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

 

[eljose79]

for your help!

Hi there,

Thank you for reaching out to the scientific community for help with your 3D poisson equation problem. I am a scientist with experience in solving similar equations using finite difference schemes, and I would be happy to guide you in the right direction.

Firstly, I would like to commend you for successfully implementing the finite difference scheme for the 2D case and finding an algorithm for a non-constant ε(r). This is a great starting point for solving the 3D case.

In terms of finding an algorithm for a non-constant ε(r) in 3D, there are a few different approaches you can take. One option is to extend the algorithm you have for the 2D case to 3D, by adding additional terms and variables to account for the third dimension.

Another approach is to use a different finite difference scheme, such as the finite volume method or the finite element method, which may be better suited for handling non-constant ε(r) in 3D. These methods involve dividing the domain into smaller volumes or elements, and solving the equation within each volume or element separately, before combining the results to obtain a solution for the entire domain.

Regarding your proposed 3D form of equation 42 from the link you provided, I would suggest checking it against the original equation to ensure that all terms are accounted for correctly. Additionally, you may want to consider using a matrix notation to represent the finite difference scheme, as it can make the implementation and solution process more efficient.

I hope this helps guide you in the right direction for solving the 3D poisson equation with a non-constant ε(r). If you have any further questions or need clarification, please don't hesitate to ask. Good luck with your research!