Download e-book for iPad: A Multigrid Tutorial by William L. Briggs

By William L. Briggs

ISBN-10: 0898714621

ISBN-13: 9780898714623

A Multigrid instructional is concise, enticing, 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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Example text

The solution (in this case, the error) has become smoother and its norm is now 36% of the initial error norm. Further relaxations on the fine grid would provide only a slow improvement at this point. This signals that it is time to move to the coarse grid. The middle right plot shows the fine-grid error after one relaxation sweep on the coarse-grid residual equation, superimposed on the initial guess. Clearly, we have achieved another reduction in the error by moving to the coarse grid; the norm of the error is now 26% of the initial error norm.

Chapter 3 Elements of Multigrid Through analysis and experimentation, we have examined some of the basic iterative methods. Our discoveries have formed the beginnings of what we might call a spectral (or Fourier mode) picture of relaxation schemes. As we proceed, more essential details of this picture will become clear. So far we have established that many standard iterative methods possess the smoothing property. This property makes these methods very effective at eliminating the high-frequency or oscillatory components of the error, while leaving the low-frequency or smooth components relatively unchanged.

Recall that w = was chosen to give preferential damping to the oscillatory components. Indeed, the smooth waves are damped very slowly, while the upper half of the spectrum (k ) shows rapid convergence. Again, this is consistent with Fig. 7. 9. 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).

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A Multigrid Tutorial by William L. Briggs


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