Multigrid method: Difference between revisions

Revision as of 23:25, 14 July 2005

Revision as of 18:34, 13 August 2005

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In all these cases, multigrid exhibits a convergence rate that is independent of the number of unknowns in the discretized system. It is therefore an optimal method. In combination with nested iteration it can solve these problems to truncation error accuracy in a number of operations that is proportional to the number of unknowns.

Multigrid can be generalized in many different ways. It can be applied naturally in a time-stepping solution of [[parabolic equation]]s, or it can be applied directly to time-dependent PDE. Research on multilevel techniques for hyperbolic equations is under way. Multigrid can also be applied to [[integral equation]]s, or for problems in [[statistical physicphysics]]s.

Other extensions of multigrid include techniques where no PDE and no geometrical problem background is used to construct the multilevel hierarchy. Such '''algebraic multigrid methods''' (AMG) construct their hierarchy of operators directly from the system matrix and thus become true black-box solvers for [[sparse matrices]].