[firedrake] Solve a Variational problem in a part of the domain
Anna Kalogirou
a.kalogirou at leeds.ac.uk
Thu Aug 13 15:36:07 BST 2015
Hi Colin,
I attached a two-page document, where the system of equations I was
talking about in the previous email is (1a)-(1e). I am puzzled with
equation (1b) really. I can also eliminate (1b) & (1d) and solve for the
remaining (1a), (1c) & (1f), but in this case I still don't know how to
deal with (1f) since there is an integral into the integral, which
contains the unknown function lambda^{n+1/2}.
Thanks, Anna.
On 13/08/15 13:47, Colin Cotter wrote:
> Hi Anna,
> Sounds interesting. Please could you provide a bit more detail?
>
> all the best
> --Colin
>
> On 13 August 2015 at 12:09, Anna Kalogirou <a.kalogirou at leeds.ac.uk
> <mailto:a.kalogirou at leeds.ac.uk>> wrote:
>
> Dear all,
>
> I have a system of equations to solve, which involves three space
> dependent functions phi, eta, lambda and two constants/scalars Z, W.
> These need to be solved simultaneously because all the equations
> involve
> at least 2 unknowns.
>
> How do I solve that, considering that one of the scalar equations
> includes a spacial integral of one of the (still unknown)
> functions? Is
> it best to define the scalars as Constants?
>
> This problem goes away when I write down the system in a standard FEM
> formulation, introducing the mass matrix etc. In this case, it is
> clear
> that I could solve for that function and then consecutively solve each
> of the remaining 4 equations. It is not that obvious how I can do that
> using Firedrake, that is why I thought I would have to solve
> simultaneously, but then I have the problem described above.
>
> Regards,
>
> Anna.
>
>
> On 06/08/15 11:04, Lawrence Mitchell wrote:
> > -----BEGIN PGP SIGNED MESSAGE-----
> > Hash: SHA1
> >
> > Hi Anna,
> >
> > On 06/08/15 10:37, Anna Kalogirou wrote:
> >> Dear all,
> >>
> >> I have a rather simple question but I would like to get some
> >> feedback from someone in the Firedrake team.
> >>
> >> I am now working on the problem which includes solving a
> >> variational problem in a part of the domain only (a water wave
> >> problem which includes a floating body).
> >>
> >> I can think of a couple of possible solutions on how to solve this:
> >> 1. Define two domains and solve the problem separately in each
> >> domain. However, I will have to deal with nonzero boundary
> >> conditions on the common boundary.
> >>
> >> 2. I prefer solving the problem in the whole domain, since most of
> >> the equations/functions are valid everywhere. Then I can define a
> >> Heavyside step function which will be 0 in one part and 1 in the
> >> part of the domain I am interested in (under the floating body). I
> >> will essentially write down a variational problem valid everywhere,
> >> but will actually be zero in a part of the domain.
> >>
> >> Is the 2nd step a good approach? The question essentially is how
> >> to split a mass matrix M_kl which is defined everywhere, and solve
> >> and integral form in a part of the domain only.
> > I think step two is a fine approach. However, note the following
> > issues at present. The way you would have to do this currently
> is as
> > follows:
> >
> > Define a DG0 field to hold your indicator function
> >
> > indicator = Function(DG0)
> >
> > # Set it to 1 in the appropriate part of the domain
> > indicator.interpolate(...)
> >
> > # Now use this extra field everywhere when defining your variational
> > # problem.
> >
> > However, all your integrals are still over the whole domain, you
> just
> > pick up lots of zeros.
> >
> > Steadily climbing our todo list (and at an increasing pace, I feel),
> > is the ability to define proper sub domains in a mesh, and then be
> > able to perform integrals over them. Until that time, I think
> > approach 2 is somewhat easier to do than approach 1.
> >
> > Cheers,
> >
> > LAwrence
> > -----BEGIN PGP SIGNATURE-----
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> > =wE25
> > -----END PGP SIGNATURE-----
> >
> > _______________________________________________
> > firedrake mailing list
> > firedrake at imperial.ac.uk <mailto:firedrake at imperial.ac.uk>
> > https://mailman.ic.ac.uk/mailman/listinfo/firedrake
>
> --
>
> Dr Anna Kalogirou
> Research Fellow
> School of Mathematics
> University of Leeds
>
> http://www1.maths.leeds.ac.uk/~matak/
> <http://www1.maths.leeds.ac.uk/%7Ematak/>
>
>
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>
>
>
>
> --
> http://www.imperial.ac.uk/people/colin.cotter
>
> www.cambridge.org/9781107663916 <http://www.cambridge.org/9781107663916>
>
>
>
>
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> firedrake at imperial.ac.uk
> https://mailman.ic.ac.uk/mailman/listinfo/firedrake
--
Dr Anna Kalogirou
Research Fellow
School of Mathematics
University of Leeds
http://www1.maths.leeds.ac.uk/~matak/
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