Y. Let F. Using the technique of edge bubble functions introduced by Verfürth [36, 37], it is shown that Hence, Y λ m, since F ω,f h F nt F[[K h u h ]] 0,F (F) λ M, u u h B, (λ M, λ m, ω,f) F (F) he proof is complete. λ M, λ (F) u u h B, λ M, u u h B,. F (F) m, ω,f (n FK n F ) (n F K n F ) (n F K n F ) + (n F K n F ). u u h B,, o analyze the local efficiency of the non-conforming error indicator ι, a particular choice must be made for s h V. Presently, one of the state-of-the-art approaches consists in considering the so-called Oswald interpolate of the discrete solution u h. For v h V h, its Oswald interpolate I Os (v h ) V h V is defined by prescribing its values at the usual Lagrange interpolation nodes on each mesh element by taking the average of the values of v h at the node, I Os (v h )(s) = s s v h (s), (3.) where s is the set of mesh elements that contain the node s and where s denotes the cardinal of that set. On boundary nodes, I Os (v h )(s) is set to zero. he Oswald interpolation operator I Os yields the following local approximation properties [, 7]: For all v h V h and for all h, v h I Os (v h ) 0, C h F [[v h ]] 0,F, (3.3) h (v h I Os (v h )) 0, C F F h,f F F h,f h F [[v h]] 0,F, (3.4) where the constant C depends on the space dimension, the polynomial degree p used to construct the space V h, and the shape-regularity parameter associated with the mesh h ; the dependency of the constant C on p has been recently explored in []. Setting s h := I Os (u h ) to evaluate ι, it is inferred using (3.) and (3.4) that ι λ M, R λ m, u u h B,, (3.5) where R = { h ; }. Clearly, the above estimate is not robust with respect to heterogeneities and/or anisotropies in the diffusion tensor. In the isotropic case, the result

11 0 A. ERN and A. F. SEPHANSEN can be improved by using weighted averages in (3.) to define the nodal values of the Oswald interpolate. he weights depend on the diffusivity and a robust bound can be inferred on ι when evaluated with this modified Oswald interpolate provided a monotonicity property of the diffusivity around vertices is assumed to hold; see [, 0, 8]. o the authors knowledge, no fully satisfactory result on a modified Oswald interpolation operator is yet available in the case of anisotropic diffusivity. We will not explore this issue further here. Finally, we point out that the local efficiency of the error indicator ι has to be weighted against the computational costs required for its evaluation. Indeed, since any reconstructed function s h V can be chosen to evaluate it and since inf u h s B, u h u B,, (3.6) s V the local efficiency properties of ι can be improved at the expense of solving more detailed local problems. Developments along this line go beyond the present scope. Remark 3.. Using a triangle inequality, the flux error indicator ζ can be split into two contributions, one associated with the jump of the diffusive flux and the other associated with the jump of the discrete solution itself, and the latter can be regrouped with the non-conforming error indicator ι. Both contributions are locally efficient and fully robust with respect to heterogeneities in the diffusivity, as shown in the proof of Proposition 3. where the quantities X and Y are bounded separately. By proceeding this way, the error upper bound is somewhat less sharp because a triangle inequality has been used, but the final form of the a posteriori error estimate takes a more familiar form Advection-diffusion-reaction In this section we turn to the general case of an advection-diffusion-reaction problem. Our purpose is to extend the a posteriori error indicators derived in Lemma 3. and in heorem 3. to this situation, with a particular emphasis on the robustness of the estimates in the high- Péclet regime in the sense of Verfürth [38]. he starting point is again the abstract estimate derived in Lemma 3. which is now applied with Z := V, Z h := V h, A S (v,w) = (K h v, h w) 0,Ω + ( µv,w) 0,Ω, (3.7) A SS (v,w) = (β h v,w) 0,Ω + (( β)v,w) 0,Ω, (3.8) and A = A S + A SS = B as defined by (.7). Observe that A S is symmetric and nonnegative on Z + Z h, that coincides with B, and that A SS is skew-symmetric on Z (but not on Z + Z h ). As a first step, we rewrite the quantity B(u u h,φ) + A SS (u h s,φ) in a more convenient form. Lemma 3.3. Let s V. For all h, define on the volumetric residual R(u h ) = f + h (K h u h ) β h u h µu h, (3.9) let J K (u h ) be defined on by (3.), and let J β (u h s) be defined such that for F, J β (u h s) F = γ β [[u h s]] + β n F {u h s} F, (3.30) where F denotes the mean value over F. hen, for all φ V, B(u u h,φ) + A SS (u h s,φ) = X + X + X 3, (3.3)

