Zero flux boundary conditions issue with Poisson equation

Good morning,

I am using version 4.3b. One of the steps in my project requires me to solve a Poisson equation with zero flux conditions at the boundaries. My geometry is a simple 2-by-2 square (domain: [-1,1]x[-1,1]). I have 2 questions regarding a tolerance error that keeps popping up.

1). When I have a source term as simple as: f=cos(2*pi*x), it says:

Failed to find a solution.
The relative error (25) is greater than the relative tolerance.
Returned solution is not converged.
- Feature: Stationary Solver 1(sol1/s1)
- Error: Failed to find a solution.

But if I replace just 1 of the side walls with a Dirichlet condition, I get the analytic solution.

I have verified the consistency condition for Poisson's equation with Neumann conditions (integral of f over the region is equal to zero, which is the same as the integral of zero (flux) over the boundary). I don't understand why COMSOL can't handle the zero flux condition.

2). When I try something even simpler with source term: f=0, I get the same error if my initial guess is u=2, but I get a valid solution when my initial guess is u=0. With zero flux on the walls, both u=2 and u=0 everywhere should be valid solutions. What is the reasoning for this error?

Thanks in advance,
-Peter