A dual singular complement method for the numerical solution of the Poisson equation with $L^2$ boundary data in non-convex domains

TL;DR

This paper introduces a dual singular complement method to improve the numerical solution of the Poisson equation with $L^2$ boundary data in non-convex domains, maintaining convergence order and validated by numerical experiments.

Contribution

A novel dual singular complement method is proposed to enhance convergence in non-convex domains for Poisson problems with $L^2$ boundary data.

Findings

Retains convergence order in convex domains

Improves convergence order in non-convex domains

Numerical experiments confirm theoretical results

Abstract

The very weak solution of the Poisson equation with $L^2$ boundary data is defined by the method of transposition. The finite element solution with regularized boundary data converges with order $1/2$ in convex domains but has a reduced convergence order in non-convex domains. As a remedy, a dual variant of the singular complement method is proposed. The error order of the convex case is retained. Numerical experiments confirm the theoretical results.