[Dune] Gaussian Quadrature order 22 and higher

[Aleksejs Fomins]

Dear Dune,

I have made a rough calculation of the order of gaussian quadrature
which we will require for currently-ultimate interpolation order of
hades. Given 5th order interpolation and 5th order basis functions, we
will require order 4+4+4 = 12 for jacobian determinant, because it is a
product of derivatives of lagrange polynomials 3 times. Also 5+5 = 10
for basis functions, because we need to integrate the products of basis
functions.

Therefore, we will require quadrature over tetrahedron of order 22.

We have performed some research online and found that people really
struggle in constructing such quadratures above order 14.

Question 1: What is the current maximal tetrahedral quadrature level
available in DUNE, and do you consider possible extending it to order 22
and beyond.

Question 2: If not, I would like to implement the curvilinear geometry
module using analytic integration instead. Meaning that I would
construct a polynomial class, which would store analytical sums of
polynomial terms, and for evaluating the volume or the integral over
volume of a tetrahedron it would analytically integrate this polynomial
class and evaluate it correspondingly. Do you think this is ok?

Kind regards,
Aleksejs Fomins

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