Goal-oriented error estimation based on equilibrated-flux reconstruction for finite element approximations of elliptic problems

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1 Goal-oriented error estimation based on equilibrated-flux reconstruction for finite element approximations of elliptic problems Igor Mozolevski, Serge Prudomme o cite tis version: Igor Mozolevski, Serge Prudomme. Goal-oriented error estimation based on equilibrated-flux reconstruction for finite element approximations of elliptic problems <al > HAL Id: al ttps://al.arcives-ouvertes.fr/al Submitted on 5 May 2014 HAL is a multi-disciplinary open access arcive for te deposit and dissemination of scientific researc documents, weter tey are publised or not. e documents may come from teacing and researc institutions in France or abroad, or from public or private researc centers. L arcive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recerce, publiés ou non, émanant des établissements d enseignement et de recerce français ou étrangers, des laboratoires publics ou privés.

2 Goal-oriented error estimation based on equilibrated-flux reconstruction for finite element approximations of elliptic problems Igor Mozolevski a,, Serge Prudomme b a Federal University of Santa Catarina UFSC) Campus Universitário, rindade, Florianópolis, SC, Brazil b Department of Matematics and Industrial ngineering École Polytecnique de Montréal C.P. 6079, Succ. Centre-ville, Montréal, Québec H3C 3A7, Canada Abstract We propose an approac for goal-oriented error estimation in finite element approximations of second-order elliptic problems tat combines te dualweigted residual metod and equilibrated-flux reconstruction metods for te primal and dual problems. e objective is to be able to consider discretization scemes for te dual solution tat may be different from tose used for te primal solution. It is only assumed ere tat te discretization metods come wit a priori error estimates and an equilibrated-flux reconstruction algoritm. A ig-order discontinuous Galerkin dg) metod is actually te preferred coice for te approximation of te dual solution tanks to its flexibility and straigtforward construction of equilibrated fluxes. One contribution of te paper is to sow ow te order of te dg metod for asymptotic exactness of te proposed estimator can be cosen in te cases were a conforming finite element metod, a dg metod, or a mixed Raviart- omas metod are used for te solution of te primal problem. Numerical experiments are also presented to illustrate te performance and convergence of te error estimates in quantities of interest wit respect to te mes size. Keywords: Finite element metod, goal-oriented estimates, quantity of interest, dual problem, asymptotically-exact error estimates Corresponding autor. el/fax: ) mail addresses: igor.mozolevski@ufsc.br Igor Mozolevski), serge.prudomme@polymtl.ca Serge Prudomme) Preprint submitted to Comput. Metods in Appl. Mec. and ng. April 30, 2014

3 1. Introduction A variety of finite element discretization scemes, suc as te mixed finite element metods, te non-conforming finite element metods, or discontinuous Galerkin dg) finite element metods, ave been developed during te past decades in order to provide better approximation properties tan tose offered by te classical finite element metod depending on te type of partial differential equations at and. ese metods ave become increasingly popular and are now widely used for solving problems of various interest in engineering and sciences. At te same time, te necessity to obtain accurate finite element approximations to given boundaryvalue problems as stimulated te development of a posteriori error estimators tat provide fully computable, reliable, and efficient error bounds in terms of te problem data and te finite element approximation. In te case of standard conforming finite element scemes, a posteriori error estimates wit respect to global energy norms are indeed well-establised, see e.g. [52, 4, 7, 8, 43, 31, 50, 53]. A posteriori error estimation for nonconforming finite element metods wit application to second-order elliptic problems as recently seen significant progress and is still te subject of sustained researc efforts, see e.g. [9, 34, 51, 55, 33, 1, 18, 42, 41, 27, 24] for dg metods and [16, 35, 2, 17, 37] for mixed finite element metods. We also refer te reader to [36, 18, 3, 28, 22] and references terein for te presentation of unifying frameworks on te topic. In practical applications, end users are owever interested in error estimates in some specific features of te true solution. ese so-called quantities of interest, eiter local or global, are represented as functionals defined on te vector space of trial solutions of te boundary-value problem. rror estimation wit respect to suc functionals is usually referred to as goal-oriented error estimation. e key ingredient for goal-oriented error estimation is te formulation of an auxiliary problem, te dual problem to te primal problem, wose solution provides necessary information for reliable estimates of te error in te goal functional. Several strategies for goal-oriented error estimation ave been proposed in te case of elliptic problems: goal-oriented error estimates based on energy norm of te errors in te primal and dual solutions were introduced in [48, 45, 46, 49] and furter developed by various autors, see for example [4, 5], and references terein, error estimates using 2

