Parameter dependent finite element analysis for ferronematics solutions

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 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 𝐻 1convergence and a ℓ-independent 𝐻 2 -bound for the global minimizers on smooth bounded 2D domains, with smooth boundary and topologically trivial Dirichlet conditions. The 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 𝐿 2 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.

Two-dimensional Ferronematics, Canonical Harmonic Maps and Minimal Connections

“…The Q-profile has a discrete set of non-orientable nematic defects and the M-profile exhibits line defects connecting these nematic defects, in the ε → 0 limit. Whilst the practical relevance of such studies remains uncertain, it is clear that strong theoretical underpinnings are much needed for systematic scientific progress in this field, and our work is a first powerful step in an exhaustive study of ferronematic solution landscapes [46] (also see recent work in [23], [39]).…”