Fourier Analysis of Modified Nested Factorization Preconditioner for Three-Dimensional Isotropic Problems

For solving large sparse symmetric linear systems, arising from the discretization of elliptic problems, the preferred choice is the preconditioned con- jugate gradient method. The convergence rate of this method mainly depends on the condition number of the preconditioner chosen. Using Fourier analy- sis the condition number estimate of common preconditioning techniques for two dimensional elliptic problem has been studied by Chan and Elman [SIAM Rev., 31 (1989), pp. 20-49]. Nested Factorization(NF) is one of the powerful preconditioners for systems arising from discretization of elliptic or hyperbolic partial differential equations. The observed convergence behavior of NF is bet- ter compared to well known ILU(0) or modified ILU. In this paper we introduce Modified Nested Factorization(MNF) which is an improvement over NF. It is proved that condition number of modified NF is O(h−1 ). An optimal value of the parameter for the model problem is derived. The condition number of modified NF predicts the condition number of NF in limiting sense when the parameter is close to zero. Moreover it is proved that condition number of NF is atleast O(h−1 ). Numerical results justify Fourier analytic method by exhibiting remarkable similarity in spectrum of periodic and Dirichlet problems