On 01/18/2014 04:12 AM, Jan Blechta wrote: > On Fri, 17 Jan 2014 16:54:27 -0800 > Nikolaus Rath <[email protected]> wrote: > >> On 01/09/2014 11:19 AM, Jan Blechta wrote: >>>>>> 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 >>>>> >> [...] >>>> >>>> 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 >> >> >> Alright, I did quite a bit of reading now (mostly Gockenbach, also >> looked into Bhatti, Reddy and Strang), and I think I got the part >> about natural/essential vs Neumann/Dirichlet boundary conditions (and >> also got a lot more understanding of the finite element method >> itself). Thanks for the pointers! >> >> >> However, I am still struggling to combine the finite element method >> with Lagrange multipliers. I think I have a good handle on Lagrange >> multipliers for constrained optimization of a scalar function or >> integral, but I fail to transfer this to FE. > > Ok, potential for Poisson problem is > > \Psi(u) = 1/2 \int |\nabla u|^2 - L(u)
Ah, the potential is the starting point. Thanks! > So if you want to minimize \Psi on V = H^1(\Omega) subject to constraint > \int u = 0, you do can try to find a minimum (u, c) \in (V \times R) of > > \Psi(u) - c \int u > My apologies if I'm slow, but why would I want to find a minimum (u,c) \in (V * R)? It seems to me that I don't want to find a specific value c -- I want a minimum u \in V \forall c. Best, -Nikolaus _______________________________________________ fenics mailing list [email protected] http://fenicsproject.org/mailman/listinfo/fenics
