Local error analysis for the Stokes equations with a punctual source term
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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.
Mathematics Subject Classification
65M60 65M15 76D07References
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