Mesh independent convergence of the modified inexact Newton method for a second order nonlinear problem

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In this paper, we consider an inexact Newton method applied to a second order nonlinear problem with higher order nonlinearities. We provide conditions under which the method has a mesh-independent rate of convergence. To do this, we are required to first, set up the problem on a scale of Hilbert spaces and second, to devise a special iterative technique which converges in a higher than first order Sobolev norm. We show that the linear (Jacobian) system solved in Newton's method can be replaced with one iterative step provided that the initial nonlinear iterate is accurate enough. The closeness criteria can ... continued below

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Kim, T; Pasciak, J E & Vassilevski, P S September 20, 2004.

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In this paper, we consider an inexact Newton method applied to a second order nonlinear problem with higher order nonlinearities. We provide conditions under which the method has a mesh-independent rate of convergence. To do this, we are required to first, set up the problem on a scale of Hilbert spaces and second, to devise a special iterative technique which converges in a higher than first order Sobolev norm. We show that the linear (Jacobian) system solved in Newton's method can be replaced with one iterative step provided that the initial nonlinear iterate is accurate enough. The closeness criteria can be taken independent of the mesh size. Finally, the results of numerical experiments are given to support the theory.

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PDF-file: 23 pages; size: 0.2 Mbytes

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  • Journal Name: Numerical Linear Algebra with Applications; Journal Volume: 13; Journal Issue: 1

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  • Report No.: UCRL-JRNL-206774
  • Grant Number: W-7405-ENG-48
  • Office of Scientific & Technical Information Report Number: 883841
  • Archival Resource Key: ark:/67531/metadc891768

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  • September 20, 2004

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  • Sept. 23, 2016, 2:42 p.m.

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  • Nov. 29, 2016, 8 p.m.

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Kim, T; Pasciak, J E & Vassilevski, P S. Mesh independent convergence of the modified inexact Newton method for a second order nonlinear problem, article, September 20, 2004; Livermore, California. (digital.library.unt.edu/ark:/67531/metadc891768/: accessed October 18, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.