Multi space reduced basis preconditioners for parametrized Stokes equations. (English) Zbl 1442.65357
Summary: We introduce a two-level preconditioner for the efficient solution of large scale saddle-point linear systems arising from the finite element (FE) discretization of parametrized Stokes equations. This preconditioner extends the Multi Space Reduced Basis (MSRB) preconditioning method proposed in [N. Dal Santo, “An algebraic least squares reduced basis method for the solution of parametrized Stokes equations”, Tech. Rep. 21. MATHICSE-EPFL (2017)]; it combines an approximated block (fine grid) preconditioner with a reduced basis (RB) solver which plays the role of coarse component. A sequence of RB spaces, constructed either with an enriched velocity formulation or a Petrov-Galerkin projection, is built. Each RB coarse component is defined to perform a single iteration of the iterative method at hand. The flexible GMRES (FGMRES) algorithm is employed to solve the resulting preconditioned system and targets small tolerances with a very small iteration count and in a very short time. Numerical test cases for Stokes flows in three dimensional parameter-dependent geometries are considered to assess the numerical properties of the proposed technique in different large scale computational settings.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |
| 35Q35 | PDEs in connection with fluid mechanics |
| 76D07 | Stokes and related (Oseen, etc.) flows |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
Keywords:
preconditioning techniques; reduced basis method; finite element method; parametrized Stokes equations; computational fluid dynamicsReferences:
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