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

[Lawrence Mitchell]

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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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