[Dune] Available quadrature orders?

[Christian Engwer]

On Fri, Mar 06, 2015 at 11:11:17AM +0100, Aleksejs Fomins wrote:
> Dear Oliver,
> 
> Thank you for your response. 
> 
> 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? What would you do if you need to integrate a higher order polynomial?


No, higher orders are constructed using tensor products. Up to order
12 we have optimal (in terms of number of points) quadrature rules.

> 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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-- 
Prof. Dr. Christian Engwer 
Institut für Numerische und Angewandte Mathematik
Fachbereich Mathematik und Informatik der Universität Münster
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