Characterizing mesh independent quadratic convergence of newton's method for a class of elliptic problems

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2 Citations (Scopus)

Abstract

The classical mesh independence principle (MIP) describes a desirable property for Newton's method, its main feature being that the discrete iterations exhibit the same quadratic convergence behavior for any mesh size, i.e., uniformly as the mesh is refined. We study the latter property for a general class of second order nonlinear elliptic boundary value problems solved by finite element discretization. For this, a more specific principle, the mesh independence principle for quadratic convergence (MIPQC), is introduced. It is proved that the MIPQC holds if and only if the elliptic equation is semilinear.

Original languageEnglish
Pages (from-to)1279-1303
Number of pages25
JournalSIAM Journal on Mathematical Analysis
Volume44
Issue number3
DOIs
Publication statusPublished - 2012

Fingerprint

Mesh Independence
Quadratic Convergence
Newton-Raphson method
Newton Methods
Elliptic Problems
Boundary value problems
Mesh
Nonlinear Elliptic Boundary Value Problem
Finite Element Discretization
Semilinear
Elliptic Equations
If and only if
Iteration
Class

Keywords

  • Finite element discretization
  • Lipschitz continuity
  • Mesh independence
  • Newton's method
  • Nonlinear elliptic problem
  • Quadratic convergence

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Computational Mathematics

Cite this

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AB - The classical mesh independence principle (MIP) describes a desirable property for Newton's method, its main feature being that the discrete iterations exhibit the same quadratic convergence behavior for any mesh size, i.e., uniformly as the mesh is refined. We study the latter property for a general class of second order nonlinear elliptic boundary value problems solved by finite element discretization. For this, a more specific principle, the mesh independence principle for quadratic convergence (MIPQC), is introduced. It is proved that the MIPQC holds if and only if the elliptic equation is semilinear.

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