Adaptive Finite Element Algorithms for the Stokes Problem: Convergence Rates and Optimal Computational Complexity

Although adaptive finite element methods (FEMs) are recognized as powerful techniques for solving mixed variational problems of fluid mechanics, usually they are not even proven to converge. Only recently, in [SINUM, 40 (2002), pp.1207-1229] Bänsch, Morin and Nochetto introduced an adaptive Uzawa FEM for solving the Stokes problem, and showed its convergence. In their paper, numerical experiments indicate (quasi-) optimal triangulations for some values of the parameters, where, a theoretical explanation of these results is still open. In this paper, we present a similar adaptive Uzawa finite element algorithm that uses a generalization of the optimal adaptive FEM of Stevenson [SINUM, 42 (2005), pp.21882217] as an inner solver. By adding a derefinement step to the resulting adaptive Uzawa algorithm, in order to optimize the underlying triangulation after each fixed number of iterations, we show that the overall method converges with optimal rates with linear computational complexity.