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Multigrid methods for elliptic problems in unbounded domains. (English) Zbl 0772.65075
This paper analyzes multigrid methods for elliptic problems in unbounded domains. To this end, one uses an approximate boundary condition (ABC) method as follows: One first replaces the unbounded domain by a bounded subdomain defined by a ball of radius centered at the origin, and the condition at infinity is replaced by a convenient approximate boundary condition.
The optimal error scheme is achieved using a finite element space by grading the mesh in such a way that the element mesh sizes become larger as the distance of the element from the origin increases. One so obtains a sparse system, and it is shown that both the conditioning and the multigrid convergence rates are independent of the radius of the ball. The ball radius in the (ABC)-method must be sufficiently large in order to reduce the truncation error due to the approximate boundary condition.

##### MSC:
 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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