Making `quadrature_rule` a public method on relevant element groups about meshmode

[inducer]

I agree with this. This would allow quadrature_rule for the derived classes to return mp.Quadrature(self.weights, self.unit_nodes, exact_to=self.order), correct?

Yep. At least as long as the polynomial space contains P^N (see the current, kinda hokey, definition of exact_to).

Though is this not well-conditioned, right? You mentioned at one point that this is equivalent to Newton-Cotes, which is... problematic for high order

Newton-Cotes with equispaced nodes is obviously terrible. Newton-Cotes with edge-clustered nodes is not that bad. (After all, Newton-Cotes with Legendre nodes is just Gaussian quadrature.)

Here's a demo: https://gist.github.com/inducer/19eb49ff7702d4bf5b12477208540731

[inducer]

Thanks for writing up a summary of this! Suppose we go with the direction that inducer/modepy#33 seems to be leaning (?) (i.e. allow feeding exact_to to the Quadrature constructor), I claim that the resulting "mass matrix quadrature" is exact to the order of the reference polynomial space, by definition of the mass matrix as long as:

• the DOFs are nodal, and
• 1 is represented exactly, by a DOF vector of all ones.

[inducer]

(Opinions?) cc @alexfikl

[thomasgibson]

I claim that the resulting "mass matrix quadrature" is exact to the order of the reference polynomial space, by definition of the mass matrix as long as:

• the DOFs are nodal, and
• 1 is represented exactly, by a DOF vector of all ones.

I agree with this. This would allow quadrature_rule for the derived classes to return mp.Quadrature(self.weights, self.unit_nodes, exact_to=self.order), correct?

Though is this not well-conditioned, right? You mentioned at one point that this is equivalent to Newton-Cotes, which is... problematic for high order 😅

[inducer]

(Note how even the weights stay positive in the demo.)

[inducer]

(But how the rule has obviously lost the 2*n accuracy of Gauss.)

[thomasgibson]

Here's a demo: https://gist.github.com/inducer/19eb49ff7702d4bf5b12477208540731

Very nice!

I've also gone ahead and started working on this issue: 44606a7