ENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS

ENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS CARLO LOVADINA AND ROLF STENBERG Abstract The paper deals with the a-posteriori error analysis of mixed finite element methods for second order elliptic equations It is shown that a reliable and efficient error estimator can be constructed using a postprocessed solution of the method The analysis is performed in two different ways; under a saturation assumption and using a Helmholtz decomposition for vector fields 1 Introduction We consider the mixed finite element approximation of second order elliptic equations with the Poisson problem as a model: 11 12 u = f in Ω R n, u = 0 The problem is written as the system 13 14 on Ω σ u = 0, div σ + f = 0, which is approximated with the Mixed method Find σ h, u h S h V h Hdiv:Ω L 2 Ω such that 15 16 σ h, τ + div τ, u h = 0 τ S h, div σ h, v + f, v = 0 v V h In the method the polynomial used for approximating the flux σ is of higher degree than that used for the displacement u, which is counterintuitive in view of 13 As a consequence, the mixed method has to be carefully designed in order to satisfy the Babuška-Brezzi conditions, cf eg [8] There are two ways of posing these conditions, both yielding the same a priori estimates The more common one is to use the Hdiv : Ω norm for the flux and the L 2 Ω norm for the displacement The other one is to use so called mesh dependent norms [3] which are close to the energy norm of the continuous problem The a posteriori error analysis of mixed methods has been performed in [1], [10] and [5] In [10] the estimate is for the Hdiv : Ω norm This is in a way unsatisfactory since the div part of the norm is trivially computable and also may dominate the error, see Remark 34 below In [5] an estimate for the L 2 -norm 1991 Mathematics Subject Classification 65N30 Key words and phrases mixed finite element methods, a-posteriori error estimates, postprocessing This work has been supported by the European Project HPRN-CT-2002-00284 New Materials, Adaptive Systems and their Nonlinearities Modelling, Control and Numerical Simulation 1

2 C LOVADINA AND R STENBERG of the flux is derived but it is, however, not optimal The reason for this is that the estimator includes the element residual in the constitutive relation 13 As the polynomial degree of approximation for the displacement is lower than that for the flux, it is clear that this residual is large The purpose of this paper is to point out a simple remedy to this Since the work of Arnold and Brezzi [2] it is known that the mixed finite element solution can be locally postprocessed in order to obtain an improved displacement Later other postprocessing has been proposed [6, 9, 7, 17, 16] On each element the postprocessed displacement is of one degree higher than the flux, which is in accordance with 13 Hence, it is natural to use it in the a posteriori estimate In this paper, we will focus on the postprocessing introduced in [17, 16] In Section 2 we develop an a-priori error analysis by recognizing that the postprocessed output can be viewed as the direct solution of a suitable modified method In Section 3 we introduce our estimator based on the postprocessed solution, and we prove its efficiency and reliability Throughout the paper we will use standard notations for Sobolev norms and seminorms Moreover, we will denote with C and C i i = 1, 2, generic constants independent of the mesh parameter h, which may take different values in different occurrences 2 A-priori estimates and postprocessing In this section we will consider the mixed methods, their postprocessing and error analysis We will also give the stability and error analysis by treating the method and the postprocessing as one method This will be useful for the a posteriori analysis We will use standard notation used in connection with mixed FE methods By C h we denote the finite element regular partitioning and by Γ h the collection of edges or faces of C h The subspaces σ h, u h S h V h Hdiv : Ω L 2 Ω are piecewise polynomial spaces defined on C h In this paper we will consider the following families of elements The results are, however, easily applicable for other families as well RTN elements the triangular elements of Raviart-Thomas [15] and their tetrahedral counterparts of Nedelec [14]; BDM elements the triangular elements of Brezzi-Douglas-Marini [9] and their tetrahedral counterparts of Brezzi-Douglas-Duran-Fortin [7] Accordingly, given an integer k 1, we define: 21 S RT N h = { τ Hdiv:Ω τ K [P k 1 K] n x P k 1 K K C h } 22 S BDM h = { τ Hdiv:Ω τ K [P k K] n K C h } 23 V RT N h = V BDM h = { v L 2 Ω v K P k 1 K K C h }, where P k 1 K denotes the homogeneous polynomials of degree k 1 For quadrilateral and hexahedral meshes there exist a wide choice of different alternatives, cf [8] By defining the following bilinear form 24 Bϕ, w; τ, v = ϕ, τ + div τ, w + div ϕ, v

