Newton Iteration for Partial Differential Equations and the Approximation of the Identity

By: Gregory E. Fasshauer, Eugene C. Gartland, Jr. and Joseph W. Jerome

It is known that the critical condition which guarantees quadratic convergence of approximate Newton methods is an approximation of the identity condition. This requires that the composition, of the numerical inversion of the Fr\'{e}chet derivative with the derivative itself, approximate the identity to an accuracy calibrated by the residual. For example, the celebrated quadratic convergence theorem of Kantorovich can be proven when this holds, subject to regularity and stability of the derivative map. In this paper, we study what happens when this condition is not evident `a priori' but is observed `a posteriori'. Through an in-depth example involving a semilinear elliptic boundary value problem, and some general theory, we study the condition in the context of dual norms, and the effect upon convergence. We also discuss the connection to Nash iteration.
This paper has appeared in Numerical Algorithms 25 (2000), 181--195, and can be viewed in the following format: