Ignacio Aymerich Ojea

Irene Drelichman, Ricardo Durán, Ignacio Ojea

We consider the approximation of Poisson type problems where the source is given by a singular measure and the domain is a convex polygonal or polyhedral domain. First, we prove the well-posedness of the Poisson problem when the source belongs to the dual of a weighted Sobolev space where the weight belongs to the Muckenhoupt class. Second, we prove the stability in weighted norms for standard finite element approximations under the quasi-uniformity assumption on the family of meshes.

Ignacio Ojea

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.