Projected gradient different from Neumann BC

[kamensky]

Note that your problem is not well-posed. A nonzero source term is only compatible with a zero-flux Neumann BC if the integral of the source term is exactly zero. Even with a compatible source term, the solution is only defined up to an arbitrary constant. My answer in the following thread explains these issues in more detail:

White spots in the plot

To expand on @nate’s answer, there are two basic well-posedness issues that come up with pure Neumann BCs. The first is compatibility of the Neumann data with the volume source term. If you have the interior PDE -\nabla\cdot(\nabla u) = f\text{ ,} integrating both sides over \Omega and applying the divergence theorem on the left implies that \int_{\partial\Omega}-\nabla u\cdot\mathbf{n} \,d\partial\Omega = \int_\Omega f\,d\Omega\text{ .} If you also have the Neumann BC of \nabla u\cdot\m…