Summary: UNIFIED ANALYSIS OF DISCONTINUOUS GALERKIN METHODS
FOR ELLIPTIC PROBLEMS
DOUGLAS N. ARNOLD, FRANCO BREZZI, BERNARDO COCKBURN§,
AND L. DONATELLA MARINI
SIAM J. NUMER. ANAL. c 2002 Society for Industrial and Applied Mathematics
Vol. 39, No. 5, pp. 1749­1779
Abstract. We provide a framework for the analysis of a large class of discontinuous methods for
second-order elliptic problems. It allows for the understanding and comparison of most of the dis-
continuous Galerkin methods that have been proposed over the past three decades for the numerical
treatment of elliptic problems.
Key words. elliptic problems, discontinuous Galerkin, interior penalty
AMS subject classification. 65N30
PII. S0036142901384162
1. Introduction. In 1973, Reed and Hill [61] introduced the first discontinuous
Galerkin (DG) method for hyperbolic equations, and since that time there has been
an active development of DG methods for hyperbolic and nearly hyperbolic problems.
Recently, these methods also have been applied to purely elliptic problems; examples
are the original method of Bassi and Rebay [10], the variations studied in [23] and
[22], and a generalization called the local discontinuous Galerkin (LDG) methods
introduced in [41] and further studied in [33], [26], and [36]. Also in the 1970s, Galerkin

Source: Arnold, Douglas N. - School of Mathematics, University of Minnesota

Collections: Mathematics