A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements

Wohlmuth, Barbara; Hoppe, Ronald H. W.

doi:10.1090/s0025-5718-99-01125-4

Cited by 83 publications

(58 citation statements)

References 30 publications

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“…Several a posteriori error estimators have been developed and analyzed for the mixed finite element methods based on the above formulation; see, for example, [2,4,7,16] for some early works and [1,[10][11][12]15] for more recent ones. The performance of an error estimator is commonly measured by the effectivity index which is the ratio of the estimated error to the actual error.…”

Section: Introductionmentioning

confidence: 99%

On the Asymptotic Exactness of an Error Estimator for the Lowest-Order Raviart-Thomas Mixed Finite Element

Kim

2013

Korean Journal of Mathematics

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Abstract.In this paper we analyze an error estimator for the lowest-order triangular Raviart-Thomas mixed finite element which is based on solution of local problems for the error. This estimator was proposed in [Alonso, Error estimators for a mixed method, Numer. Math. 74 (1996), 385-395] and has a similar concept to that of Bank and Weiser. We show that it is asymptotically exact for the Poisson equation if the underlying triangulations are uniform and the exact solution is regular enough.

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Section: Introductionmentioning

confidence: 99%

On the Asymptotic Exactness of an Error Estimator for the Lowest-Order Raviart-Thomas Mixed Finite Element

Kim

2013

Korean Journal of Mathematics

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“…A posteriori error estimates for mixed finite element methods were started in the works of Alonso [7], Braess and Verfürth [15], Carstensen [23], Hoppe and Wohlmuth [39], Achchab et al [2], Wohlmuth and Hoppe [67], Carstensen and Bartels [24], Kirby [42], El Alaoui and Ern [34], Wheeler and Yotov [66], and Lovadina and Stenberg [44]. For some discussion of these results, we refer to [62].…”

Section: Introductionmentioning

confidence: 99%

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

Vohralı́k

2010

Math. Comp.

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Abstract. We derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second-order elliptic problems. It is based on the classical primal weak formulation, the postprocessing of the potential proposed in [T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for secondorder elliptic problems, Math. Comp. 64 (1995), 943-972], and the discrete Friedrichs inequality. Our analysis in particular avoids any explicit use of the uniform discrete inf-sup condition and in a straightforward manner and under minimal necessary assumptions, known convergence and superconvergence results are recovered. The same framework then turns out to lead to optimal a posteriori energy error bounds. In particular, estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived. They give a guaranteed and fully computable upper bound on the energy error, represent error local lower bounds, and are robust under some conditions on the diffusion-dispersion tensor. They are thus suitable for both overall error control and adaptive mesh refinement. Moreover, the developed abstract framework and a posteriori error estimates are quite general and apply to any locally conservative method. We finally prove that in parallel and simultaneously in converse to Galerkin finite element methods, under some circumstances, the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the H 1 0 (Ω) space and also establish several links between mixed finite element approximations and some generalized weak solutions.

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“…In particular, the duality argument was used in [9] to derive an error estimator for the L 2 -norm of the scalar error, which requires a certain regularity on the given problem. For other types of error estimators and their comparison we refer to [22].…”

Section: Introductionmentioning

confidence: 99%

A posteriori error analysis for locally conservative mixed methods

Kim

2007

Math. Comp.

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Abstract. In this work we present a theoretical analysis for a residual-type error estimator for locally conservative mixed methods. This estimator was first introduced by Braess and Verfürth for the Raviart-Thomas mixed finite element method working in mesh-dependent norms. We improve and extend their results to cover any locally conservative mixed method under minimal assumptions, in particular, avoiding the saturation assumption made by Braess and Verfürth. Our analysis also takes into account discontinuous coefficients with possibly large jumps across interelement boundaries. The main results are applied to the P 1 nonconforming finite element method and the interior penalty discontinuous Galerkin method as well as the mixed finite element method.

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A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements

Cited by 83 publications

References 30 publications

On the Asymptotic Exactness of an Error Estimator for the Lowest-Order Raviart-Thomas Mixed Finite Element

On the Asymptotic Exactness of an Error Estimator for the Lowest-Order Raviart-Thomas Mixed Finite Element

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

A posteriori error analysis for locally conservative mixed methods

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