• Title/Summary/Keyword: Asymptotic exactness

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ON THE ASYMPTOTIC EXACTNESS OF AN ERROR ESTIMATOR FOR THE LOWEST-ORDER RAVIART-THOMAS MIXED FINITE ELEMENT

  • Kim, Kwang-Yeon
    • Korean Journal of Mathematics
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    • v.21 no.3
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    • pp.293-304
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    • 2013
  • 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.

ASYMPTOTIC EXACTNESS OF SOME BANK-WEISER ERROR ESTIMATOR FOR QUADRATIC TRIANGULAR FINITE ELEMENT

  • Kim, Kwang-Yeon;Park, Ju-Seong
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.2
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    • pp.393-406
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    • 2020
  • We analyze a posteriori error estimator for the conforming P2 finite element on triangular meshes which is based on the solution of local Neumann problems. This error estimator extends the one for the conforming P1 finite element proposed in [4]. We prove that it is asymptotically exact for the Poisson equation when the underlying triangulations are mildly structured and the solution is smooth enough.

A POSTERIORI ERROR ESTIMATORS FOR THE STABILIZED LOW-ORDER FINITE ELEMENT DISCRETIZATION OF THE STOKES EQUATIONS BASED ON LOCAL PROBLEMS

  • KIM, KWANG-YEON
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.21 no.4
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    • pp.203-214
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    • 2017
  • In this paper we propose and analyze two a posteriori error estimators for the stabilized $P_1/P_1$ finite element discretization of the Stokes equations. These error estimators are computed by solving local Poisson or Stokes problems on elements of the underlying triangulation. We establish their asymptotic exactness with respect to the velocity error under certain conditions on the triangulation and the regularity of the exact solution.

SUPERCONVERGENCE AND POSTPROCESSING OF EQUILIBRATED FLUXES FOR QUADRATIC FINITE ELEMENTS

  • KWANG-YEON KIM
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.27 no.4
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    • pp.245-271
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    • 2023
  • In this paper we discuss some recovery of H(div)-conforming flux approximations from the equilibrated fluxes of Ainsworth and Oden for quadratic finite element methods of second-order elliptic problems. Combined with the hypercircle method of Prager and Synge, these flux approximations lead to a posteriori error estimators which provide guaranteed upper bounds on the numerical error. Furthermore, we prove some superconvergence results for the flux approximations and asymptotic exactness for the error estimator under proper conditions on the triangulation and the exact solution. The results extend those of the previous paper for linear finite element methods.

HIERARCHICAL ERROR ESTIMATORS FOR LOWEST-ORDER MIXED FINITE ELEMENT METHODS

  • Kim, Kwang-Yeon
    • Korean Journal of Mathematics
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    • v.22 no.3
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    • pp.429-441
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    • 2014
  • 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.

POSTPROCESSING FOR THE RAVIART-THOMAS MIXED FINITE ELEMENT APPROXIMATION OF THE EIGENVALUE PROBLEM

  • Kim, Kwang-Yeon
    • Korean Journal of Mathematics
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    • v.26 no.3
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    • pp.467-481
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    • 2018
  • In this paper we present a postprocessing scheme for the Raviart-Thomas mixed finite element approximation of the second order elliptic eigenvalue problem. This scheme is carried out by solving a primal source problem on a higher order space, and thereby can improve the convergence rate of the eigenfunction and eigenvalue approximations. It is also used to compute a posteriori error estimates which are asymptotically exact for the $L^2$ errors of the eigenfunctions. Some numerical results are provided to confirm the theoretical results.

POSTPROCESSING FOR GUARANTEED ERROR BOUND BASED ON EQUILIBRATED FLUXES

  • KIM, KWANG-YEON
    • Journal of the Korean Mathematical Society
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    • v.52 no.5
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    • pp.891-906
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    • 2015
  • In this work we analyze a postprocessing scheme for improving the guaranteed error bound based on the equilibrated fluxes for the P1 conforming FEM. The improved error bound is shown to be asymptotically exact under suitable conditions on the triangulations and the regularity of the true solution. We also present some numerical results to illustrate the effect of the postprocessing scheme.