I'm learning (or rather should be learning) how to solve the Neumann problem using the method of finite differences. The specific PDE I'm interested in is: $$\left\{ \begin{array}{r l} - \Delta u = f & \text{in} \, \, U \\ \frac{\partial u}{\partial n} = 0 & \text{in} \, \, \partial U. \end{array} \right.$$ where $U$ is a bounded $C^{1}$ domain in $\mathbb{R}^{d}$ and $f \in C^{\infty}(\bar{U})$.

If we discretize the domain by looking at $U \cap (n^{-1} \mathbb{Z}^{d})$ and rescale to $nU \cap \mathbb{Z}^{d}$, then we know how to solve the corresponding equation for Dirichlet boundary conditions using hitting times of the simple random walk.

Is a similar approach available for the Neumann boundary condition?

  • $\begingroup$ Don't think in terms of the discrete problem--think in terms of the exact probabilistic representation of the solution to the desired problem. You can then discretize that probabilistic representation. In the Dirichlet case you just stop the Brownian motion (or whatever diffusion process) upon hitting the boundary and then count the value of the boundary function at the point where you hit the boundary. In the Neumann case you instead need to take a reflecting boundary condition for the process (so that you hit the boundary and instantly turn back around). $\endgroup$ – Ian Jul 15 '17 at 15:45
  • $\begingroup$ Is the connection between reflecting BM and the Neumann problem well known? I thought so, but when Googled it and the best thing I found was arxiv.org/pdf/1502.01319.pdf and the necessary machinery appears to be quite technical. $\endgroup$ – fourierwho Jul 15 '17 at 16:03
  • $\begingroup$ It is indeed more technical than the Dirichlet problem because you need the machinery of the local time in order to formulate everything properly. But it is still well known, for instance it was covered in one of my graduate courses. $\endgroup$ – Ian Jul 15 '17 at 16:16

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