On Solving the Singular System Arisen from Poisson Equation with Neumann Boundary Condition

Abstract

We consider solving the singular linear system arisen from the Poisson equation with the Neumann boundary condition. To handle the singularity, there are two usual approaches: one is to fix a Dirichlet boundary condition at one point, and the other seeks a unique solution in the orthogonal complement of the kernel. One may incorrectly presume that the two solutions are the similar to each other. In this work, however, we show that their solutions differ by a function that has a pole at the Dirichlet boundary condition. The pole of the function is comparable to that of the fundamental solution of the Laplace operator. Inevitably one of them should contain the pole, and accordingly has inferior accuracy than the other. According to our novel analysis in this work, it is the fixing method that contains the pole. The projection method is thus preferred to the fixing method, but it also contains cons: in finding a unique solution by conjugate gradient method, it requires extra steps per each iteration. In this work, we introduce an improved method that contains the accuracy of the projection method without the extra steps. We carry out numerical experiments that validate our analysis and arguments.