MIXED FINITE ELEMENT METHODS FOR PROBLEMS WITH ROBIN BOUNDARY CONDITIONS

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1 MIXED FINITE ELEMENT METHODS FOR PROBLEMS WITH ROBIN BOUNDARY CONDITIONS JUHO KÖNNÖ, DOMINIK SCHÖTZAU, AND ROLF STENBERG Abstract. We derive new a-priori and a-posteriori error estimates for mixed nite element discretizations of second-order elliptic problems with general Robin boundary conditions, parameterized by a non-negative and piecewise constant function ε. The estimates are robust over several orders of magnitude of ε, ranging from pure Dirichlet conditions to pure Neumann conditions. A series of numerical experiments is presented that verify our theoretical results.. Introduction. We consider the dual mixed nite element method for second order elliptic equations subject to general Robin boundary conditions. These we parameterize by ε, with natural Dirichlet conditions for ε = and essential Neumann conditions in the limit ε. For the mixed method the Neumann condition is an essential condition and could be explicitly enforced. However, we prefer to see the method implemented in the same way for all possible boundary conditions and then the Neumann condition is obtained by penalization, i.e. choosing ε "large". Let us recall that the situation for a primal (displacement) nite element method is the opposite, Neumann conditions are natural and Dirichlet essential, and the latter are penalized by choosing ε "small". For this case it is well known that the problem becomes ill-conditioned in two ways. The error estimates are not independent of ε and the stiness matrix becomes ill-conditioned. We here remark that in [8] Nitsche's method was extended to general Robin boundary conditions yielding a primal formulation avoiding ill-conditioning. The following question naturally arises now. Is the mixed method ill-conditioned near the Neumann limit? In this paper we will show that this is not the case. We will prove both a priori and a posteriori error estimates that are uniformly valid independent of the parameter ε. We also show that the stiness matrix is wellconditioned. It seems that this has not earlier been reported in the literature. Robin conditions are treated in [], but the robustness with respect to the parameter was not studied. The outline of the paper is the following. In the next section we recall the mixed nite element method. In Section 3 we derive a-priori error estimates and prove an optimal bound for the error in the ux. In Section 4 we analyze the postprocessing method of [5, 4] which enhances the accuracy of the displacement variable. In Section 5, we introduce a residual-based a-posteriori error estimator and establish its reliability and eciency. In Section 6 we consider the solution of the problem by hybridization and show that this leads to a well-conditioned linear system. A set of numerical examples are presented in Section 7 that verify the ε-robustness of our estimates. Finally, we end the paper with some concluding remarks in Section 8. Throughout the paper, we use standard notation. We denote by C, C, C etc. generic constants that are not necessarily identical at dierent places, but are always Institute of Mathematics, Helsinki University of Technology, P.O. Box, FIN-5 TKK, Finland (jkonno@math.tkk.fi). Mathematics Department, University of British Columbia, Vancouver, BC, V6T Z, Canada (schoetzau@math.ubc.ca) Institute of Mathematics, Helsinki University of Technology, P.O. Box, FIN-5 TKK, Finland (rstenber@cc.hut.fi).

2 independent of ε and the mesh size.. Mixed nite element methods. In this section, we introduce two families of mixed nite element methods for the mixed form of Poisson's equation with Robin boundary conditions... Model problem. We consider the following model problem: subject to the general Robin boundary conditions σ u = in Ω, (.) div σ + f = in Ω, (.) εσ n = u u + εg on Ω. (.3) Here, Ω R n, n =, 3, is a bounded polygonal or polyhedral Lipschitz domain, f a given load, and u and g are prescribed data on the boundary of Ω. The vector n denotes the unit outward normal vector on Ω. The boundary conditions (.3) are parameterized by the non-negative function ε. For simplicity, we assume ε to be piecewise constant on the boundary (with respect to the partition of Ω induced by a triangulation of Ω). In the limiting case ε =, we obtain the Dirichlet boundary conditions u = u on Ω. (.4) On the other hand, if ε everywhere on Ω, we recover the Neumann boundary conditions σ n = g on Ω. (.5) To cast (.)(.) in weak form, we rst note that (σ, u) satises (σ, τ ) + (div τ, u) u, τ n Ω = τ H(div, Ω), (.6) (div σ, v) + (f, v) = v L (Ω). (.7) Then we solve for u in the expression (.3) for the boundary conditions and insert the result into (.6). We nd that a ε (σ, τ ) + (div τ, u) = u + εg, τ n Ω τ H(div, Ω), (.8) with a ε (σ, τ ) dened by (div σ, v) + (f, v) = v L (Ω), (.9) a ε (σ, τ ) = (σ, τ ) + εσ n, τ n Ω. Here, we denote by (, ) the standard L -inner product over Ω, and by, Ω the one over the boundary Ω. By introducing the bilinear form B ε (σ, u; τ, v) = a ε (σ, τ ) + (div τ, u) + (div σ, v), we thus obtain the following weak form of (.)(.): nd (σ, u) H(div, Ω) L (Ω) such that B ε (σ, u; τ, v) + (f, v) = u + εg, τ n Ω (.) for all (τ, v) H(div, Ω) L (Ω). The well-posedness of (.) follows from standard arguments of mixed nite element theory [4].

