Newton iteration for partial differential equations and the approximation of the identity

被引:13
|
作者
Fasshauer, GE
Gartland, EC
Jerome, JW
机构
[1] IIT, Dept Appl Math, Chicago, IL 60616 USA
[2] Kent State Univ, Dept Math & Comp Sci, Kent, OH 44242 USA
[3] Northwestern Univ, Dept Math, Evanston, IL 60208 USA
关键词
Newton methods; partial differential equations; approximation of the identity; Nash iteration;
D O I
10.1023/A:1016609007255
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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 Frechet 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 ill the context of dual norms, and the effect upon convergence. We also discuss the connection to Nash iteration.
引用
收藏
页码:181 / 195
页数:15
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