Can Fenics also implement a more general Robin boundary condition?

Question

Dear All

I have today implemented the Robin boundary condition where
the normal derivative of u and u are related by a constant.
For my research I would be interested in the case where
the relation is a more general one such that
du/dn(x)= \int K(x,y) u(y) ds
which would correspond to a matrix relationship
du/dn = L u in the functionspace V on the boundary of the domain

regards

Moritz

Answer 1

I'm afraid that discretization of such a non-local operator
$$
u \rightarrow \int K(x,y) u(y) \;\mathrm{d}S
$$
is not feasible to be done automatically by FEniCS. This is because you need to integrate again the term $\mathrm{d}u/\mathrm{d}\mathbf{n}$ so you end up with double integral. This is not supported. You can make a better picture from previous questions:

• http://fenicsproject.org/qa/198/mapping-between-spaces-variable-transformation-functions
• http://scicomp.stackexchange.com/questions/7715/subdomain-integrals-in-fenics
• some were also on https://answers.launchpad.net/fenics

But you could succeed doing this in an iterative manner. For known $u$ you compute $\int K(x,y) u(y) \;\mathrm{d}S$ which can be certainly done with variational formulation. Then you use this quantity in BC of your original problem. You repeat these steps and hope it converges. You could probably write a custom Newton algorithm for this using automatic differentiation of FEniCS to improve convergence rate.

Comments

Dear Jan

I have again thought about the robin boundary condition and
I realized, that my kernel is actually of the form
K(x,y)=\sum_{li} Y_{lm}(x) u_{lm} Y_{lm|^*(y)
(sorry; don't know how have this type set in the preview)
Thus instead of a double integral we are dealing with two integrations.
and the corresponding matrix is a full matrix on the surface that however
can be written as a sum of outer products.
Is there a straightforward way to implement the robin condition in fenics in this case?

regards

Moritz

---

$K(x,y)=\sum_{li} Y_{lm}(x) u_{lm} Y_{lm}^*(y)$

Sorry, I don't understand your formula - what are correct indices there?

Is there a straightforward way to implement the robin condition in fenics in this case?

You will have to be more concrete. Bring us some math exploiting your formula for $K(x,y)$.

Please, use LaTeX.