6 - Multilevel methods and preconditioners [T3]: coarse grid approximation

Summary

Whenever both the multigrid method and the domain decomposition method work, the multigrid is faster.

Jinchao Xu, Lecture at University of Leicester EPSRC Numerical Analysis Summer School, UK (1998)

This paper provides an approach for developing completely parallel multilevel preconditioners.… The standard multigrid algorithms do not allow for completely parallel computations, since the computations on a given level use results from the previous levels.

James H. Bramble, et al. Parallel multilevel preconditioners. Mathematics of Computation, Vol. 55 (1990)

Multilevel methods [including multigrid methods and multilevel preconditioners] represent new directions in the recent research on domain decomposition methods … they have wide applications and are expected to dominate the main stream researches in scientific and engineering computing in 1990s.

Tao Lu, et al. Domain Decomposition Methods. Science Press, Beijing (1992)

The linear system (1.1) may represent the result of discretization of a continuous (operator) problem over the finest grid that corresponds to a user required resolution. To solve such a system or to find an efficient preconditioner for it, it can be advantageous to set up a sequence of coarser grids with which much efficiency can be gained.

This sequence of coarser grids can be nested with each grid contained in all finer grids (in the traditional geometry-based multigrid methods), or non-nested with each grid contained only in the finest grid (in the various variants of the domain decomposition methods), or dynamically determined from the linear system alone in a purely algebraic way (the recent algebraic multigrid methods).