A local error estimate for the poisson equation with a line source term

Abstract

In this paper, we show a local a priori error estimate for the Poisson equation in three space dimensions (3D), where the source term is a Dirac measure concentrated on a line. This type of problem can be found in many application areas. In medical engineering, e.g., blood flow in capillaries and tissue can be modeled by coupling Poiseuille’s and Darcy’s law using a line source term. Due to the singularity induced by the line source term, finite element solutions converge suboptimal in classical norms. However, quite often the error at the singularity is either dominated by model errors (e.g. in dimension reduced settings) or is not the quantity of interest (e.g. in optimal control problems). Therefore we are interested in local error estimates, i.e., we consider in space a L2-norm on a fixed subdomain excluding a neighborhood of the line, where the Dirac measure is concentrated. It is shown that linear finite elements converge optimal up to a log-factor in such a norm. The theoretical considerations are confirmed by some numerical tests.