Dear colleagues, I'm solving Poisson's equation with Neumann boundary conditions in rectangular area as you can see at the pic 1. I realized fully explicit algorithm, but it costs to much operations per time. Now I'm looking for economy algorithm in 2D.
Can you advice any? If you are using c++, maybe it's any open source librarys with poisonn's solver?
Fast Fourier Transformations are not allowed.
I think you want to solve Poisson's equation in a rectangle with Neumann boundary conditions on two sides or all four sides. Any way apply 5-point finite difference scheme and form a block tri-diagonal system and solve it by Gauss Seidel method. Since it is not time dependent problem, so nothing like explicit method. For detail see, G D Smith or Greenspan or L Collatz or M K Jain book.
Use parallel computing to reduce the time. It works well for this type of problems.
see also FEM in the following link with c++ codes:
https://people.sc.fsu.edu/~jburkardt/cpp_src/fem2d_poisson_sparse/fem2d_poisson_sparse.html
Lots of thanks to all colleagues!
Dear R.C.Mittal your advice is right what I was looking for.
Dear Seyed Hamed Hashemi Mehne, thanks a lot, I was planing to go parallel my whole code when it will be working ok!
Since after discretisation you will got a linear system to solve A*\phi = f, where A is symetric positive semidefinite and f is the rhsm I think you still can use preconditioned conjugate gradient method (for instance CG-ILU) .
Best
It should be noted that solutions to this problem are not unique. Take any solution and add a constant and you should get a new valid solution. That being said some numerical schemes will have difficulty with this. I would suggest a FEM with some additional constraint that the integral of the solution over the domain is some constant like 0.
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