[Dune] Available quadrature orders?

[Aleksejs Fomins]

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Dear all,

I think I get it now. I can use 2D and 3D simplex rules up to order 61, they will be optimal up to orders (12, 5), and for all higher orders the quadrature points are transformed from cuboid to simplex via duffy transform, so the integration will work the same way, but there will be more points than there could have been if the optimal quadrature was known.

Thanks,
Aleksejs

On 06/03/15 11:46, Andreas Dedner wrote:
> 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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>>
>>
>>
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> 
> 
> 
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