ENERGY NORM A POSTERIORI ESTIMATES FOR MIXED METHODS 3 the mixed method can compactly be defined as: Find σ h, u h S h V h such that 25 Bσ h, u h ; τ, v + f, v = 0 τ, v S h V h For the displacement and the flux we will use the following norms: 26 v 2 1,h = v 2 0,K + h 1 E [[v]] 2 0,E, and 27 τ 2 0,h = τ 2 0 + h E τ n 2 0,E, where n is the unit normal to E Γ h and [[v ] is the jump in v along interior edges/faces and v on edges/faces on Ω By an element by element partial integration we have 28 div τ, v τ 0,h v 1,h τ, v S h V h In the FE subspace the norm for the flux is equivalent to the L 2 -norm: 29 C τ 0,h τ 0 τ 0,h τ S h Hence, it also holds 210 div τ, v C τ 0 v 1,h τ, v S h V h With this choice of norms the Babuška-Brezzi stability condition is the following Lemma 21 There is a positive constant C such that 211 sup τ S h div τ, v τ 0 C v 1,h v V h Proof We first point out that since Vh RT N = Vh BDM the result for BDM is a consequence of that for RTN Therefore, we focus on the RTN family, first recalling that the local degrees of freedom for the flux variable are the following: 212 213 and S RT N h τ n, z E z P k 1 E, E K, τ, z K z [P k 2 K] n S BDM h Above and in the rest of the paper, we use the notation, K and, E for the L 2 inner product on the element K and on the edge/face E, respectively Hence, given v V h we can define τ S h by 214 215 τ n, z E = h 1 E [[v]], z E z P k 1 E, E Γ h, τ, z K = v, z K z [P k 2 K] n, K C h Noting that v K [P k 2 K] n, [[v]] E P k 1 E, from 214 215 we obtain 216 217 It follows that cf also 26 τ n, [[v]] E = h 1 E [[v ] 2 0,E, τ, v K = v 2 0,K

4 C LOVADINA AND R STENBERG 218 div τ, v = Using scaling arguments 214 215 imply τ, v K + τ n, [[v]] E = v 2 0,K + h 1 E [v]] 2 0,E = v 2 1,h 219 τ 0,h C v 1,h The assertion now follows from 218 and 219 From this stability estimate, the following full stability result holds Lemma 22 There is a positive constant C such that Bϕ, w; τ, v 220 sup C ϕ 0 + w 1,h τ,v S h V h τ 0 + v 1,h ϕ, w S h V h In our analysis we will exploit the interpolation operator R h : Hdiv : Ω [L s Ω] n S h, with s > 2, such that 221 div τ R h τ, v = 0 v V h, which can be constructed by using the degrees of freedom for S h, cf [15, 14, 9, 7] In addition, we will use the equilibrium property 222 div S h V h When denoting by P h : L 2 Ω V h the L 2 -projection, this implies that 223 div τ, u P h u = 0 τ S h The projection and interpolation operators satisfy the following commuting property: 224 div R h = P h div Theorem 23 There is a positive constant C such that 225 σ σ h 0 + P h u u h 1,h C σ R h σ 0 Proof By Lemma 22 there is a pair τ, v S h V h, with τ 0 + v 1,h C, such that 226 σ h R h σ 0 + u h P h u 1,h Bσ h R h σ, u h P h u; τ, v Next, 221, 223 and 224 give 227 Bσ h R h σ, u h P h u; τ, v = σ h R h σ, τ + div τ, u h P h u + div σ h R h σ, v = σ R h σ, τ σ R h σ 0 τ 0 C σ R h σ 0 The assertion then follows from the triangle inequality This gives assuming full regularity: 228 229 σ σ h 0 + P h u u h 1,h Ch k+1 σ k+1 σ σ h 0 + P h u u h 1,h Ch k σ k for BDM, for RTN

