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We are interested in the finite element solution of elliptic problems with aright-hand side of the single layer distribution type. Such problems arise whenone aims at accounting for a physical hypersurface (or line, for bi-dimensionalproblem), but also in the context of fictitious domain methods, when one aimsat accounting for the presence of an inclusion in a domain (in that case thesupport of the distribution is the boundary of the inclusion). The most popularway to handle numerically the single layer distribution in the finite elementcontext is to spread it out by a regularization technique. An alternativeapproach consists in approximating the single layer distribution by acombination of Dirac masses. As the Dirac mass in the right hand side does notmake sense at the continuous level, this approach raises particular issues. Theobject of the present paper is to give a theoretical background to thisapproach. We present a rigorous numerical analysis of this approximation, andwe present two examples of application of the main result of this paper. Thefirst one is a Poisson problem with a single layer distribution as a right-handside and the second one is another Poisson problem where the single layerdistribution is the Lagrange multiplier used to enforce a Dirichlet boundarycondition on the boundary of an inclusion in the domain. Theoretical analysisis supplemented by numerical experiments in the last section.

Approximation of single layer distributions by Dirac masses in Finite Element computations