Problem while using DG = 0

Question

Hi FEniCS users,

I am coding a Discontinuous Galerkin scheme setting as piecewise constant:

W = VectorFunctionSpace(mesh,'DG', 0)
w = TestFunction(W)
m_ = Function(W)

Rm = inner(grad(w),grad(m_))*dx

it seems the terms grad(w) and grad(m_) are the ones which are causing the following error:

This integral is missing an integration domain.
Traceback (most recent call last):
File "dg_BNf.py", line 117, in
Rm = inner(grad(w),grad(m_))dx
File "/usr/lib/python2.7/dist-packages/ufl/measure.py", line 398, in rmul
error("This integral is missing an integration domain.")
File "/usr/lib/python2.7/dist-packages/ufl/log.py", line 158, in error
raise self._exception_type(self._format_raw(message))
ufl.log.UFLException: This integral is missing an integration domain.
Aborted (core dumped)

When I change to

W = VectorFunctionSpace(mesh,'DG', 1)

the issue disappear. Any idea about how to set this properly. I need to use DG = 0 so setting DG = 1 is not a solution.

Thanks for any help

Comments

Roughly speaking, grad of piecewise constant is zero, so the form Rm is not meaningful. Unless some jump terms are added, your DG scheme would not work.

Answer 1

Be careful with the DG elements and grad. You might need to include
jump and average terms as well. Have a look at the dg-poisson demo.
(The gradient of piecewise constants are zero, only jump terms do not vannish)

Answer 2

Hello,

When using DG, you always need to consider the "dS" terms, which represents the integrating over interior facets.

Considering this example, as mentioned on the other answer, your weak formulation will look somehow like this (i haven't tested it, so, there may be a problem still.. please double check...):

 Rm = inner(grad(w),grad(m_))*dx + inner(jump(m_, n), jump(w, n))*dS - dot(avg(grad(m_)), jump(w, n))*dS - dot(jump(m_, n), avg(grad(w)))*dS

Remember that you will also have to define n:

n = FacetNormal(mesh)

More useful info here and specially here (equations 3.3).

Equation 3.3:

$$\sum_{K\in\mathcal{T}_h}\int_{\partial K} q_K \cdot n_K \phi_K\,ds=\int_\Gamma [q] \cdot {\phi}\,ds + \int_{\Gamma^0} {q} \cdot [\phi]\,ds$$

where:

$$ q= TrialFunction $$

$$ \phi = TestFunction $$

Regards,
Leonardo