By William L. Briggs
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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. (Use the fact that the eigenvalues are given by the Rayleigh quotients of the eigenvectors, = (Awfc, Wfc)/(wfc, Wfc), where wk is the eigenvector associated with 16. Properties of Gauss-Seidel. Assume A is symmetric, positive definite. (a) Show that the jth step of a single sweep of the Gauss-Seidel method applied to Au = f may be expressed as (b) Show that the jth step of a single sweep of the Gauss-Seidel method can be expressed in vector form as where is the jth unit vector.
The norm of the error is now about 8% of its initial value. The coarse-grid approximation to the error is now used to correct the fine-grid approximation. After three additional fine-grid relaxations, the 2-norm of the error is reduced to about 3% of the initial error norm. This result is plotted in the bottom right figure. The residual is once again transferred to the coarse grid and three coarse-grid relaxations follow. At this point, the 2-norm of the error is about 1% of its original value.
We need some qualitative terms for the various Fourier modes that have been discussed. The modes in the lower half of the spectrum, with wavenumbers in the range 1 k < , are called low-frequency or smooth modes. The modes in the upper half of the spectrum, with k n — 1, are called high-frequency or oscillatory modes. Having taken this excursion through Fourier modes, we now return to the analysis of the weighted Jacobi method. We established that the eigenvalues of the iteration matrix are given by What choice of w gives the best iterative scheme?
A Multigrid Tutorial by William L. Briggs