An adaptive finite element method for two-dimensional elliptic equations with line Dirac sources

TL;DR

This paper introduces an adaptive finite element method tailored for two-dimensional elliptic equations with line Dirac delta sources, utilizing a novel error estimator based on a transmission problem to achieve efficient and accurate solutions.

Contribution

It presents a new a posteriori error estimator and an adaptive algorithm that effectively handle line Dirac sources without regularization, improving solution regularity and convergence.

Findings

Achieves quasi-optimal convergence rates with adaptive refinement.

Error estimator is proven reliable and efficient.

Adaptive meshes focus refinement at singular points.

Abstract

In this paper, we propose a novel adaptive finite element method for an elliptic equation with line Dirac delta functions as a source term. We first study the well-posedness and global regularity of the solution in the whole domain. Instead of 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 with zero source term and nonzero flux jumps on line fractures. The transmission problem is defined in the same domain as the original problem excluding on line fractures, and the solution is therefore shown to be more regular. The estimator relies on meshes conforming to the line fractures and its edge jump residual essentially uses the flux jumps of the transmission problem on line fractures. The error estimator is proven to be both reliable and efficient, an adaptive finite element algorithm is proposed based on the error estimator and the bisection refinement method. Numerical tests show that quasi-optimal convergence rates are achieved even for high order approximations and the adaptive meshes are only locally refined at singular points.