### Abstract

Language | English |
---|---|

Pages | 251-278 |

Number of pages | 28 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 53 |

Issue number | 1 |

Early online date | 22 Jan 2015 |

DOIs | |

Publication status | Published - 2015 |

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### Keywords

- reduced hessian method
- saddle-point problems
- unit-vector constraints
- director models
- liquid crystals
- renormalized newton method

### Cite this

*SIAM Journal on Numerical Analysis*,

*53*(1), 251-278. https://doi.org/10.1137/130942917

}

*SIAM Journal on Numerical Analysis*, vol. 53, no. 1, pp. 251-278. https://doi.org/10.1137/130942917

**A renormalized Newton method for liquid crystal director modeling.** / Gartland, Jr, Eugene C.; Ramage, Alison.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A renormalized Newton method for liquid crystal director modeling

AU - Gartland, Jr, Eugene C.

AU - Ramage, Alison

N1 - © 2015, Society for Industrial and Applied Mathematics

PY - 2015

Y1 - 2015

N2 - We consider the nonlinear systems of equations that result from discretizations of a prototype variational model for the equilibrium director field characterizing the orientational properties of a liquid crystal material. In the presence of pointwise unit-vector constraints and coupled electric fields, the numerical solution of such equations by Lagrange-Newton methods leads to linear systems with a double saddle-point form, for which we have previously proposed a preconditioned nullspace method as an effective solver [A. Ramage and E. C. Gartland, Jr., SIAM J. Sci. Comput., 35 (2013), pp. B226–B247]. Here we propose and analyze a modified outer iteration (“Renormalized Newton Method”) in which the orientation variables are normalized onto the constraint manifold at each iterative step. This scheme takes advantage of the special structure of these problems, and we prove that it is locally quadratically convergent. The Renormalized Newton Method bears some resemblance to the Truncated Newton Method of computational micromagnetics, and we compare and contrast the two. This brings to light some anomalies of the Truncated Newton Method.

AB - We consider the nonlinear systems of equations that result from discretizations of a prototype variational model for the equilibrium director field characterizing the orientational properties of a liquid crystal material. In the presence of pointwise unit-vector constraints and coupled electric fields, the numerical solution of such equations by Lagrange-Newton methods leads to linear systems with a double saddle-point form, for which we have previously proposed a preconditioned nullspace method as an effective solver [A. Ramage and E. C. Gartland, Jr., SIAM J. Sci. Comput., 35 (2013), pp. B226–B247]. Here we propose and analyze a modified outer iteration (“Renormalized Newton Method”) in which the orientation variables are normalized onto the constraint manifold at each iterative step. This scheme takes advantage of the special structure of these problems, and we prove that it is locally quadratically convergent. The Renormalized Newton Method bears some resemblance to the Truncated Newton Method of computational micromagnetics, and we compare and contrast the two. This brings to light some anomalies of the Truncated Newton Method.

KW - reduced hessian method

KW - saddle-point problems

KW - unit-vector constraints

KW - director models

KW - liquid crystals

KW - renormalized newton method

UR - http://epubs.siam.org/loi/sjnaam

U2 - 10.1137/130942917

DO - 10.1137/130942917

M3 - Article

VL - 53

SP - 251

EP - 278

JO - SIAM Journal on Numerical Analysis

T2 - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 1

ER -