Dear Prof. Bangerth,
Thank you for your reply. My interface condition is \integral_{Interface}
p*I*n_stokes - jump(velocity_gradient) ds, where I is the interface and
jump(velocity_gradient) = vel_grad_stokes*n_stokes +
vel_grad_laplace*n_laplace. I am using this with zero boundary conditions
on pressure. The interface in the code I attached is y=0.5, with y>=0.5
stokes and below y=0.5 laplace. When I let the mesh size go to zero, even
when I prescribe zero pressure everywhere (by defining zero contribution
from pressure on my right hand side) I am still not getting zero pressure
as the mesh size goes to zero. Furthermore, as my mesh size goes to zero
my velocity converges to something that is not related to the right-hand
side I chose. Specifically, it feautures a non zero jump in the gradient
across the interface, even though I am prescribing the same right-hand side
to both sides (i.e. with zero pressure on both sides). This does not
happen if I solve just stokes everywhere or just laplace (with the same
right hand side :i.e. zero contribution from pressure).
Kind regards,
Georgios Sialounas
On Wednesday, August 22, 2018 at 5:42:34 AM UTC+1, Wolfgang Bangerth wrote:
>
>
> Georgios,
>
> > Thanks for your reply! Essentially what I was trying to do is patch
> together
> > two product spaces: ([H1_0]^2, L^2_0) in one part of the domain and
> > ([H1_0]^2, L^2_0) in another, with an interface condition to ensure
> > solvability for my problem (Stokes in one part of the domain, laplace in
> > another). The interface condition was chosen such that there would be
> no jump
> > of the velocity gradient across the interface (this was done by choosing
> an
> > L2_0 pressure that goes in the right-hand side).
>
> Can you state this interface condition in mathematical terms? In general,
> functions in H^1 can be expected to be continuous along interfaces, but
> their
> gradients can not. Indeed, their normal derivatives along the interface
> will
> only be in H^{-1/2}, and so the difference between the normal derivatives
> from
> both sides of the interface can at best be expected to be weakly
> continuous
> (i.e., when tested with a sufficiently smooth test function).
>
>
> > Unfortunately this did not
> > work in the implementation phase and I ended up getting a jump of the
> velocity
> > gradient across the interface.
>
> But is this different than the jump in the gradient between each cell and
> the
> next? There, the gradient is not continuous either. The question is what
> happens as you let the mesh size go to zero.
>
>
> > At the moment I don't know if this is a coding
> > bug, or an erroneous choice of spaces. To be honest I expected weak
> > differentiability to be enough to ensure a zero velocity gradient jump
> across
> > the interface too. As such, I was thinking how I should choose my spaces
> to
> > ensure gradient continuity across the interface to see if this would fix
> my
> > problem but I do not know how to enforce that in deal.ii. As such, any
> advice
> > on how to do that would be great.
>
> It's actually quite complicated to ensure continuity of the gradient
> strongly.
> This is why the biharmonic equation is so difficult to solve!
>
> Best
> Wolfgang
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: [email protected]
> <javascript:>
> www: http://www.math.colostate.edu/~bangerth/
>
>
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