[Dune] Available quadrature orders?

[Oliver Sander]

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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