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Summary: ASYMPTOTIC MESH INDEPENDENCE
OF NEWTON'S METHOD REVISITED
MARTIN WEISER, ANTON SCHIELA, AND PETER DEUFLHARD§
SIAM J. NUMER. ANAL. c 2005 Society for Industrial and Applied Mathematics
Vol. 42, No. 5, pp. 18301845
Abstract. The paper presents a new affine invariant theory on asymptotic mesh independence
of Newton's method for discretized nonlinear operator equations. Compared to earlier attempts, the
new approach is both much simpler and more intuitive from the algorithmic point of view. The
theory is exemplified at finite element methods for elliptic PDE problems.
Key words. asymptotic mesh independence, Newton's method, affine invariance
AMS subject classifications. 65J15, 65L10, 65N30
DOI. 10.1137/S0036142903434047
Introduction. The term "mesh independence" characterizes the observation
that finite dimensional Newton methods, when applied to a nonlinear PDE on suc-
cessively finer discretizations with comparable initial guesses, show roughly the same
convergence behavior on all sufficiently fine discretizations. The "mesh independence
principle" has been stated and even exploited for mesh design in papers by Allgower
and B¨ohmer [1] and McCormick [19]. Further theoretical investigations of the phe-
nomenon have been given in [2] by Allgower, B¨ohmer, Potra, and Rheinboldt. Those
papers, however, lacked certain important features in the theoretical characterization
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