3 .. Mixed nite element discretization. In order to discretize the variational problem (.), let K h be a regular and shape-regular partition of Ω into simplices. As usual, the diameter of an element K is denoted by h K, and the global mesh size h is dened as h = max K Kh h K. We denote by Eh the set of all interior edges (faces) of K h, and by Eh the set of all boundary edges (faces). We write h E for the diameter of an edge (face) E. Mixed nite element discretization of (.) is based on nite element spaces S h V h H(div, Ω) L (Ω) of piecewise polynomial functions with respect to K h. We will focus here on the Raviart-Thomas (RT) and Brezzi-Douglas-Marini (BDM) families of elements [,, 3,, 4]. That is, for an approximation of order k, the ux space S h is taken as one of the following two spaces S RT h = { σ H(div, Ω) σ K [P k (K)] n x P k (K), K K h }, S BDM h = { σ H(div, Ω) σ K [P k (K)] n, K K h }, (.) where P k (K) denotes the polynomials of total degree less or equal than k on K, and P k (K) the homogeneous polynomials of degree k. For both choices of S h above, the displacements are approximated in the multiplier space V h = { u L (Ω) u K P k (K), K K h }. (.) The spaces are chosen such that the following equilibrium property holds: div S h V h. (.3) The mixed nite element method now consists of nding (σ h, u h ) S h V h such that B ε (σ h, u h ; τ, v) + (f, v) = u + εg, τ n Ω (.4) for all (τ, v) S h V h. We remark that, by the equilibrium condition (.3), we have immediately the identity with P h denoting the L -projection onto V h. div σ h = P h f, (.5) 3. A-priori error estimates. In this section, we derive a-priori error estimates for the method in (.4). 3.. Stability. We begin by introducing the jump of a piecewise smooth scalar function u. To that end, let E = K K be an interior edge (face) shared by two elements K and K. Then the jump of f over E is dened by [f ] = f K f K. (3.) Next, we recall the following well-known trace estimate: for an edge (face) E of an element K, there holds h E σ,e C σ,k σ S h. (3.) Stability will be measured in mesh-dependent norms. For the uxes, we dene σ ε,h = σ + (ε + h E ) σ n,e. (3.3) E Eh 3

4 Here, we denote by,d the L -norm over a set D. In the case where D = Ω, we simply write. For the displacement variables, we introduce the norm u ε,h = u,k + E E h [u],e + h E E E h ε + h E u,e. (3.4) The continuity of the bilinear forms in the above norms follows by straightforward estimation. Lemma 3.. We have a ε (σ, τ ) σ ε,h τ ε,h, σ, τ S h, (3.5) (div σ, u) C τ ε,h u ε,h, σ S h, u V h. (3.6) Furthermore, it holds B ε (σ, u; τ, v) C ( σ ε,h + u ε,h )( τ ε,h + v ε,h ) (3.7) for all σ, τ S h and u, v V h. Proof. The bound (3.5) is a simple consequence of the Cauchy-Schwarz inequality: a ε (σ, τ ) = (σ, τ ) + εσ n, τ n Ω = (σ, τ ) + ε / σ n, ε / τ n E σ ε,h τ ε,h. E E h To prove (3.6), we use partial integration, elementary manipulations and the Cauchy- Schwarz inequality to obtain (div σ, u) = (σ, u) K + σ n K, u K σ n,k u,k + h E σ,e h E [u],e E Eh + (ε + h E ) σ n,e (ε + h E ) u,e, E E h with n K denoting the unit outward normal on K. The trace estimate (3.) and a repeated application of the Cauchy-Schwarz inequality then readily prove (3.6). Finally, the continuity bound (3.7) follows directly from (3.5) and (3.6). Next, we address the coercivity of the form a ε. Lemma 3.. There is a constant C > such that a ε (σ, σ) C σ ε,h σ S h. Proof. Since a ε (σ, σ) = σ + E E ε σ n h,e, the trace estimate (3.) readily yields the desired result. Finally, we prove the following inf-sup condition for the divergence form. Lemma 3.3. There exists a constant C > such that (div σ, u) sup C u ε,h u V h. σ S h σ ε,h 4

