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n this paper, we show a local a priori error estimate for the Poisson equation in three space dimensions, where the source term is a Dirac measure concentrated on a line. Due to the singularity induced by the line source term, finite element solutions converge suboptimal in classical norms. In this work, we focus our attention on 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.