A priori and a posteriori error analysis for semilinear problems in liquid crystals

Ruma Rani Maity, Apala Majumdar, Neela Nataraj

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Abstract

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.
Original languageEnglish
Pages (from-to)3201-3250
Number of pages50
JournalESAIM: Mathematical Modelling and Numerical Analysis
Volume57
Issue number6
DOIs
Publication statusPublished - 17 Nov 2023

Keywords

  • 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

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