ENERGY NORM A POSTERIORI ESTIMATES FOR MIXED METHODS 5 We note that these estimates contain a superconvergence result for P h u u h 1,h This, together with the fact that σ h is a good approximation of u, implies that an improved approximation for the displacement can be constructed by local postprocessing Below we will consider the method introduced in [17, 16] The postprocessed displacement is sought in a FE space V h V h For our choices, the spaces are 230 231 V BDM h = { v L 2 Ω v K P k+1 K K C h }, Vh RT N = { v L 2 Ω v K P k K K C h } Postprocessing method Find u h V h 232 P h u h = u h and such that 233 u h, v K = σ h, v K v I P h V h K The error analysis of this postprocessing is done in [17, 16] Here we proceed in a slightly different way by considering the method and the postprocessing as one method To this end we define the bilinear form 234 B h ϕ, w ; τ, v =ϕ, τ + div τ, w + div ϕ, v + w ϕ, I P h v K Then we have the following equivalence to the original problem Lemma 24 Let σ h, u h S h V h be the solution to the problem 235 B h σ h, u h; τ, v + P h f, v = 0 τ, v S h V h, and set u h = P h u h V h Then σ h, u h S h V h coincides with the solution of 15 16 Conversely, let σ h, u h S h V h be the solution of 15 16, and let u h V h be the postprocessed displacement defined by 232 233 Then σ h, u h S h Vh is the solution to 235 Proof Testing by τ, 0 S h V h in 235 gives 236 σ h, τ + div τ, u h = 0 τ S h The equilibrium property 222 implies 237 div τ, u h = div τ, u h Hence, 15 is satisfied Next, for a generic v Vh set v = P hv V h and observe that V h = P h Vh Testing in 235 with 0, v, and using the fact that P h f, v = f, v, we obtain 238 div σ h, v + f, v = 0 v V h, ie the equation 16 Conversely, let σ h, u h S h V h be the solution of 15 16, and let u h V h be defined by 232 233 Splitting a generic v V h as v = P h v + I P h v we have

6 C LOVADINA AND R STENBERG 239 B h σ h, u h; τ, v = B h σ h, u h; τ, P h v + B h σ h, u h; 0, I P h v = σ h, τ + div τ, u h + div σ h, P h v + + div σ h, I P h v + u h σ h, I P h P h v K u h σ h, I P h I P h v K = σ h, τ + div τ, u h P h f, P h v = P h f, v τ, v S h V h Therefore, σ h, u h S h V h solves 235 Next, we prove the stability In the proof we will use the following norm equivalence Lemma 25 There are positive constants C 1 and C 2, such that 240 w 1,h P h w 1,h + I P h w 1,h C 2 w 1,h and 1/2 241 C 1 w 1,h P h w 1,h + I P h w 2 0,K C2 w 1,h, for every w V h Proof We first prove 240 The estimate w 1,h P h w 1,h + I P h w 1,h follows immediately from the triangle inequality To continue, we notice that 242 P h w 1,h + I P h w 1,h 2 P h w 1,h + w 1,h We now fix an interior edge/face E, and we consider the elements K 1 and K 2 such that E = K 1 K 2 A scaling argument shows that 243 h 1 E [[P hw ]] 2 0,E + 2 P h w 2 0,K i i=1 C h 1 E [[w ]] 2 0,E + If E K is an edge/face lying in Ω, a similar argument gives 2 w 2 0,K i 244 h 1 E P hw 2 0,E + P h w 2 0,K C h 1 E w 2 0,E + w 2 0,K The estimate 245 P h w 1,h + I P h w 1,h C 2 w 1,h easily follows from 242 244 cf also 26 Hence, 240 is proved To prove 241 we first notice that 245 implies 1/2 246 P h w 1,h + I P h w 2 0,K C2 w 1,h Next, scaling arguments lead to i=1

ENERGY NORM A POSTERIORI ESTIMATES FOR MIXED METHODS 7 247 h 1 E [[w ] 2 0,E + + 2 w 2 0,K i i=1 C h 1 E [[P hw ]] 2 0,E 2 Ph w 2 0,K i + I P h w 2 0,K i, i=1 for an interior edge/face E, and to 248 h 1 E w 2 0,E + w 2 0,K C h 1 E P hw 2 0,E + P h w 2 0,K + I P h w 2 0,K, for a boundary edge/face E The estimate 1/2 C 1 w 1,h P h w 1,h + I P h w 2 0,K is a consequence of 247 248 The proof is complete Lemma 26 There is a positive constant constant C such that 249 sup τ,v S h V h B h ϕ, w ; τ, v τ 0 + v 1,h C ϕ 0 + w 1,h ϕ, w S h V h Proof Let ϕ, w S h V h be arbitrary By choosing v = v V h and using the equilibrium condition 222 we then get 250 B h ϕ, w ; τ, v = ϕ, τ + div τ, w + div ϕ, v = ϕ, τ + div τ, P h w + div ϕ, v = Bϕ, P h w ; τ, v, Hence, the stability of Lemma 22 implies that we can choose τ, v such that 251 B h ϕ, w ; τ, v ϕ 2 0 + P h w 2 1,h and 252 τ 0 + v 1,h C 1 ϕ 0 + P h w 1,h

