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 H1, 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.