I am currently working on a Masters project in which I will be looking at numerical methods for a specific PDE. The PDE that I will be considering is a Fokker-Planck equation on a unit hyper-cube.

More specifically I am studying the following PDE (in non-divergence form) $$ \frac{\partial u}{\partial t} = \frac{x_{d}(1-x_{d})}{2}\frac{\partial^2u}{\partial x_{d}^2}+\left(1-2x_{d}-M_{d}(x)\right)\frac{\partial u}{\partial x_{d}}-(1-\omega_{d})u, $$ for all $x \in [0,1]^D$ and $t\in[0,\infty)$. The boundary conditions will be zero flux except at the $(0,...,0)$ and $(1,...,1)$ corners.

where summation over $d$ is assumed on the right side of the equation. We also define $$ M_d(x)=\sum_{k=1}^{D} M_{dk}(x_k-x_d), \ \ \ \text{and}\ \ \ \omega_d = M_d(x)=\sum_{\substack{k=1\\k\neq d}}^{D} M_{dk}. $$ The flux vector of this equation is given by $$ F_d(u,x) = \frac{1}{2}\frac{\partial}{\partial x_d}\left[x_\alpha(1-x_d)u\right]+\sum_{k=1}^{D} M_{dk}(x_k-x_d)u. $$

I will be using the finite volume method (FVM) to approximate a discrete solution to this PDE. Had this been a constant coefficient problem, or even a positive coefficient problem, I would be fine. However, I am not sure how to deal with the fact that the leading order coefficients go to zero on the boundaries of our domain.

Based on my intuition, I have a hunch that we should be able to show that the solution is a smooth function in the interior of the domain with delta distributions on the boundaries. Sadly, I am not sure how to show this or if this is correct.

Any references or suggestions that you think may be helpful would be appreciated.

Thanks in advance for the help!

  • 1
    $\begingroup$ This model has been intensively studied in the literature and goes back to Wright and Fischer. In the language of semigroup theory you find an analytic treatment in some papers by P. Clement and S. Cerrai. Just to indicate a place where you can find many references. The solution is smooth in the interior because of local parabolic regularity. $\endgroup$ Mar 31, 2020 at 12:30
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    $\begingroup$ You still have an unresolved issue about boundary conditions, even in dimension 1: Sure, wherever the diffusion coefficient does not vanish you can impose zero-flux, but in dimension 1 there is clearly a problem as the whole boundary is "degenerate" in that respect. In fact the very meaning of weak solutions is unclear and you should chose a definition of weak solution according to your needs/application (HINT: the boundaries will have indeed singular Dirac terms, and one can IMPOSE extra conservation laws in order to get a unique solution in some reasonable sense) $\endgroup$ Apr 3, 2020 at 17:06


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