A posteriori error estimates in $\mathbf{W}^{1,p} \times \mathrm{L}^p$ spaces for the Stokes system with Dirac measures

TL;DR

This paper develops reliable and efficient a posteriori error estimators for the Stokes system with singular sources in $ extbf{W}^{1,p} imes ext{L}^p$ spaces, enabling optimal adaptive finite element methods in complex domains.

Contribution

It introduces new a posteriori error estimators for the Stokes system with singular sources in $ extbf{W}^{1,p} imes ext{L}^p$ spaces and demonstrates their reliability and efficiency in adaptive finite element methods.

Findings

Error estimators are reliable and locally efficient.

Adaptive methods achieve optimal convergence rates.

Numerical experiments confirm theoretical results.

Abstract

We design and analyze a posteriori error estimators for the Stokes system with singular sources in suitable $\mathbf{W}^{1,p}\times \mathrm{L}^p$ spaces. We consider classical low-order inf-sup stable and stabilized finite element discretizations. We prove, in two and three dimensional Lipschitz, but not necessarily convex polytopal domains, that the devised error estimators are reliable and locally efficient. On the basis of the devised error estimators, we design a simple adaptive strategy that yields optimal experimental rates of convergence for the numerical examples that we perform.