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IMA Journal of Numerical Analysis Advance Access originally published online on December 11, 2006
IMA Journal of Numerical Analysis 2007 27(4):741-764; doi:10.1093/imanum/drl036
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© The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A posteriori estimates for approximations of time-dependent Stokes equations

Fotini Karakatsani{dagger}

Department of Mathematics, University of Crete, 71409 Heraklion, Crete, Greece

Charalambos Makridakis{ddagger}

Department of Applied Mathematics, University of Crete, 71409 Heraklion, Crete, Greece and Institute of Applied and Computational Mathematics, FORTH, 71110 Heraklion Crete, Greece

{dagger} Email: fotini{at}math.uoc.gr

{ddagger} Email: makr{at}tem.uoc.gr

Received on 16 September 2004. Revised on 3 April 2006.


  Abstract

In this paper, we derive a posteriori error estimates for space-discrete approximations of the time-dependent Stokes equations. By using an appropriate Stokes reconstruction operator, we are able to write an auxiliary error equation, in pointwise form, that satisfies the exact divergence-free condition. Thus, standard energy estimates from partial differential equation theory can be applied directly, and yield a posteriori estimates that rely on available corresponding estimates for the stationary Stokes equation. Estimates of optimal order in L{infty}(L2) and L{infty}(H1) for the velocity are derived for finite-element and finite-volume approximations.

Key Words: a posteriori error estimators; finite elements; finite volumes; time-dependent Stokes problem; discrete divergence-free spaces


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