TY - JOUR

T1 - Newton iteration for partial differential equations and the approximation of the identity

AU - Fasshauer, Gregory E.

AU - Gartland, Eugene C.

AU - Jerome, Joseph W.

PY - 2000/1/1

Y1 - 2000/1/1

N2 - 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 Fréchet 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 reg ularity 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.

AB - 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 Fréchet 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 reg ularity 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.

KW - Approximation of the identity

KW - Nash iteration

KW - Newton methods

KW - Partial differential equations

UR - http://www.scopus.com/inward/record.url?scp=0034562122&partnerID=8YFLogxK

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U2 - 10.1023/a:1016609007255

DO - 10.1023/a:1016609007255

M3 - Article

AN - SCOPUS:0034562122

VL - 25

SP - 181

EP - 195

JO - Numerical Algorithms

JF - Numerical Algorithms

SN - 1017-1398

IS - 1-4

ER -