By William L. Briggs
A Multigrid educational is concise, attractive, and obviously written. Steve McCormick is the single man i do know which may pull off instructing in spandex. simply ensure you sit down within the again row.
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Extra info for A Multigrid Tutorial
1) and substitute. Assuming for the moment that Cj = 0, we find that the amplitudes are related by (Exercise 7) To find the smoothing factor from the complex amplification factor, it is easiest to plot |G( )|, as shown in Fig. 2. A bit of analysis reveals that A subtle point could be made here. The amplification factor, G( ), gives the (complex) eigenvalues of the Gauss-Seidel iteration matrix, not on a bounded domain with specified boundary conditions, but on an infinite domain. This calculation differs from the eigenvalue calculation of Chapter 2, in which the eigenvalues for a bounded domain were found to be real.
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. 3. Weighted Jacobi iteration. (a) Starting with the component form of the weighted Jacobi method, show that it can be written in matrix form as v(1) = [(1 — LU}!
There is also a good physical explanation for why smooth error modes are so resistant to relaxation. 2) that stationary linear iterations can be written in the form Subtracting this equation from the exact solution u, the error at the next step is We see that changes in the error are made with spatially local corrections expressed through the residual. If the residual is small relative to the error itself, then changes in the error will be correspondingly small. At least for the model problems we have posed, smooth error modes have relatively small residuals (Exercise 19), so the error decreases slowly.
A Multigrid Tutorial by William L. Briggs