| Issue |
ESAIM: M2AN
Volume 57, Number 6, November-December 2023
|
|
|---|---|---|
| Page(s) | 3201 - 3250 | |
| DOI | https://doi.org/10.1051/m2an/2023056 | |
| Published online | 17 November 2023 | |
A priori and a posteriori error analysis for semilinear problems in liquid crystals
1
Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, 00076 Helsinki, Finland
2
Department of Mathematics And Statistics, University of Strathclyde, 16 Richmond St, Glasgow G1 1XQ, UK
3
Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
4
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
* Corresponding author: neela@math.iitb.ac.in; nataraj.neela@gmail.com
Received:
14
August
2022
Accepted:
15
June
2023
In this paper, we develop a unified framework for the a priori and a posteriori error control of different lowest-order finite element methods for approximating the regular solutions of systems of partial differential equations under a set of hypotheses. The systems involve cubic nonlinearities in lower order terms, non-homogeneous Dirichlet boundary conditions, and the results are established under minimal regularity assumptions on the exact solution. The key contributions include (i) results for existence and local uniqueness of the discrete solutions using Newton–Kantorovich theorem, (ii) a priori error estimates in the energy norm, and (iii) a posteriori error estimates that steer the adaptive refinement process. The results are applied to conforming, Nitsche, discontinuous Galerkin, and weakly over penalized symmetric interior penalty schemes for variational models of ferronematics and nematic liquid crystals. The theoretical estimates are corroborated by substantive numerical results.
Mathematics Subject Classification: 65N30 / 35J15 / 65N12 / 76A15 / 82D30
Key words: Conforming FEM Nitsche’s method / discontinuous Galerkin and WOPSIP methods / a priori and a posteriori / error analysis / non-linear elliptic PDEs / non-homogeneous Dirichlet boundary conditions / nematic liquid crystals / ferronematics
© The authors. Published by EDP Sciences, SMAI 2023
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