Computable a Posteriori $L_\infty $-Error Bounds for the Approximate Solution of Two-Point Boundary Value Problems
Mary Anne Mccarthy, Richard A. Tapia · SIAM Journal on Numerical Analysis · 1975
In this paper we use the general theory of Newton’s method for operator equations with functional constraints in Banach spaces recently developed by Tapia and the Kantorovich theorem to construct $c^1 $-approximations to the solution and its derivative of the nonlinear two-point boundary value problem and computable upper bounds for the $L_\infty $-norm of the error of both approximations. We also show that in many cases, it is actually possible to calculate lower bounds for the $L_\infty $-norm of the error. Several numerical examples for both boundary value and initial value problems are included. These examples demonstrate the quality of the error bounds. Tapia’s theory is used to transform the two-point boundary value problem into a nonlinear integral equation which avoids the use of Green’s functions and admits the nonlinear presence of $y'$. Numerous authors have attempted similar approaches using Green’s functions, but none in the generality we present. The success of our approach is very satisfying not only because it solves an important problem, but also because it demonstrates a useful application of Tapia’s general theory of Newton’s method for constrained problems.