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

Kreuzer, Christian

doi:10.1090/S0025-5718-2011-02524-X

Analysis of an adaptive Uzawa finite element method for the nonlinear Stokes problem
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by Christian Kreuzer;

Math. Comp. 81 (2012), 21-55

DOI: https://doi.org/10.1090/S0025-5718-2011-02524-X

Published electronically: May 11, 2011

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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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