[firedrake] Solve a Variational problem in a part of the domain

[Colin Cotter]

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> 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:
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> > 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
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> >
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> > firedrake at imperial.ac.uk
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>
> --
>
>   Dr Anna Kalogirou
>   Research Fellow
>   School of Mathematics
>   University of Leeds
>
>   http://www1.maths.leeds.ac.uk/~matak/
>
>
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-- 
http://www.imperial.ac.uk/people/colin.cotter

www.cambridge.org/9781107663916
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