A posteriori error analysis for locally conservative mixed methods

Kim, Kwang-Yeon

doi:10.1090/s0025-5718-06-01903-x

Cited by 65 publications

(53 citation statements)

References 24 publications

(34 reference statements)

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“…[ 35], whereas a guaranteed error upper bound in the lowest-order Crouzeix-Raviart case can be obtained along the lines of Destuynder and Métivet [37], see Ainsworth [2], Kim [52,53], or [76]. Different flux equilibrations exist and tight links hold between them, see [46].…”

Section: Introductionmentioning

confidence: 99%

“…For the discontinuous Galerkin method, first guaranteed upper bounds by locally equilibrated fluxes and H 1 (Ω)-conforming reconstructed potentials have been obtained by Ainsworth [3], Kim [52,53], Cochez-Dhondt and Nicaise [32], Ainsworth and Rankin [10], and [41,43,42], see also the references therein. Similar results for mixed finite elements can be found in Kim [52,54], Ainsworth [4], Ainsworth and Ma [6], and [76,78]. All the cited references typically prove local efficiency as well.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Polynomial-Degree-Robust A Posteriori Estimates in a Unified Setting for Conforming, Nonconforming, Discontinuous Galerkin, and Mixed Discretizations

Ern¹,

Vohralı́k²

2015

SIAM J. Numer. Anal.

158

196

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We present equilibrated flux a posteriori error estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed finite element discretizations of the two-dimensional Poisson problem. Relying on the equilibration by mixed finite element solution of patchwise Neumann problems, the estimates are guaranteed, locally computable, locally efficient, and robust with respect to polynomial degree. Maximal local overestimation is guaranteed as well. Numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous Galerkin scheme.Key words: a posteriori error estimate, equilibrated flux, unified framework, robustness, polynomial degree, conforming finite element method, nonconforming finite element method, discontinuous Galerkin method, mixed finite element method IntroductionA posteriori error estimates in the conforming finite element setting have already received a large attention. In particular, following the concept of Prager and Synge [64], cf. also Synge [72], Aubin and Burchard [13], and Hlaváček et al. [50], and invoking fluxes in the H(div, Ω) space, guaranteed upper bounds on the error can be obtained. A general functional framework delivering guaranteed upper bounds, independent of the numerical method, has been derived by Repin [66,67,68]. It does not rely on Galerkin orthogonality neither on local equilibration and accommodates an arbitrary flux reconstruction. The idea of using a local residual equilibration procedure for the normal face fluxes reconstruction has been proposed by Ladevèze [55], Ladevèze and Leguillon [56], Kelly [51], Ainsworth and Oden [7,8], and Parés et al. [61,62]. In this context, guaranteed upper bounds typically require solving infinite-dimensional element problems, which, in practice, are approximated. On the other hand, an essential property achieved by means of local equilibration procedures is local efficiency, meaning that the derived estimators also represent local lower bounds of the error, up to a generic constant. This appears to be crucial in view of local mesh refinement, as well as in order to obtain robustness in singularly perturbed problems. Cheap local flux equilibrations leading to a fully computable guaranteed upper bound have been obtained by Destuynder and Métivet [38]. Later, mixed finite element solutions of local Neumann problems posed over patches of (sub)elements, where one minimizes locally the estimator contributions, were proposed, see Luce and Wohlmuth [57], Braess and Schöberl [19],and [77,30,79]. As a matter of fact, lifting the normal face fluxes of the equilibrated residual method to H(div, Ω) immediately yields equilibrated fluxes, cf. Nicaise et al. [59]. Then, both a guaranteed bound and local efficiency are obtained. For computational comparisons of some of these approaches in the lowest-order case, see Carstensen and Merdon [28].The theory in the nonconforming setting, where the discrete solution (potential) is not in the energy space H 1 (Ω), appears to be less developed....