12 A posteriori error estimates for weighted interior penalty methods with X = ((I Π h )R(u h ),φ Π h φ) 0,, (3.3) X = n n F (J K (u h ),φ Π h φ ) 0,F, (3.33) X 3 = F [((I Π h )(β h (u h s)),φ Π h φ) 0, + ( β(u h s),φ Π h φ) 0, ] + F F h (J β (u h s),[[π h φ]]) 0,F. (3.34) Proof. Let φ V. Using B(u,φ) = (f,φ) 0,Ω and integrating by parts, we infer B(u u h,φ) = (R(u h ),φ) 0, esting the discrete equations with Π h φ yields (n t F[[K h u h ]],φ) 0,F. F F i h F F h (γ[[u h ]] n t F {K h u h } ω + β n F {u h },[[Π h φ]]) 0,F + ((µ β)u h,π h φ) 0,Ω = (f,π h φ) 0,Ω. Combining the two above equations and proceeding as in the proof of Lemma 3. for the diffusive term leads to B(u u h,φ) = X + X + Using the relation F F h (γ β [[u h ]],[[Π h φ]]) 0,F F F h (β n F [[u h ]], {Π h φ}) 0,F. (( β)(u h s),π h φ) 0, (β h (u h s),π h φ) 0, + (β n F [[u h ]], {Π h φ}) 0,F + (β n F {u h s},[[π h φ]]) 0,F = 0, F F h F F h and adding A SS (u h s,φ) as evaluated from (3.8), (3.3) is inferred. Note that the upwind related term J β (u h s) can be evaluated as a mean value over each face because it is tested against a piecewise constant function and that the mean value of β h (u h s) can be taken off on each element because it is tested against φ Π h φ. Remark 3.3. he idea of evaluating the upwind related term as a mean value over each face has been proposed by Vohralík [4]. Since for any function ψ L (F), ψ F 0,F ψ 0,F, this modification can only sharpen the a posteriori error estimate. he next step is to control φ Π h φ for φ V in terms of the energy norm φ B. o obtain bounds that behave satisfactorily when the Péclet number is large, a sharper version of inequalities (3.9) (3.0) needs to be used. Observing that on all h, φ Π h φ 0, φ 0, and letting ( m = min C p h λ m,, µ m, ), (3.35)

13 A. ERN and A. F. SEPHANSEN the bound (3.9) can be sharpened as follows: φ Π h φ 0, m φ B,. (3.36) Furthermore, owing to the trace inequality v H (), v 0, C [h v 0, + v 0, v 0, ], (3.37) where the constant C depends on the space dimension, the polynomial degree p, and the shape-regularity of the mesh h, (3.0) can be sharpened as follows: φ Π h φ 0, C [h m + λ 4 where we have set C m, m ] φ B, C λ 4 m, m φ B,, (3.38) = C ( + C 4 p ). (3.39) Estimate (3.38) will be used to bound the term X introduced in Lemma 3.3. However, this estimate turns out not be sharp enough when bounding the last term in X 3. In this case, we will use the trace inequality φ h V h, φ h 0, ρ h φ h 0,, (3.40) and we define for all F F h, ( ) m F = min max (C h λ m, ), max (ρ h (F) (F) µ m, ). (3.4) Finally, let κ µ,β, = β L () µ m,. Lemma 3.4. Let s V. he following holds sup B(u u h,φ) + A(u h s,φ) φ V φ B = where the residual error indicator η is the diffusive flux error indicator ζ is ( (η + ζ + ι ) ), (3.4) η = m (I Π h )R(u h ), (3.43) ζ = C λ 4 and the non-conforming error indicator ι is m, m ι = m (I Π h )(β h (u h s)) 0, + κ µ,β, u h s 0, + J K(u h ) 0,, (3.44) F m F J β (u h s) 0,F. (3.45)

