An efficient MILU preconditioning for solving the 2D Poisson equation with Neumann boundary condition

MILU preconditioning is known to be the optimal one among all the ILU-type preconditionings in solving the Poisson equation with Dirichlet boundary condition. It is optimal in the sense that it reduces the condition number from O(h⁻²), which can be obtained from other ILU-type preconditioners, to O(h⁻¹). However, with Neumann boundary condition, the conventional MILU cannot be used since it is not invertible, and some MILU preconditionings achieved the order O(h⁻¹) only in rectangular domains. In this article, we consider a standard finite volume method for solving the Poisson equation with Neumann boundary condition in general smooth domains, and introduce a new and efficient MILU preconditioning for the method in two dimensional general smooth domains. Our new MILU preconditioning achieved the order O(h⁻¹) in all our empirical tests. In addition, in a circular domain with a fine grid, the CG method preconditioned with the proposed MILU runs about two times faster than the CG with ILU.