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This paper outlines an energy-minimization finite-element approach to thecomputational modeling of equilibrium configurations for nematic liquidcrystals under free elastic effects. The method targets minimization of thesystem free energy based on the Frank-Oseen free-energy model. Solutions to theintermediate discretized free elastic linearizations are shown to existgenerally and are unique under certain assumptions. This requires provingcontinuity, coercivity, and weak coercivity for the accompanying appropriatebilinear forms within a mixed finite-element framework. Error analysisdemonstrates that the method constitutes a convergent scheme. Numericalexperiments are performed for problems with a range of physical parameters aswell as simple and patterned boundary conditions. The resulting algorithmaccurately handles heterogeneous constant coefficients and effectively resolvesconfigurations resulting from complicated boundary conditions relevant inongoing research.

An Energy-Minimization Finite-Element Approach for the Frank-Oseen Model of Nematic Liquid Crystals: Continuum and Discrete Analysis