### 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 language | English |
---|---|

Pages (from-to) | 1279-1303 |

Number of pages | 25 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 44 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2012 |

### Fingerprint

### 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

**Characterizing mesh independent quadratic convergence of newton's method for a class of elliptic problems.** / Karátson, J.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Characterizing mesh independent quadratic convergence of newton's method for a class of elliptic problems

AU - Karátson, J.

PY - 2012

Y1 - 2012

N2 - 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.

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.

KW - Finite element discretization

KW - Lipschitz continuity

KW - Mesh independence

KW - Newton's method

KW - Nonlinear elliptic problem

KW - Quadratic convergence

UR - http://www.scopus.com/inward/record.url?scp=84861396461&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84861396461&partnerID=8YFLogxK

U2 - 10.1137/100817589

DO - 10.1137/100817589

M3 - Article

VL - 44

SP - 1279

EP - 1303

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 3

ER -