4 te dual-weigted residual metod were proposed in [25, 10, 8]; functional a posteriori error estimates were developed in [43, 50]; estimates based on te gradient-recovery metod were considered in [39, 38, 47, 44, 40]; finally, goal-oriented estimates for discontinuous Galerkin metods in te case of second-order elliptic problems were derived in [32]. e general approac to obtain goal-oriented error estimates consists, on one and, in deriving an error representation involving te residual functional and te exact dual adjoint) solution, and, on te oter and, in constructing a sufficiently accurate approximation of te adjoint solution in order to obtain a fully computable and reliable error estimate along wit local refinement indicators. In te case of te classical conforming finite element metod, suc approximation to te adjoint solution is usually calculated on a refinement of te mes used for te primal solution, wit te same polynomial degree, or, preferably, on te same mes, but wit a iger polynomial degree. In tis paper, we propose an alternative approac to goal-oriented estimation by considering an error representation tat does not uses te ortogonality property and is amenable to different types of discretization of te primal and dual problems. We only suppose ere tat te metod used to discretize te primal problem produces piecewise polynomial solutions from wic equilibrated fluxes can be reconstructed tis is not a restrictive property) and satisfies standard a priori error estimates in te L 2 and energy norms. Using an error representation similar to te one used in te dual-weigted residual metod, we use te flux-equilibration tecnique to decompose te error into a computable error estimator and a iger-order remainder. It is well known, see e.g. [5], tat in order to obtain an efficient error estimator effectivity indices remain close to unity), te dual problem must be approximated using a iger-order approximation tan te one used in te finite element approximation of te primary problem. We propose in tis paper, in order to approximate te dual solution, tat a ig-order discontinuous Galerkin metod be used and applied on te same mes as tat used for te discretization of te primal problem. e coice of te dg metod seems natural owing to its flexibility in using non-uniform ig-order polynomials. Furtermore, te dg metod benefits from te fact tat it is locally conservative; it implies tat te construction of equilibrated fluxes, needed for te evaluation of te proposed error estimator, is rater straigtforward, see e.g. [36, 26, 19, 12]. Finally, we sow tat, depending on te approximation properties of te primal metod and problem data, te order of te dg metod, used to solve te dual problem, can be cosen in suc a 3

5 way tat te respective error estimator is asymptotically exact. e paper is organized as follows. Section 2 introduces te model primal) problem, te corresponding dual problem, and preliminary notation for goal-oriented error estimation. Section 3 presents a suitable error representation for te goal functional based on te use of equilibrated fluxes in terms of te finite element solutions to te primal and dual problems. We also sow ow te error representation can be decomposed into a fully computable error estimator and a iger-order term, wic can be evaluated using 1) a priori estimates wit respect to te primal and dual discretization scemes and 2) metods for reconstruction of equilibrated fluxes. We briefly introduce in Section 4 te symmetric version of te dg metod. We ten sow ow te order of te dg metod, used for te discretization of te dual problem, can be cosen to guarantee asymptotical exactness of te error estimator wen discretizing te primal problem by ig-order dg metods, conforming finite element metods, and mixed finite element metods, as described in Sections 5, 6, and 7, respectively. Finally, we present some numerical examples to demonstrate te performance of te estimators in Section 8 before concluding in Section Model problem In tis work, we are primarily interested in general linear boundary-value problems defined in terms of elliptic second-order partial differential equations, but te results could be extended to more general situations. For te sake of clarity in te presentation, we sall restrict ourselves to a simple model problem. Let Ω R 2 be a polygonal domain wit boundary Ω. We consider te omogeneous Diriclet boundary-value problem wose solution u satisfies D u) = f in Ω, 1) u = 0 on Ω. were coefficient D = Dx), x Ω, is a piecewise constant strictly positive scalar function and f L 2 Ω). e weak formulation of te problem reads: Find u H0 1 Ω) suc tat Bu,v) = Fv), v H1 0 Ω) 2) 4

6 were B and F are te bilinear and linear forms on H0 1 Ω), respectively, defined as: Bu,v) = D u v, 3) Ω Fv) = fv. 4) Ω We sall suppose ere tat te elliptic regularity property olds true for te model problem; i.e. tere exist a unique weak solution u H 2 Ω) to 2) and a constant C > 0 tat only depends on Ω, suc tat u H 2 Ω) C f L 2 Ω). 5) Note tat te conditions for elliptic regularity are ensured if te polygonal domain Ω is assumed convex; in more general cases, te regularity of te solution usually depends on te internal angles between te boundary edges of Ω, see [30]. In te remainder of te paper, we will denote by C a generic constant tat may depend on te problem data and oter model parameters. is dependence will be made explicit in te text wen necessary. We assume tat we sall be interested in te linear goal functional Q defined on L 2 Ω) suc tat: Qv) = qv, 6) were q L 2 Ω) denotes te Riesz representer of Q. We introduce te corresponding dual problem, in strong form as: Ω D p) = q in Ω, p = 0 on Ω, 7) and in weak form as: Find p H 1 0Ω) suc tat Bv,p) = Qv), v H 1 0Ω). 8) We again empasize tat, owing to te elliptic regularity and te fact tat q L 2 Ω), we ave tat p H 2 Ω). It is also straigtforward to sow from 2) and 8) tat Fp) = Bu,p) = Qu). 9) 5

