On Mon, 13 Jan 2014 14:28:39 -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
>
> I have started trying to implement this, but I'm running into some
> problems. I have written the following code (which does not yet impose
> the extra constraint on the test functions):
>
> from __future__ import division, print_function, absolute_import
> from dolfin import *
> set_log_level(PROGRESS)
> mesh = UnitSquareMesh(30,30)
> V = FunctionSpace(mesh, 'Lagrange', 1)
> u = TrialFunction(V)
> v = TestFunction(V)
> jt = Expression('sqrt(pow(x[0], 2) + pow(x[1], 2))')
> a = jt * grad(v)[0] * dx
> L = u * dot(FacetNormal(mesh), grad(v)) * ds
> bopt = Function(V)
> solve(a == L, bopt,
> solver_parameters = {'linear_solver': 'cg',
> 'preconditioner': 'ilu'})
>
> but the call to solve() fails with:
As error message says, your left-hand side should be the bilinear form,
your surface integral. Volume integral is linear form, the right-hand
side. But your problem is singular, interior degrees of freedom of your
unknown are undetermined - this is why you need Lagrange multiplier,
which acts in a symmetric way to trial function and test function.
Jan
>
> > *** Error: Unable to define linear variational problem a(u, v) ==
> > L(v) for all v. *** Reason: Expecting the left-hand side to be a
> > bilinear form (not rank 1). *** Where: This error was encountered
> > inside LinearVariationalProblem.cpp. *** Process: 0
> > ***
> > *** DOLFIN version: 1.3.0
> > *** Git changeset:
>
>
> The message is pretty clear, but I don't know what to do about it. The
> equation that I want to solve simply doesn't have a TrialFunction in
> L.. I guess I need some trick here?
>
>
> Best,
> Nikolaus
>
> _______________________________________________
> fenics mailing list
> [email protected]
> http://fenicsproject.org/mailman/listinfo/fenics
_______________________________________________
fenics mailing list
[email protected]
http://fenicsproject.org/mailman/listinfo/fenics
