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

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

Eberhard Baensch
Weierstrass Institute for Applied Analysis and Stochastics
Mohrenstrasse 39, 10117 and Freie Universitaet Berlin, Germany

Pedro Morin
Instituto de Matematica Aplicada del Litoral
Guemes 3450
3000 Santa Fe, Argentina

Ricardo H. Nochetto
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA


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.