# How to solve singular non symmetric poisson equation with Neumann boundary condtions?

I am trying to solve 2D Poisson equations with Neumann boundary conditions. When the mesh is uniform, Poisson equation is singular and symmetric, so the method listed in Null Space Projection for Singular Systems works well. But when the mesh is non uniform and sharp immersed boundary method is implemented to treat the curved boundary, discretized Poisson equation is singular and asymmetric based on finite difference method. In this case, the pervious method seems not working well. So can someone help me and tell me how to solve this kind of problem, or just provide me some references? Thanks a lot!

• Do you have all pure neumann boundary conditions? If so have you fixed the potential at a single point to combat the singularity of the system? – Godric Seer Apr 14 '14 at 14:07
• Have you tried explicitly projecting out of the left null space of the Poisson operator? If $u_n^T A = 0$, then any vector $b$ such that $u_n^T b = 0$ will be in the range space of the Poison operator $A$. In the case of a symmetric discretization, $u_n$ is the vector of ones. Finding $u_n$ for an unsymmetric discretization may not be easy, but your particular discretization may guide you. – Sumedh Joshi Apr 14 '14 at 14:36
• Hi, Godric. Yes, I have all pure neumann BCs, and I have tried to fix a single point so that the Poisson matrix is non singular, but the unphysical solution will be generated around that point. And the accuracy of the whole computational domain may be contaminated. Do you have some suggestions on solving this problem? Thanks! – Simon Apr 15 '14 at 8:53
• Hi, Sumedh. Actually I know the null space of $A$,$u_n$ is still the vector of ones, although $A$ is asymmetric. And $Au_n=0$, but $u^T_nA$ is not zero. Could you please give me some suggestions and tell me what should I do for this singular non symmetric singular system? I am sorry that I am not good at mathematics. Can we assume that $A=[10,0,−5,−5,0;0,8,0,−4,−4;0,−3,6,−3,0;−2,−2,0,4,0;0,−1,−1,0,2]$ and $b=[30,-16,-24,-2,12]$? And the solution should be $x=[2,1,-3,1,5] +a[1,1,1,1,1]$. Thanks! – Simon Apr 15 '14 at 13:30