A robust smoother for convection-diffusion problems with closed characteristics (Q1840760)

The author focusses attention on the robust property of a block Gauss-Seidel-type smoother. A downwind ordering is used in the inner part of the domain to obtain a lower triangular matrix for the discretization of the convection. The feedback introduced by the cyclic boundary conditions is handled by a block Gauss elimination. For the inversion of the resulting (in general dense) Schur complement, a modification of the frequency filtering iteration is used. It is proved that the resulting smoother is both robust with respect to the strength of the convection and the amount of crosswind diffusion. In a work in preparation, the author shows that the robust smoothing property cannot be complemented by a robust approximation property using a monotone coarse grid discretization. However, a monotone discretization can be found which provides a sufficient approximation to make robust multigrid convergence feasible. The model problem, \(-\varepsilon\Delta u+{\partial u\over\partial x}=b\) in \(]0,1[^2\) and \(u=0\) on the Dirichlet boundary, simulates a convection-diffusion problem with cyclic convection using a linear convection on the unit square with cyclic boundary conditions. The reduced problem has closed characteristics. A full upwind finite difference scheme is used to discretize the problem. Additionally to the strength of the convection, an arbitrary amount of crosswind-diffusion can be added to the discrete level. Numerical results illustrate the convergence and robustness of the proposed method.