A Mixed Approach to the Poisson Problem with Line Sources

Abstract

In this work we consider the dual-mixed variational formulation of the Poisson equation with a line source. The analysis and approximation of this problem is nonstandard, as the line source causes the solutions to be singular. We start by showing that this problem admits a solution in appropriately weighted Sobolev spaces. Next, we show that given some assumptions on the problem parameters, the solution admits a splitting into higher- and lower-regularity terms. The lower-regularity terms are here explicitly known and capture the solution singularities. The higher-regularity terms, meanwhile, are defined as the solution of an associated mixed Poisson equation. With the solution splitting in hand, we then define a singularity removal--based mixed finite element method in which only the higher-regularity terms are approximated numerically. This method yields a significant improvement in the convergence rate when compared to approximating the full solution. In particular, we show that the singularity removal--based method yields optimal convergence rates for lowest-order Raviart--Thomas and discontinuous Lagrange elements.