Non-nested multi-grid solvers for mixed divergence-free Scott–Vogelius discretizations

Abstract

We apply the general framework developed by John et al. in Computing 64:307–321, 2000 to analyze the convergence of multi-level methods for mixed finite element discretizations of the generalized Stokes problem using the Scott–Vogelius element. The Scott–Vogelius element seems to be promising since discretely divergence-free functions are divergence-free pointwise. However, to satisfy the Ladyzhenskaya–Babuska–Brezzi stability condition, we have to deal in the multi-grid analysis with non-nested families of meshes which are derived from nested macro element triangulations. Additionally, the analysis takes into account an optional symmetric stabilization operator which suppresses spurious oscillations of the velocity provoked by a dominant reaction term. Usually, the generalized Stokes problems appears in semi-implicit splitting schemes for the unsteady Navier–Stokes equations, but the symmetric part of a stabilized discrete Oseen problem can be reguarded as a discrete generalized Stokes problem likewise.