Sharp maximum norm error estimates for finite element approximations of the Stokes problem in $2$-D

Authors:
R. Durán, R. H. Nochetto and Jun Ping Wang

Journal:
Math. Comp. **51** (1988), 491-506

MSC:
Primary 65N30; Secondary 65N15, 76-08, 76D99

DOI:
https://doi.org/10.1090/S0025-5718-1988-0935076-9

MathSciNet review:
935076

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Abstract: 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 eixstence of a local projection operator ${\Pi _h}$ satisfying \[ \int _\Omega \operatorname {div}({\mathbf {v}} - {\Pi _h}{\mathbf {v}})q\;d{\mathbf {x}} = 0,\quad \forall {\mathbf {v}} \in {{[H_0^1(\Omega )]}^2},\forall q \in {M_h},\] 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.

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Keywords:
Finite element method,
Stokes equation

Article copyright:
© Copyright 1988
American Mathematical Society