Moment based reffes

[JordiManyer]

So we've been having a look at the current implementation of Raviart-Thomas and Nedelec. It has become clear that the current implementation

• has issues with high-order, that have to do with how we choose our basis/prebasis when computing the moments.
• has issues related to scaling in mixed formulations.
• is outright wrong when using high-order approximations for the geometry.
• is very hard to read, modify and extend.

Moreover, there has been interest lately in implementing more types of moment-based discretisations like Crouzeix-Raviart.

We believe the best way to solve all these issues at once is to create some extendable machinery to easily create moment-based FEs.

TODOs:

• Create the moment-based machinery
• Create the pullback machinery
• Replicate RT and ND in 2D/3D with oriented meshes
• Replicate RT and ND in 2D/3D with non-oriented meshes
• Benchmark against current implementations

New RefFEs:

• Crouzeix-Raviart
• Mardal-Tai-Winther
• Arnold-Winther

Other: Polynomial bases

The solution to the first issue requires extending Jacobi-type polynomial bases. Although it could be done blindly, I think we could try to merge a lot of code together. In the future we might want to have even more basis of polynomials, which would require a lot of copied code.

• Extend Monomial bases to accomodate Jacobi and other types of polynomials
• Benchmark extensively

[codecov]

Codecov Report

Attention: Patch coverage is 28.44037% with 468 lines in your changes missing coverage. Please review.

Project coverage is 87.25%. Comparing base (06de5f7) to head (b9b8fc4).

Files with missing lines	Patch %	Lines
src/Polynomials/ChebyshevPolynomialBases.jl	0.00%	341 Missing ⚠️
src/Polynomials/QGradJacobiPolynomialBases.jl	37.65%	101 Missing ⚠️
src/ReferenceFEs/RaviartThomasRefFEs.jl	60.00%	18 Missing ⚠️
src/Polynomials/JacobiPolynomialBases.jl	0.00%	4 Missing ⚠️
src/ReferenceFEs/MomentBasedReferenceFEs.jl	0.00%	4 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1048      +/-   ##
==========================================
- Coverage   88.88%   87.25%   -1.64%     
==========================================
  Files         195      198       +3     
  Lines       23570    24126     +556     
==========================================
+ Hits        20951    21050      +99     
- Misses       2619     3076     +457     

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.