A Comparison of Parallel Block Multi-level Preconditioners for the Incompressible Navier–stokes Equations

Over the past several years, considerable effort has been placed on developing efficient solution algorithms for the incompressible Navier–Stokes equations. The effectiveness of these methods requires that the solution techniques for the linear subproblems generated by these algorithms exhibit robust and rapid convergence; These methods should be insensitive to problem parameters such as mesh size and Reynolds number. This study concerns a preconditioner derived from a block factorization of the coefficient matrix generated in a Newton nonlinear iteration for the primitive variable formulation of the system. This preconditioner is based on the approximation of the Schur complement operator using a technique proposed by Kay, Loghin, and Wathen [11] and Silvester, Elman, Kay, and Wathen [18]. It is derived using subsidiary computations (solutions of pressure Poisson and convection–diffusion– like subproblems) that are significantly easier to solve than the entire coupled system, and a solver can be built using tools, such as smooth aggregation multigrid for the subproblems. We discuss a computational study performed using MPSalsa, a stabilized finite element code, in which parallel versions of these preconditioners from the pressure convection-diffusion preconditioners are compared with an overlapping Schwarz domain decomposition preconditioner. Our results show nearly ideal convergence rates for a wide range of Reynolds numbers on two-dimensional problems with both enclosed and in/out flow boundary conditions on both structured and unstructured meshes.