I want to numerically (with Matlab) solve Poisson's equation : $ \frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = f(x,y)$ On a rectangular domain using experimental data.From experimental data let's say that I have a $ [1:m \times 1:n] $ matrix that represents the right hand side of the equation. From experimental data I have access to $ \nabla(u)$ which is also a $ [1:m \times 1:n] $ matrix.My question is : Is it mathematically/physically correct to reduce my mesh to a size of $ [2:m-1 \times 2:n-1] $ and to take $ \nabla(u)(1,:) $, $ \nabla(u)(n,:)$, $ \nabla(u)(:,1)$, $ \nabla(u)(:,n)$ as my Neumann boundary conditions ? Or am I missing something ?
Boundary conditions for solving Poisson's Equation with Experimental Data
computational physics;boundary conditions
I eventually found out how to procede.I found the work of Dr. De Kat, whose method was exactly the one I wanted to implement. It is explained in the pdf file here  http://repository.tudelft.nl/view/ir/uuid:be00fb31-71cb-4859-ab79-67879caadc3c/ (starting page 49)To him the best way is to have a region of the flow where Bernoulli's equation can be applied but he does use the Navier-Stokes equation which yields the pressure gradient as his neumann boundary conditions. At least one point must be exactly known in order to procede to a shift of the data after computation.