5 Proof. The proof is an extension of that of [9, Lemma.]. Since S RT h S BDM h, we only have to prove the condition in the Raviart-Thomas case. We recall that, on an element K, the local degrees of freedom for the RT family are given by the moments σ n K, z E z P k (E), E K, (σ, z) K z [P k (K)] n. Let now u V h be arbitrary. We then dene σ S RT h by setting on each element K: σ n K, z E = h E [u], z E σ n K, z E = ε + h E u, z E z P k (E), E E h, E K, z P k (E), E E h, E K, (σ, z) K = ( u, z) K z [P k (K)] n. Choosing z = [u] P k (E) and z = u [P k (K)] n gives σ n K, [u] E = h E [u],e, σ n K, [u] E = ε + h E u,e, E E h, E K, E E h, E K, (σ, u) K = u,k. Then we employ partial integration over each element and apply the dening moments for σ: (div σ, u) = (σ, u) K + σ n K, u K = u,k + [u],e + u (3.8),E h E ε + h E = u ε,h. E E h E E h Moreover, an explicit inspection of the degrees of freedom readily yields σ ε,h C u ε,h. (3.9) Identity (3.8) and the bound (3.9) give the desired inf-sup condition. By combining continuity (Lemma 3.), coercivity (Lemma 3.) and the inf-sup condition (Lemma 3.3), we readily obtain the following stability result. Lemma 3.4. There is a constant C > such that B(σ, u; τ, v) sup C( σ ε,h + u ε,h ) (σ, u) S h V h. (τ,v) S h V h τ ε,h + v ε,h 5

6 3.. Error estimates. We are now ready to derive a-priori error estimates. To that end, let (σ, u) be the solution of (.), and (σ h, u h ) the mixed nite element approximation of (.4). Let R h : H(div, Ω) S h be the RT or BDM interpolation operator [4]. It satises as well as the commuting diagram property (div (σ R h σ), v) = v V h, (3.) div R h σ = P h div σ, (3.) see, e.g., [4]. Moreover, we note that the equilibrium property (.3) implies (div τ, u P h u) = τ S h. (3.) Remark 3.5. In order for R h σ to be well-dened, some extra regularity is required for σ; see [4]. Since the regularity assumptions of Theorem 3.7 below are more than sucient, we neglect this issue in the following. Proposition 3.6. There is a constant C > such that σ h R h σ ε,h + u h P h u ε,h C σ R h σ. Proof. By the stability result in Lemma 3.4 there exists (τ, v) S h V h such that τ ε,h + v ε,h C and σ h R h σ ε,h + u h P h u ε,h B ε (σ h R h σ, u h P h u; τ, v). Using the consistency of the mixed method and properties (3.), (3.), we obtain B ε (σ h R h σ, u h P h u; τ, v) = a ε (σ h R h σ, τ ) + (div τ, u h P h u) + (div (σ h R h σ), v) = a ε (σ R h σ, τ ) + (div τ, u P h u) + (div (σ R h σ), v) = (σ R h σ, τ ) + ε (σ R h σ) n, τ n E. E E h Then the dening moments for RT or BDM interpolation yield (noting that ε is edgewise (facewise) constant) ε (σ R h σ) n, τ n E =, (3.3) so that Thus, we conclude that E E h B ε (σ h R h σ, u h P h u; τ, v) = (σ R h σ, τ ). σ h R h σ ε,h + u h P h u ε,h σ R h σ τ C σ R h σ, which completes the proof. 6

7 In the sequel, we denote by k the standard Sobolev norm of order k. The following theorem is the main result of this section. Theorem 3.7. We have the approximation bound σ h R h σ ε,h + P h u u h ε,h { Ch k σ k for RT elements, Ch k+ σ k+ for BDM elements. (3.4) Moreover, we have the following optimal a-priori error estimate for the L -error in the ux: σ σ h { Ch k σ k for RT elements, Ch k+ σ k+ for BDM elements. (3.5) Proof. The bound (3.4) is an immediate consequence of Proposition 3.6 and the approximation properties of R h, see, e.g., [4]. The error estimate (3.5) follows readily from the triangle inequality, the consistency bound in Proposition 3.6 and the approximation properties of R h. Remark 3.8. We point out that the quantity P h u u h ε,h in (3.4) is superconvergent. As in [9], this fact allows us to enhance the displacement approximation via local postprocessing; see Section 4 below. We further emphasize that the constant C in the error bound (3.5) is independent of ε. Hence, the L -norm error estimate for the uxes is ε-robust. 4. Postprocessing. In this section, we introduce a local postprocessing for the displacement and prove an optimal error estimate in the postprocessed displacement. 4.. Postprocessing method. Let u h be the displacement obtained by the mixed method (.4). The postprocessed displacement u h is sought in the augmented space Vh V h dened as { { u L (Ω) u Vh K P k (K), K K h } for RT elements, = (4.) { u L (Ω) u K P k+ (K), K K h } for BDM elements. The postprocessed displacement u h is dened on each element K by the conditions P h u h = u h, (4.) ( u h, v) K = (σ h, v) K v (I P h )V h K, (4.3) where we recall that P h is the L -projection onto V h. In order to analyze the error in the postprocessed displacement u h, we introduce the modied bilinear form Bε,h(σ, u ; τ, v ) = B ε (σ, u ; τ, v ) + ( u σ, (I P h )v ) K. (4.4) Then we will consider the modied variational problem: nd (σ h, u h ) S h V h such that B ε,h(σ h, u h; τ, v ) + (P h f, v) = u + εg, τ n Ω (4.5) for all (τ, v ) S h Vh. The following proposition relates the solution of the modied problem (4.5) to that of the original problem (.4). Its proof is exactly the same as the one for the standard mixed methods considered in [9, Lemma.4]. 7

