On 4/1/22 14:40, Jau-Uei Chen wrote:
It is not clear to me how modifying FEFaceQ can even achieve the first
step (highlighted in yellow) you describe. It seems that FEFace_Q is
universally defined on *the unit face* and cannot see the structure of
triangulation. Hence, the finite element space is always defined on the
each face of an element when we are doing "distribute_dofs".
Correct.
Are you
referring that we can use DSSY element (given in the paper) to construct
the discontinuous function defined on the reference domain of a face to
achieve the first step?
No, what I meant is that there are other contexts in which one wants to
manually add constraints, and the situation in the linked paper is one.
In your case, you would want to add additional constraints for all faces
that are *not* in fact refined. This is because on these faces you want
to bring the space back down to linear and continuous across the entire
face.
An possible way I imagine is to re-defined FEFace_Q in the following way
to complete the first step:
(assume degree = 1, so fcn 1 and fcn 2 are Lagrange polynomial with
degree 1)
fcn1 fcn2
*------------**------------* <- basis function with supporting points *
o - - - - - - - - - - - - - o <- geometric line segment in reference
domain with vertices o
Yes, this is exactly how it should look.
However, it seems not possible since the construction of a finite
element space relies on inheritance of "FE_PolyFace" and such
discontinuous function is not polynomial.
No, the inheritance is not necessary. FE_PolyFace is just a convenience
but not a requirement. At the end of the day, you need to implement the
interface of the FiniteElement class, and how you do this is up to you.
Different FE_* classes do this in different ways, depending on the
structure of their finite element spaces. You can definitely implement
non-polynomial finite elements -- you just need to implement all of the
required virtual functions.
(I should also mention that because your shape functions are only
piecewise polynomial, whenever you integrate terms on these faces you
need to use iterated quadrature formulas. QIterated is your friend.)
Best
W.
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