Multigrid applied to singular perturbation problems

摘要

The solution of the singular perturbation problem −ε″u + b(x)′u = f, 0 < x < 1, with 1⪢ ε> 0, u(0) = u0, u(1) = u1 by a multigrid algorithm is considered. Theoretical and experimental results for a number of different discretizations are presented. The theoretical and observed rates agree with the results developed in an earlier work of Kamowitz and Parter. In addition, the rate of convergence of the algorithm when the coarse grid operator is the natural finite difference analogue of the fine grid operator is presented. This is in contrast to the case in the previous work where the Galerkin choice (IHhLhIhH) was used for the coarse grid operators.