Local stability and a renormalized Newton Method for equilibrium liquid crystal director modeling

E.C. Gartland, Jr, Alison Ramage

Research output: Working paper

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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 numerical
solution 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 is
complicated 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 Method
bears some resemblance to the Truncated Newton Method of computational micromagnetics, and we compare and contrast the two.
Original languageEnglish
Place of PublicationGlasgow
PublisherUniversity of Strathclyde
Number of pages19
Publication statusPublished - 2012

Publication series

NameStrathclyde Mathematics Research Report
PublisherUniversity of Strathclyde
No.9

Keywords

  • liquid crystals
  • reduced hessian method
  • saddle-point problems
  • unit-vector constraints
  • director models
  • local stability
  • renormalized newton method
  • equilibrium liquid crystal
  • director modeling

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