8 C LOVADINA AND R STENBERG Next, 210 and Schwarz inequality give 253 B h ϕ, w ; 0, I P h w = div ϕ, I P h w + w ϕ, I P h w K C 2 ϕ 0 I P h w 1,h + w, I P h w K = C 2 ϕ 0 I P h w 1,h + + I P h w 2 0,K P h w, I P h w K C 2 ϕ 0 + P h w 1,h I Ph w 1,h + I P h w 2 0,K We now notice that I P h w is L 2 orthogonal to the piecewise constant functions; therefore, a scaling argument shows that 1/2 254 I P h w 1,h C 3 I P h w 2 0,K For α > 0, we obtain from 253 and 254 255 B h ϕ, w ; 0, I P h w 1 C2 ϕ 0 + P h w 2 α 1,h 2α 2 I P hw 2 1,h + I P h w 2 0,K C2 ϕ 0 + P h w 2 1,h 1 2α Choosing α > 0 sufficiently small, we get 256 B h ϕ, w ; 0, I P h w C 4 + 1 αc2 3 I P h w 2 2 0,K I P h w 2 0,K ϕ 2 0 P h w 2 1,h Combining 251 and 256, with δ > 0 to be chosen, we have 257 B h ϕ, w ; τ, v + δi P h w 1 δc 4 ϕ 2 0 + P h w 2 1,h + δc4 I P h w 2 0,K Next, by 241 we have 258 P h w 2 1,h + δ I P h w 2 0,K C 5 w 2 1,h

ENERGY NORM A POSTERIORI ESTIMATES FOR MIXED METHODS 9 From 252 and 240 we have 259 τ 0 + v + δi P h w 1,h τ 0 + v 1,h + δ I P h w 1,h C 1 ϕ 0 + P h w 1,h + δ I Ph w 1,h C 6 ϕ 0 + w 1,h Choosing δ = 1/2C 4, estimate 249 is proved by combining 257 259 Theorem 27 The following a priori error estimate holds σ σ h 0 + u u h 1,h C σ R h σ 0 + inf v V h u v 1,h Proof From Lemma 26 it follows that there is ϕ, w S h Vh, with ϕ 0 + w 1,h C, such that 260 σh R h σ 0 + u h v 1,h Bh σ h R h σ, u h v ; ϕ, w Next, from the definition of B h and the equations 13 14 it follows that 261 B h σ, u; ϕ, w + f, w = 0 Hence it holds 262 B h σ h R h σ, u h v ; ϕ, w = B h σ R h σ, u v ; ϕ, w + f P h f, w Writing out the right hand side we have 263 B h σ R h σ, u v ; ϕ, w + f P h f, w = σ R h σ, ϕ + div ϕ, u v + div σ R h σ, w + u v σ R h σ, I P h w K + f P h f, w The commuting property 224 gives 264 div σ R h σ, w = f P h f, w Hence, the third and the last term on the right hand side of 263 cancel The other terms are directly estimated 265 266 and using 241 267 σ R h σ, ϕ σ R h σ 0 ϕ 0 C σ R h σ 0, div ϕ, u v C ϕ 0 u v 1,h C u v 1,h u v σ R h σ, I P h w K C u v 1,h + σ R h σ 0 w 1,h C u v 1,h + σ R h σ 0 The assertion then follows by collecting the above estimate and using the triangle inequality For our choices of spaces we obtain the estimates with the assumption of a sufficiently smooth solution

