*Multigrid* (MG) methods are fast
linear iterative solvers
based on the multilevel or multi-scale paradigm.
The typical application for multigrid is
in the numerical solution of elliptic partial differential equations
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 conditions. Multigrid 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 Lame-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 equations, 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 equations, or for problems in statistical physics.

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*.

The multigrid workbench visualizes the performance of a prototype multigrid algorithm. (See also about the multigrid workbench).

More information about multigrid methods on the net is available from the multigrid hierarchy and MGNet, which contains also an extensive list of references.

Ulrich Ruede , Thu Feb 2 21:05:15 MEZ 1995

Updated by Craig C. Douglas