Elliptic Problem Solvers by Schultz
By Schultz
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Example text
3). 19) U - LU , which is the discrepancy exhibited when the true differential solution U is substituted into those equations. More precisely it is easy to see that 2 h 2h - h . We can, therefore, use 2h to estimate Th , and hence to derive natural grid-refinement criteria. 17) by t-exi tro~at,~sn. are described in the following sections. 7 Chains of Problems and Time-dependent Problems We often need to solve not just one isolated problem but a sequence of (many) similar problems, depending on some parameter.
Indeed, parallel PDE solvers have been studied quite extensively. See for example the surveys of Ortega and Voigt (1977) and Heller (1978). Most of these studies are based, however, on solution techniques much slower (on sequential machines) than multigrid methods. The latter, moreover, are highly parallelizable. Each of their processes can simultaneously be made at all grid points. The multigrid gain on large-scale parallel machines is sometimes not as great as on sequential or vector machines (since coarse grids may not use all processors), but the potential is still very high.
Conjugate gradient method. choice of Dk Perhaps because of uncertainty about the and the build-up of roundoff errors, Richardson's method lay dormant until its potentialities were made clear by M. R. Hestenes and E. Stiefel (1953). They invented a "conjugate gradient" algorithm for computing recursively the u m having the smallest residual in the Krylov subspace constructed above. Their algorithm is based on the following lemma. minimizes r LEMMA. If u j K Ic K 2 c K3 c • • • , and then sal j u m+1 =u is A-orthogonal to K = b - Au in Kj , where .