2
$\begingroup$

The PDE is being solved over a cube. Four of the faces are periodic, and one set of opposing faces have no-flow Neumann boundary conditions:

$\nabla^2 u = f(x,y,z)$

$\frac{\partial u}{\partial z} = 0$ at $z = 0$ and $z = h$

I am able to solve a similar problem with full Neumann boundary conditions (assuming the compatibility condition for $f(x,y,z)$ is satisfied) by arbitrarily selecting one node and setting it to $0$ as suggested here How to numerically solve the Poisson equation given Neumann boundary conditions?

So, based on this I have 2 questions:

  1. Would this method also work for mixed periodic/Neumann boundary conditions and if so why? I can't seem to find an answer. I believe periodic boundary conditions also have a family of solutions valid up to an additive constant $C$.

  2. Surely the choice of which node you choose to set to $0$ has an affect on the final solution. Therefore, are there any rules of thumb on how to go about this? Or am I understanding the problem wrong?

I think this is a generally applicable problem, so any answers would be appreciated.

$\endgroup$
0
$\begingroup$

$\nabla^2$ is separable in space: $$\nabla^2 = \nabla^2_x + \nabla^2_y + \nabla^2_z$$ where $ \nabla^2_x + \nabla^2_y$ are with the periodic condition, and $\nabla_z^2$ be with the replicate condition. Notice that for discrete 1D Laplacian operator $[1, -2, 1]$, replicate and symmetric boundary conditions are the same. So along $z$ you can safely assume symmetric boundary condition.

Given these information, you can do Fourier transform along $x$ and $y$ to get the inverse of $\nabla^2_x + \nabla^2_y$, and discrete cosine transforms along $z$ to get the inverse of $\nabla^2_z$.

Denote $\mathcal{F}_x$ and $\mathcal{F}_y$ as Fourier transforms along $x$ and $y$, $\mathcal{D}_z$ as discrete cosine transform along $z$, your solution can be obtained as: $$u = \mathcal{D}_z^{-1}\left\{\mathcal{F}^{-1}_y\left\{\mathcal{F}^{-1}_x \left\{ \frac{\mathcal{D}_z\{\mathcal{F}_y\{\mathcal{F}_x\{f\}\}\}}{\mathcal{D}_z\{\mathcal{F}_y\{\mathcal{F}_x\{\nabla^2\}\}\}} \right\}\right\}\right\}$$

As for your questions:

  1. I think it is Yes.
  2. For this (direct) approach, you have direct point-to-point division in the spectral domain. To avoid singularity at the DC term (the mean value), you can set any non-zero number. It only affects the mean of the solution.
$\endgroup$
  • $\begingroup$ This is actually very helpful WDC. Thank you! $\endgroup$ – CapillarySale May 8 '18 at 19:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.