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

The solution of the Stokes problem with a punctual force in source term is not H(1)xL(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 L-2-norm by Koppl andWohlmuth. Here we show a quasi-optimal local convergence in H-1 x L-2-norm for a Pk/Pk-1-finite element method, k 2, and for the P(1)b/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.