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Publication 03-CNA-005

Convergence of the Discontinuous Galerkin Methods for Discontinuous Solutions

Noel J. Walkington
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

Abstract: We consider linear first order scalar equations of the form $p_t + div(pv)+ap=f$ with appropriate initial and boundary conditions. It is shown that approximate solutions computed using the discontinuos Galerkin method will converge in $L^2[0,T;L^2(\Omega)]$ when the coefficients $v$ and $a$ and data $f$ satisfy the minimal asusmptions required to establish existence and uniqueness of solutions. In particular, $v$ need not be Lipschitz, so characteristics of the equation may not be defined, and the solutions being approximated may not have bounded variation.

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