Lucas:
these two tutorials, I didn't see anything about periodic boundary
conditions, however. Two potential ways I can think to enforce periodic
boundary conditions would be (in 1D, for notational simplicity):
1. u(0) = u(L), du/dx (0) = du/dx (L)
2. u(0) = u(L), d^2 u/dx^2 (0) = d^2 u/dx^2 (L)
For the former case, it appears that you would have to extend the method
from step-47 or step-82 to accommodate these conditions somehow. In the
latter case, it seems feasible that you could write it as two coupled
Laplace equations (supposing something like what is explained in the
step-47 note holds here).
My question is: is that right? And if so, how do the solutions gotten by
imposing these boundary conditions differ? For an analytic solution, it
seems like these would be equivalent (periodic boundary conditions
assert periodicity of derivatives at all orders, yes?) so I'm not sure
what to make of it.
Like you, I suspect that you can do it either way to achieve
periodicity, and that you can choose the way that is most convenient for
your formulation. But I am not certain, nor do I know literature that
would have covered the question. It is, however, something that can be
experimentally determined via the method of manufactured solutions.
I do think that it is not crazy to go with the assumption that either
approach mentioned above will work.
Just as an aside, the actual problem that I'm trying to solve involves a
triharmonic operator. I suspect the situation is similar (although of
course the methods in the tutorials wouldn't immediately generalize),
but if there's something qualitatively different that's obvious, I'd
like to know about that.
I don't think there is a good reason to believe that the approaches for
the biharmonic and triharmonic equations should be substantially different.
I do think that it would be fun to have your triharmonic program be part
of the code gallery one day!
Best
W.
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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