A weighted setting for the numerical approximation of the Poisson problem with singular sources
Provides theoretical foundations for finite element methods for Poisson problems with singular sources, which is incremental for numerical analysis.
The paper proves well-posedness of Poisson problems with singular sources in weighted Sobolev spaces and establishes stability of standard finite element approximations under quasi-uniform meshes.
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.