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Polynomial-Degree-Robust A Posteriori Estimates in a Unified Setting for Conforming, Nonconforming, Discontinuous Galerkin, and Mixed Discretizations

Ern¹,

Vohralı́k²

2015

SIAM J. Numer. Anal.

158

196

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“…A posteriori error estimation for nonconforming finite element methods with application to second-order elliptic problems has recently seen significant progress and is still the subject of sustained research efforts, see e.g. [9,34,51,55,33,1,18,42,41,27,24] for dG methods and [16,35,2,17,37] for mixed finite element methods. We also refer the reader to [36,18,3,28,22] and references therein for the presentation of unifying frameworks on the topic.…”

Section: Introductionmentioning

confidence: 99%

“…Advantages of the dG method are that high-order elements in any space dimension can be easily implemented and that it is locally conservative, which yields in a straightforward manner equilibrated fluxes. In fact, it has been shown to provide cheap and local reconstruction algorithms [35,26,19,12].…”

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confidence: 99%

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

Mozolevski

Prudhomme

2015

Computer Methods in Applied Mechanics and Engineering

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We propose an approach for goal-oriented error estimation in finite element approximations of second-order elliptic problems that combines the dualweighted residual method and equilibrated-flux reconstruction methods for the primal and dual problems. The objective is to be able to consider discretization schemes for the dual solution that may be different from those used for the primal solution. It is only assumed here that the discretization methods come with a priori error estimates and an equilibrated-flux reconstruction algorithm. A high-order discontinuous Galerkin (dG) method is actually the preferred choice for the approximation of the dual solution thanks to its flexibility and straightforward construction of equilibrated fluxes. One contribution of the paper is to show how the order of the dG method for asymptotic exactness of the proposed estimator can be chosen in the cases where a conforming finite element method, a dG method, or a mixed RaviartThomas method are used for the solution of the primal problem. Numerical experiments are also presented to illustrate the performance and convergence of the error estimates in quantities of interest with respect to the mesh size.

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“…On the other hand, the discontinuous Galerkin method becomes recently very popular and is a very efficient tool for the numerical approximation of reaction-convection-diffusion problems for instance. Some a posteriori error analysis were performed recently, let us quote [1,2,7,10,14,16,18,20,21,26] for pure diffusion (or diffusion-dominated) problems and [9,13,16] for singularly perturbed problems (i.e., dominant advection or reaction); for Maxwell system see for instance [17]. In all these papers, no convergence results are proved and to our knowledge, only the recent paper of Karakashian and Pascal [19] provides a convergence result for a purely diffusion problem.…”

Section: Introductionmentioning

confidence: 99%

Adaptive finite element methods for elliptic problems: Abstract framework and applications

Nicaise

Cochez-Dhondt

2010

ESAIM: M2AN

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Abstract. We consider a general abstract framework of a continuous elliptic problem set on a Hilbert space V that is approximated by a family of (discrete) problems set on a finite-dimensional space of finite dimension not necessarily included into V . We give a series of realistic conditions on an error estimator that allows to conclude that the marking strategy of bulk type leads to the geometric convergence of the adaptive algorithm. These conditions are then verified for different concrete problems like convectionreaction-diffusion problems approximated by a discontinuous Galerkin method with an estimator of residual type or obtained by equilibrated fluxes. Numerical tests that confirm the geometric convergence are presented.Mathematics Subject Classification. 65N30, 65N15, 65N50.

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A posteriori error analysis for locally conservative mixed methods

Cited by 65 publications

References 24 publications

Polynomial-Degree-Robust A Posteriori Estimates in a Unified Setting for Conforming, Nonconforming, Discontinuous Galerkin, and Mixed Discretizations

Polynomial-Degree-Robust A Posteriori Estimates in a Unified Setting for Conforming, Nonconforming, Discontinuous Galerkin, and Mixed Discretizations

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

Adaptive finite element methods for elliptic problems: Abstract framework and applications

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