Manchester Applied Mathematics and Numerical Analysis Seminars Spring 2000

May 3, 2000, 2.30 pm

Room OF/B9, Oddfellows Hall, Grosvenor Street

An iterative solver for the Navier-Stokes equations in two and three dimensions.

Dr. David Kay, School of Mathematical Sciences, University of Sussex

An efficient method for solving the sparse linear system of equations arising from finite element discretisations of the Navier-Stokes or Oseen equations is given.

Due to features such as boundary layers in the true solution of these equations, it may be necessary to reduce the element size in proportion to any decrease in viscosity. Furthermore, use of Krylov subspace iterative methods to solve such systems produces convergence rates that are dependent on both element size, viscosity and time step size. Hence, a preconditioner that has little or no dependence on either mesh size, viscosity and timestep size is required.

In this work we present a preconditioner which produces convergence rates that have only mild dependence on viscosity and no dependence on the mesh size. The preconditioner is successfully applied tomixed finite elements, including both stable and stabilised triangular two dimensional elements and a stable three dimensional element.

It is shown that a practical implementation of the preconditioner can be achieved using multigrid as an inexact solve on the preconditioner. With just a couple of simple multigrid cycles applied to the preconditioner, convergence rates similar to those of the exact inversion of the preconditioner are obtained.

For further info contact either Matthias Heil (mheil@ma.man.ac.uk), Mark Muldoon (M.Muldoon@umist.ac.uk)or the seminar secretary (Tel. 0161 275 5800).