Algebraic multigrid for stabilized finite element discretizations of the Navier–Stokes equations

Abstract

An algebraic multigrid method for the solution of stabilized finite element discretizations of the Euler and Navier Stokes equations on generalized unstructured grids is described. The method is based on an elemental agglomeration multigrid strategy employing a semi-coarsening scheme designed to reduce grid anisotropy. The viscous terms are discretized in a consistent manner on coarse grids using an algebraic Galerkin coarse grid approximation in which higher-order grid transfer operators are constructed from the underlying triangulation. However, the combination of higher-order transfer operators and Galerkin rediscretization diminishes the stability of stabilized inviscid operators on coarse grids and a modification is proposed to alleviate this problem. A generalized line implicit relaxation scheme is also described where the lines are constructed to follow the direction of strongest coupling. Applications are demonstrated for convection–diffusion, Euler, and laminar Navier–Stokes. The results show that the convergence rate is largely unaffected by mesh size over a wide range of Reynolds (Peclet) numbers.