FEM representation in Firedrake and mass matrix

[casanovamaxime]

Hello everyone, i had some question about the finite element representation in Firedrake and mass matrix,
let supose we hame a continuous field represented with the finite element method u = Ui Phi, with Ui the coefficient and Phi the basis function,

When we declare u = Function(Q), with Q a given function space, u store only the Ui ?
Equally, v= TestFunction(Q), v store only the Phi ?

Also, when we get the gradient by adjoint method ,
with J given cost function
Jhat = ReducedFunctional(J)
param =minimize(Jhat)
grad = Jhat.derivative()

we can effectively check that the directional gradient with respect to a direction d is consistent by finite difference :
grad_id_i = ( J(param + eps d) - J(param - eps d) )/2eps,
does the gradient contains the mass matrix ? because when i do grad_id_i its like the standart euclidean scalar product so i wonder how and where is considered the mass matrix Mij = PhiPhj

thanks for your answers!

[dham]

Hi Maxime,

In response to 1, the way our object structure works, the finite element basis is represented by a FunctionSpace object (for technical reasons the actual class you usually see is a WithGeometry, but that's an internal detail). The FunctionSpace holds a reference to the mesh and to the finite element in reference coordinates. By combining these it has enough information to compute quantities such as integrals over the domain.

A Function then stores a vector of coefficients and a reference to the FunctionSpace. It therefore has access to both the $U_i$ and the $\phi_i$ when needed.

An Argument (a TestFunction or TrialFunction) is effectively just a reference to the FunctionSpace combined with the information about whether this is the number 0 or number 1 argument. This is used to determine which are the rows and which are the columns when a bilinear form is assembled into a matrix, for example.

In response to number 2, Jhat.derivative() returns (very confusingly) the gradient. I.e. the Function which is the L2 Riesz representer of the derivative. I am trying to change this presently when I get some coding time. In the meantime, Jhat.derivative(options={'riesz_representation': None}) will get you the derivative (which is a Cofunction). This interface is awful and should change.

[casanovamaxime]

thanks for the quick reply and the answers! Unfortunately the "Jhat.derivative(options={'riesz_representation': None})" throw me an not-implemented error..

[Ig-dolci]

What is the not-implemented error message?

[casanovamaxime]

Sorry for the very late answer!
The "Jhat.derivative(options={'riesz_representation': None})" throw : "NotImplementedError: Unknown Riesz representation None"

I think dham refered to "Jhat.derivative(options={'riesz_representation': 'l2'})" which allow the check manually the directional derivative with respect to l2