[Dune] Available quadrature orders?

[Andreas Dedner]

Hi.
It might simply help to google Duffy transform(ation). That provides the 
idea of how to
go from a tensor product quadrature rule on a cube to a quadrature on a 
simplex - but the
resulting quadrature will be far from optimal w.r.t. the number of points.
Andreas


On 06/03/15 10:21, Oliver Sander wrote:
> Hi Aleksejs,
>
>> So I have found the file
>> dune/geometry/quadraturerules/simplexquadrature.hh
>> In which it is stated that the highest quadrature order for a triangle is 12 and for a tetrahedron is 5. Is this correct?
> IIRC these are the highest available special-purpose simplex rules.  You can get higher-order ones,
> but those will be constructed by conical multiplication.
>
>> What would you do if you need to integrate a higher order polynomial?
> a) Simply request a higher-order formula from the quadrature rules cache.
>
> const QuadratureRule<double,dim>& quad = QuadratureRules<double,dim>::rule(it->type(), your_order_here);
>
> If this does not work, then
>
> b) Add more rules to dune-geometry
>
> The book by Stroud contain a lot of them, but I only have a print copy.
>
> Look on the web for higher order quad rules. E.g., A library of such rules is at
> http://www.cs.kuleuven.ac.be/~nines/research/ecf/ecf.html
>
> best,
> Oliver
>
>
>> I would gladly read the book you have sent. Do you happen to have a digital version? I've found it on amazon for $148, but that is a bit too much I think, ignoring the fact that it is in US :)
>>
>> Regards,
>> Aleksejs
>>
>>
>>
>>
>> On 06/03/15 09:55, Oliver Sander wrote:
>>>> 1) Does Dune possess only 1D quadrature rules, or 2D and 3D also?
>>> No.  Dune specializes on partial differential equations in one-dimensional domains,
>>> and therefore only 1d quadrature rules are needed.
>>>>   I understand that a higher dimensional quadrature can be obtained from a tensor product of lower-dimensional quadratures, but this question is about quadratures specialised for reference element geometries.
>>>>
>>>> 2) If yes, for which reference elements, and to what order are these quadrature rules available.
>>> See dune-geometry/dune/geometry/quadraturerules
>>>> 3) If no, how does one usually integrate over, say, triangle, using a tensor product quadrature? Does one simply set the value of the function to 0 for all sample points outside the entity? How does that affect the accuracy of integration?
>>>>
>>> https://openlibrary.org/books/OL4583400M/Approximate_calculation_of_multiple_integrals
>>>> 4) Finally, what is known about using quadrature rules for integrating non-polynomial integrands? Can one estimate the quadrature order necessary to integrate, say, sqrt(x) to a given accuracy?
>>> https://openlibrary.org/books/OL4583400M/Approximate_calculation_of_multiple_integrals
>>> Cheers,
>>> Oliver
>>
>>
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