8 Proposition 4.. Let (σ h, u h ) S h Vh be the solution of problem (4.5) and set u h = P h u h. Then (σ h, u h ) S h V h is the solution of the original problem (.4). Conversely, if (σ h, u h ) S h V h is the solution of the original problem (.4) and u h is the postprocessed displacement obtained from u h, then (σ h, u h ) S h Vh is the solution of problem (4.5). In order to show the stability of the modied method (4.5), we shall rst state the following useful result whose proof is nearly identical to that of [9, Lemma.5]. Lemma 4.. There exist constants C >, C > such that for every u Vh there holds as well as u ε,h P h u ε,h + (I P h )u ε,h C u ε,h, (4.6) C u ε,h P h u ε,h + ( (I P h )u,k ) / C u ε,h. (4.7) Since (I P h )u is L -orthogonal to constant functions, there exists a third constant C 3 > such that (I P h )u ε,h C 3 ( (I P h )u,k ) /. (4.8) With exactly the same arguments as in [9, Lemma.6], we then have the following inf-sup stability result for the modied bilinear form B ε,h. Proposition 4.3. There exists a constant C > such that Bε,h sup (σ, u ; τ, v ) (τ,v ) S h Q τ h ε,h + v C( σ ε,h + u ε,h ) (σ, u ) S h Vh. (4.9) ε,h 4.. Error in the postprocessed displacement. Now we state and prove a- priori error estimates for the postprocessed displacement u h. As before, let (σ, u) be the solution of (.), and (σ h, u h ) the postprocessed nite element approximation of (4.5). We now have the following result. Theorem 4.4. There holds: σ h R h σ ε,h + u u h ε,h C σ R h σ + inf u V h u u ε,h As a consequence, we have the ε-robust error estimate σ σ h + u u h ε,h { Ch k u k+ for RT elements, Ch k+ u k+ for BDM elements. Proof. Let u Vh. From Proposition 4.3 it follows that there is a tuple (τ, v ) S h Vh such that τ ε,h + v ε,h C and σ h R h σ ε,h + u h u ε,h Bε,h(σ h R h σ, u h u ; τ, v ). 8

9 From the denition of the method (4.5), we then have B ε,h(σ h R h σ, u h u ; τ, v ) = B ε,h(σ R h σ, u u ; τ, v ) + (f P h f, v ) = a ε (σ R h σ, τ ) + (div τ, u u ) + (div (σ R h σ h ), v ) + (f P h f, v ) + ( (u u ) (σ R h σ), (I P h )v ) K. Due to the commuting diagram property (3.) and the equation (.), div σ = f, there holds so that (div (σ R h σ), v ) = (div σ P h div σ, v ) = ( f + P h f, v ), B ε,h(σ h R h σ, u h u ; τ, v ) = a ε (σ R h σ, τ ) + (div τ, u u ) + ( (u u ) (σ R h σ), (I P h )v ) K. As in the proof of Proposition 3.6, we use (3.3) and get a ε (σ R h σ, τ ) = (σ R h σ, τ ) C σ R h σ. Moreover, by integration by parts as in the continuity proof of Lemma 3., we have (div τ, u u ) C τ ε,h u u ε,h C u u ε,h. Furthermore, by Lemma 4. the last term can be bounded by ( (u u ) (σ R h σ), (I P h )v ) K C( σ R h σ + u u ε,h ) v ε,h C( σ R h σ + u u ε,h ). Since v Vh was arbitrary, the rst assertion is proved. The error estimate is now an immediate consequence of the rst bound, the triangle inequality and standard approximation properties. 5. A-posteriori estimates. We now derive a residual-based a-posteriori estimator for the postprocessed solution (σ h, u h ). We point out that using the postprocessed solution is vital for obtaining an estimator whose local residual terms are properly matched with respect to their convergence properties [9]. 5.. Error estimator. For an element K, we dene the local error indicators η,k = u h σ h,k, η,k = h K f P h f,k. For an interior edge (face) E Eh, we introduce the jump indicator η E = h E [u h ],E. For a boundary edge (face) E Eh, we dene η E = ε + h E ε(σ h n g h ) + u h u,e 9

