Abstract:
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 quasi-optimality, 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 quasi-optimality and for higher order finite elements
optimal a priori estimates on a family of quasi-uniform meshes in a $L^2$-semi-norm.
The semi-norm is defined as a $L^2$-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 a priori bounds are obtained recursively by using Aubin--Nitsche
techniques, local error estimates, interior regularity results and weighted Sobolev norms.
Note that for the proof of these results no graded meshes
are required and that we work on a family of quasi-uniform meshes. Numerical
tests in two and three space dimensions confirm our theoretical results.