Communications in Mathematical Sciences

Volume 21 (2023)

Number 4

An energy stable finite difference scheme for the Ericksen–Leslie system with penalty function and its optimal rate convergence analysis

Pages: 1135 – 1169



Kelong Cheng (School of Science, Civil Aviation Flight University of China, Guanghan, Sichuan, China)

Cheng Wang (Department of Mathematics, University of Massachusetts, North Dartmouth, Mass., U.S.A.)

Steven M. Wise (Department of Mathematics, University of Tennessee, Knoxville, Tenn., U.S.A.)


A first-order-accurate-in-time, finite difference scheme is proposed and analyzed for the Ericksen–Leslie system, which describes the evolution of nematic liquid crystals. For the penalty function to approximate the constraint $\lvert d \rvert = 1$, a convex-concave decomposition for the corresponding energy functional is applied. In addition, appropriate semi-implicit treatments are adopted for the convection terms, for both the velocity vector and orientation vector, as well as the coupled elastic stress terms. In turn, all the semi-implicit terms can be represented as a linear operator of a vector potential, and its combination with the convex splitting discretization for the penalty function leads to a unique solvability analysis for the proposed numerical scheme. Furthermore, a careful estimate reveals an unconditional energy stability of the numerical system, composed of the kinematic energy and internal elastic energies. More importantly, we provide an optimal rate convergence analysis and error estimate for the numerical scheme. In addition, a nonlinear iteration solver is outlined, and the numerical accuracy test results are presented, which confirm the optimal rate convergence estimate.


Ericksen–Leslie system with the penalty function, convex-concave decomposition, unique solvability, energy stability, staggered mesh points, optimal rate convergence analysis

2010 Mathematics Subject Classification

35K55, 65M06, 65M12, 76D05

Received 23 March 2021

Received revised 26 August 2022

Accepted 14 September 2022

Published 24 March 2023