integration per cell

Question

I'd like to compute the integral of an Expression per cell, i.e.,
$$
E_i = \int_{T_i} e.
$$

I see that in the basic AMR example demo/undocumented/adaptive-poisson/python/demo_adaptive-poisson.py, manual integration is done by evaluating the right-hand sides f at all cell.midpoint()s.

Is there a "proper" way to handle things (i.e., accurate for higher-than-linear elements)?

Answer 1

Yes, just multiply the Expression with a piecewise constant TestFunction and assemble it into a Vector. Something like this:

e = <some Expression>
DG = FunctionSpace(mesh, "DG", 0)
v = TestFunction(DG)
L = v*e*dx
b = assemble(L)
E = b.array()

or in one line just this:

E = assemble(TestFunction(FunctionSpace(mesh, "DG", 0))*e*dx).array()