An adaptive Uzawa FEM for Stokes: Convergence without the Inf-Sup
- Bänsch, Eberhard
- Morin, Pedro
- Nochetto, Ricardo H.
2010 Mathematics Subject Classification
- 65N12 65N15 65N30 65N50 65Y20
- A posteriori error estimators, adaptive mesh refinement, convergence, data oscillation, performance, quasi-optimal meshes
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.
- SIAM J. Num. Anal. 40, no. 4, 1207-1229 (2002)