Multigrid method

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.