Analysis and Application of Multigrid Preconditioners for Singularly Perturbed Boundary Value Problems

This paper analyzes multigrid preconditioners designed to accelerate the convergence of various singularly perturbed boundary value problems for which the coefficients of low-order derivative terms are multiplied by a large parameter K. The multigrid preconditioner is not applied to the given operator, but rather to a simpler operator (such as the Laplacian) for which its implementation can be carried out much more efficiently. The preconditioner is shown to be effective for all K ≥ 0 provided the coarsest grid size is suitably chosen in terms of K. The multigrid algorithm does not solve the coarse-grid equations but consists of, at most, a few relaxation sweeps on this grid level. The main results are proved for a W cycle and a "variable" V cycle using Fourier analysis. Numerical experiments indicate the validity of these results for a standard V cycle as well.