A robust and parallel Multigrid Method for Convection Diffusion Equations

Michael Bader

Institut für Informatik, TU München, 80290 München, Germany


We present a multigrid method that is based on the combination of recursive substructuring of the computational domain and the discretisation on hierarchical bases and generating systems. The substructuring approach makes the algorithm inherently parallel. The use of hierarchical generating systems allows an optimal complexity of the resulting method. The robustness of the resulting method for convection problems is achieved by a partial elimination between certain "coarse grid unknowns".

This additional elimination is performed such that in the resulting matrices of the local systems of equations any couplings between coarse grid unknowns on the separator and coarse grid unknowns on the subdomain boundary are zero. The coarse grid unknowns themselves are chosen with respect to the physical properties of the underlying differential equation.

The resulting coarse grid meshes are not uniformly coarsened but are refined on the edges and separators of the coarse grid cells. This refinement of the coarse grid unknowns is chosen such that the number of coarse grid unknowns grows with the square root of the total number of boundary and separator unknowns. This choice is due to the fact that for convection diffusion problems the isolines of the transported matter are typically parabolic.

We will present some results for the application of this method to several convection diffusion problems. For these benchmark problems the algorithm shows an O(N) complexity with respect to the required computing time and storage. In particular, the number of iterations needed does not depend on the geometry of the flow field. Moreover it is also independent of the strength of convection as long as a certain amount of diffusion is present. More precisely, it requires that on the finest grid a certain mesh Peclet number is not exceeded.

The algorithm can also be applied to convection problems on complicated computational domains with reasonably complex boundaries and obstacles. It is especially suited to discretisation techniques that are based on a quadtree or octree representation of the computational domain.