By William L. Briggs
A Multigrid instructional is concise, attractive, and obviously written. Steve McCormick is the one man i do know which could pull off educating in spandex. simply ensure you take a seat within the again row.
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Extra info for A Multigrid Tutorial
Letting I hvh — v 2h , we have that A Multigrid Tutorial 37 We now have a well-defined way to transfer vectors between fine and coarse grids. Therefore, we can return to the correction scheme and make it precise. To do this, we define the following two-grid correction scheme. Two-Grid Correction Scheme • Relax v\ times on Ahuh = fh on £lh with initial guess vh. • Compute the fine-grid residual rh = fh — Ahvh and restrict it to the coarse grid by r2h = I hrh. • Solve A2he2h=r2h on Q 2h . • Interpolate the coarse-grid error to the fine grid by eh = /2/le2/l and correct thefine-gridapproximation by vh *—vh + eh.
This experiment uses an initial guess consisting of two modes with k = 2 and k = 16. After 10 relaxation sweeps, the high-frequency modulation on the long wave has been nearly eliminated. However, the original smooth component persists. We have belabored the discussion of the weighted Jacobi method because it is easy to analyze and because it shares many properties with other basic relaxation schemes. In much less detail, let us look at the Gauss-Seidel iteration. We can show (Exercise 14) that the Gauss-Seidel iteration matrix for the model problem (matrix A) has eigenvalues These eigenvalues, which are plotted in Fig.
Full multigrid is a technique for integrating them so that they can work together in a way that overcomes these limitations. The result is a very powerful algorithm. Exercises 1. Aliasing. Show that the kth mode on a grid h with n — 1 interior points appears as the (n — k)th mode on 2h when n/2 < k < n. 2. An important equivalence. Consider a stationary, linear method of the form v v + B-1(f — Av) applied to the problem Au = f. Au = f with an arbitrary initial guess v = V0. What are the error and residual associated with V0?
A Multigrid Tutorial by William L. Briggs