A posteriori error estimates with point sources in fractional Sobolev spaces

TL;DR

This paper develops residual-type a posteriori error estimators for Poisson's equation with point sources in fractional Sobolev spaces, demonstrating their reliability, efficiency, and optimal adaptive error decay in 2D polygonal domains.

Contribution

It introduces novel a posteriori error estimators tailored for fractional Sobolev spaces with point sources, validated through numerical experiments.

Findings

Estimators are reliable and locally efficient in 2D domains.

Adaptive algorithms using these estimators achieve optimal error decay.

Numerical tests confirm the effectiveness of the proposed methods.

Abstract

We consider Poisson's equation with a finite number of weighted Dirac masses as a source term, together with its discretization by means of conforming finite elements. For the error in fractional Sobolev spaces, we propose residual-type a posteriori estimators with a specifically tailored oscillation and show that, on two-dimensional polygonal domains, they are reliable and locally efficient. In numerical tests, their use in an adaptive algorithm leads to optimal error decay rates.