The Combination Technique for the Sparse Grid Solution of PDE's on Multiprocessor Machines

We present a new method for the solution of partial di erential equations. In contrast to the usual approach which needs in the 2-D case O(h 2 n ) grid points, our combination technique works with only O(h 1 n ld(h 1 n )) grid points, where h n denotes the employed grid size. The accuracy of the obtained solution deteriorates only slightly from O(h 2 n ) to O(h 2 n ld(h 1 n )) for a su ciently smooth solution. Additionally, the new method is perfectly suited for parallelization. On a machine with ld(h 1 n ) processors we get in practice an overall parallel complexity of only O(h 1 n ). The method can be generalized to higher dimensions. Then, the gain is expected to be even more dramatic. For the 2-dimensional case, we report the results of numerical experiments obtained on a Transputer system and on the CRAY Y-MP. 1. The combination technique. We consider a partial di erential equation