I am sorry about my unclear question.
In finite element method, we transform the integral from the physical to
the reference coordinate system.
Thus to compute \int \varphi_i(x) \varphi_j(x) dx in a physical cell, I
will need:
\varphi_i(x_q), \varphi_j(x_q), and JxW in reference cell to compute the
integral by quadrature formula.
The term |det J(x_q)| is to change dx in physical coordinates into d( \hat
x) in reference coordinates so that we can compute the integral in
reference cell.
However, in my code fe_values[vec[block_index_i]].value(i, q_point) is the
shape function in physical coordinate system.
If I use:
sum of quadrature point { fe_values[vec[block_index_i]].value(i, q_point) *
fe_values[vec[block_index_j]].value(j, q_point) *fe_values.JxW(q_point) }
It means that I am using the shape functions in physical coordinate system
to compute the integral instead of those in reference one.
Thus, I think I am wrong here, isn't it?
As my shape function is a vector function, its values in reference
coordinates are different from those in physical coordinates.
However, the FE_NedelecSZ and FE_Nedelec classes do not provide
fe.shape_value() and fe.shape_value_component() because the shape function
is not primitive.
Therefore, I am trying to get the shape function in reference cell to put
in my code.
I hope that this question is clear enough for you to help me.
Thank you very much.
Best regards,
Pham Ngoc Kien
Vào Th 4, 10 thg 4, 2019 vào lúc 12:56 Wolfgang Bangerth <
[email protected]> đã viết:
> On 4/9/19 8:51 PM, Phạm Ngọc Kiên wrote:
> > I am a little bit confusing with the integral over all cell :
> > *\int \varphi_i(x) *\varphi_j(x) * dx *on physical cell is approximated
> > by computing *\sum { \varphi_hat_i(x_hat) *\varphi_hat_i(x_hat)
> *JxW(x_hat) }
> > *on reference cell.
> > With FEValues we get *\varphi_i(x) *, if we compute the above integral
> as:
> > *\sum { \*varphi_i(x)* *\*varphi_j(x)* *JxW(x_hat) }*
> > then the value of the integral changes because shape functions defined
> on
> > reference cell are different from those on real cell.
> >
> > Is there any reason for the integral calculation*\sum { \*varphi_i(x)*
> > *\*varphi_j(x)* *JxW(x_hat) }?*
>
> I have to admit that I don't understand the question. The integral you are
> computing is
>
> \int \varphi_i(x) \varphi_j(x) dx
>
> which is approximated using quadrature via
>
> \sum_q \varphi_i(x_q) \varphi_j(x_q) w_q
>
> where w_q = |det J(x_q)| \hat w_q = JxW. So this approximation happens in
> real
> space, not on the reference cell. How the individual components are
> computed
> (via the reference cell) is not important for the approximation.
>
> Best
> WB
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: [email protected]
> www: http://www.math.colostate.edu/~bangerth/
>
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