Multiple Boundary conditions on one boundary

On Fri, Nov 18, 2011 at 09:15:42PM -0000, M Hadigol wrote:
> New question #179286 on FEniCS Project:
> https://answers.launchpad.net/fenics/+question/179286
>
> Hi folks,
>
> My question is that how we can apply both Neumann and Dirichlet boundary conditions on one boundary. I am solving a second order one dimensional PDE with two Neumann boundary conditions (zero flux on boundaries). Since we do not have a Dirichlet boundary, the solution does not have a unique solution. To overcome this issue, I apply a Dirichlet boundary condition on one the boundaries. But I still need zero net flux on both boundaries. By applying Dirichlet boundary condition on one of the boundaries, automatically, the Neumann boundary condition cancels out and lead to an incorrect solution (no more zero slope on this boundary). How can I apply the Dirichlet boundary condition by not canceling out the Neumann boundary condition.
>
> Any help is deeply appreciated.

Add the constraint \int u dx = 0 to your system using a Lagrange
multiplier. To see how this is done, take a look at the demo

  demo/undocumented/neumann-poisson/

in DOLFIN.

An easier but more dubious approach is to add a term

  eps*u*v*dx

to your bilinear form where eps is a small number (say 1e-7).

--
Anders

Thank you Anders. Actually I was looking at this demo you added. The point is that my constraint is not \int u dx = 0, it is u = 0 at one arbitrary point in 1D domain. I did not get that how you ended up with the bilinear form in neumann-poisson. Would you please let me know how you derived that, so I can change the constraint you used to my desired one.

Another quick question: Can't we simply apply the additional constraint, i.e. u = 0 at the middle of the domain, similar to what we do for Dirichlet BC on boundaries?

Thank you so much for your time.

Thank you so much Anders. The "pointwise" flag solved my problem.

Mohammad

Thanks Anders Logg, that solved my question.