Volume 55, 2021Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
|Page(s)||S879 - S907|
|Published online||26 February 2021|
Optimal a priori error estimates in weighted Sobolev spaces for the Poisson problem with singular sources
Departamento de Matemática Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, Argentina
* Corresponding author: firstname.lastname@example.org
Accepted: 1 September 2020
We study the problem -Δu=f, where f has a point-singularity. In particular, we are interested in f = δx0, a Dirac delta with support in x0, but singularities of the form f ~ |x − x0|−s are also considered. We prove the stability of the Galerkin projection on graded meshes in weighted spaces, with weights given by powers of the distance to x0. We also recover optimal rates of convergence for the finite element method on these graded meshes. Our approach is general and holds both in two and three dimensions. Numerical experiments are shown that verify our results, and lead to interesting observations.
Mathematics Subject Classification: 35J05 / 65N30 / 65N12 / 65N15 / 65Y20
Key words: Weighted Sobolev spaces / a priori error estimates / finite elements
© EDP Sciences, SMAI 2021
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