In numerical analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in problems exhibiting multiple scales of behavior. For example, many basic relaxation methods exhibit different rates of convergence for short- and long-wavelength components, suggesting these different scales be treated differently, as in a Fourier analysis approach to multigrid.[1] MG methods can be used as solvers as well as preconditioners.
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. The coarse problem, while cheaper to solve, is similar to the fine grid problem in that it also has short- and long-wavelength errors. It can also be solved by a combination of relaxation and appeal to still coarser grids. This recursive process is repeated until a grid is reached where the cost of direct solution there is negligible compared to the cost of one relaxation sweep on the fine grid. 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.[2]
Multigrid methods can be applied in combination with any of the common discretization techniques. For example, the finite element method may be recast as a multigrid method.[3] In these cases, multigrid methods are among the fastest solution techniques known today. In contrast to other methods, multigrid methods are general in that they can treat arbitrary regions and boundary conditions. They do not depend on the separability of the equations or other special properties of the equation. They have also been widely used for more-complicated non-symmetric and nonlinear systems of equations, like the Lamé equations of elasticity or the Navier-Stokes equations.[4]
^Roman Wienands; Wolfgang Joppich (2005). Practical Fourier analysis for multigrid methods. CRC Press. p. 17. ISBN 978-1-58488-492-7.
^U. Trottenberg; C. W. Oosterlee; A. Schüller (2001). Multigrid. Academic Press. ISBN 978-0-12-701070-0.
^Yu Zhu; Andreas C. Cangellaris (2006). Multigrid finite element methods for electromagnetic field modeling. Wiley. p. 132 ff. ISBN 978-0-471-74110-7.
^Shah, Tasneem Mohammad (1989). Analysis of the multigrid method (Thesis). Oxford University. Bibcode:1989STIN...9123418S.
In numerical analysis, a multigridmethod (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are...
techniques. For example, the finite element method may be recast as a multigridmethod. In these cases, multigridmethods are among the fastest solution techniques...
Applications, Volume 449, 1-16, 2014. John C. Urschel. A Space-Time MultigridMethod for the Numerical Valuation of Barrier Options, Communications in Mathematical...
parallel across the system methods like waveform relaxation. Parareal can be derived as both a multigridmethod in time method or as multiple shooting along...
manufacturer Mercedes-AMG, a division of Mercedes-Benz AM General Algebraic multigridmethod for solving differential equations Amagat (abbreviated amg), a unit...
University Park. He is known for his work on multigridmethods, domain decomposition methods, finite element methods, and more recently deep neural networks...
Meshes are also coarsened, removing elements for efficiency. The multigridmethod does something similar to refinement and coarsening to speed up the...
are for ODEs Stretched grid method — for problems solution that can be related to an elastic grid behavior. Multigridmethod — uses a hierarchy of nested...
O ( n ) {\displaystyle O(n)} solution can also be computed using multigridmethods. In computational fluid dynamics, for the solution of an incompressible...
values, contrast that with the similar but computationally cheaper multigridmethod which propagates error-correction estimates down and allows for different...
Lloyd Shapley 1964 – Heapsort developed by J. W. J. Williams 1964 – multigridmethods first proposed by R. P. Fedorenko 1965 – Cooley–Tukey algorithm rediscovered...
method relies on the SIMPLEC algorithm applied to a collocated grid arrangement. The pressure correction equation is solved using a multigridmethod for...
non-zero entries). Sparse linear solution methods, such as sparse factorization, conjugate-gradient, or multigridmethods can be used to solve these systems...
benchmarks should feature new parallel-aware algorithmic and software methods, genericness and architecture neutrality, easy verifiability of correctness...
from simple finite difference schemes to the more mature multigrid and finite element methods. Many interesting problems in science and engineering are...
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