In this paper, we provide an analysis for the performance of a standard multigrid method with a line smoother for solving the finite element equation of an anisotropic problem on a rectangular domain. We shall consider the anisotropic equation of the form
For this anisotropic problem, the standard finite element solution has a ``poor'' approximation property and hence the coarse grid solve in the multigrid algorithm is not effective in reducing the smooth components of the errors. It is known that the standard multigrid algorithm with a Jacobi or a Gauss-Seidel smoother does not provide a uniform reduction in the error. The remedy is to use a smoother, such as the line Jacobi smoother, that is effective in reducing components of error in a larger spectrum. For example, numerical experiments and heuristic arguments show that the multigrid method with line Jacobi smoother has a uniform reduction in the error. In this paper, we provide a rigorous justification of this heuristic idea.
This is a joint work with James H. Bramble.