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

In this paper, we study an adaptive finite element method for the elliptic equation with line Dirac delta functions as a source term.We investigate the regularity of the solution and the corresponding transmission problem to obtain the jump of normal derivative of the solution on line fractures. To handle the singularity of the solution, we adopt the meshes that conform to line fractures, and propose a novel a posteriori error estimator, in which the edge jump residual essentially use the jump of the normal derivative of the solution on line fractures. The error estimator is proven to be both reliable and efficient, finally an adaptive finite element algorithm is proposed based on the error estimator and the bisection refinement method. Numerical tests are presented to justify the theoretical findings.