Optimality of a Standard Adaptive Finite Element Method for the Stokes Problem

Abstract

We prove that the a standard adaptive algorithm for the Taylor--Hood discretization of the stationary Stokes problem converges with optimal rate. This is done by developing an abstract framework for quite general problems, which allows us to prove general quasi-orthogonality proposed in [C. Carstensen et al., Comput. Math. Appl., 67 (2014), pp. 1195--1253]. This property is the main obstacle towards the optimality proof and therefore is the main focus of this work. The key ingredient is a new connection between the mentioned quasi-orthogonality and $LU$-factorization of infinite matrices.