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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr


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Online
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1609-9389
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Volume 2, Issue 3

Issues

On Superconvergence of a Gradient for Finite Element Methods for an Elliptic Equation with the Nonsmooth Right–hand Side

Alexander Zlotnik
  • Moscow Power Engineering Institute, Department of Mathematical Modelling Krasnokazarmennaya 14, 111250 Moscow, Russia.
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Abstract

The elliptic equation under the nonhomogeneous Dirichlet boundary condition in 2D and 3D cases is solved. A rectangular nonuniform partition of a domain and polylinear finite elements are taken. For the interpolant of the exact solution u, a priori error estimates are proved provided that u possesses a weakened smoothness. Next error estimates are in terms of data. An estimate is established for the right–hand side f of the equation having a generalized smoothness. Error estimates are derived in the case of f which is not compatible with the boundary function. The proofs are based on some propositions from the theory of functions. The corresponding lower error estimates are also included; they justify the sharpness of the estimates without the logarithmic multipliers. Finally, we prove similar results in the case of 2D linear finite elements and a uniform partition.

Keywords: elliptic equation; nonsmooth right-hand side; finite element method; gradient superconvergence; lower error estimates

About the article

Received: 2002-07-17

Revised: 2002-09-23

Accepted: 2002-10-01

Published in Print:


Citation Information: Computational Methods in Applied Mathematics Comput. Methods Appl. Math., Volume 2, Issue 3, Pages 295–321, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.2478/cmam-2002-0018.

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© Institute of Mathematics, NAS of Belarus. This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. BY-NC-ND 4.0

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J. A. Ferreira and R. D. Grigorieff
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