Parameter dependent finite element analysis for ferronematics solutions

TL;DR

This paper analyzes the asymptotic behavior of energy minimizers in ferronematics models as the elastic constant vanishes and develops finite element methods for approximating solutions, with theoretical and numerical validation.

Contribution

It provides new asymptotic analysis results for global minimizers and establishes finite element error estimates for the nonlinear PDE system in ferronematics.

Findings

Strong $H^1$-convergence of minimizers as $ ext{ell} o 0$

Existence and uniqueness of discrete solutions with error bounds

Numerical experiments confirming theoretical convergence rates

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, $\ell \rightarrow 0$ where the re-scaled elastic constant $\ell$ is inversely proportional to the domain area. The first results concern the strong $H^1$-convergence and a $\ell$-independent $H^2$-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 $L^2$ norms with $\ell $- 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 $\ell$.