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

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.

21 pages, 11 figures

References in corpus (2)

• Error estimates in weighted Sobolev norms for finite element immersed interface methods
• A priori error estimates of regularized elliptic problems

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