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Minor improvements to Bilinear Form Integrators page #310

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@nmnobre nmnobre commented Feb 17, 2025

Hi @tzanio,

The initial motivation for this PR was clarifying the requirement for H1 (vs L2) when the gradient operator is required by the bilinear integrator. We say L2 spaces have no derivative operator, but (please do correct me if I'm wrong) if mapping by value (but not by integral - why?), we do define a gradient operator. Now, I understand this gradient is local only (i.e. within an element), i.e. it's not defined in the weak sense (i.e. across elements) like in H1, and I expect this is the reason for the H1 requirement. So then I thought that maybe we should add a note explaining this, but it looks like we already say something to this effect: "Many of these operators will work with either H1 or L2 basis functions but some that require a gradient operator should be used with H1.". Is the "should" in this sentence trying to capture what I just tried to explain?

So, at the moment, the contents of this PR are fairly trivial, but we could expand a bit more on the usage of these integrators in L2 if you think that'd be appropriate.

In addition, while going down the rabbit hole of why we define a gradient when the elements are value- but not integral-mapped, I also came across this table. Some collections for which the basis type is an option are hard-coded to a specific type, e.g. compare L2_[DIM]_[ORDER] and L2_T[BTYPE]_[DIM]_[ORDER]: they're both down as (open) Gauss-Legendre. Is this on purpose - if so, why?

Cheers,
-Nuno

@tzanio tzanio requested review from mlstowell and pazner February 19, 2025 00:27
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