How does MFEM define the DOF functional / interpolation?

[Antoinemarteau]

The basis-functions documentation page suggests that the described polynomials are the actual (element local) shape functions, such that a FE function is a linear combination of them (once mapped and made global, etc.). This would mean that the interpolation into the FE space changes depending on the choice of polynomial, because a given function in the space will have different coefficient in different bases.

Lot's of FE libraries work differently: they define DOF (functionals), and compute the shape-function basis as the dual of the DOF basis as a change of basis from a given basis of the space (still, the choice of given initial/pre-basis is wide).

What is MFEM prescribing internally, the shape-functions (first paragraphs) or the DOFs (second) ?

[v-dobrev]

Hi @Antoinemarteau,

In MFEM's FiniteElement base class, the primal methods are the basis-function-evaluation-methods like CalcShape, CalcDShape, CalcVShape, etc. In many cases, the basis function definitions are derived from a set of functionals -- e.g. for elements derived from NodalFiniteElement these are the values of the function at a given point on the reference element -- we refer to these points as "nodes". However, in general, we allow bases which are not derived from functionals, for example the positive (Bernstein) elements like H1Pos_QuadrilateralElement.

Hope this helps,
Veselin

[Antoinemarteau]

Hi @v-dobrev,

Thank you for your answer, I understand that MFEM implements both for elements based on nodal-evaluation functionals/DOFs.

Regarding curl and div conforming elements, have you also implemented both (for example using generalized Bernstein polynomial for vector-valued spaces, introduced by Arnold et. al. or Ainsworth et. al. ) ?
If yes, do you use a change of basis of functionals in order to interpolate in this case?

Thanks again,
Antoine

[v-dobrev]

Hi Antoine,

For both H(curl) and H(div) spaces, we have just one FiniteElement class per element type (like triangle, quad, tet, etc). These are Nedelec and Raviart-Thomas elements, for H(curl) and H(div), respectively. As DOFs for these spaces we use point values dotted with tangential or normal vectors, respectively. This set of DOFs is different from the ones typically used in the literature, however, the FE spaces are identical.

Cheers,
Veselin