Multigrid Methods for Elliptic Problems in Unbounded Domains

Abstract

The goal of this work is to study multigrid methods in connection with the numerical solution of elliptic problems in the exterior of a bounded domain. The numerical method consists of approximating the original problem by one on a truncated domain of diameter R and imposing a simple local approximate boundary condition on the outer boundary. The resulting problem is discretized using the finite element method. R must be made sufficiently large to reduce the truncation error (due to the approximate boundary condition) to the level of the discretization error. This results in a very large number of unknowns (increasing like $0(R^3 )$ in three dimensions), when a quasi-uniform mesh is used. In previous work by the author [Math. Comp., 36 (1981), pp. 387–404], it was shown that optimal error estimates hold with the number of unknowns independent of R using a mesh grading procedure in which the size of the elements are systematically increased as their distance from the origin increases. In the present paper it is shown that the multigrid convergence rate is independent of R using a mesh grading of this kind (with the number of unknowns increasing like $\log R$). It is also shown that the optimal error estimates in [Math Comp., 36 (1981), pp. 387–404] can be extended to datum with unbounded support. On the other hand, the number of multigrid iterations is bounded by $0(R^2 )$ when a quasi-uniform mesh is used. For three-dimensional problems, the computational cost is bounded by $0(R^5 )$ using a quasi-uniform mesh and by $0(\log R)$ using the graded mesh. The multigrid analysis is formulated and analyzed in a variational framework using weighted Sobolev spaces.