7 Let, > 0, be a family of sape-regular triangular meses on Ω, see e.g. [14], were = max ) denotes te mes size and ) is te diameter of mes element. We assume ere tat te edges of two neigboring elements will perfectly matc wit eac oter, in te sense tat te meses will be free of so-called anging nodes. For a triangular mes, wedenoteby tesetofalledgesintemes; canbefurterdecomposed into te set of interior edges i and te set of boundary edges, tat is = i. By definition, an edge is said to be an interior edge if tere exist two triangles, + suc tat = + and is said to be a boundary edge if tere exists one and only one triangle suc tat = Ω. For a polygonal domain Ω, we ave Ω =. For any function v L 2 Ω), assumed sufficiently smoot to admit a trace on all possibly different on eiter side of an edge i ), let us denote te jump of v along by [[v]] := v v +, i, [[v]] := v,, 10) were v ± denotes te trace on te respective side of. For i, we denote by n te unit normal vector to pointing from toward +, wereas for, we set n equal to te unit external normal n to Ω. e orientation of n for interior faces is cosen in accordance wit te definition of te jump; in suc a case, te arbitrariness in te coice of and + in te definition of te interior edge is irrelevant. We also introduce te standard aritmetic average of v along as {{v}} = 1 2 v +v + ), i, {{v}} = [[v]] = v,. 11) Finally, let P k ) denote te vector space of polynomials on of degree k or less. We ten introduce te finite element space V k as: V k := {v L 2 Ω) : v P k ), }. 12) For a given mes, let us denote by u V k an arbitrary approximation of te exact solution u to te primal problem 2). e objective in tis paper is to study te approximation error in te goal functional = Qu) Qu ). 13) 6

8 At tis stage, it is not necessary to specify te numerical sceme wit wic te approximate solution is obtained. Since te goal functional considered ere is assumed linear and continuous on L 2 Ω), it is sufficient to suppose tat u converges to te exact solution u in L 2 Ω) in order for Qu ) to be a reasonable approximation of Qu). We will make use of tefollowing definition interemainder of tepaper: Definition 1. e error = Qu) Qu ) in te goal functional is said to be properly of order l > 0 wit respect to te discrete solution u if tere exist constants C > 0 and C > 0, tat depend on u and Q only, suc tat for sufficiently small > 0. C l Qu) Qu ) C l 14) We propose below an asymptotically exact estimator of te error in te goal functional 13) based on te condition tat te error be properly of order l > 0 wit respect to a discrete solution family {u } >0. 3. Goal-oriented error representation In tis section, we briefly review te notion of equilibrated fluxes and present a goal-oriented error representation in terms of reconstructed equilibration fluxes. We first recall te space Hdiv, Ω) of vector-valued functions Hdiv,Ω) = {t [L 2 Ω)] 2 : t L 2 Ω)}, 15) and te Raviart-omas finite element space of order m N 0 : R m ) = {t Hdiv,Ω) : t [P m )] 2 +x P m ), }. 16) Definition 2. Let u V k, k N 0, and let l = max{0,k 1}. Let π l : L 2 Ω) V l denote te L2 -ortogonal projection operator. A vector t u ) R l ), reconstructed from te approximation u, is said to be an equilibrated flux wit respect to Problem 1) if t u ) = π l f). 17) Note tat we will use te notation t l u ) = t u ) wen te order of te Raviart-omas space in wic te flux is reconstructed needs to be mentioned explicitly. 7

9 e concept of equilibrated fluxes as been widely used to construct accurate a posteriori error estimates for conforming finite element approximations since suc estimates are exempt of unknown constants, see for example[4, 11]. In fact, various flux reconstruction tecniques ave been developed over te years in te case of elliptic problems and some of te tecniques ave been extended to non-conforming conservative metods [36, 26, 12], to first-order conforming finite element metod [13], or to te finite volume and finite difference metods [54]. We propose ere to investigate ow reconstructed equilibrated fluxes could be used in goal-oriented error estimation. We start by deriving te following error representation wit respect to te goal functional, using 9), te dual solution p to problem 7), and integration by parts, = Qu) Qu ) = Fp) Qu ) ) = fp+ D p u Ω Ω = [ fp D u p ) + n D p ) u ]. 18) Introducing te jump 10) in u across edge, te sum of te edge integrals can be recast as: = n D p ) [[u ]]. 19) fp D u p ) + Note tat for an arbitrary t R l ), above relation can be rewritten, by simple addition and subtraction of te term t p, as: = t D u ) p = [ fp+t p ) n D p ) [[u ]] f t ) p+ t n ) p t D u ) p+ ] n D p ) [[u ]]. 8

10 Since, owing to te omogeneous Diriclet boundary condition on te dual solution and te continuity of te normal component of t, [ ] t n)p = [[t n]]p = 0 we ten obtain: = ) f t p + n D p ) [[u ]]. t +D u ) p We are now ready to establis te following error representation in terms of te goal functional. eorem 1. Let u V k, k N 0, be an approximation of te solution u H0Ω) 1 to te primal problem 1) and let p V m, m > k, be an approximation of te solution p H0 1Ω) H2 Ω) to te dual problem 7). Suppose tat u and p admit reconstructed equilibrated fluxes t u ) R l ), l = max{0,k 1} and t p ) R m 1, respectively. en te error in te goal functional Q can be represented as = Qu) Qu ) = ηu,p ;t u ),t p ))+Ru,p;u,p ), 20) were te error estimator η is defined as: η = ηu,p ;t u ),t p )) = f π l f)) p + n t p ))[[u ]], and te remainder term is: Ru,p,f;u,p ) = + f π l f))p p ) t u )+D u ) D 1 t p ) t u )+D u ) D 1 D p+t p ) ) n D p+t p )) ) [[u ]]. 21) 22) 9

