MATHICSE Technical Report : Multi space reduced basis preconditioners for parametrized Stokes equations

In this work we introduce a two-level preconditioner for the efficient solution of large scale saddlepoint linear systems arising from the finite element (FE) discretization of parametrized Stokes equations.The proposed preconditioner extends the Multi Space Reduced Basis (MSRB) preconditioning methodproposed in [12], and relies on the combination of an approximated block (fine grid) preconditioner witha reduced basis solver, which plays the role of coarse component. A sequence of RB spaces, constructedeither with an enriched velocity formulation or a Petrov-Galerkin projection, is built. As a matter offact, each RB coarse component is tailored to perform a single iteration of the iterative method at hand.The exible GMRES (FGMRES) algorithm is employed to solve the resulting preconditioned systemand targets small tolerances with a very small iteration count and in a very short time. Numerical testcases dealing with Stokes flows in three dimensional parameter-ependent geometries are consideredto assess the numerical performance of the proposed technique in different large scale computationalsettings. A detailed comparison with both the current state of the art of i) standard RB methodsand ii) preconditioning techniques for Stokes equations highlights the better efficiency of the proposedmethodology.