Guaranteed a posteriori error estimator for mixed finite element methods of elliptic problems

Kim, Kwang-Yeon

doi:10.1016/j.amc.2012.04.084

Cited by 18 publications

(14 citation statements)

References 22 publications

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“…We remark that guaranteed upper bounds for the global error a −1/2 (σ − σ h ) 0,Ω were studied in [1,8,13]. …”

Section: Lower Boundsmentioning

confidence: 99%

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Hierarchical Error Estimators for Lowest-Order Mixed Finite Element Methods

Kim

2014

Korean Journal of Mathematics

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Abstract. In this work we study two a posteriori error estimators of hierarchical type for lowest-order mixed finite element methods. One estimator is computed by solving a global defect problem based on the splitting of the lowest-order Brezzi-Douglas-Marini space, and the other estimator is locally computable by applying the standard localization to the first estimator. We establish the reliability and efficiency of both estimators by comparing them with the standard residual estimator. In addition, it is shown that the error estimator based on the global defect problem is asymptotically exact under suitable conditions.

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“…We remark that guaranteed upper bounds for the global error a −1/2 (σ − σ h ) 0,Ω were studied in [1,8,13]. …”

Section: Lower Boundsmentioning

confidence: 99%

“…In the past two decades much effort has been devoted to development of a posteriori error estimators for mixed finite element methods of (2); see, for example, [1,2,4,5,6,7,8,10,11,13,14] and references therein. We also refer to [16] for comparison of four different kinds of error estimators.…”

Section: Introductionmentioning

confidence: 99%

Hierarchical Error Estimators for Lowest-Order Mixed Finite Element Methods

Kim

2014

Korean Journal of Mathematics

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“…In order to study how to choose the values of parameters a and b, we consider the Poisson equation (1) on the unit square Ω = [0, 1] 2 with the exact solution[33] …”

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confidence: 99%

Adaptive finite element analysis of elliptic problems based on bubble-type local mesh generation

Zhang

Nie

2015

Journal of Computational and Applied Mathematics

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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%

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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Guaranteed a posteriori error estimator for mixed finite element methods of elliptic problems

Cited by 18 publications

References 22 publications

Hierarchical Error Estimators for Lowest-Order Mixed Finite Element Methods

Hierarchical Error Estimators for Lowest-Order Mixed Finite Element Methods

Adaptive finite element analysis of elliptic problems based on bubble-type local mesh generation

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

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