Multigrid method: Difference between revisions

Revision as of 08:10, 10 March 2006

Revision as of 08:27, 10 March 2006

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'''Multigrid''' ('''MG''') '''methods''' in [[numerical analysis]] are fasta lineargroup iterativeof solvers[[algorithm]]s basedfor onsolving the[[differential multilevelequations]] orusing multi-scalea paradigm[[hierarchy]] of [[discretization]]s. The typical application for multigrid is in the numerical solution of [[elliptic operator|elliptic]] [[partial differential equation]]s in two or more dimensions.

MG can be applied in combination with any of the common discretization techniques. In these cases, multigrid is among the fastest solution techniques known today. In contrast to other methods, multigrid is general in that it can treat arbitrary regions and [[boundary condition]]s. It does not depend on the separability of the equations or other special properties of the equation. MG is also directly applicable to more complicated, non-symmetric and nonlinear systems of equations, like the [[Lamé system]] of [[elasticity]] or the [[Navier-Stokes equations]].

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 physics]].

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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]].

== Convergence rate ==

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InThis allapproach thesehas cases,the multigridadvantage exhibitsover aother convergence ratemethods that isit independentoften ofscales linearly with the number of unknownsdiscrete innodes the discretized systemused. ItThat is: therefore an optimal method. In combination with nested iteration itIt can solve these problems to truncationa errorgiven accuracy in a number of operations that is proportional to the number of unknowns.

In [[mathematics]], more specifically in [[numerical analysis]], '''multigrid methods''' are a group of [[algorithm]]s for solving [[differential equations]] using a [[hierarchy]] of [[discretization]]s. This approach has the advantage over other methods that it scales linearly with the number of discrete nodes used.

In order for the multigrid methods to be applicable, one needs to make several assumptions. Assume that one has a differential equation which can be solved approximately (with a given accuracy) on a grid <math>i</math> with a given grid point

density <math>N_i</math>. Assume furthermore that a solution on any grid <math>N_i</math> may be obtained with a given

effort <math>W_i = \rho K N_i</math> from a solution on a coarser grid <math>i+1</math> with grid point