Optimal a priori error estimates for an elliptic problem with Dirac right-hand side (Q2927831)

This paper is concerned with the accuracy of finite element approximations of elliptic boundary value problems where the solution is not an \(H^1\)-function, and thus the standard a priori error estimates do not yield an optimal order. The problem under consideration is the Poisson equation in which the right hand side is a linear combination of a smooth function and some Dirac measures concentrated at some points from the domain on which the equation is posed. The usual quasi-uniform meshes (without any use of the graded meshes) are considered. The authors show that the convergence order is quasi-optimal (up to a log factor) for the case of piecewise linear finite elements and is optimal for the case of higher-order finite elements in some convenient \(L^2\)-seminorm. This \(L^2\)-seminorm is defined in some manner which excludes the locations of the delta source terms. Numerical tests in two and three space dimensions are presented to support the theoretical results.