14 A posteriori error estimates for weighted interior penalty methods 3 Proof. Let φ V such that φ B =. We bound the three terms X, X and X 3 introduced in Lemma 3.3. Owing to (3.36) and (3.38), it is clear that X + X (η + ζ ) φ B,. Decompose X 3 into X 3 = X 3, +X 3, where X 3, denotes the sum over elements and where X 3, denotes the sum over faces. Observing that φ Π h φ 0, = φ 0, and using again (3.36), we obtain X 3, (m (I Π h )(β h (u h s)) 0, + κ µ,β, u h s 0, ) φ B,. o bound X 3,, let F F h. On the one hand, owing to (3.0), (J β (u h s),[[π h φ]]) 0,F = (J β (u h s),[[π h φ φ]]) 0,F (J β (u h s),π h φ φ) 0,F (F) On the other hand, owing to (3.40), (J β (u h s),[[π h φ]]) 0,F Hence, and therefore, J β (u h s) 0,F max (C (F) (F) (J β (u h s),π h φ ) 0,F J β (u h s) 0,F max ( C (F) (J β (u h s),[[π h φ]]) 0,F m F J β (u h s) 0,F X 3, he conclusion is straightforward. ( F h λ m, ) h µ m, ) (F) m F J β (u h s) 0,F ) φ B,. (F) (F) φ B,, φ B,. φ B,. heorem 3.. Pick any s h V and define the non-conforming error indicator ι as ι = u h s h B,, (3.46) and let ι be evaluated from (3.45) using s h. hen, u u h B ( Proof. Apply Lemmata 3. and 3.4. (η + ζ + ι ) ) + ( (ι ) ). (3.47)

15 4 A. ERN and A. F. SEPHANSEN Remark 3.4. he non-conforming error indicators ι and ι non-conforming error indicator ι by setting can be regrouped into a single ι = 4(ι ) + (ι ). (3.48) hen, (3.47) becomes u u h B ( (η + ζ ) which is less sharp but has a more familiar form. ) + ( ι ), (3.49) We now investigate the local efficiency of the above error indicators η, ζ and ι. Here, x y indicates the inequality x cy with positive c independent of the mesh and of the parameters K, β, and µ. Again, the data f is assumed to be a polynomial; otherwise, the usual data oscillation term has to be added to the estimates. As in the pure diffusion case, we will not take advantage of the presence of the operator (I Π h ) in η and in the first term of ι to derive the bounds below. Proposition 3.3. For all h, where η m [λ M, h + min(α,,α, )] u u h B,, (3.50) α, = µ L () + β L (), α, = µ β L () + β L ()h. µ λ µ m, m, Proof. Let h, let b be a suitable local bubble function in vanishing on and set ν = b R(u h ). hen, m, R(u h ) 0, (R(u h ),ν ) 0, = (K h (u u h ), h ν ) 0, + (µ(u u h ),ν ) 0, + (β h (u u h ),ν ) 0, λ M, h u u h B, R(u h ) 0, + min (α,,α, ) u u h B, R(u h ) 0,, where the min is obtained by integrating by parts or not the advective derivative. he conclusion is straightforward. Proposition 3.4. For all h, ζ λ 4 m, m N ( + m α, )m λ 4 m, u u h B,. (3.5) Proof. Let h. Observe that ζ λ 4 m, m F δ F h F [[u h]] F + λ 4 m, m F ω,f n t F[[K h u h ]] F X + Y,

