Multigrid

Fattebert J-L and Bernholc J 2000 Towards grid-based 0(N) density-functional theory methods optimized nonorthogonal orbitals and multigrid acceleration Phys. Rev. B 62 1713-22... 

S.F. McCormick, Editor, Multigrid Methods , SIAM, Philadelphia, 1987... 

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... 

A method for smoothing the residual obtained on the fine grid in order to compute the corresponding residual on the coarse grid. In the terminology of the multigrid method, this step is called restriction. 

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. 

McCeomick, S. F., a Multigrid Tutorial, 2nd ed.. Society for Industrial and Applied Mathematics, Philadelphia (2000). 

W. Hackbusch, Multigrid Methods and Applications. Springer-Verlag, Berlin, 1985. 

In the last decade, most new algorithms, schemes, solvers, and preconditioners have found their way into most commercial software packages. Multigrid solvers are also available. Furthermore, all CFD vendors have developed powerful pre- and post processing routines. 

F. Schmidt, et al.. Adaptive multigrid methods for the vectorial Maxwell eigenvalue... 

To be successful in solving applied and mostly differential problems numerically, we must know how to implement our physico-chemical based differential equations models inside standard numerical ODE solvers. The numerical ODE solvers that we use in this book are integrators that work only for first-order differential equations and first-order systems of differential equations. [Other DE solvers, for which we have no need in this book, are discretization methods, finite element methods, multigrid methods etc.]... 

Terzopoulos D 1986 Image analysis using multigrid relaxation methods. IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI-8(2), 129-139. 

J. Fish, W. Chen Discrete-to-continuum bridging based on multigrid principles. Comp. Meth. App. Mech. Eng. 193, 1693-1711 (2004)... 

Wesseling P., An Introduction to Multigrid Methods, Wiley, New York (1992)... 

Fio. 4. Types of multiscale modeling and solution strategies. Hybrid models (one model at each scale) apply well when there is separation of scales (onion or nested-type models). When there is lack of separation of scales, mesoscale models need to be developed where the same technique (e.g., MD or MC) is accelerated. Alternatively, multigrid (heterogeneous) hybrid models can be employed where the unresolved degrees of freedom are determined from a finer scale model and passed to a coarser scale model. 

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