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Localized pointwise a posteriori error estimates for gradients of piecewise linear finite element approximations to second-order quasilinear elliptic problems

Abstract

Two types of pointwise a posteriori error estimates are presented for gradients of finite element approximations of second-order quasilinear elliptic Dirichlet boundary value problems over convex polyhedral domains $\Omega$ in space dimension n \ge 2. We first give a residual estimator which is equivalent to \|\nabla(u-u_h)\|_{L_\infty(\Omega) up to higher-order terms. The second type of residual estimator is designed to control \nabla(u-u_h)locally over any subdomain of $\Omega$. It is a novel a posteriori counterpart to the localized or weighted a priori estimates of [Sch98]. This estimator is shown to be equivalent (up to higher-order terms) to the error measured in a weighted global norm which depends on the subdomain of interest. All estimates are proved for general shape-regular meshes which may be highly graded and unstructured. The constants in the estimates depend on the unknown solution u in the nonlinear case, but in a fashion which places minimal restrictions on the regularity of u.

MSC codes

  1. 65N30
  2. 65N15

Keywords

  1. finite element methods
  2. quasilinear elliptic problems
  3. a posteriori error estimation
  4. pointwise error analysis

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Information & Authors

Information

Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 494 - 514
ISSN (online): 1095-7170

History

Published online: 25 July 2006

MSC codes

  1. 65N30
  2. 65N15

Keywords

  1. finite element methods
  2. quasilinear elliptic problems
  3. a posteriori error estimation
  4. pointwise error analysis

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