Optimal A Priori Error Estimates for an Elliptic Problem with Dirac Right-Hand Side

It is well known that finite element solutions for elliptic PDEs with Dirac measures as source terms converge, due to the fact that the solution is not in $H^1$, suboptimal in classical norms. A standard remedy is to use graded meshes, then quasioptimality, i.e., optimal up to a log-factor, for low order finite elements can be recovered, e.g., in the $L^2$-norm. Here we show for the lowest order case quasioptimality and for higher order finite elements optimal order a priori estimates on a family of quasi-uniform meshes in an $L^2$-seminorm. The seminorm is defined as an $L^2$-norm on a fixed subdomain which excludes the locations of the delta source terms. Our motivation in the use of such a norm results from the observation that in many applications the error at the singularity is dominated by the model error, e.g., in dimension reduced settings or is not the quantity of interest, e.g., in optimal control problems. The quasi-optimal and optimal order a priori bounds are obtained recursively by using Aubin--Nitsche techniques, local Wahlbin-type error estimates, interior regularity results, and weighted Sobolev norms. For the proof of these results no graded meshes are required, it is sufficient to work on a family of quasi-uniform meshes. Numerical tests in two and three space dimensions confirm our theoretical results.