A low-rank update for relaxed Schur complement preconditioners in fluid flow problems

Abstract

The simulation of fluid dynamic problems often involves solving large-scale saddle-point systems. Their numerical solution with iterative solvers requires efficient preconditioners. Low-rank updates can adapt standard preconditioners to accelerate their convergence. We consider a multiplicative low-rank correction for pressure Schur complement preconditioners that is based on a (randomized) low-rank approximation of the error between the identity and the preconditioned Schur complement. We further introduce a relaxation parameter that scales the initial preconditioner. This parameter can improve the initial preconditioner as well as the update scheme. We provide an error analysis for the described update method. Numerical results for the linearized Navier–Stokes equations in a model for atmospheric dynamics on two different geometries illustrate the action of the update scheme. We numerically analyze various parameters of the low-rank update with respect to their influence on convergence and computational time.