On solving the singular system arisen from Poisson equation with Neumann boundary condition (Q334346)

Solving the Poisson equation with Neumann boundary conditions gives after discretization a singular linear system. The singularity can be removed by either fixing a value of the solution in a point \(Q\) of the domain \(\Omega\), or by projecting the right-hand side on the column space of the system and solving for a solution orthogonal to the kernel of the system. The purpose of the paper is to prove that the two solutions thus obtained differ by a function that has a pole at \(Q\). The norm of the difference is estimated by analyzing the problem in both \(L^2\) and \(L^\infty\). The result is that the second method is more accurate. However, it is possible to modify the first method to get the same solution. This analysis is done for a standard 5-point discretization in a rectangular domain. Numerical experiments confirm these findings.