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

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