Multigrid method

Multigrid (MG) methods in numerical analysis are algorithms 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. MG methods can be used as solvers as well as preconditioners.These steps can be used as shown in the MATLAB style pseudo code for 1 iteration of V-Cycle Multigrid:The following represents F-Cycle Multigrid. This multigrid cycle is slower than V-Cycle per iteration but does result in faster convergence.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: Multigrid (MG) methods in numerical analysis are algorithms 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. 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. 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. 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. There are many variations of multigrid algorithms, but the common features are that a hierarchy of discretizations (grids) is considered. The important steps are: There are many choices of multigrid methods with varying trade-offs between speed of solving a single iteration and the rate of convergence with said iteration. The 3 main types are V-Cycle, F-Cycle, and W-Cycle. For a discrete 2D problem, F-Cycle takes 83% more time to compute than a V-Cycle iteration while a W-Cycle iteration takes 125% more. If the problem is setup in a 3D domain, then a F-Cycle iteration and a W-Cycle iteration take about 64% and 75% more time respectively than a V-Cycle iteration ignoring overheads. Typically, W-Cycle produces similar convergence to F-Cycle. 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. In cases where the system has a high condition number, the correction procedure is modified such that only a fraction of the prolongated coarser grid solution is added onto the finer grid.