[Dune] Question on the example problem from the tutorial

[Oliver Sander]

Hi Genki,

> Could you briefly explain the math behind the following lines? Why can you estimate the appropriate quadrature order by these calculations? Also, why are they slightly different from each other?

> int order = 2 * (localFiniteElement.localBasis().order()-1); (P.18, "For DUNE ver. 2.7", 2020)

the integrand is \nabla \phi_i \nabla \theta_j.  Both shape functions \phi_i
and \theta_j are polynomials of order [...].order().  Taking the gradient
reduces the order by one.  That makes [...].order()-1 for each of the two
factors.  And as you are integrating the product you have to add the orders.
That explains the "2*".

> int order = 2∗(dim∗localFiniteElement.localBasis().order()−1); (P.19 "For DUNE ver. 2.4", 2016)

To my own surprise I don't even have that document anymore.  Without digging
in the git log I guess that the example was using a grid with quadrilateral
elements back then.  In that case the polynomial order of a Lagrange shape function
is "dim*[...].order()".

Best,
Oliver

> 
> 
> Best regards,
> 
> Genki Unagami
> 
> 
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