Optimal adaptivity for a standard finite element method for the Stokes problem
Provides the first optimality proof for a standard adaptive algorithm for the Stokes problem, solving a long-standing theoretical gap for computational fluid dynamics.
Proved that a standard adaptive finite element method for the Stokes problem achieves optimal convergence rates by establishing a general quasi-orthogonality property via a novel connection to LU-factorizations of infinite matrices.
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.