A Mixed Approach to the Poisson Problem with Line Sources

TL;DR

This paper introduces a novel mixed finite element method for the Poisson equation with line sources, effectively handling solution singularities and achieving optimal convergence rates by separating singular and regular solution components.

Contribution

It develops a singularity removal approach that splits the solution into regular and singular parts, enabling improved numerical approximation of the regular component.

Findings

The method achieves optimal convergence rates for Raviart-Thomas elements.

The solution admits a splitting into higher and lower regularity terms.

The approach significantly improves convergence compared to full solution approximation.

Abstract

In this work we consider the primal mixed variational formulation of the Poisson equation with a line source. The analysis and approximation of this problem is non-standard 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 its own 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 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.