Support averaging operators (fixes #107)

[wence-]

Done by just integrating the averaged expression over the relevant
entity.

Needs a matching UFL merge.

[wence-]

That's here: https://bitbucket.org/fenics-project/ufl/pull-requests/89/cellavg-facetavg-are-no-longer-terminal/diff

[wence-]

This does the wrong thing for cell_avg(inner(f, f))*dx where f is VFS right now. I think I'm just being dense.

[wence-]

Think I fixed it, the XXX average is (sum_q w_q expr_q) / (reference_entity_volume), not just sum_q w_q expr_q.

[blechta]

For affine-transformed cell, yes. For bendy cells each infinitesimal reference cell volume contributes differently to physical volume... Not sure which case is Q1 cell.

[wence-]

I don't think so. Here's my reasoning. For a function f, the mean over some compact domain U is 1/Vol(U) \int f dU. Which, with quadrature, is (\sum_q w_q f_q) / (\sum_q w_q).

Now consider integrating cell_avg(f)*dx.

This is \int_\Omega \bar{f} dx = \sum_e \int_e \bar{f} dx_e.

Now I pull back to reference space:

\sum_e \int_e \bar{f} dx_e = \sum_e \int_E \bar{\tilde{f}} \det J dX

and \bar{\tilde{f}} = \int_E \tilde{f} dX / Vol(dX) (because we're already in reference space).

I think the only thing I assumed here was that averaging commutes will pullback.

Does this make sense?

[wence-]

This is still totally borked. Promotes bugs in tsfc optimisation passes (possibly I'm doing something wrong). Or doesn't work at all.

[blechta]

I don't think that the assumption is correct. We have

\bar{f} = \frac{ \int_e f dx }{ \int_e dx } = \frac{ \int_E \hat{f} |\det J| dX }{ \int_E |\det J| dX }

This is clearly not equal to

\frac{ \int_E \hat{f} dX }{ \int_E dX }

unless \det J is constant in E.

[wence-]

Thanks. I've come to that conclusion too. The latest commits do this "right", but still have issues elsewise. So I'm holding off any merging for now.

[miklos1]

I like that you could implement this in such a small diff.