Discontinuous Galerkin approximations to elliptic and parabolic problems with a Dirac line source

Department of Computational and Applied Mathematics, Rice University, Houston, TX, USA

^* Corresponding author: riviere@caam.rice.edu; riviere@rice.edu

Abstract

The analyses of interior penalty discontinuous Galerkin methods of any order k for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by deriving a priori error estimates in the L² norm and in weighted energy norms. In addition, we prove almost optimal local error estimates in the energy norm for any approximation order. Further, almost optimal local error estimates in the L² norm are obtained for the case of piecewise linear approximations whereas suboptimal error bounds in the L² norm are shown for any polynomial degree. For the time-dependent case, convergence of semi-discrete and of backward Euler fully discrete scheme is established by proving error estimates in L² in time and in space. Numerical results for the elliptic problem are added to support the theoretical results.