On the convergence theory of adaptive mixed finite element methods for the Stokes problem

Abstract

In the first part of this paper, we establish a conditional optimality result for an adaptive mixed finite element method for the stationary Stokes problem discretized by the standard Taylor-Hood elements, under the assumption of the so-called general quasi-orthogonality. Optimality is measured in terms of a modified approximation class defined through the total error, as is customary since the seminal work of Cascon, Kreuzer, Nochetto and Siebert. The second part of the paper is independent of optimality results, and concerns interrelations between the modified approximation classes and the standard approximation classes (the latter defined through the energy error). Building on the tools developed in the papers of Binev, Dahmen, DeVore, and Petrushev, and of Gaspoz and Morin, we prove that the modified approximation class coincides with the standard approximation class, modulo the assumption that the data is regular enough in an appropriate scale of Besov spaces.