Issue |
ESAIM: M2AN
Volume 57, Number 2, March-April 2023
|
|
---|---|---|
Page(s) | 693 - 716 | |
DOI | https://doi.org/10.1051/m2an/2022083 | |
Published online | 27 March 2023 |
Variational and numerical analysis of a Q-tensor model for smectic-A liquid crystals
1
College of Meteorology and Oceanography, National University of Defense Technology, Changsha, P.R. China
2
Mathematical Institute, University of Oxford, Oxford, UK
* Corresponding author: jingmin.xia@nudt.edu.cn
Received:
19
February
2022
Accepted:
30
September
2022
We analyse an energy minimisation problem recently proposed for modelling smectic-A liquid crystals. The optimality conditions give a coupled nonlinear system of partial differential equations, with a second-order equation for the tensor-valued nematic order parameter Q and a fourth-order equation for the scalar-valued smectic density variation u. Our two main results are a proof of the existence of solutions to the minimisation problem, and the derivation of a priori error estimates for its discretisation of the decoupled case (i.e., q = 0) using the C0 interior penalty method. More specifically, optimal rates in the H1 and L2 norms are obtained for Q, while optimal rates in a mesh-dependent norm and L2 norm are obtained for u. Numerical experiments confirm the rates of convergence.
Mathematics Subject Classification: 76A15 / 49J10 / 35B45 / 65N12 / 65N30
Key words: C0 interior penalty method / a priori error estimates / finite element methods / smectic liquid crystals
© The authors. Published by EDP Sciences, SMAI 2023
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