# An Adaptive Uzawa FEM for the Stokes Problem:
Convergence without the Inf-Sup Condition

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SIAM J. Numer. Anal., 40 (2002), 1207--1229.
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Eberhard Baensch

Weierstrass Institute for Applied Analysis and Stochastics

Mohrenstrasse 39, 10117
and Freie Universitaet Berlin, Germany

baensch@wias-berlin.de
Pedro Morin

Instituto de Matematica Aplicada del Litoral

Guemes 3450

3000 Santa Fe, Argentina

pmorin@math.unl.edu.ar
Ricardo H. Nochetto

Department of Mathematics

University of Maryland, College Park

College Park, MD 20742, USA

rhn@math.umd.edu
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Abstract

We introduce and study an adaptive finite element
method for the Stokes system
based on an Uzawa outer iteration to update the pressure and an
elliptic adaptive inner iteration for velocity. We show linear
convergence in terms of the outer iteration counter
for the pairs of spaces consisting
of continuous finite elements of degree $k$ for velocity
whereas for pressure the elements can be either discontinuous of
degree $k-1$ or continuous of degree $k-1$ and $k$.
The popular Taylor-Hood family is
the sole example of stable elements included in the theory,
which in turn relies on the stability of the continuous problem
and thus makes no use
of the discrete inf-sup condition.
We discuss the realization and complexity of the elliptic
adaptive inner solver, and
provide consistent computational evidence that the resulting
meshes are quasi-optimal.