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[Graham Rowlands <question159192@xxxxxxxxxxxxxxxxxxxxx>]

Question #159192 on DOLFIN changed:
https://answers.launchpad.net/dolfin/+question/159192

Graham Rowlands posted a new comment:
Nathan,

Thanks for your thoughtful response. I've more or less arrived at
similar conclusions, and appreciate the suggestion to relax at an
elevated diffusion level. I assume the number of available steady states
is system dependent, but those I'm working with should have a well
defined global energy minimum. All in all it seems like a well-managed
time-dependent technique is problem the most promising.

I guess I'm slightly unsure how you added the reflective boundary
condition. When solving in the time-dependent case I've been using
Dirichlet conditions at the boundaries (with seemingly minor loss in
overall probability). The Crank-Nicolson scheme uses the following
forms:

a = v*u*dx + 0.5*(   -dt*dot(A*u,grad(v))*dx +  dt*( dot(u*dot(B,div(B)),grad(v)) + 0.5*dot(dot(B*B.T,grad(u)),grad(v))  )*dx    )
L = v*u0*dx - 0.5*(  -dt*dot(A*u0,grad(v))*dx +  dt*( dot(u0*dot(B,div(B)),grad(v)) + 0.5*dot(dot(B*B.T,grad(u0)),grad(v))  )*dx )

Where u0 is the result from the previous time step. For the reflective
boundaries are you simply taking L += g*v*ds, where g is valued 0.0
along the boundaries? Am I missing your meaning?

You are correct though, FPEs seem somewhat confounding to FE solvers in
certain respects (though I lack familiarity with both the equations and
the solvers.) I'm curious about what systems you are studying, btw.

Thanks,
Graham

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