On Thu, 9 Jan 2014 11:02:04 -0800
Nikolaus Rath <[email protected]> wrote:
> Hi Garth,
>
> On 01/09/2014 10:49 AM, Garth N. Wells wrote:
> > On 2014-01-09 18:40, Nikolaus Rath wrote:
> >> Hello,
> >>
> >> I would like to solve the following equation (which does not
> >> directly come from a PDE):
> >>
> >> \int dV f(x) * \partial_r g(x) = \int dA u(x) * \partial_n g(x)
> >>
> >> f(x) is known, u(x) is unknown, and the equation should hold for
> >> any g(x) that satisfy Laplace's equation.
> >>
> >>
> >> In other words, I'm looking for a weight function u(x), such that
> >> the surface integral of the normal derivative of any g(x)
> >> (weighted by u) gives the same result as the volume integral of
> >> the radial derivative of
> >> g(x) (weighted by the known function f(x)).
> >>
> >> Is it possible to do this with FEniCS?
> >> It seems that the equation itself is easy to express in UFL, but I
> >> am not sure how do deal with the fact that there are no boundary
> >> conditions, and that any trial function g(x) needs to satisfy
> >> Laplace's equation.
> > Yes. Look at the demo
> >
> > demo/documented/neumann-poisson
> >
> > Garth
>
> Thanks for the quick reply!
>
> So if I understand correctly, I don't need to do anything special for
> the boundary conditions because Dolfin assumes Neumann by default, and
> Neumann conditions are only reflected in L.
This is not true - DOLFIN does not assume. It is a property of this
variational problem.
>
> However, even after going through the example, I'm not sure how I can
> tell FEniCS to use only trial functions satisfying Laplace's equation.
> In the example, the constraint on the test functions seems to fix just
> the constant offset. It's not clear to me to extend this to something
> more complicated, where the constraint itself takes the form of a PDE.
> Do you think you could explain in a bit more detail?
You change the constraint to Laplace problem, and the space R for
Lagrange multiplier and corresponding test function to some appropriate
FE subspace of Sobolev space W_0^{1,2}. Hint:
http://en.wikipedia.org/wiki/Lagrange_multipliers_on_Banach_spaces
Jan
>
> Best,
> Nikolaus
>
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