Treatment of 3D diffusion problems with discontinuous coefficients and Dirac curvilinear sources

Abstract

Three-dimensional diffusion problems with discontinuous coefficients and unidimensional Dirac sources arise in a number of fields. The statement we pursue is a singular-regular expansion where the singularity, capturing the stiff behavior of the potential, is expressed by a convolution formula using the Green kernel of the Laplace operator. The correction term, aimed at restoring the boundary conditions, fulfills a variational Poisson equation set in the Sobolev space $H^1$, which can be approximated using finite element methods. The mathematical justification of the proposed expansion is the main focus, particularly when the variable diffusion coefficients are continuous, or have jumps. A computational study concludes the paper with some numerical examples. The potential is approximated by a combined method: (singularity, by integral formulas, correction, by linear finite elements). The convergence is discussed to highlight the practical benefits brought by different expansions, for continuous and discontinuous coefficients.