10 where g h is the L -projection of g onto P k (E) for RT elements and onto P k (E) for BDM elements. To also take into account the approximation of g, we introduce the set E h,+ = { E E h ε E > }. (5.) of all boundary edges (faces) E with a non-vanishing ε E. For a boundary edge (face) E Eh,+, we then introduce the indicator related to the approximation of g by setting η g,e = h E g g h,e. Summing up these local indicators, our error estimator is given by η = ( η,k + η,k) + ηe + ηe + η g,e. (5.) E E h E E h E E h,+ Remark 5.. Note that, for ε =, the indicator η g,e can be omitted in the denition of η. The resulting estimator then coincides with the ones derived in the papers [9, 3] for homogeneous and inhomogeneous Dirichlet boundary conditions, respectively. The eciency of the indicators η,k and η E is given by the following lower bounds (note that the indicators η,k and η E,g are data approximation terms). We denote by by Π h the L -projection on the boundary edges, onto P k (E) for RT elements and onto P k (E) for BDM elements. Proposition 5.. There holds: η,k (u u h),k + σ σ h,k, K K h, η E [u u h ],E he, E E h, η E ε (σ h R h σ) n,e + u u h,e + min{r (u ), R (σ, g)} E E ε + he h,+, η E u u h,e he E E h \ E h,+. Here the terms R and R are dened as R (u ) = u Π h u,e ε + he, R (σ, g) = ε (σ R h σ) n,e + ε g g h,e. Proof. The rst bound follows simply from the triangle inequality, the second from the fact that [u] = over E Eh. To prove the third assertion, we note that, by the dening moments for R h, we have R h σ n = Π h (σ n) on all boundary edges E E h,+. Thus, applying the L -projection Π h to the boundary conditions in (.3) we see that ε(r h σ n g h ) = Π h (u u).

11 Therefore, the function to be evaluated in the residual η E can be written as ε(σ h n g h ) + u h u = ε(σ h n g h ) + u h u ε(r h σ n g h ) + Π h (u u) = ε(σ h R h σ) n (u Π h u ) (u u h). Taking the L -norm and applying the triangle inequality yields the estimate with R (u ) for any edge η E E h,+ with non-vanishing ε E. On the other hand, we can directly insert the exact boundary condition into the term to be estimated, yielding ε(σ h n g h ) + u h u = ε(σ h n g h ) + u h ε(σ n g) u = ε(σ h R h σ) n + (u h u) ε(σ R h σ) n + ε(g g h ). Using the triangle inequality and the relation ε/ ε + h ε gives the third assertion with the term R (σ, g). Finally, if ε E = for a boundary edge E, we have u = u on E, which yields immediately the fourth estimate. Remark 5.3. In conjunction with our a-priori results, Proposition 5. indicates that all the indicators converge with at least the same order and are thus properly matched. In particular, due to Proposition 3.6 it holds ε (σ h R h σ) n,e ε + he ε (σ h R h σ) n,e C σ R h σ and thus the term converges independently of ε. Similarly, for the terms inside the minimum we have optimal convergence rate for R for h E < ε and for R for the case h E ε. Thus, the boundary estimator is fully ε-robust. This is conrmed numerically in Section 7 below. 5.. Reliability. To derive an upper bound for the a-posteriori estimator η in (5.), we denote by ( σ, ũ) the solution of the perturbed problem where we replace g by g h : for all (τ, v) H(div, Ω) L (Ω). Since B ε ( σ, ũ; τ, v) + (f, v) = u + εg h, τ n Ω (5.3) g h, τ n E = g, τ n E τ S h, E E h, it is clear that the nite element approximations (σ h, u h ) are in fact also approximations to ( σ, ũ). We will make use of the following saturation assumption [8, 9]: Let K h/ be a uniformly rened subtriangulation of K h, obtained by dividing each simplex K K h into n elements. We denote by σ h/ and u h/ the ux and postprocessed displacement obtained on the ner mesh K h/. The saturation assumption can now be formulated as follows. Assumption 5.4 (Saturation assumption). There exists a constant β < such that σ σ h/ + ũ u h/ ε,h/ β ( σ σ h + ũ u h ε,h ). The following result establishes the reliability of the estimator η.

12 Theorem 5.5. Suppose that Assumption 5.4 holds. Then there exists a constant C > such that σ σ h + u u h ε,h Cη. (5.4) Proof. We proceed in several steps. Step : Let (σ, u) and ( σ, ũ) denote the solutions of (.) and (5.3), respectively. The dierence u ũ in the displacement then satises the second-order equation: (u ũ) = in Ω, ε (u ũ) n + (u ũ) = ε(g g h ) on Ω. Multiplying this equation by u ũ and integrating by parts the left-hand side, we obtain (u ũ) (u ũ) n, u ũ E =. E E h,+ Using the boundary condition, we conclude that (u ũ) + E E h,+ ε u ũ,e = E E h,+ g g h, u ũ E. Let P be the L -projection onto the piecewise constants. For any edge E E h,+ with E K, we now use the denition of g h and standard approximation results to get g g h, u ũ E = g g h, u ũ P (u ũ) E Ch E g g h,e (u ũ),k. We thus readily obtain (u ũ) + E E h,+ ε u ũ,e C E E h,+ η g,e. The denition of the norm ε,h, the inequality (ε E +h E ) ε E for all E E h,+, and the fact that u ũ E = on all edges (faces) with ε E = yield σ σ + u ũ ε,h Cη. (5.5) Step : From the triangle inequality and the bound (5.5), we obtain σ σ h + u u h ε,h Cη + σ σ h + ũ u h ε,h. It is thus sucient to bound the error of the nite element approximation (σ h, u h ) to the perturbed solution ( σ, ũ) in (5.3). From Assumption 5.4, we conclude that σ σ h + ũ u h ε,h ) ( σ h/ σ h + u h/ β u h ε,h/. Thus, it remains to prove that there is a constant C > such that ( σ h/ σ h + u h/ u h ε,h/ ) C η. (5.6)