16 A posteriori error estimates for weighted interior penalty methods 5 and let us bound X and Y by the right-hand side of (3.5). (i) Bound on X. Owing to (3.) and the definition of δ F, X λ 4 m, m F (F) δ F h F λ m, u u h B, m λ 4 m, (λ M, λ F (F) m λ 4 m, N h M, δ F) u u h B,, h F u u h B, since λ M, λ M, δ F. Owing to the obvious bound h m λ 4, it is inferred that X m, is bounded by the right-hand side of (3.5). (ii) Bound on Y. Let F. Following the ideas of Verfürth [38], let b F be a suitable bubble function with support in F and let l F be the lifting of (n t F [[K hu h ]])b F in (F) with cut-off parameter on each (F). hen, Observe that and that θ = m C p h n t F[[K h u h ]] 0,F (n t F[[K h u h ]],l F ) 0,F, B(u u h,l F ) Furthermore, since l F 0, h θ λ m,, nt F[[K h u h ]] 0,F m λ 4 m, nt F[[K h u h ]] 0,F, l F 0, h θ nt F[[K h u h ]] 0,F m λ 4 m, nt F[[K h u h ]] 0,F. B(u u h,l F ) = (R(u h ),l F ) 0, (F) + (n t F[[K h u h ]],l F ) 0,F, (F) (R(u h ),l F ) 0, (F) and since h 4 B(u u h,l F ), whence m λ (λ M, m λ 4 (F) (F) (F) m, m λ 4 n t F[[K h u h ]] 0,F m, + m λ 4 R(u h ) 0, l F 0, [λ M, h [λ M, h m, α, ) u u h B, n t F[[K h u h ]] 0,F. + min(α,,α, )] u u h B, l F 0, + α, ]m λ 4 m, u u h B, n t F[[K h u h ]] 0,F, m,, it is inferred that (R(u h),l F ) 0, (F) can be bounded as (F) (λ M, m λ 4 m, + m λ 4 m, α, ) u u h B,.

17 6 A. ERN and A. F. SEPHANSEN As a result, Y λ 4 m, m F (F) ω,f (λ M, m λ 4 λ 4 m, m (λ M, λ F (F) λ 4 m, m ( N he conclusion is straightforward. + m α, )m m, + m λ 4 m, α, ) u u h B, m, ω,f)( + m α, )m λ 4 m, u u h B,. λ 4 m, u u h B, Finally, we investigate the local efficiency of the non-conforming error estimator ι. o this purpose, we pick s h = I Os (u h ). As discussed at the end of 3., a modified Oswald interpolation operator can be considered in the case of isotropic and heterogeneous diffusivity with a monotonicity property around vertices to sharpen the result. Proposition 3.5. Set s h = I Os (u h ). Let h. hen, ( ι λ M, + h µ L () + m β L () + h κ µ,β, + λ R F m F β L (F)h m, u u h B,. (3.5) Proof. Let h. Observe first that using (3.3) (3.4), u h s h B, (λ M, + h µ L () ) λ R m, u u h B,, where R = { h ; }. Furthermore, still using (3.3) (3.4), the first two terms in ι (see (3.45)) are bounded by (m β L () + h κ µ,β, ) λ R m, u u h B,, and it remains to bound the last term, namely F m F J β (u h s h ) 0,F. For F, it can be shown that for all v h V h, v h I Os (v h ) 0,F [[v h ]] 0,F. F F h,f F Applying this estimate with v h := u h, the conclusion is straightforward. o illustrate by a simple example, assume that β and µ are of order unity, that β is solenoidal (or that its divergence is uniformly bounded by µ locally), and that the diffusion is homogeneous and isotropic, i.e., K = ǫi d with real parameter 0 < ǫ and where I d denotes the identity matrix in R d. hen, m = min(h ǫ,), α, = + ǫ, α, = + h, and it is readily verified that all the constants appearing in the upper bounds for η, ζ, and ι are of the form ( + ǫ min(h ǫ,)), which corresponds to the result derived in [38] for continuous finite elements with vanishing, isotropic, and homogeneous diffusion. F )

