Multigrid (MG) methods in numerical analysis are a group of algorithms for solving differential equations using a hierarchy of discretizations. The 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. 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. 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. 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 partial differential equations. 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 finite element method becomes multigrid by choosing linear wavelets as the basis.
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Algorithm
There are many variations of multigrid algorithms, but the common features are that a hierarchy of discretisations (grids) is considered. The important steps are:
- 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.
- Prolongation – interpolating a correction computed on a coarser grid into a finer grid.
Convergence rate
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 <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>. Here, <math>\rho = N_{j+1} / N_j < 1</math> is the ratio of grid points on "neighboring" grids and is assumed to be constant throughout the grid hierarchy, and <math>K</math> 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 <math>k</math>:
- <math>W_k = W_{k+1} + \rho K N_k</math>
And in particular, we find for the finest grid <math>N_1</math> that
- <math>W_1 = W_2 + \rho K N_1</math>
Combining these two expressions (and using <math>N_{k} = \rho^{k-1} N_1</math>) gives
- <math>W_1 = K N_1 \sum_{p=0}^n \rho^p </math>
Using the geometric series, we then find (for finite <math>n</math>)
- <math>W_1 < K N_1 \frac{1}{1 - \rho}</math>
that is, a solution may be obtained in <math>O(N)</math> time.
See also
- Numerical analysis
- Partial differential equation
- Finite element method
- Finite difference
- Spectral method
- Domain decomposition method
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
