DG shape function pros and cons

[lindsayad]

Inspired by the question on idaholab/moose#23986 (comment). What are pros and cons of L2_LAGRANGE vs L2_HIERARCHIC vs MONOMIAL? I am going to create a list here:

L2_LAGRANGE

Pros:

• Implementation simplicity

Cons:

• limited to a max of a third order basis

L2_HIERARCHIC

Pros:

• Arbitrarily high order

Cons:

MONOMIAL

Pros:

• Arbitrarily high order
• less global dofs for meshes with non-simplicial elements

Cons:

• Not as good of an error constant when cross-terms are dropped, e.g. for a P=1 basis you don't get the bilinear term xy that you do with LAGRANGE/HIERARCHIC
• Worse condition number, particularly at high polynomial order

[jwpeterson]

I think that L2_HIERARCHIC are going to have better conditioning at high order than MONOMIAL. Not sure about L2_LAGRANGE in that regard.

[dknez]

I agree, MONOMIAL is pretty much "worst case" for condition number IIRC. If your order isn't too high, then that doesn't matter much, though.

[dknez]

The reason I added L2_LAGRANGE is to enable "nodal DG methods". There is a textbook from Hesthaven about this approach, for example. The main (only?) advantage of this compared to other approaches is simplicity of implementation.

[jwpeterson]

Yeah, computing the nodal_soln for an Elem is trivially easy for L2_LAGRANGE, for instance. We use them for stress postprocessing.

[roystgnr]

MONOMIAL has the same number of DoFs (for the same p level) on simplicial elements, but fewer DoFs on any mesh that has quads/prisms/pyramids/hexes.

It gets that by dropping the "cross terms" - e.g. for dim=2 p=2 you get x², y², xy, but you don't get x²y or y²x or x²y².

In CG problems people generally think those cross terms are worth it - you get a better convergence constant and slightly faster integration and the cost is slightly more bandwidth in your sparsity pattern - but it's not as obvious in DG. I'd actually be somewhat interested to see experiments. If it was worthwhile, it'd be possible to create a FEFamily that covers the same polynomial space as MONOMIAL but has a better-conditioned basis.

[lindsayad]

Is there any equivalence between (L2_)HIERARCHIC and (L2_)LAGRANGE for given element types/polynomial order? Just glancing cursorily at the source code it looks like there is equivalence in 1D for first order ... but that's about it...?

[roystgnr]

The function spaces are the same at any order, but the bases are different for p>1 on any element type.

[jwpeterson]

@lindsayad you may want to add the FEXYZ to your comparison list as well. It is basically the same thing as MONOMIAL but the phi values depend on each element's physical coords, and there is some benefit to this extra work, though I can't remember exactly what at the moment.

[lindsayad]

Can maintainers edit others' posts? ... I guess I can answer my own question on that: yes