The purpose of this research is to study efficient iterative methods (especially domain decomposition and multigrid methods) for convection- dominated problems.
We present our approaches in two directions. One is a specific type of block iterative method called the ``cross-wind-block (CWB) iterative method''. We divide the computational domain into cross-wind blocks and perform the block iterative algorithm. A specific monotone scheme, the EAFE scheme, has been adopted as the main discretization method in our computation and analysis. Also, to effectively divide the domain into cross-wind blocks and order them in the downwind direction, we apply graph theory to our mesh and the stiffness matrix, and a special depth-first search algorithm, Tarjan's ordering algorithm, is used. The convergence of the CWB method is particularly good for the problems with non-recirculating flows. The solution for the monotone schemes, though with only first order accuracy, may be applied to help construct preconditioners or good initial guesses for higher order schemes, such as streamline diffusion and SUPG schemes.
The other approach is to discretize the problem with long-thin elements at its boundary layers. This method is effective in resolving the spurious oscillations usually occur near the boundary layers with the standard Galerkin methods. One application of this long-thin element method is to be used as a preconditioner for spectral methods.