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'''Multigrid''' ('''MG''') '''methods''' in [[numerical analysis]] are a group of [[algorithm]]s for solving [[differential equations]] using a [[hierarchy]] of [[discretization]]s. The idea is similar to [[extrapolation]] between coarser and finer grids. 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 (physics)|elasticity]] or the [[Navier-Stokes equations]].
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 [[partial differential equation]]s. Research on multilevel techniques for [[hyperbolic equation]]s is under way. Multigrid can also be applied to [[integral equation]]s, or for problems in [[statistical physics]].
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