Convergence and instability in PCG methods for bordered systems

• Published in: Computers & mathematics with applications (1987) Vol. 30; no. 12; pp. 101 - 109
• Main Authors: Kraut, G.L., Gladwell, I.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.12.1995; Elsevier
• Subjects: Bordered almost block diagonal systems; Boundary value problems; Exact sciences and technology; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Ordinary differential equations; Preconditioned conjugate gradients; Sciences and techniques of general use; Ordinary differential equations; Preconditioned conjugate gradients; Boundary value problems; Bordered almost block diagonal systems; Conjugate gradient method; Discretization; Finite difference equation; Differential equation; Two point boundary value problem; Instability; Matrix decomposition; Preconditioning; Numerical convergence
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/0898-1221(95)00177-Z

Bordered almost block diagonal systems arise from discretizing a linearized first-order system of n ordinary differential equations in a two-point boundary value problem with nonseparated 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 2 n + 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.