**2. Multigrid Methods**

The MGMs are proving themselves as very successful tools for solving the SLAE associated with discretization of partial differential equations (PDEs).

The main idea of the MGM has been proposed by R.P. Fedorenko in [16]. Then, A. Brandt [17], W. Hackbusch [18], and other researchers showed the efficiency of the multigrid approach and extended Fedorenko's idea.

The multigrid technique is based on two principles: error smoothing and coarse grid correction. The smoothing property is fundamental for the MGM. It is connected with fast damping high-frequency Fourier components of an initial error in decomposition on the basis from eigenvectors.

There exist two approaches in the MGM: geometric multigrid and algebraic multigrid methods.

Geometric multigrid methods were critical to the early development of the MGM and still play an important role today. Nevertheless, there are classes of problems for which geometric techniques are too difficult to apply or cannot be used at all. These classes can be solved by the algebraic multigrid (AMG) methods, as introduced in [19,20].

The MGM is not a fixed algorithm. Rather, there is a multigrid technique that defines its scope. The efficiency of the MGM depends on the adjustment of its components to the considered problem [21]. The key to this is the correct choice of its components and effective interaction between smoothing and coarse-grid correction [22]. We need to use special iteration methods as smoothers for the MGM and nonstandard course-grid correction to a good approximation of the smooth error components.

The smoothing method is the central component of the multigrid algorithm; it is the most dependent part of the MGM on the type of the problem being solved. The role of smoothing methods is that they should not so much reduce the total error as smooth it (namely, suppress the high-frequency harmonics of the error) so that the error can be well approximated on a coarse grid.

Standard smoothing methods are linear iteration methods, for example, the Gauss–Seidel method. An alternative is the following methods:


The MGMs can be used as solvers as well as preconditioners. The MGMs have been widely used for complicated nonsymmetric and nonlinear systems, like the Lame equations of elasticity or the Navier–Stokes problems.
