@techreport{1b3d12cc61234f38b0d9d6f381ebcab0,
title = "Local stability and a renormalized Newton Method for equilibrium liquid crystal director modeling",
abstract = "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 numericalsolution of such equations by Lagrange-Newton methods leads to problems 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., submitted]. The characterization of local stability of solutions iscomplicated by the double saddle-point structure, and here we develop efficiently computable criteria in terms of minimum eigenvalues of certain projected Schur complements. We also propose 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 Methodbears some resemblance to the Truncated Newton Method of computational micromagnetics, and we compare and contrast the two.",
keywords = "liquid crystals, reduced hessian method, saddle-point problems, unit-vector constraints, director models, local stability, renormalized newton method, equilibrium liquid crystal, director modeling",
author = "{Gartland, Jr}, E.C. and Alison Ramage",
year = "2012",
language = "English",
series = "Strathclyde Mathematics Research Report ",
publisher = "University of Strathclyde",
number = "9",
type = "WorkingPaper",
institution = "University of Strathclyde",
}