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Sharp maximum norm error estimates for finite element approximations of the Stokes problem in 2-D. (English) Zbl 0699.76038

Summary: This paper deals with finite element approximations of the Stokes equations in a plane bounded domain \(\Omega\), using the so-called velocity-pressure mixed formulation. Quasi-optimal error estimates in the maximum norm are derived for the velocity, its gradient and the pressure fields. The analysis relies on abstract properties which are in turn a consequence of the existence of a local projection operator \(\Pi_ h\) satisfying \(\int_{\Omega}div(v-\Pi_ hv)qdx=0,\) for all \(v\in [H^ 1_ 0(\Omega)]^ 2,\) for all \(q\in M_ h,\) where \(M_ h\) is the finite element space associated with the pressure. Several examples for which this operator can be constructed locally illustrate the theory.

MSC:

76D07 Stokes and related (Oseen, etc.) flows
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65B05 Extrapolation to the limit, deferred corrections
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