Regularity and finite element approximation for two-dimensional elliptic equations with line Dirac sources

TL;DR

This paper investigates elliptic equations with line Dirac sources in 2D, establishing regularity results and developing finite element methods that achieve optimal convergence despite solution singularities.

Contribution

It introduces new regularity results in weighted Sobolev spaces and proposes finite element algorithms with optimal convergence for line Dirac source problems.

Findings

Regularity results in weighted Sobolev spaces for solutions

Finite element algorithms with optimal convergence rates

Numerical tests confirming theoretical predictions

Abstract

We study the elliptic equation with a line Dirac delta function as the source term subject to the Dirichlet boundary condition in a two-dimensional domain. Such a line Dirac measure causes different types of solution singularities in the neighborhood of the line fracture. We establish new regularity results for the solution in a class of weighted Sobolev spaces and propose finite element algorithms that approximate the singular solution at the optimal convergence rate. Numerical tests are presented to justify the theoretical findings.