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Pointwise error estimates of the local discontinuous Galerkin method for a second order elliptic problem

Author: Hongsen Chen
Journal: Math. Comp. 74 (2005), 1097-1116
MSC (2000): Primary 65N30, 65N15, 65N12; Secondary 41A25, 35B45, 35J20
Published electronically: July 16, 2004
MathSciNet review: 2136995
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Abstract: In this paper we derive some pointwise error estimates for the local discontinuous Galerkin (LDG) method for solving second-order elliptic problems in $R^N$ ($N\geq 2$). Our results show that the pointwise errors of both the vector and scalar approximations of the LDG method are of the same order as those obtained in the $L^2$ norm except for a logarithmic factor when the piecewise linear functions are used in the finite element spaces. Moreover, due to the weighted norms in the bounds, these pointwise error estimates indicate that when at least piecewise quadratic polynomials are used in the finite element spaces, the errors at any point $z$ depend very weakly on the true solution and its derivatives in the regions far away from $z$. These localized error estimates are similar to those obtained for the standard conforming finite element method.

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Additional Information

Hongsen Chen
Affiliation: Department of Mathematics, University of Wyoming, Laramie, Wyoming 82070

Keywords: Local discontinuous Galerkin method, pointwise error estimate, maximum norm, elliptic problem
Received by editor(s): December 7, 2003
Received by editor(s) in revised form: February 21, 2004
Published electronically: July 16, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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