Local error analysis for the Stokes equations with a punctual source term

Abstract

The solution of the Stokes problem with a punctual force in source term is not $H^1 \times \mathbb {L}^2$ and therefore the approximation by a finite element method is suboptimal. In the case of Poisson problem with a Dirac mass in the right-hand side, an optimal convergence for the Lagrange finite elements has been shown on a subdomain which excludes the singularity in $\mathbb {L}^2$-norm by Köppl and Wohlmuth. Here we show a quasi-optimal local convergence in $H^1 \times \mathbb {L}^2$-norm for a $\mathbb {P}_{k}/\mathbb {P}_{k-1}$-finite element method, $k \geqslant 2$, and for the $\mathbb {P}_{1}{\mathrm{b}}/\mathbb {P}_{1}$. The error is still analysed on a subdomain which does not contain the singularity. The proof is based on local Arnold and Liu error estimates, a weak version of Aubin–Nitsche duality lemma applied to the Stokes problem and discrete inf-sup conditions. These theoretical results are generalized to a wide class of finite element methods, before being illustrated by numerical simulations.