Convert Divergence equation into matrix form

[AGerami]

Dear All,
I am working on electrical modeling which I cannot change divergence to matrix form to solve it with finite difference or finite element, anybody can help me?

J Current density
E Electrical fi eld
I Current source

J = σE (1)
J = ρ^(-1)1 E (2)
E = −∇v(x,y,z) (3)
∇ · J = I δ(x − xp) δ(y − yp) δ(z − zp) (4)
∇ · (ρ^(-1) ∇v(x,y,z)) = −I δ(x − xp) δ(y − yp) δ(z − zp) (5)

question is how to solve equation (5) with FD and / or FE methods?
Appreciate your help.

 

[gtoguy97]

The solution to equation (5) can be solved using the finite difference (FD) or finite element (FE) methods. The FD method involves discretizing the derivatives of the function v(x,y,z) and approximating them with a central difference formula. The FE method involves dividing the domain into small elements and approximating the derivatives of the function v(x,y,z) using a weighted average of the values at the nodes of the element. Both methods can then be used to solve the equation numerically.