finite element with shape functions having delta_ij property at quadrature points

[Simon]

Dear all,

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. 

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.

Any input is greatly appreciated!

Best

Simon

[Wolfgang Bangerth]

> 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.

--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[Simon Wiesheier]

"You've already found this: Both the FE_Q and FE_DGQArbitraryNodes classes have
constructors that create shape functions based on this information."

The latter is the element I was looking for. The problem  with the former is that the first and last quadrature point has to be located at zero and one, respectively - a condition which is not satisfied in my case. That said, FE_DGQArbitraryNodes is the right element for me, because I do not use this element to produce a global (dis)continuous field but only do a local approximation on each cell individually. In particular, I do not need to create a DoFHandler for this element or do calls like constraints.distribute_local_to_global(). 

Although FE_DGQArbitraryNodes works fine for me, I asked myself why the constructor of FE_Q wants a quadrature object whos first and last qp is at zero and one, respectively? One can not use this constructor for the family of QGauss objects, just to mention one instance. 

Best 

Simon 

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[peterrum]

>  I asked myself why the constructor of FE_Q wants a quadrature object whos first and last qp is at zero and one, respectively?

Because it is a continuous element and dofs at 0 and 1 are assigned to shared entities (vertices, lines, quads).

Peter