A Multigrid Tutorial by William L. Briggs

By William L. Briggs

A Multigrid educational is concise, enticing, and obviously written. Steve McCormick is the single man i do know that could pull off educating in spandex. simply ensure you sit down within the again row.

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C) Show that the eigenvector associated with is Wk,j = cos sin ( . 15. Richardson iteration. (a) Recall that for real vectors u, v, the inner product is given by (u, v) = uTv and ||u|| = (u, u). Furthermore, if A is symmetric positive definite, A Multigrid Tutorial 29 then || A||2 = p(A), the spectral radius of A. Richardson's iteration is given by where r(0) = f — Av (0) is the residual. Show that when A has a constant diagonal, this method reduces to the weighted Jacobi method. (b) Show that the error after one sweep of Richardson's method is governed by (c) If the eigenvalues of A are ordered 0 < < 2 < ••• < and the smallest eigenvalues correspond to the smooth modes, show that Richardson's method has the smoothing property.

Second, all of the quantities in the above procedure are well defined except for A2h. For the moment, we take A2h simply to be the result of discretizing the problem on 2h. Finally, the integers v\ and v are parameters in the scheme that control the number of relaxation sweeps before and after visiting the coarse grid. They are usually fixed at the start, based on either theoretical considerations or on past experimental results. It is important to appreciate the complementarity at work in the process.

We do not describe this type of data management in any further detail, as the advances in these languages occur so rapidly that any discussion would soon be outdated! We describe a data structure for a simpler FORTRAN-like language. Multigrid codes "grew up" in such an environment and many people learn to write multigrid codes using MATLAB or a similar prototyping language with more restrictive data structures. With these languages, there seems to be general agreement that the solutions and right-side vectors on the various grids should be stored contiguously in single arrays.

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