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The cascadic multigrid method for elliptic problems. (English) Zbl 0873.65107

The numerical method discussed here may be viewed as multigrid without correction cycles. Consequently, more iterates are needed on the coarse grids than would be used in traditional multigrid methods, and the stopping criterion on each grid is one of the issues faced in this paper. The authors obtain estimates of the rate of convergence and of the computational complexity for their method, with conjugate gradient and with symmetric Gauss-Seidel smoothers. They also consider an adaptive version, with the gridding dependent on a dynamic estimate of the smoothness of the solution. Example computations illustrate the effectiveness of the adaptive method for the Laplace equation on a domain with a re-entrant corner, specifically, a square with a slit.

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
65F10 Iterative numerical methods for linear systems
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65Y20 Complexity and performance of numerical algorithms
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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