Discontinuous Galerkin Approximations to Elliptic and Parabolic Problems with a Dirac Line Source

TL;DR

This paper analyzes interior penalty discontinuous Galerkin methods for elliptic and parabolic problems with Dirac line sources, establishing convergence and error estimates for various approximation orders, supported by numerical validation.

Contribution

It provides the first comprehensive convergence analysis and error estimates for DG methods applied to problems with line Dirac sources, including both steady-state and time-dependent cases.

Findings

Proves convergence of DG methods for elliptic and parabolic problems with Dirac line sources.

Derives a priori error estimates in L2 and weighted energy norms.

Numerical results confirm theoretical error bounds.

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.