Further developments in Polytope and RefFE for high order FEM about gridap.jl

[fverdugo]

• Think about more general monomial generation for high order FEM

This work needs to be done in the other repo TensorPolynomialBases.jl. I can do this. For scalar-valued polynomials it is quite clear what anisotropic order is. But for vector, and tensor valued ones? What do we need, 1. or 2. ?

1. Different order in each direction, but idem for all components. E.g., in 2D, $u_1(x_1,x_2)$ and $u_2(x_1,x_2)$ have the same order in each direction.

2. A more general case, where the order in each direction also depends on the component. E.g., in 2D, $u_1(x_1,x_2)$ and $u_2(x_1,x_2)$ can have different orders in each direction.

Implementing 1. is straight forward. Implementing 2. requires some more work but can also be done.

[santiagobadia]

The funcionality 2 is needed to implement Nedelec or RT elements. E.g., Qk,k+1xQk+1,k. It can only be used for n-cubes. For Tets even more complicated, check Olm's article.

[fverdugo]

For n-cubes, I can implement the monomials needed for Nedelec or RT for arbitrary dimensions.

However, for n-simplices, do we want to implement the general formula in Olm's paper? or we need to implement only the formula for 2D and the formula for 3D?

[fverdugo]

For n-cubes, I can implement the monomials needed for Nedelec or RT for arbitrary dimensions.

Done!

@santiagobadia I have implemented the monomial basis needed for Nedelec for n-cubes for arbitrary dims. It can be build with the constructor GradMonomialBasis (I have called like this since it corresponds to the gradient of the monomials of an isotropic Q-space, but we can find another name...)

using Gridap
p = Point{2,Int}[(2,3),(5,7)]
T = VectorValue{2,Float64}
b = GradMonomialBasis(T,3) # Q_{2,3} \times Q_{3,2}
evaluate(b,p)
evaluate(∇(b),p)

You will need to instantiate your environment