# I Solving the 3D Poisson Equation Using Finite Difference/Volume

#### Jimmy and Bimmy

Hi,

I'm attempting to solve the 3D poisson equation

∇ ⋅ [ ε(r) ∇u ] = -ρ(r)

Using a finite difference scheme.

The scheme is simple to implement in 3D when ε(r) is constant, and I have found an algorithm that solves for a non-constant ε(r) in 2D. But I am having trouble finding an algorithm that can handle a non-constant ε(r) in 3D.

Here is the link for the 2D case: www.ece.utah.edu/~ece6340/LECTURES/Feb1/Nagel 2012 - Solving the Generalized Poisson Equation using FDM.pdf

Would anyone be able to guide me to a 3D case, or help me determine the 3D form of equation 42 in the above link? Below is my guess

a0 = ε(i,j,k) + ε(i,j-1,k) + ε(i,j-1,k-1) + ε(i,j,k-1) + ε(i-1,j,k) + ε(i-1,j-1,k) + ε(i-1,j-1,k-1) + ε(i-1,j,k-1)
a1 = 1/4 [ ε(i,j,k) + ε(i,j-1,k) + ε(i,j,k-1) + ε(i,j-1,k-1) ]
a2 = 1/4 [ ε(i,j-1,k) + ε(i-1,j-1,k) + ε(i-1,j-1,k-1) + ε(i,j-1,k-1) ]
a3 = 1/4 [ ε(i-1,j,k) + ε(i-1,j-1,k) + ε(i-1,j,k-1) + ε(i-1,j-1,k-1) ]
a4 = 1/4 [ ε(i,j,k) + ε(i-1,j,k) + ε(i-1,j,k-1) + ε(i,j,k-1) ]
a5 = 1/4 [ ε(i,j,k) + ε(i-1,j,k) + ε(i-1,j-1,k) + ε(i,j-1,k) ]
a6 = 1/4 [ ε(i,j,k-1) + ε(i-1,j,k-1) + ε(i-1,j-1,k-1) + ε(i,j-1,k-1) ]

V(i,j,k) = 1/a0 [ a1V(i+1,j,k) + a2V(i,j-1,k) + a3V(i-1,j,k) + a4V(i,j+1,k) + a5V(i,j,k+1) + a6V(i,j,k-1) - Q(i,j)/ε0]

Thanks

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#### Greg Bernhardt

Admin
Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

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