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

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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