Regularity and an adaptive finite element method for elliptic equations with Dirac sources on line cracks

Abstract

In this paper, we consider an adaptive finite element method for solving elliptic equations with line Dirac delta functions as the source term. Instead of using a local $H^{-1}$ local indicator, or regularizing the singular source term and using the classical residual-based a posteriori error estimator, we propose a novel a posteriori estimator based on an equivalent transmission problem. This equivalent problem is defined in the same domain as the original problem but features a zero source term and nonzero flux jumps along the line cracks, leading to a more regular solution. The a posteriori error estimator relies on meshes that conform to the line cracks, and its edge jump residual essentially incorporates the flux jumps of the transmission problem on these cracks. The proposed error estimator is proven to be both reliable and efficient. We also introduce an adaptive finite element algorithm based on this error estimator and the bisection refinement method. Numerical tests demonstrate that quasi-optimal convergence rates are achieved for both low-order and high-order approximations, with the associated adaptive meshes primarily refined at a finite number of singular points in the domain.