Multigrid method

They are an example of a class of techniques called multiresolution methods, very useful in problems exhibiting multiple scales of behavior.

The main idea of multigrid is to accelerate the convergence of a basic iterative method (known as relaxation, which generally reduces short-wavelength error) by a global correction of the fine grid solution approximation from time to time, accomplished by solving a coarse problem.

This multigrid cycle typically reduces all error components by a fixed amount bounded well below one, independent of the fine grid mesh size.

The typical application for multigrid is in the numerical solution of elliptic partial differential equations in two or more dimensions.

[4] There are many variations of multigrid algorithms, but the common features are that a hierarchy of discretizations (grids) is considered.

However, in cases of convection-diffusion problems with high Péclet numbers, W-Cycle can show superiority in its rate of convergence per iteration over F-Cycle.

The choice of smoothing operators are extremely diverse as they include Krylov subspace methods and can be preconditioned.

Any geometric multigrid cycle iteration is performed on a hierarchy of grids and hence it can be coded using recursion.

Since the function calls itself with smaller sized (coarser) parameters, the coarsest grid is where the recursion stops.

Similarly the procedures can modified as shown in the MATLAB style pseudo code for 1 iteration of W-cycle multigrid for an even superior rate of convergence in certain cases: This approach has the advantage over other methods that it often scales linearly with the number of discrete nodes used.

[citation needed] A multigrid method with an intentionally reduced tolerance can be used as an efficient preconditioner for an external iterative solver, e.g.,[7] The solution may still be obtained in

Multigrid preconditioning is used in practice even for linear systems, typically with one cycle per iteration, e.g., in Hypre.

Such imposed SPD constraints may complicate the construction of the preconditioner, e.g., requiring coordinated pre- and post-smoothing.

Originally described in Xu's Ph.D. thesis[9] and later published in Bramble-Pasciak-Xu,[10] the BPX-preconditioner is one of the two major multigrid approaches (the other is the classic multigrid algorithm such as V-cycle) for solving large-scale algebraic systems that arise from the discretization of models in science and engineering described by partial differential equations.

The BPX-preconditioner is known to be naturally more parallel and in some applications more robust than the classic V-cycle multigrid method.

[12] Research on multilevel techniques for hyperbolic partial differential equations is underway.

[13] Multigrid methods can also be applied to integral equations, or for problems in statistical physics.

[15][16] For example, one use of wavelets is to reformulate the finite element approach in terms of a multilevel method.

Practically important extensions of multigrid methods include techniques where no partial differential equation nor geometrical problem background is used to construct the multilevel hierarchy.

[19] Such algebraic multigrid methods (AMG) construct their hierarchy of operators directly from the system matrix.

In classical AMG, the levels of the hierarchy are simply subsets of unknowns without any geometric interpretation.

While classical AMG was developed first, a related algebraic method is known as smoothed aggregation (SA).

In an overview paper[21] by Jinchao Xu and Ludmil Zikatanov, the "algebraic multigrid" methods are understood from an abstract point of view.

They developed a unified framework and existing algebraic multigrid methods can be derived coherently.

Also, they proved that, under appropriate assumptions, the abstract two-level AMG method converges uniformly with respect to the size of the linear system, the coefficient variation, and the anisotropy.

The well known Parareal parallel-in-time integration method can also be reformulated as a two-level multigrid in time.

Nearly singular problems arise in a number of important physical and engineering applications.

Simple, but important example of nearly singular problems can be found at the displacement formulation of linear elasticity for nearly incompressible materials.

There were many works to attempt to design a robust and fast multigrid method for such nearly singular problems.