18 A posteriori error estimates for weighted interior penalty methods 7 4. Numerical results In this section, the present a posteriori error estimators are assessed on two test cases. he first one is a pure diffusion problem with heterogeneous isotropic diffusion; its aim is to verify numerically the sharpness of the diffusion flux error indicator ζ when evaluated with the proper weights. he second test case is an advection diffusion-reaction problem with homogeneous diffusion; its aim is to verify the behavior of the a posteriori error estimates in the low- and high-péclet regimes. We have always taken = 4 and θ = in (.) and (.0), respectively. he corresponding dg method is the so-called Symmetric Weighted Interior Penalty method analyzed recently in [3]. Moreover, we have set p =, i.e., used piecewise linears. In all cases, the non-conforming error indicators have been evaluated using the standard Oswald interpolate of the discrete solution; see (3.). 4.. Heterogeneous diffusion We consider the following test problem proposed in [33]. he domain Ω = (, ) (, ) is split into four subregions: Ω = (0,) (0,), Ω = (,0) (0,), Ω 3 = (,0) (,0), and Ω 4 = (0,) (,0). he source term f is zero. he diffusion tensor is isotropic, i.e., of the form K = ǫ i I with constant value within each subregion. Letting ǫ = ǫ 3 = 00 and ǫ = ǫ 4 =, the exact solution written in polar coordinates is with α = and u Ωi = r α (a i sin(αθ) + b i cos(αθ)), (4.) a = b = , a = b = , a 3 = b 3 = , a 4 = b 4 = Non-homogeneous Dirichlet boundary conditions as given by (4.) are enforced on Ω. he exact solution possesses a singularity at the origin, and its regularity depends on the constant α, namely u H α (Ω). he expected convergence order of the error in the L -norm is α, while the expected convergence order in the energy norm is α. able 4. presents the results on a series of quasi-uniform unstructured triangulations (that are compatible with the above partition of the domain Ω). he last line of this table displays the convergence orders evaluated on the last two meshes. he convergence orders for the error both in the L -norm and in the energy norm are in good agreement with the theoretical predictions. he same conclusion is reached for the a posteriori error estimators based on ζ and ι (observe that in the present case, η = 0 because f = 0 and p = ). Note that u h s h B is actually lower than the actual error norm u u h B, which indicates that although the lower bound (3.6) can be invoked to guarantee the efficiency of the non-conforming error estimators, there may be functions in V V h (here the Oswald interpolate of the discrete solution) that are actually closer to the discrete solution than is the exact solution. Furthermore, the column labelled est in able 4. reports the total a posteriori error estimator derived in heorem 3., and the column labelled eff reports the efficiency of the estimator, namely the ratio of the a posteriori error estimator to the actual approximation error. he efficiency is about 4 on all meshes. Notice that all

