Efficiently construct FEM/FVM matrix

Question

This is a typical use case for FEM/FVM equation systems, so is perhaps of broader interest. From a triangular mesh à la

I would like to create a scipy

Answer 1

So, in the end this turned out to be the difference between COO's and CSR's sum_duplicates (just like @hpaulj suspected). Thanks to the efforts of everyone involved here (particularly @paul-panzer), a PR is underway to give tocsr a tremendous speedup.

SciPy's tocsr does a lexsort on (I, J), so it helps organizing the indices in such a way that (I, J) will come out fairly sorted already.

For for nx=4, ny=2 in the above example, I and J are

[1 6 3 5 2 7 5 5 7 4 5 6 0 2 2 0 1 2 1 6 3 5 2 7 5 5 7 4 5 6 0 2 2 0 1 2 5 5 7 4 5 6 0 2 2 0 1 2 1 6 3 5 2 7 5 5 7 4 5 6 0 2 2 0 1 2 1 6 3 5 2 7]
[1 6 3 5 2 7 5 5 7 4 5 6 0 2 2 0 1 2 5 5 7 4 5 6 0 2 2 0 1 2 1 6 3 5 2 7 1 6 3 5 2 7 5 5 7 4 5 6 0 2 2 0 1 2 5 5 7 4 5 6 0 2 2 0 1 2 1 6 3 5 2 7]

First sorting each row of cells, then the rows by the first column like

cells = numpy.sort(cells, axis=1)
cells = cells[cells[:, 0].argsort()]

produces

[1 4 2 5 3 6 5 5 5 6 7 7 0 0 1 2 2 2 1 4 2 5 3 6 5 5 5 6 7 7 0 0 1 2 2 2 5 5 5 6 7 7 0 0 1 2 2 2 1 4 2 5 3 6 5 5 5 6 7 7 0 0 1 2 2 2 1 4 2 5 3 6]
[1 4 2 5 3 6 5 5 5 6 7 7 0 0 1 2 2 2 5 5 5 6 7 7 0 0 1 2 2 2 1 4 2 5 3 6 1 4 2 5 3 6 5 5 5 6 7 7 0 0 1 2 2 2 5 5 5 6 7 7 0 0 1 2 2 2 1 4 2 5 3 6]

For the number in the original post, sorting cuts down the runtime from about 40 seconds to 8 seconds.

Perhaps an even better ordering can be achieved if the nodes are numbered more appropriately in the first place. I'm thinking of Cuthill-McKee and friends.