By William L. Briggs
A Multigrid educational is concise, enticing, and obviously written. Steve McCormick is the one man i do know that could pull off educating in spandex. simply ensure you sit down within the again row.
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Extra info for A Multigrid Tutorial
Therefore, \k(R G } gives the convergence rate, not for the kth mode of A, but for the kth eigenvector of RG. 8: Weighted Jacobi method with (a)w>= 1 and (b)w;= | applied to the one-dimensional model problem with n = 64 points. The initial guesses consist of the modes wk for 1 k 63. The graphs show the number of iterations required to reduce the norm of the initial error by a factor of 100 for each Wk. Note that for w = , the damping is strongest for the oscillatory modes (32 k 63). 9: Weighted Jacobi method with w = | applied to the one-dimensional model problem with n = 64 points and with an initial guess consisting of (a) w3, (b) W16, and (c) (w2 + w16)/2.
A Multigrid Tutorial 27 Exercises 1. Residual vs. error. Consider the two systems of linear equations given in the box on residuals and errors in this chapter. Make a sketch showing the pair of lines represented by each system. Mark the exact solution u and the approximation v. Explain why, even though the error is the same in both cases, the residual is small in one case and large in the other. 2. Residual equation. Use the definition of the algebraic error and the residual to derive the residual equation Ae = r.
This time the actual approximations are plotted. The weighted Jacobi method with (w — | is applied to the same model problem on a grid with n — 64 points. 9(a) shows the error with wavenumber k = 3 after one relaxation sweep (left plot) and after 10 relaxation sweeps (right plot). This smooth component is damped very slowly. 9(b) shows a more oscillatory error (k = 16) after one and after 10 iterations. The damping is now much more dramatic. Notice also, as mentioned before, that the weighted Jacobi method preserves modes: once a k = 3 mode, always a k = 3 mode.