Wolfgang,
> It doesn't have to be. For example, for FE_Q, we also build on
> TensorProductPolynomials but the ordering is not lexicographic. So it would
> still be of interest to document the order of shape functions if you end up
> finding out what it is!
Noted. So I have verified this with the following code.
#include <deal.II/base/quadrature_lib.h>
#include <deal.II/base/point.h>
#include <deal.II/fe/fe_dgq.h>
#include <iostream>
#include <cmath>
#include <vector>
using namespace dealii;
int main(int argc, char **argv)
{
const int degree = 4;
const int n_poly1 = degree+1;
const FE_DGQLegendre<1> fe1(degree);
const FE_DGQLegendre<3> fe3(degree);
const QGauss<3> quad(degree+1);
const int n_qp = quad.size();
const std::vector<Point<3>>& points = quad.get_points();
const std::vector<double>& weights = quad.get_weights();
for(int k=0; k<n_poly1; k++){
for(int j=0; j<n_poly1; j++){
for(int i=0; i<n_poly1; i++){
int index_3d = i + j*n_poly1 + k*n_poly1*n_poly1;
// compute error between 'index_3d'-th polynomial of 'fe3' and the product
of
// i-th, j-th and k-th polynomials of 'fe1'
double error = 0;
for(int q=0; q<n_qp; q++){
error += weights[q]*std::pow(
fe3.shape_value(index_3d, points[q]) - (
fe1.shape_value(i, Point<1>(points[q][0]))*
fe1.shape_value(j, Point<1>(points[q][1]))*
fe1.shape_value(k, Point<1>(points[q][2]))
),
2
);
}
std::cout << "Tensor indices: " << i << " " << j << " " << k << ", "
<< "3d index: " << index_3d << ", error: " << error << "\n";
}
}
}
}
And the output shows all errors to be 0! If this is ok, I will create a pr
with a patch to the documentation shortly.
I don't recall whether there is an easy way to achieve what you are looking
> for, but some elements definitely do something like what you are trying to
> achieve. For example, the difference between FE_Q and FE_DGQ is, in
> essence,
> just a change of basis where the basis functions are permuted. Similarly,
> the
> difference between FE_Q and FE_QHierarchical is similar to the nodal ->
> modal
> change you are interested in. Finally, there is also the case of the FE_Q
> constructor that receives a Quadrature object as argument and that then
> computes a basis change.
> You might want to look into how all of these are implemented. Most of this
> kind of functionality exists in some kind of helper function that might be
> useful to you.
Thanks a lot for providing these details. For now, I have done this by
myself, but I will keep this in mind.
Thanks very much!
On Tue, 25 May 2021 at 19:35, Wolfgang Bangerth <[email protected]>
wrote:
>
> Vachan
>
> > Thanks a lot for your reply. What I actually need is a change of basis
> from
> > Lagrange polynomials (nodal) to Legendre polynomials (modal). I then
> want to
> > know the coefficients of certain modes.
> >
> > So, if there is any straightforward way to do this in dealii, I would
> proceed
> > with that. I, could not find any such functions and hence planned on
> doing it
> > manually, using the shape functions. However, even using any inbuilt
> functions
> > would only address my issue partly if the ordering is not clear.
>
> I don't recall whether there is an easy way to achieve what you are
> looking
> for, but some elements definitely do something like what you are trying to
> achieve. For example, the difference between FE_Q and FE_DGQ is, in
> essence,
> just a change of basis where the basis functions are permuted. Similarly,
> the
> difference between FE_Q and FE_QHierarchical is similar to the nodal ->
> modal
> change you are interested in. Finally, there is also the case of the FE_Q
> constructor that receives a Quadrature object as argument and that then
> computes a basis change.
>
> You might want to look into how all of these are implemented. Most of this
> kind of functionality exists in some kind of helper function that might be
> useful to you.
>
> Best
> W.
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: [email protected]
> www: http://www.math.colostate.edu/~bangerth/
>
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