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

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 canadapt standard preconditioners to accelerate their convergence. We consider a multiplicative low-rank cor-rection for pressure Schur complement preconditioners that is based on a (randomized) low-rank approxi-mation of the error between the identity and the preconditioned Schur complement. We further introducea relaxation parameter that scales the initial preconditioner. This parameter can improve the initial pre-conditioner 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 twodifferent geometries illustrate the action of the update scheme. We numerically analyze various parametersof the low-rank update with respect to their influence on convergence and computational time. MSC codes. 65F08, 65F10, 65N22, 65F55