A weighted setting for the numerical approximation of the Poisson problem with singular sources

TL;DR

This paper develops a weighted finite element approach for accurately approximating Poisson problems with singular sources on convex polygonal domains, ensuring well-posedness and stability in weighted norms.

Contribution

It introduces a framework for analyzing Poisson problems with singular sources using weighted Sobolev spaces and proves stability of finite element methods in this setting.

Findings

Well-posedness established for sources in dual weighted Sobolev spaces.

Finite element approximations are stable in weighted norms.

Applicable to convex polygonal and polyhedral domains.

Abstract

We consider the approximation of Poisson type problems where the source is given by a singular measure and the domain is a convex polygonal or polyhedral domain. First, we prove the well-posedness of the Poisson problem when the source belongs to the dual of a weighted Sobolev space where the weight belongs to the Muckenhoupt class. Second, we prove the stability in weighted norms for standard finite element approximations under the quasi-uniformity assumption on the family of meshes.