10 C LOVADINA AND R STENBERG Corollary 28 There are positive constants C such that 268 σ σ h 0 + u u h 1,h Ch k+1 u k+2 269 σ σ h 0 + u u h 1,h Ch k u k+1 for BDM, for RTN 3 A-posteriori estimates We define the following local error indicators on the elements 31 η 1,K = u h σ h 0,K, η 2,K = h K f P h f 0,K, and on the edges 32 η E = h 1/2 E [[u h]] 0,E Using these quantities, the global estimator is 33 η = η 2 1,K + η2,k 2 + η 2 E 1/2 The efficiency of the estimator is given by the following lower bounds, which directly follow from 13 using the triangle inequality, and from 32 noting that [[u]] = 0 on each edge E Theorem 31 It holds η 1,K u u h 0,K + σ σ h 0,K, 34 η E = h 1/2 E [[u u h]] 0,E As far as the estimator reliability is concerned, below we will use two different techniques 31 Reliability via a saturation assumption The first technique to prove the upper bound is based on the following saturation assumption We let C h/2 be the mesh obtained from C h by refined each element into 2 n n = 2, 3 elements For clarity all variables in the spaces defined on C h will be equipped with the subscript h whereas h/2 will be used for those defined on C h/2 Accordingly, we let σ h/2, u h/2 S h/2 Vh/2 be the solution to 35 B h/2 σ h/2, u h/2 ; τ h/2, vh/2 + P h/2f, vh/2 = 0 τ h/2, vh/2 S h/2 Vh/2 As already done in [5], we make the following assumption for the solutions of 235 and 35 Saturation assumption There exists a positive constant β < 1 such that 36 σ σ h/2 0 + u u h/2 1,h/2 β σ σ h 0 + u u h 1,h Since it holds 37 u u h 1,h u u h 1,h/2 we also have 38 σ σ h/2 0 + u u h/2 1,h/2 β σ σ h 0 + u u h 1,h/2 Using the triangle inequality we then get 39 σ σ h 0 + u u h 1,h/2 1 σh/2 σ h 0 + u h/2 1 β u h 1,h/2

ENERGY NORM A POSTERIORI ESTIMATES FOR MIXED METHODS 11 By again using 37 we obtain 310 σ σ h 0 + u u h 1,h 1 σh/2 σ h 0 + u h/2 1 β u h 1,h/2 We now prove the following result Theorem 32 Suppose that the saturation assumption 36 holds Then there exists a positive constant C such that 311 σ σ h 0 + u u h 1,h Cη Proof By 310 it is sufficient to prove the following bound 312 σ h/2 σ h 0 + u h/2 u h 1,h/2 Cη By Lemma 26 applied to the finer mesh C h/2, there is τ h/2, v h/2 S h/2 V h/2, with τ h/2 0 + v h/2 1,h/2 C, such that σh σ h/2 0 + u h u h/2 1,h/2 313 Using the fact that B h/2 σ h σ h/2, u h u h/2 ; τ h/2, v h/2 314 σ h/2, τ h/2 + div τ h/2, u h/2 = 0 we have 315 B h/2 σ h σ h/2, u h u h/2 ; τ h/2, v h/2 = σ h σ h/2, τ h/2 + div τ h/2, u h u h/2 + div σ h σ h/2, v h/2 + /2 u h u h/2 σ h σ h/2, I P h/2 v h/2 K = σ h, τ h/2 + div τ h/2, u h + div σ h σ h/2, v h/2 + /2 u h σ h, I P h/2 v h/2 K, We now notice that it holds cf 29 316 C τ h/2 0,h τ h/2 0 τ h/2 0,h τ h/2 S h/2 Therefore, using 316 and 31 33, we obtain 317 σ h, τ h/2 + div τ h/2, u h = σ h u h, τ h/2 K + τ h/2 n, [[u h ] E σ h u h 0,K τ h/2 0,K + τ h/2 n 0,E [[u h ] 0,E η τ h/2 0,h ηc τ h/2 0 Cη Similarly for the last term in 315 we get using 240 318 u h σ h, I P h/2 vh/2 K Cη I P h/2 vh/2 1,h/2 /2 Cη v h/2 1,h/2 Cη

