Improve the notation for boundary integrators/operators

[nmnobre]

Hi Tzanio,

I think commit 41878ad won't be controversial as it simple aligns the notation with what you already wrote for the H(div) continuous boundary operators in the Bilinear Form Integrators page.

~Commit 9680418 might be. But, unless you are compensating for the sign within VectorFEBoundaryTangentLFIntegrator, which doesn't sound like something you would do(?), I think we do need to flip the sign in the docs?~ EDIT: We aren't compensating for the sign within VectorFEBoundaryTangentLFIntegrator (which I think is a good thing 🙂), so we definitely need the sign flip in the docs. I think the confusion might've arisen because it looks like the units tests we have for VectorFEBoundaryTangentLFIntegrator are all projections, where the domain and boundary operators appear both on the right-hand side with the same sign. The docs cover the more general case though, i.e. solving for $u$ or $\vec{u}$, where that isn't the case.

Cheers, -N

[tzanio]

@mlstowell, can you take a look?

[mlstowell]

I agree that these boundary terms are confusing. In fact people are often tempted to place them on the left hand side and treat the $u$ appearing in them as an unknown. Perhaps we should go a step further and replace $u$ in the boundary terms with something like $u{bc}$. We could then add a concise statement such as "Where $u{bc}$ is a known function."

With this addition I think this PR would be ready to go.

[tzanio]

ping: @nmnobre

[nmnobre]

Apologies Tzanio (@tzanio), I was busy with teaching last week and had some catch-up to do yesterday. :) Since last time, I've 1) added Mark's (@mlstowell) suggestions and 2) consolidated the notation around unit vectors all around. I've generated the docs locally and it looks like everything is rendering appropriately, but I'd appreciate at least a second pair of eyes.

Cheers, -N

[tzanio]

Thanks @nmnobre !