13 Step 3: We show (5.6). To that end, we employ the inf-sup condition in Proposition 4.3 over the ner spaces and conclude that there is (τ, v ) S h/ Q h/ such that and τ ε,h/ + v ε,h/ C (5.7) C( σ h/ σ h + u h/ u h ε,h/ ) B ε,h/ (σ h/ σ h, u h/ u h; τ, v ). From linearity and the denition of the postprocessed method, we obtain B ε,h/ (σ h/ σ h, u h/ u h; τ, v ) = (P h/ f, v ) + u + εg, τ n Ω B ε,h/ (σ h, u h; τ, v ) = (P h/ f, v ) + u + εg, τ n Ω (σ h, τ ) εσ h n, τ n Ω (div τ, u h) (div σ h, v ) ( u h σ h, (I P h/ )v ) K. / To simplify this identity, we use that div σ h = P h f, see (.5). Moreover, we integrate by parts the term (div τ, u h ) over the elements K K h: (div τ, u h) = ) (( u h, τ ) K τ n K, u h K. Rearranging the terms, we conclude that where C( σ h/ σ h + u h/ u h ε,h/ ) T + T + T 3 + T 4 + T 5, (5.8) T = (σ h u h, τ ), T = ε(σ h n g h ) + u h u, τ n Ω, T 3 = (P h/ f P h f, v ), T 4 = τ n K, u h K\ Ω, T 5 = By (5.7), the term T can be bounded by T ( / ( u h σ h, (I P h/ )v ) K. η,k ) τ Cη. To bound T, we use the Cauchy-Schwarz inequality and the fact that ε is piecewise constant. We obtain T ( ηe ) ( (ε + h E ) τ n ),E E E h Cη ( E E h (ε + h E ) τ n,e ) Cη. E E h/ 3

14 To estimate T 3, we use exactly the same arguments as in equations (3.9)(3.) of [9] to get T 3 C ( h K f P h f,k) Cη. The term T 4 can be rewritten as T 4 = τ n, [u h ] E. E E h Using the Cauchy-Schwarz inequality and the polynomial trace inequality (3.) over the ner mesh K h/, it can then be bounded by T 4 ( h E τ ) (,E h E [u h ] ),E E E h Cη ( E E h h E τ ),E E E h/ Cη τ Cη. Finally, due to (5.7) and (4.7), we get T 5 ( ) ( η,k / (I P h/ )v,k ) (5.9) Cη v ε,h/ Cη. (5.) Referring to (5.6), (5.8) and the above bounds for T through T 5 completes the proof. 6. A remark on hybridization. It is a well-known procedure for mixed nite elements to simplify the solution of the algebraic system resulting from equations (.6) (.7) by introducing a Lagrange multiplier to enforce the normal continuity of the ux variable σ h [, 5]. We introduce the multiplier spaces Mh BDM = { m L (E) m P k (E), E Eh, m E =, E Eh }. (6.) Mh RT = { m L (E) m P k (E), E Eh, m E =, E Eh }. (6.) Now it can be easily shown, that the normal continuity of the ux σ h is equivalent to the requirement σ h n, p K =, p M h. (6.3) Thus, the original nite element problem (.4) can be formulated as follows [6]: nd (σ h, u h, m h ) S h V h M h such that a ε (σ h, τ ) + (div τ, u h ) + τ n, m h K = u + εg, τ n Ω, (6.4) (div σ h, v) + (f, v) =, (6.5) σ h n, p K =, (6.6) 4

