[Dune] Gaussian Quadrature order 22 and higher

[Aleksejs Fomins]

Dear Martin,

Thank you very much for your replies.

I have read about problems with round-off errors with high order
quadratures in other sources as well.

I have chosen to implement the exact integral using analytic polynomial
integration.

Regards,
Aleksejs

On 07/30/2014 10:28 AM, Martin Nolte wrote:
> 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
>>
>>
>>
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>>
> 

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