Multiscale Finite Element Method Based on Local Preconditioning for Compressible Flows at Low Mach Numbers

In this work we present a nonlinear multiscale finite element method coupled with the Weiss-Smith/Choi-Merkle (WSCM) local preconditioner for solving steady compressible flows at low Mach numbers. The multiscale formulation is based on the strategy of separating scales, in which the subgrid scale space is spanned by bubble functions, allowing to use a static condensation procedure in the local matrix system to define the resolved scale problem. The resulting numerical formulation is completed by adding an artificial viscosity operator in all scales of the discretization. We evaluate the multiscale formulation coupled with the WSCM preconditioner comparing it with the non-preconditioned case. The numerical experiments show that this numerical methodology yields good results.