A posteriori error estimates for the Stokes problem with singular sources

TL;DR

This paper develops reliable and efficient a posteriori error estimators for finite element solutions of the Stokes problem with singular sources, enabling optimal adaptive mesh refinement in complex domains.

Contribution

It introduces new a posteriori error estimators for the Stokes problem with singular sources, applicable in non-convex Lipschitz domains, and demonstrates their effectiveness in adaptive algorithms.

Findings

Estimators are proven reliable and locally efficient.

Adaptive strategy achieves optimal convergence rates.

Numerical examples confirm theoretical results.

Abstract

We propose a posteriori error estimators for classical low-order inf-sup stable and stabilized finite element approximations of the Stokes problem with singular sources in two and three dimensional Lipschitz, but not necessarily convex, polytopal domains. The designed error estimators are proven to be reliable and locally efficient. On the basis of these estimators we design a simple adaptive strategy that yields optimal rates of convergence for the numerical examples that we perform.