15 for all (τ, v, p) S h V h M h. Due to the property (6.3), the pair (σ h, u h ) retrieved from equations (6.4)(6.6) coincides with the solution of the original discrete problem (.4). Thus we can apply the same postprocessing scheme as proposed earlier, even if we use hybridization to solve the initial system. The corresponding algebraic system is of the form (A + εã)σ + Bu + Cm = εg + u B T σ = f C T σ =, where (σ, u, m) are the coecient vectors of the solution. We can solve for σ easily using element-by-element inversion of the block diagonal matrix A + εã. On elements in the interior of the domain, we have and on elements with at least one edge on the boundary σ = A ( Bu Cm), (6.7) σ = (A + εã) (εg + u Bu Cm) (6.8) Thus, we only get an ε-dependent problem on the elements touching the boundary of the domain. Furthermore, the matrix à corresponding to the part σ n, τ n E has nonzero components only corresponding to the boundary degrees of freedom. Since these degrees of freedom do not couple to the interelement Lagrange multipliers, one has no ε-dependence in the condition number of the nal linear system for the variables (u, m). In addition, the resulting linear system for (u, m) will be symmetric and positive denite. 7. Numerical results. In this section, we present a series of numerical tests. The main focus is on showing that the proposed mixed nite element method is ε- robust, both in the sense of the a-priori estimates in Theorem 4.4 and the a-posteriori estimates in Theorem 5.5. We use two test cases, the rst with a smooth solution and the second with a singular one. For the constant in estimate of Theorem 5.5 we choose C =. Furthermore, the data approximation terms are neglected, thus the estimator is smaller than the actual error in all of the results presented. This is particularly clearly visible in the test case with non-smooth boundary data. 7.. Smooth solution. In the rst test case, we use a smooth solution to retrieve the convergence rates predicted by our theoretical results. We consider the rectangular domain Ω = (, ) and choose the load in problem (.)(.) so that the displacement u(x, y) is given by the smooth function u(x, y) = sin(x) sinh(y) + C, (7.) and the ux by σ = u. The constant C is chosen so as to ensure a zero-mean displacement, i.e., we take C = (cos() )(cosh() ). The boundary data are computed from u and σ by setting u = u and g = σ n. We then enforce the Robin boundary conditions (.3) for several values of ε. We test the proposed mixed method both for rst-order (k = ) and second-order (k = ) BDM elements. We use uniformly rened triangular meshes of mesh size h /N, N being the number of degrees of freedom of the discretization. 5

16 5 Estimator, BDM True error, BDM Estimator, BDM True error, BDM 5 Estimator, BDM True error, BDM Estimator, BDM True error, BDM Error Error Number of degrees of freedom Number of degrees of freedom Fig. 7.. Convergence of the smooth solution with ε =. 6 8 Estimator, BDM True error, BDM Estimator, BDM True error, BDM Fig Convergence of the smooth solution with ε = 4. 5 Estimator, BDM True error, BDM Estimator, BDM True error, BDM Error Error Number of degrees of freedom Number of degrees of freedom Fig. 7.. Convergence of the smooth solution with ε =. Fig Convergence of the smooth solution with ε =. In Figures 7. through 7.4, we plot the errors σ σ h + u u h ε,h and the values of the a-posteriori estimator η in (5.) with respect to the number of degrees of freedom N for ε =,, 4,, respectively. The slopes in the logarithmic scale in the gures are half of the actual convergence rates. In all the curves, we see convergence of order k + in the mesh size, in agreement with Theorem 4.4. Moreover, the curves clearly conrm the reliability and eciency of the estimator η; see Theorem 5.5 and Proposition 5.. Notably, we also get optimal order of convergence for the estimator η in the test case with ε =, where the situation ε = h is encountered. Evidently, the convergence is completely independent of the value of ε, and the magnitude of the errors does not depend on ε either. In particular, the performance of the a posteriori estimator truly is ε-independent. 7.. Singular solution. In the second example, we consider again the domain (, ). In polar coordinates (r, Θ) about the origin, the displacement is chosen to be u(r, Θ) = r β sin(βθ) + C, (7.) and the ux is σ = u. Here, the parameter β denes the exact regularity of u and can be used to model singular behavior at the origin. The constant C is once again 6

17 dened such that u will have zero mean value. For this solution, we have u H +β (Ω) and subsequently σ [H β (Ω)] n ; see [7]. The boundary data are computed from u and σ as before. In the following test, we set β =.67, corresponding to a highly singular displacement. We use only the lowest-order (k = ) BDM elements, since raising the polynomial degree would give no advantage due to the insucient regularity of the solution. We compute the solutions on adaptively rened meshes for various values of ε. To create the mesh sequences, we have chosen to rene all elements in which the elemental indicator exceeds 3 percent of the maximal value. Contribution from an individual edge estimators is divided evenly to the elements sharing the edge or face. To ensure sucient renement, we further halve the renement threshold until at least percent of all elements are rened. For comparison, the computations are also performed using uniform mesh renement with an equivalent number of degrees of freedom. In Figures 7.5 through 7.8, we show the errors σ σ h + u u h ε,h and the values of the estimator η obtained for this problem with the values ε =,, 4, Estimator, adaptive True error, adaptive Estimator, uniform True error, uniform 3 Estimator, adaptive True error, adaptive Estimator, uniform True error, uniform 5 4 Error 6 Error Number of degrees of freedom Number of degrees of freedom Fig Convergence of the singular solution with ε =. 3 4 Estimator, adaptive True error, adaptive Estimator, uniform True error, uniform Fig Convergence of the singular solution with ε = Estimator, adaptive True error, adaptive Estimator, uniform True error, uniform 5 5 Error Error Number of degrees of freedom Number of degrees of freedom Fig Convergence of the singular solution with ε =. Fig Convergence of the singular solution with ε = 8. In the case of uniform mesh renement, the convergence rates are now limited by the regularity of the solution. They are of the order h β as expected. On the other hand, the adaptive mesh renement strategy clearly is able to retrieve the convergence to some extent. More importantly, even for the irregular boundary data considered here, both the a-priori and a-posteriori estimates are observed to remain 7