11 Remark 1. e error estimator η can be easily decomposed into a sum of local element error contributions. Indeed, let us note tat t p ) n[[u ]] = χ )n t p ))[[u ]], 23) were χ denotes te edge indicator function on, tat is χ ) = 1/2, if i, and χ = 1, if. erefore, in tat case, te decomposition would take te form: η = η ) = η O )+η )+η H ) 24) wit η O ) = η ) = η H ) = f π l f)) p t u )+D u ) D 1 t p ) χ )[[u ]]n t p )) Here η O ) represents te data oscillation in element wit respect to te primal problem, weigted by te dual approximate solution; te flux estimator η ) measures te deviation of te discrete gradient D u from te reconstructed flux t u ) Hdiv, ); and η H ) measures te deviation of u from H0 1 Ω), bot weigted by te reconstructed equilibrated flux of te dual approximate solution. Note tat te error estimator η is fully computable once te flux reconstructions for t u ) and t p ) are establised. Remark 2. Depending on te properties of te data in te primal problem, te error estimator 21) may take a simpler form; e.g. te oscillation term disappears for f V l. Remark 3. Depending on te discretization metod used, te finite element approximation u may weakly or strongly) satisfy additional continuity properties along te interior edges and specific boundary conditions on te boundary edges of te mes. Values of te different components of te error estimator will tus strongly depend on te coice of te numerical metod used in te approximation of te primal problem. For example, if conforming finite element metods are used, η H = 0. 10

12 In te remainder of te paper, we sow ow te error representation 21) can provide an asymptotically exact goal-oriented estimator of te error in approximations of te primal problem. One crucial ingredient of te estimator is te coice of te finite element metod for te approximation of te dual solution. If one prefers to keep te same mes to solve te dual problem as tat used for te primal finite element solution, wile retaining an efficient error estimator, iger-order metods sould be employed, see e.g. [5]. Furtermore, it is desirable tat te discretization metod for te dual problem allows for computationally inexpensive equilibrated-flux reconstruction. Based on tese facts, we propose ere to use a ig-order discontinuous Galerkin discretization metod to approximate te dual solution. Advantages of te dg metod are tat ig-order elements in any space dimension can be easily implemented and tat it is locally conservative, wic yields in a straigtforward manner equilibrated fluxes. In fact, it as been sown to provide ceap and local reconstruction algoritms [35, 26, 19, 12]. 4. e dg metod We briefly introduce in tis section te symmetric interior penalty dg metod and present some basic results on te stability and approximation propertiesoftemetod. Wefirst introducetebilinear formb : V k Vk R as B u,v ) = D u v {{n D u }}[[v ]] + {{n D v }}+γ [[v ]])[[u ]], 25) were γ > 0 denotes te interior penalty parameter to be specified later. For te primal boundary value problem 1), te symmetric interior penalty dg metod is defined as follows see e.g. [20]): Find u V k suc tat B u,v ) = Fv ), v V k. 26) Since symmetric dg metod is adjoint consistent, te corresponding dual problem reads: Find p V m suc tat B v,p ) = Qv ), v V m. 27) 11

13 For completeness, we present below some basic analytical results referring to te symmetric interior penalty dg metod; owing to te symmetry of te metod and of te primal and dual problems, we can limit te analysis to te primal problem only. Firstly, let us note tat, for sufficiently large penalty parameters, problem 26) as a unique solution, see e.g. [20]. We introduce ere a computable lower bound γ ) on γ, as presented in [6] for a general setting, tat is sufficient to sow existence and uniqueness of te dg metod. erefore, we suppose from now on tat te penalty parameters satisfy γ γ ). 28) Convergence properties of ig-order dg metods are summarized in te following teorem, see e.g. [20]. eorem 2. Let us suppose tat Ω is a convex polygonal domain in R 2 and tat te solution to Problem 1) belongs to H k+1 Ω). Let, > 0, be a family of matcing sape regular triangular meses on Ω and assume tat parameters γ satisfy te stability condition 28). en, te following estimates old for te dg solution to 26) wit a positive constant C tat depends only on te sape-regularity of : u u C k u H k+1 Ω) 29) u u L 2 Ω) C k+1 u H k+1 Ω), 30) were te dg energy norm is defined on V k as v 2 = v 2 L 2 K)) 2 + γ [[v ]] 2 L 2 ). In te following sections, we will sow tat, depending on te approximation properties of te primal metod and on te problem data, te order of te dg metod, used to solve te dual problem, can be cosen in suc a way tat te respective error estimator is asymptotically exact. We will consider te following discretizations of te primal problem: ig-order discontinuous Galerkin metod, te first-order conforming finite element metod, and te lowest order mixed Raviart-omas finite element metod. Oter non-conforming metods, finite volume and finite difference metods can be considered in a similar fasion using corresponding flux reconstruction algoritms, see e.g. [54, 29]. 12

