Hi, dear friends,

I am trying to solve the Poisson equation on a retangular by using Neumann and Dirichlet boundary conditions. In the numerical method there is no mesh. We work with points distributed in the geometry. My question is this: "How to apply the Neumann boundary condition?"

Thank you a lot, Kemelli

How do you discretize the Poisson equation, i.e. how do you describe derivatives using the control points? You must know that,... and your question should answer itself.

I can think of two approaches, which may be called strong and weak implementations.

Strong 1 You already have a formula for the derivatives. Plug them into the Neumann condition and solve for the boundary value.

Strong 2 If you are using least squares for approximating derivatives, use a Lagrange multiplier to include the Neumann condition.

Weak If you are using least squares, square the Neumann condition and add it to the quantity you are minimizing.