Preconditioners for elliptic problems via non-uniform meshes

In this paper, we concerned with the preconditioned iterative methods for the solution of elliptic problems with Dirichlet or Robbins boundary condition via non-uniform meshes over an unit domain. We construct and analyze several preconditioners for the coefficient matrix derived from the finite difference discretization via non-uniform meshes. We prove that the preconditioners can be chosen so that the condition number of the preconditioned system can be reduced from O(n2) to O(1). The analysis is supported by numerical experiments.