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Analysis of an adaptive Uzawa finite element method for the nonlinear Stokes problem

Author: Christian Kreuzer
Journal: Math. Comp. 81 (2012), 21-55
MSC (2010): Primary 65N30, 65N12, 35J60
Published electronically: May 11, 2011
MathSciNet review: 2833486
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Abstract: We design and study an adaptive algorithm for the numerical solution of the stationary nonlinear Stokes problem. The algorithm can be interpreted as a disturbed steepest descent method, which generalizes Uzawa’s method to the nonlinear case. The outer iteration for the pressure is a descent method with fixed step-size. The inner iteration for the velocity consists of an approximate solution of a nonlinear Laplace equation, which is realized with adaptive linear finite elements. The descent direction is motivated by the quasi-norm which naturally arises as distance between velocities. We establish the convergence of the algorithm within the framework of descent direction methods.

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Additional Information

Christian Kreuzer
Affiliation: Fakultät für Mathematik, Universität Duisburg-Essen, Forsthausweg 2, Duisburg, Germany 47057
MR Author ID: 833122
ORCID: 0000-0003-2923-4428

Keywords: Convergence, adaptive finite elements, p-Stokes, p-Laplacian, quasi norm, Uzawa algorithm, nonlinear pde
Received by editor(s): December 16, 2009
Received by editor(s) in revised form: January 22, 2011
Published electronically: May 11, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.