THE EFFECTIVE CONDITION NUMBER APPLIED TO ERROR ANALYSIS OF CERTAIN BOUNDARY COLLOCATION METHODS

The boundary collocation method (BCM) is widely used in the engineering community for the numerical solution of linear boundary value problems. It is commonly stated that the results computed by this method are affected by large rounding errors. This conclusion is based on a simple worst-case analysis using the conventional condition number for linear systems. In this paper we present a refined error analysis for a class of boundary value problems, using the concept of effectively well-conditioned systems. This analysis leads to much smaller a priori bounds for the rounding error in the computed BCM solution. These smaller bounds imply that the numerically computed solution expresses - much more accurately than previously supposed - properties of the ideal BCM solution (computed with infinite precision). Therefore, the intrinsic properties of the BCM can be investigated through numerical computations to a far larger extent than previously supposed, thus allowing for a numerical diagnostic investigation of the BCM. For example, when the BCM is applied to problems with certain geometries, it is observed that the approximation does not converge. Using our new, smaller error bounds, we conclude that the lack of convergence of the BCM solution for these geometries is due to the BCM itself and not to rounding errors.