Error estimation based on locally weighted defect for boundary value problems in second order ordinary differential equations

Abstract

We investigate efficient asymptotically correct a posteriori error estimates for the numerical approximation of two-point boundary value problems for second order ordinary differential equations by piecewise polynomial collocation methods. Our error indicators are based on the defect of the collocation solution with respect to an associated exact difference scheme and backsolving using a cheap, low order finite-difference scheme. We prove high asymptotical correctness of this error indicator and illustrate the theoretical analysis by numerical examples.