14 5. Goal-oriented error estimates for dg metods We consider ere te symmetric dg metod of order k N for te discretization of Problem 1). We construct equilibrated fluxes t k 1 u ) R k 1 in terms of te dg solution u V k, depending on te value of k, as follows see e.g. [36, 21]): k 1 : n t k 1 u ) q = {{n D u }}+γ [[u ]] ) q, k 2 : t k 1 u ) r = q P k 1 ), ; 31) D u r + χ )n Dr [[u ]], r P k 2 )) d,. 32) Note tat te flux reconstruction using 31)-32) is computationally inexpensive and is relatively easy to implement as an element-wise procedure. Definition 3. A function f L 2 Ω) is said to ave oscillations of order l N 0 on V k if tere exists a constant C, tat depends on f and on te sape regularity of, suc tat f π k f L 2 Ω) C l, > 0. 33) For example, it follows from te optimality of te ortogonal projection tat a function f H s Ω) as oscillations of order l = min{k +1,s} on V k. eorem 3. Let f L 2 Ω) in Problem 1) ave oscillations of order k 1 in V k 1. Let u H k+1 Ω) and u V k be te solutions to 1) and 26), respectively. Moreover, let t u ) R k 1 be te equilibrated flux wit respect to u given by 31)-32). en tere exists a constant C > 0, tat depends on te problem data {D,f} and on te sape-regularity of, suc tat t k 1 u )+D u [L 2 Ω)] 2 Ck. 34) Proof: quality 17) follows from 31)-32) and 26) by straigtforward calculations. It is also known see e.g. [36, 21]) tat te following estimate olds for t k 1 u ) given by 31)-32): t k 1 u )+D u [L 2 Ω)] 2 u u + f π k 1 f L 2 Ω). 13

15 erefore, inequality 34) follows from above inequality owing to te a priori estimate 29) and oscillation estimate 33). e next teorem provides te key result to investigate exactness of te error estimator 24) for te primal dg metods. eorem 4. Let f L 2 Ω) in Problem 1) ave oscillations of order k 1 in V k 1 and let u H k+1 Ω) be te solution to 1). Let p H m+1 Ω) be te solution to te dual problem 7), wit m > k, and suppose tat te Riesz representer q of te goal functional as oscillations of order m 1 in V m 1. Let u V k and p V m be te dg finite element solutions to 26) and 27), respectively. en, tere exists a constant C, depending only on te sape regularity of, te data {D,f,q} in te primal and dual problems, and te exact solutions {u, p}, suc tat Ru,p,f;u,p ) C k+m. 35) Proof: Using te Caucy-Scwarz inequality, te oscillation estimate for f, and te a priori estimate for te dual solution, tere olds f π k 1 f))p p ) f π k 1 f) L 2 Ω) p p L 2 Ω) C k 1 m+1 = C k+m. We infer from eorem 3 tat te reconstructed equilibrated flux t m 1 p ) wit respect to te dual dg solution satisfies t m 1 p )+D p) L 2 Ω)) 2 Cm. 36) It follows tat t k 1 u )+D u ) p+d 1 t m 1 p )) t k 1 u )+D u [L 2 Ω)] 2D 1 D p+t m 1 p ) [L 2 Ω)] 2 C k m. 14

16 Now, owing to te smootness of te exact solution and using te divergence teorem, one gets n D p+t m 1 p )) ) [[u ]] = = n D p t m 1 p )) ) [[u u ]] D p t m 1 p )) u u ) + q π m 1 q))u u ). Finally, using eorem 3, te a priori estimates for te dg solutions, and te data oscillation estimate for te dual problem, one concludes tat n D p+t m 1 p )) ) [[u ]] C k+m 37) wic completes te proof. eorem 5. Let te conditions of eorem 4 old and let us suppose tat te goal functional error is properly of order 2k. en error estimator 21) is asymptotically exact, tat is, ηu,p ;t k 1 u ),t m 1 p )) lim = Qu) Qu ) Proof: From eorem 4, we infer tat Ru,p,f;u,p ) Qu) Qu ) cm k wit m > k, so tat ηu,p ;t k 1 u ),t m 1 p )) lim 0 + Qu) Qu ) [ = lim 1 Ru,p,f;u ],p ) = Qu) Qu ) Remark 4. e order doubling in te rate of convergence for te approximation of te goal functional is typical in te case of te symmetric version of te discontinuous Galerkin metod, see e.g. [32]. 15

