Optimal adaptivity for a standard finite element method for the Stokes problem

TL;DR

This paper proves that a standard adaptive finite element algorithm for the stationary Stokes problem converges optimally by establishing a new abstract framework and connecting quasi-orthogonality with LU-factorizations.

Contribution

It introduces a novel abstract framework for indefinite problems and links quasi-orthogonality to LU-factorizations to prove optimal convergence of adaptive methods.

Findings

Proved optimal convergence of adaptive Taylor-Hood discretization for Stokes.

Developed a new connection between quasi-orthogonality and LU-factorizations.

Established a general framework applicable to indefinite problems.

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 indefinite problems which allows us to prove general quasi-orthogonality proposed in [Carstensen et al., 2014]. 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 $LU$-factorizations of infinite matrices.