On convergence of an unconditional stable numerical scheme for Q-tensor flow based on invariant quardratization method

TL;DR

This paper proves the convergence of a stable numerical scheme for Q-tensor flows in nematic liquid crystals, utilizing the Invariant Energy Quadratization method to ensure accuracy and stability.

Contribution

It introduces a convergence analysis for a new numerical scheme based on the Invariant Energy Quadratization method for Q-tensor flows, establishing uniform estimates and strong solution convergence.

Findings

Uniform $H^2$ estimates for numerical solutions

Convergence to a strong solution of the Q-tensor equation

Equivalence of the auxiliary variable to the original energy functional

Abstract

We present convergence analysis towards a numerical scheme designed for Q-tensor flows of nematic liquid crystals. This scheme is based on the Invariant Energy Quadratization method, which introduces an auxiliary variable to replace the original energy functional. In this work, we have shown that given an initial value with $H^2$ regularity, we can obtain a uniform $H^2$ estimate on the numerical solutions for Q-tensor flows and then deduce the convergence to a strong solution of the parabolic-type Q-tensor equation. We have also shown that the limit of the auxiliary variable is equivalent to the original energy functional term in the strong sense.