17 6. Goal-oriented error estimates for finite element metods We now consider te case in wic Problem 1) is discretized by te lowest-order conforming finite element metod: Find u V suc tat D u v = fv, v V, 38) Ω were V = V 1 H1 0 Ω). Note tat te gradient u is a piecewise constant vector field tat may be discontinuous along te interelement edges. Following [11, 13], we consider reconstructed equilibrated fluxes in te form t u ) = D u σu ), were σu ) represents some correction to te broken flux D u suc tat t u ) belongs to R 0 ). Suc a correction σu ) is actually sougt in te space R 1 ) = { s [L 2 Ω)] 2 : s [P 0 )] 2 +x P 0 ), } and sould satisfy te conditions div σu ) = Π 0 f), [[σu ) n ]] = [[D u n ]] i, in order to guarantee tat t u ) be a zero-order reconstructed flux. e existence of suc σu ) is proven in [13] in a more general situation and a constructive proof in te two-dimensional case is presented in [11]. e proof uses te ypercircle metod for flux equilibration in te star patces ω i associated wit all vertices i in. An estimate for tis correction is provided in te following teorem see [11], p. 184): eorem 6. ere exists a constant C > 0 tat depends only on te sape regularity of suc tat σu ) L 2 Ω)) 2 C u u L 2 Ω)) 2 + f π0 f) L 2 Ω)). 39) eorem 7. Let u H 2 Ω) be te solution to 1) and let p H 3 Ω) be te solution to te dual problem 7). Suppose tat te Riesz representer q of te goal functional as first-order oscillations in V 1. Let u V be te finite element solution to 38) and p V 2 be te second-order dual dg finite element solution to 27). en, tere exists a constant C, depending only on te sape regularity of, te data {D,f,q} in te primal and dual problems, and te exact solutions {u,p}, suc tat Ω Ru,p,f;u,p ) C 3. 40) 16

18 Proof: e first term in 22) is estimated as f π 0 f))p p ) C 3, since, owing to te elliptic regularity, one as p p L 2 Ω) C 3. From 36), 39), and an energy norm a priori error estimate for te finite element solution, we obtain by applying Caucy-Scwarz: t u )+D u ) p+d 1 t 1 p )) = σu ) p+d 1 t 1 p )) C 3. is completes te proof since te tird term in te remainder vanises owing to te consistency of te metod. eorem 8. Let u V be te finite element solution to 38) and suppose tat te goal functional error is properly of order 2. en, under te conditions of eorem 7, te error estimator ηu,p ;t u ),t 1 p )) = f π 0 f)) p is asymptotically exact. σu ) D 1 t 1 p ) 41) 7. Goal-oriented error estimates for te mixed R 0 finite element In tis section, we consider te mixed form of te primal problem 1). e weak formulation ten reads: Find u,r) L 2 Ω) Hdiv,Ω) suc tat D 1 r q u q = 0 Ω Ω v r = fv Ω 17 Ω q Hdiv,Ω) v L 2 Ω). 42)

19 It is well known, see e.g. [15], tat tere exists a unique solution u,r) to te mixed problem and tat te mixed formulation 42) is equivalent to 2) wit u H 1 0Ω) and r = D u. In addition, we ave te following a priori estimates: u L 2 Ω) C f L 2 Ω) r Hdiv,Ω) C f L 2 Ω) wereconstantc depends onlyonparameterdanddomainω. Weintroduce te finite element space: Q 0 ) = {q R 0 ) : q n = 0 on } 43) e lowest-order Raviart-omas finite element metod for te mixed formulation 42) reads: Find u,r ) P 0 R 0 ) suc tat D 1 r q u q = 0 q Q 0 ) Ω Ω v r = fv v P 0. Ω Ω 44) eorem 9. Let te Riesz representer q of te goal functional ave oscillations of first-order in V 1. Let u,r ) P 0 R 0 ) be te solution to 44) and let p V 2 be te second-order dual dg finite element solution to 27). en, tere exists a constant C, depending only on te sape regularity of, te data {D,f,q} in te primal and dual problems, and te exact solutions {u,r),p}, suc tat Ru,r),D,f,q;u,r ),p ) C 2. 45) Proof: e remainder 22) is estimated as follows. First, we determined te zerot-order reconstructed flux r wit respect to u. We ten ave f π 0 f))p p ) C 2, 46) owing to te suboptimal) estimate p p L 2 Ω) C 2. 47) 18

20 Next, from te elliptic regularity, we ave u H 2 Ω) and r H 1 Ω)) 2 so tat te following a priori error estimates old, see e.g. [15], erefore r r L 2 Ω)) 2 C r L 2 Ω)) 2, u u L 2 Ω) C r L 2 Ω)) 2 + u L 2 Ω)) 2 ). wic yields Noting tat r +D u L 2 Ω)) 2 = r L 2 Ω)) 2 r r L 2 Ω)) 2 + r L 2 Ω)) 2 C1+) r L 2 Ω)) 2, r +D u ) p+d 1 t 1 p )) C 2. n D p+t 1 p ))[[u ]] D p+t 1 p )) u + C 2 u L 2 Ω)) 2 +C u u L 2 Ω) C 2 q πq))u u 1 ) allows us to conclude. eorem 10. Let u,r ) P 0 R 0 ) be te solution to 44) and let te error in te goal functional be assumed properly of first order. en, under te conditions of eorem 9, te error estimator ηu,p ;r,t 1 p )) = f πf))p 0 is asymptotically exact. + r D 1 t 1 p ) n t 1 p )[[u ]] 48) 19

