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Draft: Feature/linear transformed finite elements

Maik Porrmann requested to merge maik.porrmann/dune-localfunctions:feature/linear-transformed-finite-elements into master

This MR is a draft for infrastructure for non-affine (non-piola) equivalent finite elements. As such, it provides implementations of the generic reference finite elements for the cubic Hermite (1d - 3d) on simplices, quadratic Morley triangle, quintic Argyris triangle, and the cubic ArnoldWinther element. Generic here means, that the implementations model the definition of the respective FE on the reference Domain, which in the above cases is not affine equivalent to some real finite element. The transformation necessary to obtain affine equivalent finite elements is implemented in dune-functions, see dune-functions MR. The previous development happened as an external module with contains both dune-localfunctions and dune-functions aspects.

Together these two MRs raise a number of questions, also listed here like:

  • How to handle interpolation with FEs that involve derivative evaluations as DOFs? See also dune-function issue
  • How to encode what can be cached?
  • How to collect and pass nonlocal information to LocalFiniteElements? Examples would be edge orientation, derivative directions and grid-dependent scalings
  • Are the generic FEs, like in this MR, meaningful in dune-localfunctions or should they be kept together with their transformations?
  • How to implement matrix-valued finite elements and what are the precise LocalFiniteElementTraits?
Edited by Maik Porrmann

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