Numerical issues for discretized PDEs with more than two spatial dimensions

Numerical issues for discretized PDEs with more than two spatial dimensions 

Consider a BVP involving the Poisson equation in three dimensions 

We set a uniform grid of points over the domain and substitute finite difference approximations for each second derivative to obtain for each interior point (x , yj, zj ) a linear equation 

Nz are the numbers of grid points in the x, y, and z directions respectively. We stack all unknowns in a single vector of dimension NxNyNz by assigning to each grid point (x , y, zi) the unique label 

We then have the linear ecpiation for each interior grid point 

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