12 C LOVADINA AND R STENBERG When estimating the term div σ h σ h/2, vh/2 in 315 we recall that div σ h = P h f and div σ h/2 = P h/2 f, and that P h, P h/2 are L 2 -projection operators Therefore, we have div σ h σ h/2, v h/2 = P h/2f P h f, v h/2 319 = P h/2f f, v h/2 + f P hf, v h/2 = P h/2 f f, v h/2 P h/2v h/2 + f P hf, v h/2 P hv h/2 Next, we use the following interpolation estimates, which are easily proved by standard scaling arguments cf [5, Lemma 31]: where v h/2 P hv h/2 0,K Ch K v h/2 1,h/2,K, K C h, v h/2 2 1,h/2,K = K i v h/2 2 0,K i + E i h 1 E i [[v h/2 ] 2 0,E i Here K i K are the elements of C h/2 and E i are the edges of Γ h/2 lying in the interior of K This gives f P h f, vh/2 P hvh/2 C h 2 K f P h f 2 1/2 v 320 0,K h/2 1,h/2 C h 2 K f P h f 2 1/2 0,K Cη We also have 321 P h/2 f f, v h/2 P h/2v h/2 f P h/2 f 0,K vh/2 P h/2vh/2 0,K /2 C h K f P h/2 f 0,K vh/2 0,K /2 C h 2 K f P h/2 f 2 1/2 v 0,K h/2 1,h/2 /2 C h 2 K f P h/2 f 2 1/2 0,K /2 C h 2 K f P h/2 f 2 1/2 0,K Since, by the properties of L 2 -projection operators, it holds from 321 we obtain f P h/2 f 0,K f P h f 0,K K C h, 322 P h/2 f f, vh/2 P h/2vh/2 C h 2 K f P h f 2 1/2 0,K Cη By collecting the estimates 317 320 and 322, from 315 we get 323 B h/2 σ h σ h/2, u h u h/2 ; τ h/2, vh/2 Cη The assertion now follows from 313

ENERGY NORM A POSTERIORI ESTIMATES FOR MIXED METHODS 13 We have presented the above proof since this is rather general and can be used for other problems as well In [13] we use it for a plate bending method 32 Reliability via a Helmholtz decomposition Now, let us give another proof of the estimator reliability, not relying on the saturation assumption Theorem 33 Suppose that Ω R 2 is a simply connected domain Then there exists a positive constant C such that 324 σ σ h 0 + u u h 1,h Cη Proof We use the techniques of [11] and [10] We first notice that σ σ h, ϕ 325 σ σ h 0 = sup ϕ L 2 Ω ϕ 0 For a generic ϕ L 2 Ω, we consider the L 2 -orthogonal Helmholtz decomposition see, eg [12]: 326 ϕ = ψ + curl q, ψ H 1 0 Ω, q H 1 Ω/R, with 327 ϕ 0 = ψ 2 0 + curl q 2 0 1/2 Therefore, from 325 327 we see that it holds σ σ h, ψ σ σ h, curl q 328 σ σ h 0 sup + sup ψ H0 1Ω ψ 1 q H 1 Ω/R q 1 Given ψ H 1 0 Ω, from 14 and 16 it follows that 329 div σ σh, P h ψ = 0 Hence, we have 330 σ σ h, ψ = div σ σ h, ψ As a consequence, we get cf 31 = div σ σ h, ψ P h ψ 1/2 ψ 1 C h 2 K div σ σ h 2 0,K 1/2 ψ 1 C h 2 K f P h f 2 0,K 331 σ σ h, ψ 1/2 sup C h 2 ψ H0 1Ω ψ K f P h f 2 0,K = C 1 η 2 2,K 1/2

14 C LOVADINA AND R STENBERG To continue, let I h q be the Clément interpolant of q in the space of continuous piecewise linear functions see [4], for instance satisfying 1/2 332 q I h q 1 + h 1 E q I hq 0,E 2 C q 1 Noting that curl I h q S h, and div curl I h q = 0, from 13 and 15 we get 333 σ σ h, curl I h q = 0 Therefore, using 332, one has 334 σ σ h, curl q = σ σ h, curlq I h q = u σ h, curlq I h q = σ h, curlq I h q = σh u h, curlq I h q + K u h, curlq I h q K 1/2 q 1 C σ h u h 2 0,K + u h, curlq I h q K Furthermore, an integration by parts and standard arguments and 332 give u h, curlq I h q K = u h t, q I h q K = [[ u h t]], q I h q E 335 1/2 1/2 h E [ u h t]] 2 0,E h 1 E q I hq 2 0,E 1/2 q 1 C h 1 E [[u h ] 0,E 2 From 334 and 335 we obtain see 31 and 32 336 σ σ h, curl q C q 1 sup q H 1 Ω/R Using 331 and 336 we deduce σ h u h 2 0,K + 1/2 = C η1,k 2 + ηe 2 337 σ σ h 0 C η1,k 2 + η2,k 2 + η 2 E We now estimate the term u u h 1,h We first recall that h 1 E [[u h]] 2 0,E 1/2 338 u u h 1,h = u u h 2 0,K + h 1 E [u u h ] 2 0,E 1/2 1/2

