Umut Tabak <[email protected]> writes: > Preconditioner side: my experience was that one should be really lucky > to get a good preconditioner which is really really rare, as mentioned, > especially for ill-conditioned problems, almost impossible. If my > condition number estimate is above, say, 1e4 1e5, I do not expect much > from iterative methods,
Ill-conditioning is a red herring. For example, FMG can solve well-behaved problems with 1e12 condition number in one cycle (about 5 "work units"). OTOH, very well-conditioned problems with eigenvalues encircling the origin converge extremely slowly (these are nonsymmetric). Anyway, some SPD industrial problems see poor performance with AMG, BDDC, and similar otherwise-scalable methods due to discretization or physical features that elude the heuristics used to produce good coarse spaces. Sometimes these problems can be formulated in more solver-friendly ways. Other times, custom methods would be needed. Or the methods could converge well, but only with high grid complexity (coarse spaces that do not decay in size fast enough).
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