6 - A Fully-Coupled Finite Element Algorithm, Using Direct and Iterative Solvers, for the Incompressible Navier-Stokes Equations

Summary

Abstract

This chapter presents a Finite Element solution method for the incompressible Navier- Stokes equations, in primitive variables form. To provide the necessary coupling between continuity and momentum, and enhance stability, a pressure dissipation in the form of a Laplacian is introduced into the continuity equation. The recasting of the problem variables in terms of pressure and an “auxiliary” velocity demonstrates how the effects of the pressure dissipation can be eliminated, while retaining its stabilizing properties. The method can also be interpreted as a Helmholtz decomposition of the velocity vector.

The governing equations are discretized by a Galerkin weighted residual method and, because of the modification to the continuity equation, equal interpolation for all the unknowns is permitted. Newton linearization is used and, at each iteration, the linear algebraic system is solved in a fully-coupled manner by direct or iterative solvers. For direct methods, a vector-parallel Gauss elimination method is developed that achieves execution rates exceeding 2.3 Gigaflops, i.e. over 86% of a Cray YMP-8 current peakperformance. For iterative methods, preconditioned conjugate gradient-like methods are studied and good performances, competitive with direct solvers, are achieved. Convergence of such methods being sensitive to preconditioning, a hybrid dissipation method is proposed, with the preconditioner having an artificial dissipation that is gradually lowered, but frozen at a level higher than the dissipation introduced into the physical equations.

Convergence of the Newton-Galerkin algorithm is very rapid. Results are demonstrated for two-and three-dimensional incompressible flows.