Convergence and Instability in PCG Methods for Bordered Systems

Bordered almost block diagonal systems arise from discretizing a linearized first-order system of n ordinary differential equations in a twopoint boundary value problem with non-separated boundary conditions. The discretization may use spline collocation, finite differences, or multiple shooting. After internal condensation, if necessary, the bordered almost block diagonal system reduces to a standard finite difference structure, which can be solved using a preconditioned conjugate gradient method based on a simple matrix splitting technique. This preconditioned conjugate gradient method is "guaranteed" to converge in at most 2n + 1 iterations. We exhibit a significant collection of two-point boundary value problems for which this preconditioned conjugate gradient method is unstable and hence convergence is not achieved. Keywords: Boundary Value Problems, Ordinary Differential Equations, Preconditioned Conjugate Gradients, Bordered Almost Block Diagonal Systems author for correspond..