Parallel adaptive solution of the Stokes and Oseen problems on unstructured 3D meshes

Abstract

This chapter presents a method for a parallel adaptive solution of the Stokes and Oseen problems. The main blocks of this method are the parallel adaptive mesh generator and the parallel solver for discrete problems with block-diagonal preconditioners. Both blocks are independent of each other and are constructed to be as much independent from the underlying problem as possible. The problem-independent parallel iterative solver is based on the block diagonal preconditioning of the system matrix. The first three diagonal blocks are preconditioners for the velocity blocks. These preconditioners are based on the domain decomposition (DD) method with the black box AMG subdomain solvers and the BSOR smoother in the space of aggregated vectors. The preconditioners for the three velocity blocks are based on a domain decomposition method with a black box sub domain solver and a smoother in a space of aggregated vectors. The fourth block is a preconditioner for the pressure Schur complement. For the Stokes problem, it is just the approximate inverse of the mass matrix associated with the pressure grid. For the Oseen problem, the preconditioned preconditioner is premultiplied with the velocity operator projected onto the pressure grid and the inverse of the discrete Laplacian on the pressure grid. The data required by the preconditioners are confined to the mesh data. Efficiency of this method is demonstrated with numerical experiments. The algorithms are independent of the underlying boundary value problem.