[Dune] Lagrange local basis on pyramids, in dune-localfunctions

[Simon Praetorius]

Sorry, but I think I've found the answer by myself after studying some 
papers about the topic:
- It is impossible of choosing polynomial shape functions if we want to 
preserve the conformity with other elements
- Some approaches to overcome this problem is to use rational basis 
functions, some groups use a splitting of the pyramid into two tetrahedra

As reference I used:
Higher-Order Finite Elements for Hybrid Meshes using new Nodal Pyramidal 
Elements, Journal of Scientific Computing (2010), Volume 42

Best,
Simon

Am 29.05.2016 um 15:28 schrieb Oliver Sander:
> Hi Simon,
>
> the implementation was originally taken from the old UG3 code.  I do not know of any
> particular reason why our implementation should by piecewise linear, and I would not
> object to changing it.
>
> Best,
> Oliver
>
> On 29.05.2016 14:45, Simon Praetorius wrote:
>> Hello,
>> While implementing the partial() method in dune-localfunction for several localBasis, I came across the pyramid basis functions. Those are implemented as piecewise linear/quadratic functions and I want to ask: why?
>>
>> For me, the natural way would be to implement it as bilinear/biquadratic x linear/quadratic. The drawback is, that the polynomial order then increases, but to be exact, the polynomial order 1, as given currently for the pyramidp1 localBasis would only be enough for quadrature formulas, if the quadrature is also implemented piecewise over the sub-tetrehedra and then added up.
>>
>> The combination of the current pyramid elements with hexahedral Qk elements would also not fit perfectly.
>>
>> In contrast to this, the prism elements are implemented as bilinear x linear functions and thus work well.
>>
>> Have I missed something, e.g. is the smooth pyramid element implemented in some other class that I have overseen?
>>
>> I would suggest to either replace the current pyramid element with the smooth ones, or add another pyramid finite element additionally if the piecewise smooth elements are in use somewhere.
>> The evaluateJacobian test in the test-localfe implementation could then also test for the pyramid elements instead of removing the test for only these elements.
>>
>> Simon Praetorius
>>
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