A normalized implicit conjugate gradient method for the solution of large sparse systems of linear equations

Linear systems arising from the finite difference approximation of a general self-adjoint elliptic partial differential equation on the unit square and cube are considered. The regular and banded structure of the coefficient matrices allows an approximate factorization in which only the elements along the diagonal, the co-diagonal and the r outermost diagonals of the band are included. This approximate factorization is chosen as the basis to yield a normalized system to which the method of conjugate gradients is implicitly applied. Due to the fact that a good approximate inverse of the coefficient matrix is used, the convergence of the new method-the normalized implicit conjugate gradient (NICG) method is very much improved.