18 fully ε-independent, thus conrming the theoretical results First refinement.5 After 7 refinements Fig Mesh density after one step, in the middle of the renement procedure, and on the nal mesh with ε = First refinement.5 After 7 refinements Fig. 7.. Mesh density after one step, in the middle of the renement procedure, and on the nal mesh with ε = First refinement.5 After 7 refinements Fig. 7.. Mesh density after one step, in the middle of the renement procedure, and on the nal mesh with ε = 4. The singularity of the solution in (7.) lies in the origin and we expect the adaptive meshes to be strongly rened into this corner. To ascertain the ε-independence of the adaptive renement procedure, we plot the mesh densities for dierent values of the parameter ε at dierent stages of the renement procedure. The plots are normalized with respect to the largest element size such that the largest element size equals unity, and are shown using a logarithmic scale. From Figures 7.9 through 7., we see that our adaptive mesh renement procedure clearly yields equivalent mesh density distributions for dierent values of ε. This is what is desired for this problem, since the renement should only be driven by the irregularity of the solution. These numerical tests indicate that the residual-based estimators introduced in this paper are indeed appropriate for controlling adaptive renement. 8

19 8. Conclusions. In this paper, we have analyzed mixed nite element methods for the dual mixed form of the Poisson problem with general Robin-type boundary conditions. We have extended a well-known postprocessing technique to this case. As a consequence, we obtain optimal error estimates where the convergence rates for the broken H -errors in the postprocessed displacements correctly match the ones for the L -errors in the uxes. Furthermore, the postprocessing allows us to design a properly functioning residual-based error indicator. A key feature of our analysis is that both the a-priori and a-posteriori error estimates are fully ε-robust. The numerical results verify our theoretical results. They show the optimality and sharpness of our estimates and conrm their robustness. They also show that the error indicator can be employed to drive adaptive renement for all values of ε. REFERENCES [] D. Arnold and F. Brezzi, Mixed and nonconforming nite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Anal. Numér., 9 (985), pp. 73. [] F. Brezzi, J. Douglas, Jr., R. Durán, and M. Fortin, Mixed nite elements for second order elliptic problems in three variables, Numer. Math., 5 (987), pp [3] F. Brezzi, J. Douglas, Jr., and L. D. Marini, Two families of mixed nite elements for second order elliptic problems, Numer. Math., 47 (985), pp [4] F. Brezzi and M. Fortin, Mixed and hybrid nite element methods, vol. 5 of Springer Series in Computational Mathematics, Springer-Verlag, New York, 99. [5] F. Brezzi, Jr. J. Douglas, and L.D. Marini, Two families of mixed nite elements for second order elliptic problems, Numer. Math., 47 (985), pp [6] B.X. Fraeijs de Veubeke, Displacement and equilibrium models in the nite element method, in Stress Analysis, O.C. Zienkiewicz and G. Hollister, eds., Wiley, 965. [7] E. Gekeler, Mathematische Methoden zur Mechanik, Springer-Verlag, Berlin Heidelberg, 6. [8] M. Juntunen and R. Stenberg, Nitsche's method for general boundary conditions, Math. Comp., 78 (9), pp [9] C. Lovadina and R. Stenberg, Energy norm a posteriori error estimates for mixed nite element methods, Math. Comp., 75 (6), pp [] J.-C. Nédélec, Mixed nite elements in R 3, Numer. Math., 35 (98), pp [] P.-A. Raviart and J. M. Thomas, A mixed nite element method for nd order elliptic problems, in Mathematical aspects of nite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 975), Springer, Berlin, 977, pp Lecture Notes in Math., Vol. 66. [] J.E. Roberts and J.-M. Thomas, Mixed and hybrid nite element methods, in Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions, eds., vol. II: Finite Element Methods (Part ), North-Holland, 99, pp [3] M. Rüter and R. Stenberg, Errorcontrolled adaptive mixed nite element methods for second-order elliptic equations, Comp. Mech., 4 (8), pp [4] R. Stenberg, Some new families of nite elements for the Stokes equations, Numer. Math., 56 (99), pp [5] R. Stenberg, Postprocessing schemes for some mixed nite elements, RAIRO Modél. Math. Anal. Numér., 5 (99), pp