Optimal preconditioners on solving the Poisson equation with Neumann boundary conditions

MILU preconditioner is well known [16,3] to be the optimal choice among all the ILU-type preconditioners in solving the Poisson equation with Dirichlet boundary conditions. However, it is less known which is an optimal preconditioner in solving the Poisson equation with Neumann boundary conditions. The condition number of an unpreconditioned matrix is as large as O(h⁻²), where h is the step size of grid. Only the optimal preconditioner results in condition number O(h⁻¹), while the others such as Jacobi and ILU result in O(h⁻²). We review Relaxed ILU and Perturbed MILU preconditioners in the case of Neumann boundary conditions, and present empirical results which indicate that the former is optimal in two dimensions and the latter is optimal in two and three dimensions. To the best of our knowledge, these empirical results have not been rigorously verified yet. We present a formal proof for the optimality of Relaxed ILU in rectangular domains, and discuss its possible extension to general smooth domains and Perturbed MILU.