19 8 A. ERN and A. F. SEPHANSEN able 4.: Heterogeneous diffusion with parameter α = 0.3 h u u h 0,Ω u u h B ( ζ ) ( ι ) est. eff. 9.43e- 4.99e e- 4.4e e- 3.63e order able 4.: Heterogeneous diffusion with parameter α = 0.54 h u u h 0,Ω u u h B ( ζ ) ( ι ) est. eff. 9.43e-.35e-3.06e e e- 8.9e-4 8.9e e e-.95e-4 6.7e e order the constants in the estimators are explicitly evaluated. It is interesting to compare the results of able 4. to those obtained using the more conventional dg method based on arithmetic averages (i.e., weights equal to on all faces) and a penalty term γ K,F equal to the arithmetic mean of the normal diffusivities on each face. In this case, the efficiency is equal to 8, i.e., 7 times larger. We have also examined a similar test case with a less singular solution corresponding to milder contrasts in the diffusion, namely ǫ = ǫ 3 = 5 and ǫ = ǫ 4 =. In this case, the exact solution is still given by (4.) with α = and a = b = , a = b = , a 3 = b 3 = , a 4 = b 4 = able 4. presents the results. he conclusions are similar to those reached with the previous test case. he efficiency is between 5 and 6 on all meshes, and thus takes comparable values to those taken in the previous test case, confirming the robustness of the estimates with respect to diffusion heterogeneities. If the more conventional dg method with arithmetic averages is used instead, the efficiencies are about 7, hinting at a dependency on diffusion heterogeneities. 4.. Advection-diffusion-reaction Consider the domain Ω = (0,) (0,), the advection field β = (,0) t, the reaction coefficient µ =, and an isotropic homogeneous diffusion tensor K = ǫi. We run tests with ǫ = and ǫ = 0 4 to examine the difference between dominant diffusion and dominant advection regimes. Since the diffusion is homogeneous and isotropic, the SWIP method coincides with the more conventional Interior Penalty dg method. he source term f is designed so that the exact solution is ( u(x,y) = 0.5 tanh ( 0.5 x γ )). (4.)

20 A posteriori error estimates for weighted interior penalty methods 9 able 4.3: Advection-diffusion with ǫ = h u u h B ( η ) ( ζ ) ( ι ) ( ι ) est. eff. 9.43e- 7.47e- 3.45e-0.37e e- 9.83e e- 4.04e-.74e-0.05e-.79e- 7.07e e-.05e- 8.69e- 3.83e- 3.64e e order able 4.4: Advection-diffusion with ǫ = e-4 h u u h B ( η ) ( ζ ) ( ι ) ( ι ) est. eff. 9.43e-.57e- 6.6e- 8.46e-3.93e e-3.55e e- 9.34e e- 7.64e-3.9e e-3.70e e-.96e-3.08e- 5.e-3 6.6e-.40e e- 83 order Here, the parameter γ = 0.05 controls the thickness of the internal layer at x = 0.5. On the left and right boundaries of Ω (x = 0 and x = ), non-homogeneous Dirichlet boundary conditions as given by (4.) are enforced, while on the lower and upper boundaries (y = 0 and y = ), homogeneous Neumann conditions are enforced. In able 4.3 we present the results for the dominant diffusion regime. he estimator and the error converge at the same order, and the global efficiency is comparable with that obtained for a pure diffusion problem. he dominant contributions to the total a posteriori error estimate are the residue and the diffusive flux error indicators. When the advection becomes dominant, the error u u h B converges at.5 (because it is dominated by the L -contribution), while the total a posteriori error estimate (see column labelled est ) maintains the order of convergence equal to one, as can be seen in able 4.4. his is because the cut-off coefficients m and the like are equal to one with dominant advection. As a result, the global efficiency increases (roughly as h ) as the mesh is refined. he trend will only be reversed once the mesh is sufficiently fine to resolve the diffusion. We notice that the dominant error indicators here are the non-conforming error indicator ι and the residue η, as expected. 5. Conclusions In this work, we have proposed and analyzed a posteriori energy-norm error estimates for weighted interior penalty dg approximations to advection-diffusion-reaction equations with heterogeneous and anisotropic diffusion. All the constants in the error upper bounds have been specified, so that the present estimates can be used for actual control over the error in practical simulations. Local lower error bounds in which all the dependencies on model parameters are explicitly stated, have been derived as well. In the case of pure diffusion, full robustness is achieved with respect to diffusion heterogeneities owing to the use of suitable diffusiondependent weights to formulate the consistency terms in the dg method. his feature has been verified numerically and stands in contrast to the results obtained with more conventional interior penalty dg approximations. Furthermore, diffusion anisotropies enter the lower error bounds only through the square root of the condition number of the diffusion tensor on a given mesh cell and its neighbors. he current state-of-the-art available results have been used to

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