Below are some results on a 3D problem, on a Linux cluster at UTEP. On this problem, the iterative method (ISOLVE=3) was much faster than the others, but for many, more difficult, problems, the iterative method converges very slowly, or not at all, and then the direct methods (ISOLVE=1 is a sparse direct solver, ISOLVE=2 is a frontal method, ISOLVE=6 is a parallel band solver) are the only options.

Uxx+Uyy+Uzz=3U in the unit cube.
ISOLVE procs 13x13x13 grid 20x20x20 grid 20x20x20 grid
17576 unknowns 64000 unknowns
rel. err. = 5.E-8 rel. err. = 9.E-9
1 1 117 seconds 2566 seconds 517 MW memory/node
2 1 432 (large) 22
3 1 10 68 13
2 7 42 13
4 6 36 7
8 6 37 4
16 7 39 2
6 1 276 14953 643
2 106 1893 327
4 87 1104 164
8 52 653 82
16 43 377 41

Uxx+Uyy+Uzz=3U in the unit cube.
ISOLVE	procs	13x13x13 grid	20x20x20 grid	20x20x20 grid
		17576 unknowns	64000 unknowns
		rel. err. = 5.E-8	rel. err. = 9.E-9
1	1	117 seconds	2566 seconds	517 MW memory/node
2	1	432	(large)	22
3	1	10	68	13
	2	7	42	13
	4	6	36	7
	8	6	37	4
	16	7	39	2
6	1	276	14953	643
	2	106	1893	327
	4	87	1104	164
	8	52	653	82
	16	43	377	41

For PDE2D, multiple processors are primarily useful for 3D problems, because on 2D problems the sparse direct solver is usually (though not always) faster on one processor than the parallel methods on many processors. However, they are useful for 2D problems when all eigenvalues of an eigenproblem are calculated, as illustrated by the example below.

Uxx+Uyy=λU in the unit circle.
procs 21x51 grid
4284 unknowns
1.0E-7 rel. err. in first eigenvalue
1 3453 seconds
2 1524
4 1044
8 795
16 1171

procs	21x51 grid
	4284 unknowns
	1.0E-7 rel. err. in first eigenvalue
1	3453 seconds
2	1524
4	1044
8	795
16	1171