Pham,
> When testing my codes with FENedelec and FENedelecSZ elements, I saw that
> the
> shape functions change versus the mesh size.
> I think that for each edge of a cell, the shape functions for the degree of
> freedom related to that edge are scaled with the inverse edge length.
> For example, assuming I have a cell that is a cube, whose edge length is 1,
> the shape function at the quadrature point located on the edge is {0, 0, 1}.
> For the smaller cell whose edge length is 0.5, the shape function values is
> {0, 0, 2}.
Yes. This can be because of two things:
* Shape functions are transformed by the Piola or a related transform that
multiplies shape functions by a measure related to the size of the cell.
* Shape functions can be defined not as point values at specific points,
but as *integrals* over an edge or a face. So if the degree of freedom
is that the integral of a shape function (or its tangential component)
over an edge needs to be 1.0, then the *height* of the function needs to
increase by a factor of 2 if the size of the cell is reduced by a factor of
2.
> In my limited opinion, the shape function is defined in unit cell over
> [0,1]^3, and the shape function should not change when the real cell change
> its size.
This is not necessarily so. It depends on the finite element and the
corresponding transform, see above.
> This problem leads to the change of my solution when using the finer mesh
> compared to that of coarser one.
What is the problem you are encountering?
If you refine the mesh and the height of the shape functions increase by a
factor, then the entries of the solution vector will just go down by a factor
of 2. The product of shape function times solution coefficients will then
remain the same.
Best
W.
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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