This talk is concerned with the design of robust and efficient iterative methods for solving advection-diffusion equations. Specifically, we consider the stabilisation of discrete finite element approximations on uniform grids which do not resolve boundary layers. Such stabilisation is necessary when the ratio of convection to diffusion is large, in order to prevent oscillations in the discrete solutions.
The particular stabilisation method featured in this talk is the commonly-used streamline diffusion approach, where a certain amount of additional diffusion is added in the direction of the flow streamlines in order to damp these oscillations. One issue which immediately arises is how much diffusion should be added: adding too little will not damp the oscillations sufficiently, but adding too much will result in an overly smooth and inaccurate solution. In the first part of the talk, we will use Fourier analysis to establish an appropriate value for the streamline diffusion parameter in terms of producing accurate solutions.
In addition, we face the equally important question of how to solve the resulting matrix equation efficiently. The large size of the linear systems arising from discretisations of practical problems, particularly in three dimensions, means that iterative solution methods are often the only feasible option. In the second part of the talk, we consider the choice of streamline diffusion parameter from the point of view of ensuring fast convergence of smoothers based on the standard GMRES iteration, and show that there is a symbiotic relationship between `optimal' solution approximation (in the above sense) and efficient GMRES performance.