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

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.