Finite - Difference Methods and the Eigenvalue Problem for Nonselfadjoint

In this paper we analyze the convergence of a centered finite-difference approximation to the nonselfadjoint Sturm-Liouville eigenvalue problem 2[u\ = [a(x) u'Y b{x)u' + c(x)u = \u, 0 < x < 1, ( } «(0) = u(l) = 0 where S has smooth coefficients and a(x) ï a« > 0 on [0, 1]. We show that the rate of convergence is 0(Az2) as in the selfadjoint case for a scheme of the same accuracy. We also establish discrete analogs of the Sturm oscillation and comparison theorems. As a corollary we obtain the result (2) hmsup i 22 f < °° where Ax = \/{M + 1) is the mesh size and Ap, V" are the characteristic pairs of L, the M X M matrix which approximates C, and V? is normalized so that \\Vp\\i = 1.