On the convergence of multigrid methods for flow problems
We prove two theorems on the residual damping in multigrid methods when solving convection dominated diffusion equations and shock wave problems, discretized by the streamline diffusion finite element method. The first theorem shows that a \(V\)-cycle, including sufficiently many pre and post smoothing steps, damps the residual in \(L^{\text{loc}}_1\) for a constant coefficient convection problem with small diffusion in two space dimensions, without the assumption that the coarse grid is sufficiently fine. The proof is based on discrete Green's functions for the smoothing and correction operators on a uniform unbounded mesh aligned with the characteristic. The second theorem proves a similar result for a certain continuous version of a two-grid method, with isotropic artificial diffusion, applied to a two-dimensional Burgers shock wave problem.
convection dominated diffusion equations
streamline diffusion finite element method
two-dimensional Burgers shock wave problem
scientific article; zbMATH DE number 1306004