Biharmonic equation with periodic BCs

[Lucas Myers]

Hi folks,

Given the explanations in step-47 and step-82, it appears that the particular boundary conditions associated with the biharmonic equation can make it relatively easy to solve (e.g. two coupled Laplace equations) or really difficult (having to use the interior penalty method or the lifting operator method). In skimming 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.

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.

Any help is appreciated!

Thanks,

Lucas

[Wolfgang Bangerth]

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.

--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[Lucas Myers]

Hi Wolfgang,

Thanks for the feedback! I have also not found any literature dealing with this periodic boundary conditions problem -- I doubt I will try to get to the bottom of it experimentally unless I run into technical issues solving my particular problem. With both the lifting method and interior penalty method, it appears that you need knowledge of adjacent cells in assembly; I suspect this will be difficult (or at least annoying) with periodic boundaries if it's not already implemented in deal.II.

For the triharmonic equation, this is the only thing I've been able to find. It seems quite similar to step-82.

In any case, I will try the method of introducing auxiliary variables (so that it's just a bunch of coupled Laplace equations).

Thanks again for the help!

- Lucas

[Wolfgang Bangerth]

On 3/22/23 10:03, Lucas Myers wrote:
>
> Thanks for the feedback! I have also not found any literature dealing with
> this periodic boundary conditions problem -- I doubt I will try to get to the
> bottom of it experimentally unless I run into technical issues solving my
> particular problem. With both the lifting method and interior penalty method,
> it appears that you need knowledge of adjacent cells in assembly; I suspect
> this will be difficult (or at least annoying) with periodic boundaries if it's
> not already implemented in deal.II.

Not difficult, that's all already there. Look at step-45.

> For the triharmonic equation, this is the only thing I've been able to find

> <https://nam10.safelinks.protection.outlook.com/?url=https%3A%2F%2Farxiv.org%2Fpdf%2F1804.03793.pdf&data=05%7C01%7CWolfgang.Bangerth%40colostate.edu%7Cb3c80d79ce8c4a781f3d08db2aef01a8%7Cafb58802ff7a4bb1ab21367ff2ecfc8b%7C0%7C0%7C638150979102895314%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=YRYTbpdtoyWdXhXdAUnasaVqsRd0x0kvTwMmwYCinq0%3D&reserved=0>. It seems quite similar to step-82.

>
> In any case, I will try the method of introducing auxiliary variables (so that
> it's just a bunch of coupled Laplace equations).

Yes, that would certainly be easiest if your boundary conditions allow it (and
periodic boundary conditions *should* allow it).