Multigrid with Overlapping Patches
نویسندگان
چکیده
SUMMARY Solving boundary value problems with optimal eeciency requires adaptivity and multilevel techniques. In 7] an implementation of the AFACx algorithm (cf. 9]) is presented that is based on rectangular cartesian grids. This implementation does not allow for overlap of grids that lie on the same level of reenement. We investigate the case where these grids can overlap. A standard technique for overlapping grids is the Schwarz algorithm (cf. 13] and 14]). Some ways of using the Schwarz algorithm in a standard multigrid scheme are presented. Also, a problem that arises in some situations with non-aligned overlapping grids is described. This situation comes up in a natural way when the Schwarz algorithm is used as a relaxation scheme within a multilevel algorithm. We identify the reason for the bad convergence and show that by more sophisticated interpolation the diiculties can be overcome. Then we present a multiplicative Schwarz algorithm for a large number of grids that has a high potential for parallelization. Finally we give some numerical results for the FACx algorithm with overlapping grids on each reenement level. The implementation of the described codes uses C++ and the array class libraries A++ and P++ (cf. 5],,6] and 12]). Using the A++/P++ programming environment it was possible to move from a serial code to a parallel code within a few days.
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