[Dune] Gaussian Quadrature order 22 and higher

[Martin Nolte]

Hi Aleksijs,

well, there is no problem in generating a quadrature for order 22 or higher. 
Just the requirement of a Gaussian one is a tough nut.

If you just want a Quadrature, you can use the TensorProductQuadrature, which 
will in theory work for any geometry type and any order. However, I don't know 
about round-off errors with such high orders.

Alternatively, you can construct a Lagrange basis of order 22 and the 
corresponding interpolation. Then, evaluate the exact integral of the basis 
functions and you have a quadrature (with 0.5*23*22 quadrature points, of course).

Best,

Martin

On 07/30/2014 10:00 AM, Aleksejs Fomins wrote:
> 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
>
>
>
> _______________________________________________
> Dune mailing list
> Dune at dune-project.org
> http://lists.dune-project.org/mailman/listinfo/dune
>

-- 
Dr. Martin Nolte <nolte at mathematik.uni-freiburg.de>

Universität Freiburg                                   phone: +49-761-203-5630
Abteilung für angewandte Mathematik                    fax:   +49-761-203-5632
Hermann-Herder-Straße 10
79104 Freiburg, Germany