On 12/05/2016 09:47 AM, Oded Yaakobi wrote:
I am working on generalizing a model that originally described only the
interior domain by the equations that I wrote on my Friday’s notes. I am not
familiar with anyone who solved the combined set of equations for the interior
and exterior domains. However, a simpler problem should be common in other
situations where the droplet is made of a single phase. In these cases, a
matching condition in the form of Eq. 10 in my notes should be satisfied (with
\Psi=0). In recent days I have surveyed the literature to learn how people
deal with these situations, but still haven’t found an answer.
Oded -- I guess finding a way of implementing your model is then part of the
research you have to do.
I'm going to add this as a note: You can think of these interface conditions
as boundary conditions for one side if you knew the solution on the other side
of the interface; of as a boundary condition for the other side if you knew
the solution on this side. As a consequence, only certain kinds of interface
conditions actually make sense. For example, if you had a Laplace equation on
both sides of an interface, an interface condition of the kind
u|_\Omega_1 = u|_\Omega_2
make sense because it corresponds to Dirichlet boundary conditions for one of
the solutions if the other one were known. Likewise, the following all make sense:
partial_n u|_\Omega_1 = partial_n u|_\Omega_2
partial_n u|_\Omega_1 = u|_\Omega_2
(u + alpha partial_n u)|_\Omega_1 = u|_\Omega_2
...
because they correspond to Neumann or Dirichlet or Robin boundary conditions.
But, the following does not make any sense:
grad u|_\Omega_1 = grad u|_\Omega_2
That's because for the Laplace equation, you can't impose the gradient at the
boundary, only its normal component.
Of course, the types of interface conditions that are allowed are also the
ones that are typically easy to implement -- Neumann and Robin conditions go
into the weak formulation, Dirichlet conditions go into a ConstraintMatrix. If
the interface conditions you have do not easily fit into this pattern, then
that does not *automatically* mean that you formulation is ill-posed and may
not have a solution, but it would certainly make me *question* whether it is
ill-posed, and if necessary *prove* that it is well-posed before I would try
to implement it. There is nothing worse than spending the time implementing a
model for which it later turns out that there are no solutions :-)
Cheers
W.
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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