Abstract: A method of high order approximate LU decomposition is adopted to give the preconditioner for the coefficient matrix of 3-D elliptic partial differential equation when the normal 7-point scheme is used. Numerical results of the preconditioned conjugate gradient (PCG) solving a tipycal problem are given.The concepts of the order, the order matrix and the P-order influence areas given in this paper can be applied to a general sparse non-singular and diagonally dominant linear system of equations. The results of different meshsize show that 3-D PCG has a superlinear convergence rate. In the case of higher orders the superlinearity of the PCG convergence rate is stronger even more. Since the number of non-zero diagonals in the preconditioner increases rapidly when the order of the PCG method increases, it is not beneficial to use the preconditioner of order P if P is larger than one. The preeonditioners of order zero and order one are recommended.

Key words: preconditioned conjugate gradient, 7-point scheme, LU decomposition