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
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