An algebraic least squares reduced basis method for the solution of nonaffinely parametrized Stokes equations

32/2018

An algebraic least squares reduced basis method for the solution of nonaffinely parametrized Stokes equations

Sunday 3rd June 2018

Dal Santo, N.; Deparis, S.; Manzoni, A.; Quarteroni, A.

In this paper we propose a new, purely algebraic, Petrov-Galerkin reduced basis (RB) method to solve the parametrized Stokes equations, where parameters serve to identify the (variable) domain geometry. Our method is obtained as an algebraic least squares reduced basis (aLS-RB) method, and improves the existing RB methods for Stokes equations in several directions. First of all, it does not require to enrich the velocity space, as often done when dealing with a velocity-pressure formulation, relying on a Petrov-Galerkin RB method rather than on a Galerkin RB (G-RB) method. Then, it exploits a suitable approximation of the matrix-norm in the definition of the (global) supremizing operator. The proposed method also provides a fully automated procedure to assemble and solve the RB problem, able to treat any kind of parametrization, and we rigorously prove the stability of the resulting aLS-RB problem (in the sense of a suitable inf-sup condition). Next, we introduce a coarse aLSRB (caLSRB) method, which is obtained by employing an approximated RB test space, and further improves the efficiency of the aLSRB method both offline and online. We provide numerical comparisons between the proposed methods and the current state-of-art G-RB methods. The new approach results in a more convenient option both during the offline and the online stage of computation, as shown by the numerical results.