21 8. Numerical examples We sall consider two classes of bencmark problems: one wit large data oscillations and one wit no data oscillation. e first problem, following[19], is te omogeneous boundary-value problem 1) wit D = 1 in Ω = 0,1) 2, for wic te loading term f is given suc tat te exact solution reads ux,y) = 10 4 x1 x)y1 y)exp 100 x 0.75) 2 +y 0.75) 2)). 49) e goal functional is cosen for te first problem as: Qu) = 1 u 50) ω wit ω ω = {x,y) Ω : 1.5 x+y 1.75}. 51) In oter words, te goal is to calculate te average of te solution over te stripω, namelyteregionneartemaximumoftesolution, wereterigtand side f, te solution u, and te gradient of u exibit large oscillations. e exact value of te quantity of interest was evaluated analytically to be Qu) = e second bencmark problem is given by te omogeneous boundaryvalue problem 1) in Ω = 0,1) 2 wit uniform load f = 10 3, for wic te oscillations vanis. e goal functional is given by 50) evaluated over te subdomain ω = [0.75,1] [0.75,1], 52) wic corresponds to te averaged solution in te upper-rigt corner were te solution exibits large gradient. A representation of te exact solution in terms of series was used to calculate te exact value of te functional. e objective of te numerical experiments is to study te beavior of te contributions to te error estimator η 21) in te case of tree finite element metods described above. For eac problem, we describe te numerical results tat confirm te asymptotical exactness of te suggested error estimator and demonstrate te potential of suc an estimator in goal-oriented mes adaptation. In all examples presented below, meses are adapted using a refinement strategy based on te metod proposed by Dörfler [23], wereby te elements in a minimal set M, suc tat θ Mη) 2 η) 2, 20

22 are refined. e refinement parameter θ = 0.75 was used in te first bencmark problem wile θ = 0.5 was considered in te second problem. lements are refined using te longest edge bisection tecnique and additional refinements of te mes are considered in order to eliminate anging nodes. Finally, te quality of te error estimator η = ηu,p ;t u ),t p )) will be assessed in terms of te effectivity index: I η = ηu,p ;t u ),t p )) Qu) Qu ) 53) evaluated on sequences of uniformly and adaptively refined meses dg metod We consider ere te first-order and second-order symmetric dg metods for te discretization of te primal and dual problems and reconstruct te equilibrated fluxes in R 0 and R 1, respectively. We sow in Figure 1a) te effectivity index of estimator η for a sequence of uniform and adaptive mes refinements for te first test problem. For uniform refinement, te effectivity index converges to unity as te mes size decreases, wic confirms asymptotic exactness of te error estimator. In Figure 1b) te true error in te goal functional approximation as well as te error estimator and its different components, are sown for a sequence of adaptive mes refinements. o illustrate te effect of error cancelation wen summing up te tree contributions, te positive and negative parts of eac component are sown in black and red, respectively. Moreover, given a quantityf i.e.eorη), wedenotebyf + = max{f,0}resp.f = max{ f,0})te positiveresp.negative)partoff. Wecanseetatteerrorestimateremains very close to te true error offering accurate information for mes adaptation. For suc a problem wit large data oscillations, te error estimate is essentially quantified by te difference between te oscillation contribution η O and te flux contribution η, wic bot dominate in absolute value but wit opposite sign, and is corrected by te relatively smaller η H component. In te case of te second test problem, te effectivity index is sown in Figure 2)a) for uniform and adapted sequences of meses and te exact error e, error estimator η and its components η O, η, and η H are sown in Figure 2)b). In te absence of oscillations in te problem data, te component η O 0 witin macine precision) as expected, and te flux error estimate η is te dominant contribution to te error estimator. 21

23 ffectivity index Uniform mes Adapted mes Relative error e + e η + η η O + η O η + η η H + η H a) DOF b) DOF Figure 1: First bencmark problem using te dg metod for te solution of te primal problem: a) effectivity index I η ; b) exact error e in quantity of interest, error estimator η, and contributions η O, η, and η H to error estimate η on a sequence of adapted meses negative values are sown in red, positive values in black) Conforming finite element metod In te case of te first test problem, we consider te first-order conforming finite element metod for te primal problem and dg metods of various orders for te dual problem. e equilibrated fluxes for te primal problem was constructed using te ypercircle metod presented in [11]. In Figure 3a), we present te effectivity index of estimator η for a sequence of uniform and adaptive mes refinements. In te case of uniform refinements, we use dg metods of order first, second, and tird for te calculation of te error estimates. As one can observe from te figure, te first-order dg metod does not provide an asymptotically exact error estimator te effectivity index tends to 1.9 on fine meses), but asymptotic exactness is recovered in accordance wit eorem 8 in te case of te second- and tird-order dg metods. We sow in Figure 3b) te exact error e in te goal functional approximation as well as te error estimate η and te different contributions calculated using te second-order dg metod) for a sequence of adaptive mes refinements. e component η H vanises ere, so te difference between te oscillation estimate η O and te flux estimate η, wic are of te same order but of opposite signs, accurately captures te goal functional 22

