¿Qué pasa con tres o todas las fronteras Neumann en 2D?

Vamos a suponer $latex n=3$ para reducir el tamaño de las matrices. Empezamos suponiendo que conocemos: $latex frac{partial}{partial x}|_{0,0,}u, frac{partial}{partial x}|_{0,1}u, frac{partial}{partial x}|_{0,2}u$ $latex frac{partial}{partial y}|_{0,0}u, frac{partial}{partial y}|_{1,0}u$ $latex frac{partial}{partial y}|_{0,2}u, frac{partial}{partial y}|_{1,2}u$ $latex u|_{2,0}, u|_{2,1}, u|_{2,2}$ Discretizamos: $latex frac{u_{-1,0}-2u_{0,0}+u_{1,0}}{h^2} + frac{u_{0,-1}-2u_{0,0}+u_{0,1}}{h^2} = f_{0,0}$ $latex frac{u_{-1,1}-2u_{0,1}+u_{1,1}}{h^2} + frac{u_{0,0}-2u_{0,1}+u_{0,2}}{h^2} = f_{0,1}$ $latex frac{u_{-1,2}-2u_{0,2}+u_{1,2}}{h^2} + frac{u_{0,1}-2u_{0,2}+u_{0,3}}{h^2} = […]