Anisotropic finite elements for elliptic problems with singular data
This work provides a rigorous convergence analysis for finite elements with singular data, addressing a known bottleneck in numerical analysis for problems with measure-valued sources.
The paper proves that optimal finite element convergence rates for elliptic problems with singular measures (point Dirac delta or line-supported) can be achieved using properly graded isotropic or anisotropic meshes, with numerical experiments confirming the theoretical results.
We study the problem $-Δu = γ$, where $γ$ is a singular measure, with support on a curve or a point. We prove that optimal rates of convergence for the finite element method can be obtained using properly graded meshes. In particular, we consider isotropic graded meshes when $γ$ is a point Dirac delta, and anisotropic graded meshes when $γ$ is a measure supported on a segment. Numerical experiments are shown that verify our results, and lead to interesting observations.