24 ffectivity index Uniform mes Adapted mes Relative error e + e η + η η O + η O η + η η H + η H a) DOF b) DOF Figure 2: Second bencmark problem using te dg metod for te solution of te primal problem: a) effectivity index I η ; b) exact error e in quantity of interest, error estimator η, and contributions η O, η, and η H to error estimate η on a sequence of adapted meses negative values are sown in red, positive values in black). error. In te case of te second bencmark problem, we use te same finite element metod for te primal problem as before and restrict ourselves to te second-order symmetric dg metod for te dual problem. Flux reconstruction is performed as in te previous test problem. e numerical results, sown in Figure 4), are as expected since in tis case bot η O 0 and η H Raviart-omas mixed finite element We solve te first bencmark problem using R 0 mixed finite elements, calculate te error estimate 48) on a sequence of uniform meses using te first- and second-order symmetric dg metods, and reconstruct equilibrated fluxes in R 0 and R 1, respectively. e results are sown in Figure 5a). eorem 10 is not valid inte first case and we cansee tat te effectivity index fails to converge to unity as expected. Wen te approximation order of bottedg metodused forte solutionof tedual problemand flux reconstruction metod increases, te error estimate becomes again asymptotically exact. e exact error and te error indicator wit its components, calcu- 23

25 ffectivity index Relative error e + e eta + eta η O + η O η + η a) DOF Uniform mes, dg10 Uniform mes, dg21 Uniform mes dg32 Adapted mes, dg21 b) DOF Figure 3: First bencmark problem using te conforming finite element metod for te solution of te primal problem: a) effectivity index I η ; b) exact error e in quantity of interest, error estimator η, and contributions η O, η, and η H to error estimate η on a sequence of adapted meses negative values are sown in red, positive values in black). lated using te second-order symmetric dg metod and equilibrated fluxes reconstructed in R 1, are sown in Figure 5b) on a sequence of adapted meses. e flux estimate η and jump estimate η H are in tis case very close to eac oter in absolute value of te order O1)) but are of opposite signs. It is interesting to see tat te error e is essentially driven ere by te oscillation error η O. e second problem is solved using R 0 mixed finite elements as before. Here, te error estimate 48) is also evaluated using te second-order symmetric dg metod and reconstructed fluxes in R 1. e results are presented in Figure 6. In absence of data oscillations, η O O 0 η O O is actually not sown in Figure 6b)), and te error estimate η amounts to te te difference between te flux and te jump estimates. ven in tis case, we can observe tat te error estimate η is asymptotically exact as demonstrated in Figure 6a). 24

26 ffectivity index Relative error e + e eta + eta η + η 0.9 a) 0.85 Uniform mes, dg10 Uniform mes, dg DOF b) DOF Figure 4: Second bencmark problem using te conforming finite element metod for te solution of te primal problem: a) effectivity index I η ; b) exact error e in quantity of interest, error estimator η, and contributions η O, η, and η H to error estimate η on a sequence of adapted meses negative values are sown in red, positive values in black). 9. Conclusions We ave presented a flexible approac for te computation of a posteriori error estimates wit respect to quantities of interest in te case of elliptic problems. e main ingredients of te metod are te calculation of igerorder approximation of te dual problem by te discontinuous Galerkin metod and te construction of equilibrated fluxes in Raviart-omas finite element spaces. e advantage of using a dg metod for te dual problem lies in its simplicity to consider ig-order approximations and to reconstruct equilibrated fluxes from its solution. e error estimator in te quantity of interest is decomposed into tree contributions: 1) an error estimate due to te data oscillations in te primal problem, 2) an error estimate tat measures te difference between te reconstructed fluxes and te finite element fluxes, and 3) a contribution from te jump of te finite element solution at te interface of te elements, all terms being weigted by te discontinuous Galerkin solution to te dual problem. Witin tis framework, te primal problem can be approximated by any finite element metod as long as it satisfies standard a priori error estimates and its solution can be subjected to 25

27 ffectivity index Relative error e + e η + η η O + η O η + η η H + η H a) 0.5 R 0, dg21, uniform mes R 0, dg10, uniform mes R 0, dg21, adapted mes DOF b) DOF Figure 5: First bencmark problem using te mixed R 0 metod for te solution of te primal problem and dg metod for te dual problem: a) effectivity index I η ; b) exact errorein quantity of interest, errorestimator η, and contributions η O, η, and η H to error estimate η on a sequence of adapted meses negative values are sown in red, positive values in black). te reconstruction of equilibrated fluxes. We ave sown tat, depending on te convergence properties of te underlying metod, te order of te dual dg metod can be cosen in suc a way tat te resulting error estimator be asymptotically exact. Numerical experiments, using eiter te dg metod, te conforming finite element metod, or te mixed Raviart-omas metod for te solution of te primal problem, clearly confirm tat te proposed error estimator as effectivity indices close to unity and is asymptotically exact. Acknowledgements e first autor gratefully acknowledges te partial support by CNPq, Brazil. e second autor is grateful for te support by a Discovery Grant from te Natural Sciences and ngineering Researc Council of Canada. He is also a participant of te KAUS SRI center for Uncertainty Quantification in Computational Science and ngineering. 26

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