Parameter dependent finite element analysis for ferronematics solutions

Ruma Rani Maity, Apala Majumdar, Neela Nataraj

Research output: Contribution to journalArticlepeer-review

Abstract

This paper focuses on the analysis of a free energy functional, that models a dilute suspension of magnetic nanoparticles in a two-dimensional nematic well. The {\it first part} of the article is devoted to the asymptotic analysis of global energy minimizers in the limit of vanishing elastic constant, ℓ→0 where the re-scaled elastic constant ℓ is inversely proportional to the domain area. The first results concern the strong H1-convergence and a ℓ-independent H2-bound for the global minimizers on smooth bounded 2D domains, with smooth boundary and topologically trivial Dirichlet conditions. The {\it second part} focuses on the discrete approximation of regular solutions of the corresponding non-linear system of partial differential equations with cubic non-linearity and non-homogeneous Dirichlet boundary conditions. We establish (i) the existence and local uniqueness of the discrete solutions using fixed point argument, (ii) a best approximation result in energy norm, (iii) error estimates in the energy and L2 norms with ℓ- discretization parameter dependency for the conforming finite element method. Finally, the theoretical results are complemented by numerical experiments on the discrete solution profiles, the numerical convergence rates that corroborates the theoretical estimates, followed by plots that illustrate the dependence of the discretization parameter on ℓ.
Original languageEnglish
Pages (from-to)127-155
Number of pages36
JournalComputers and Mathematics with Applications
Volume103
Early online date8 Nov 2021
DOIs
Publication statusPublished - 1 Dec 2021

Keywords

  • ferronematics
  • composite system energy optimization
  • convergence of minimizers
  • finite element method
  • error estimates
  • numerical experiments

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