Multigrid iteration

In CED, a number of different iterative solvers for linear algebraic systems have been applied. Two of the most successful and most widely used methods are conjugate gradient and multigrid methods. The basic idea of the conjugate gradient method is to transform the linear equation system Eq. (38) into a minimization problem... 

In contrast to the conjugate gradient method, the multigrid method is rather a general framework for iterative solvers than a specific method. The multigrid method exploits the fact that the iteration error... 

In one cycle of the multigrid method, first a few iterations are performed on the fine grid in order to obtain a comparatively smooth iteration error. After that the obtained residual is restricted to the coarse grid, where further iterations are performed in order to damp out the long-wave components of the solution error. Subsequently the coarse-grid solution is interpolated to the fine grid and the solution on the fine grid is updated. 

Multigrid methods have proven to be powerful algorithms for the solution of linear algebraic equations. They are to be considered as a combination of different techniques allowing specific weaknesses of iterative solvers to be overcome. For this reason, most state-of-the-art commercial CFD solvers offer the multigrid capability. 

Holst, M., R. E. Kozack, F. Saied and S. Subramaniam. (1994b). Treatment of electrostatic effects in proteins multigrid-based Newton iterative method for solution of the full nonlinear Poisson-Boltzmann equation. Proteins. 18 231-45. 

Multigrid acceleration of the Gauss-Seidel point-iterative method is currently used in many commercial CFD codes to solve the system of algebraic equations resulting from the discretization of the governing equations. For this reason, the basic principles and nomenclature must be known by the users of commercial codes and in particular for researchers that are making their own codes. 

The general numerical method of solution is described in Sec. V. In the Debye-Hiickel approximation, all the equations are linear. First, the equilibrium potential is solved second, the ionic potential and the velocity field are alternately obtained in an iteration loop. These three systems are solved by means of classical iteration schemes such as the conjugate gradient for the potentials and a multigrid technique for the velocity. General estimates of the expected precision are given. 

In the previous section, we discussed the basic theory of the classic iterative solution to elliptic problems. The multigrid method allows for a dramatic performance improvement of standard iterative approaches such as the SOR method. The basic principles of its operations are briefly introduced in the following section. 

The simplest version of the multigrid algorithm is the so-called two-grid iteration employing only two grid levels. In the ith iteration, the procedure starts from the approximation v of u in Eq. [27], and the following five steps are performed ... 

Finally, it should be noted that the multigrid method can be used as either an iterative process or as a direct solver (the so-called full multigrid or nested iteration method ). 

It Is Important to realise that MLATs are not solvers In their own right, but ate merely convergence accelerators and as such require a solution technique such as Gauss-Seldel or some other Iterative procedure. Another Important feature, although not one dealt with In this paper Is the possible application of multigrids... 

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