ENERGY NORM A POSTERIORI ESTIMATES FOR MIXED METHODS 15 and we notice that cf 32 1/2 339 h 1 E [u 1/2 u h ] 2 0,E = h 1 E [[u h]] 2 0,E = η 2 E 1/2 We have u u h 2 0,K = u u h, u u h K = σ u h, u u h K 340 by which we obtain = σ σ h, u u h + σ K h u h, u u h K σ σ h 0,K + σ h u h 0,K u u h 0,K, 341 u u h 0,K σ σ h 0,K + σ h u h 0,K Hence we infer 1/2 342 u u h 2 0,K σ σh 0 + σ h u h 2 0,K Using 337 and recalling 31, from 342 we get 1/2 343 u u h 2 0,K C η1,k 2 + η2,k 2 + Therefore, joining 339 and 343 we obtain 344 u u h 1,h C η1,k 2 + η2,k 2 + From 337 and 344 we finally deduce see 33 η 2 E 345 σ σ h 0 + u u h 1,h C η1,k 2 + η2,k 2 + We end the paper by the following 1/2 η 2 E η 2 E 1/2 1/2 1/2 = Cη Remark 34 On the estimate in the Hdiv:Ω-norm In the paper we have repeatedly used the fact that by the equilibrium property 222 we have div σ σ h = P h f f and hence div σ σ h 0 = f P h f 0 is a quantity that is directly computable from the data to the problem For the BDM spaces it furthermore holds that for a general loading and a smooth solution it holds f P h f 0 = Oh k, whereas σ σ h 0 = Oh k+1, and hence this trivial component in the Hdiv:Ω norm can dominate the whole estimate References [1] A Alonso, Error estimators for a mixed method, Numer Math, 74 1994, pp 385 395 [2] D N Arnold and F Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél Math Anal Numér, 19 1985, pp 7 32 [3] I Babuška, J Osborn, and J Pitkäranta, Analysis of mixed methods using mesh dependent norms, Math Comp, 35 1980, pp 1039 1062

16 C LOVADINA AND R STENBERG [4] D Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, 2001 [5] D Braess and R Verfürth, A posteriori error estimators for the Raviart-Thomas element, SIAM J Numer Anal, 33 1996, pp 2431 2444 [6] J H Bramble and J Xu, A local post-processing technique for improving the accuracy in mixed finite-element approximations, SIAM J Numer Anal, 26 1989, pp 1267 1275 [7] F Brezzi, J Douglas, Jr, R Durán, and M Fortin, Mixed finite elements for second order elliptic problems in three variables, Numer Math, 51 1987, pp 237 250 [8] F Brezzi and M Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, 1991 [9] F Brezzi, J Douglas Jr, and L Marini, Two families of mixed finite elements for second order elliptic problems, Numer Math, 47 1985, pp 217 235 [10] C Carstensen, A posteriori error estimate for the mixed finite element method, Math Comp, 66 1997, pp 465 476 [11] E Dari, R Durán, C Padra, and V Vampa, A posteriori error estimators for nonconforming finite element methods, RAIRO Modél Math Anal Numér, 30 1996, pp 385 400 [12] V Girault and P Raviart, Finite Element Methods for Navier-Stokes Equations Theory and Algorithms, Springer-Verlag, 1986 [13] C Lovadina and R Stenberg, A posteriori error analysis of the linked interpolation technique for plate bending problems, SIAM J Numer Anal, Accepted for publication [14] J-C Nédélec, A new family of mixed finite elements in R 3, Numer Math, 50 1986, pp 57 81 [15] P Raviart and J Thomas, A mixed finite element method for second order elliptic problems, in Mathematical Aspects of the Finite Element Method Lecture Notes in Math 606, Springer- Verlag, 1977, pp 292 315 [16] R Stenberg, Some new families of finite elements for the Stokes equations, Numer Math, 56 1990, pp 827 838 [17] R Stenberg, Postprocessing schemes for some mixed finite elements, RAIRO Modél Math Anal Numér, 25 1991, pp 151 167 Dipartimento di Matematica, Università di Pavia and IMATI-CNR, VIa Ferrata 1, Pavia 27100, Italy E-mail address: carlolovadina@unipvit Institute of Mathematics, Helsinki University of Technology, PO Box 1100, 02015 TKK, Finland E-mail address: rolfstenberg@tkkfi