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

TL;DR

This paper provides a rigorous theoretical analysis and numerical validation of approximating single layer distributions by Dirac masses in finite element methods, addressing issues of regularization and practical applications.

Contribution

It introduces a new theoretical framework for approximating single layer distributions with Dirac masses in finite element computations, including error analysis and application examples.

Findings

The approximation by Dirac masses is theoretically justified.

Numerical experiments confirm the effectiveness of the approximation.

Applications include hypersurface problems and fictitious domain methods.

Abstract

We are interested in the finite element solution of elliptic problems with a right-hand side of the single layer distribution type. Such problems arise when one aims at accounting for a physical hypersurface (or line, for bi-dimensional problem), but also in the context of fictitious domain methods, when one aims at accounting for the presence of an inclusion in a domain (in that case the support of the distribution is the boundary of the inclusion). The most popular way to handle numerically the single layer distribution in the finite element context is to spread it out by a regularization technique. An alternative approach consists in approximating the single layer distribution by a combination of Dirac masses. As the Dirac mass in the right hand side does not make sense at the continuous level, this approach raises particular issues. The object of the present paper is to give a theoretical background to this approach. We present a rigorous numerical analysis of this approximation, and we present two examples of application of the main result of this paper. The first one is a Poisson problem with a single layer distribution as a right-hand side and the second one is another Poisson problem where the single layer distribution is the Lagrange multiplier used to enforce a Dirichlet boundary condition on the boundary of an inclusion in the domain. Theoretical analysis is supplemented by numerical experiments in the last section.