I am solving a problem in 2d using FE_Q(2) elements and a gauss quadrature
rule with (fe.degree +1) quadrature points in each co-ordinate direction, that
is, I have in total nine quadrature points. My question pertains to the following:
At each cell, I need to approximate a field whose sampling (support) points
are the quadrature points belonging to reduced integration, i.e, there are
four quadrature points in my case and the four (shape) functions approximating
my field should be designed as follows:
N_j (xi_k) = delta_{jk} ,
where xi_k are the coordinates of the quadrature points. So I need four
(shape) functions, each of which is one at one of the four quadrature points
and zero at the three others.
You've already found this: Both the FE_Q and FE_DGQArbitraryNodes classes have
constructors that create shape functions based on this information.
That said, my ansatz is given by (the coefficents a(xi) are of course known)
f(xi) = a(xi_1) N_1(xi) + a(xi_2) N_2(xi) + a(xi_3) N_3(xi) + a(xi_4) N(xi)
and I need to evaluate the function f(xi) at the *nine* quadrature points.
What is the best way to do set up the FEValues object?
I have seen that there is a constructor for the FE_Q element which takes a
Quadratute<1> object. I guess this would help me to define the (shape)
functions pertaining to the field f(xi), but I think I can not evaluate this
field at the nine quadrature points, because (i) their local coordinates are
of course different in the new FEValues object and (ii) second, I would have
to insert negative local coordinates for a set of them.
Maybe I do not even need a FEValues object for my purpuse. As I said, I only
need to do the approximation f(xi) and evaluate it at the nine quadrature points.
The construction of a finite element field and its evaluation at quadrature
points are two different things. Let's say you use one of the classes above to
create a finite element with the delta-property you seek. Then you create a
DoFHandler with it that describes a finite element field on the entire mesh.
To evaluate it at certain points, you'd just create a FEValues object as
always, which allows you to obtain the values of shape functions and of the
finite element field that results from a solution vector, at the quadratrure
points of interest.
Best
W.
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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