Multigrid method

Multigrid (MG) methods in numerical analysis are a group of algorithms for solving differential equations using a hierarchy of discretizations. The main idea of multigrid is to accelerate the convergence of a base iterative method by correcting, from time to time, the solution globally by solving a coarse problem. This idea is similar to extrapolation between coarser and finer grids. The typical application for multigrid is in the numerical solution of elliptic partial differential equations in two or more dimensions.^[1]

Multigrid methods can be applied in combination with any of the common discretization techniques. In these cases, multigrid methods are among the fastest solution techniques known today. In contrast to other methods, multigrid methods are general in that they can treat arbitrary regions and boundary conditions. They do not depend on the separability of the equations or other special properties of the equation. They are also directly applicable to more-complicated non-symmetric and nonlinear systems of equations, like the Lamé system of elasticity or the Navier-Stokes equations.

Multigrid methods can be generalized in many different ways. They can be applied naturally in a time-stepping solution of parabolic partial differential equations, or they can be applied directly to time-dependent partial differential equations. Research on multilevel techniques for hyperbolic partial differential equations is under way. Multigrid methods can also be applied to integral equations, or for problems in statistical physics.

Other extensions of multigrid methods include techniques where no partial differential equation nor 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 finite element method may be recast as a multigrid method by choosing linear wavelets as the basis.

Algorithm

There are many variations of multigrid algorithms, but the common features are that a hierarchy of discretizations (grids) is considered. The important steps are:^[2][3]

• Smoothing – reducing high frequency errors, for example using a few iterations of the Gauss–Seidel method.
• Restriction – downsampling the residual error to a coarser grid.
• Interpolation or Prolongation' – interpolating a correction computed on a coarser grid into a finer grid.

Computational cost

This approach has the advantage over other methods that it often scales linearly with the number of discrete nodes used. That is: It can solve these problems to a given accuracy in a number of operations that is proportional to the number of unknowns.

Assume that one has a differential equation which can be solved approximately (with a given accuracy) on a grid ${\displaystyle i}$  with a given grid point density ${\displaystyle N_{i}}$ . Assume furthermore that a solution on any grid ${\displaystyle N_{i}}$  may be obtained with a given effort ${\displaystyle W_{i}=\rho KN_{i}}$  from a solution on a coarser grid ${\displaystyle i+1}$ . Here, ${\displaystyle \rho =N_{j+1}/N_{j}<1}$  is the ratio of grid points on "neighboring" grids and is assumed to be constant throughout the grid hierarchy, and ${\displaystyle K}$  is some constant modeling the effort of computing the result for one grid point.

The following recurrence relation is then obtained for the effort of obtaining the solution on grid ${\displaystyle k}$ :

${\displaystyle W_{k}=W_{k+1}+\rho KN_{k}}$ 

And in particular, we find for the finest grid ${\displaystyle N_{1}}$  that

${\displaystyle W_{1}=W_{2}+\rho KN_{1}}$ 

Combining these two expressions (and using ${\displaystyle N_{k}=\rho ^{k-1}N_{1}}$ ) gives

${\displaystyle W_{1}=KN_{1}\sum _{p=0}^{n}\rho ^{p}}$ 

Using the geometric series, we then find (for finite ${\displaystyle n}$ )

${\displaystyle W_{1}<KN_{1}{\frac {1}{1-\rho }}}$ 

that is, a solution may be obtained in ${\displaystyle O(N)}$  time.

In-line sources

1. U Trottenberg, CW Oosterlee, A Schüller (2001). Multigrid. Academic Press. ISBN 012701070X.
2. MT Heath (2002). "§ 11.5.7". Scientific Computing: An Introductory Survey. McGraw-Hill Higher Education. p. 478 ff. ISBN 007112229X.
3. P Wesseling (1992). An Introduction to Multigrid Methods. Wiley. ISBN 0471930830.

General references

• Achi Brandt, Multi-Level Adaptive Solutions to Boundary-Value Problems, Math. Comp, 1977(31), 333-390 (jstor link).
• William L. Briggs, Van Emden Henson, and Steve F. McCormick, A Multigrid Tutorial, Second Edition, SIAM, 2000 (book home page), ISBN 0-89871-462-1

See also

• Numerical analysis
• Partial differential equation