Assume I have a system of at least (but generally only) $N+1$ points in an $N$-dimensional space ($N > 3$ is possible). At each of these points $x_i, i=1,...,N+1$ I know an initial potential/concentration/quantity $P_i$. I can evaluate the possible connections between all points. I also have an isotropic diffusion coefficient $D$. I am now interested in calculating diffusion between these three points (with mass conservation). I have sketched an example case in 2-D.

This problem sounds rather easy, but I quickly realized that it isn't. Classic finite difference approaches require (to my knowledge) a structured gid, which I don't have. Finite volume could handle unstructured grids, but I cannot afford tessellation in high dimensions (say, $N > 100$). For both approaches I would have calculated the first potential derivative along each line (e.g., $x_1$-->$x_2$), and assumed that the same derivative half a distance outside (dashed lines) is zero. Unfortunately, I don't think this assumption works if I have more than $N+1$ points. Finite elements might be an option, but I have yet to find a tutorial which would cover such a case.

Do you have any idea how to solve this system?

Edit: Some additional clarification concerning my goal: I want to simulate the equilibration between the $N+1$ vertices. I am only interested in solutions at the vertices themselves, and diffusion would occur along the edges. The specific equation I want to solve is the heat equation:

$$\frac{dP_i}{dt}=K\Delta P_i$$

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    $\begingroup$ Are you interested in computing your quantity at intermediate points? $\endgroup$
    – nicoguaro
    Mar 31 '19 at 13:06
  • $\begingroup$ No, just at the vertices is fine! $\endgroup$
    – J.Galt
    Mar 31 '19 at 16:49
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    $\begingroup$ I don't think I'm clear about the exact setup. Where does diffusion act? Along the edges of the polytope described by your $N+1$ vertices? In the interior of the polytope? If it's just along the edges, then you have a bunch of 1d problems coupled at the vertices; since there is an analytic solution for the 1d heat equation, it all reduces to a set of ODEs. In any case, it would be useful to specify which set of equations you want to solve and where. $\endgroup$ Apr 1 '19 at 23:31
  • $\begingroup$ @WolfgangBangerth, thank you for the comment and sorry for the lack of clarity - I will add some additional information. Yes, I am only interested in diffusion along the edges - I want to simulate the equilibration between these $N+1$ points. The equation I want to solve is the heat equation at the vertices, which requires second derivatives. The approach you suggest sounds intriguing - could you give me a quick hint on how to approach this solution? $\endgroup$
    – J.Galt
    Apr 2 '19 at 9:48
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    $\begingroup$ I think I still don't understand. You say $P_i$ is the potential at vertex $i$. That's a time-dependent function. Then how do you define $\Delta P_i$? $\Delta$ is a spatial operator, but $P_i$ does not depend on any spatial variables. $\endgroup$ Apr 2 '19 at 14:14

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