Parallel multilevel algorithms for solving the incompressible Navier-Stokes equations with nonconforming finite elements in three dimensions.

Hubertus Oswald

INF 293, 69120 Heidelberg, Germany


This paper presents results of a numerical study for unsteady three-dimensional, incompressible flow. A finite element multigrid method is used in combination with a operator splitting technique and upwind discretization for the convective term. A nonconforming element pair, living on hexahedrons, which is of order O(h^2/h) for velocity and pressure, is used for the spatial discretization. The second order fractional-step-theta-scheme is employed for the time discretization. For this approach we present the parallel implementation of a multigrid code for MIMD computers with message passing and distributed memory. The parallelization uses the grid decomposition: The decomposition of the algebraic quantities is arranged according to the structure of the discretization. The grouping of gridpoints in subdomains automatically leads to a block structuring of the system matrix. Multiplicative multigrid methods as stand-alone iterations and additive multilevel methods as preconditioners are considered. We present a very efficient implementation of Gauss-Seidel resp. SOR smoothers, which have the same amount of communication as a Jacobi smoother.