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

Maity, Ruma Rani and Majumdar, Apala and Nataraj, Neela (2023) A priori and a posteriori error analysis for semilinear problems in liquid crystals. ESAIM: Mathematical Modelling and Numerical Analysis. ISSN 0764-583X (In Press) (https://doi.org/10.1051/m2an/2023056)

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Abstract

In this paper, we develop a unified framework for the a priori and a posteriori error control of different lowestorder 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.

ORCID iDs

Maity, Ruma Rani, Majumdar, Apala ORCID logoORCID: https://orcid.org/0000-0003-4802-6720 and Nataraj, Neela;
  • Item type:Article
    ID code:86371
    Dates:
    Date
    Event
    15 June 2023
    Published
    15 June 2023
    Accepted
    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, Analysis, Modelling and Simulation, Analysis, Applied Mathematics, Mathematics(all), Numerical Analysis
    Subjects:Science > Mathematics > Analysis
    Department:Faculty of Science > Mathematics and Statistics
    Depositing user: Pure Administrator
    Date deposited:03 Aug 2023 09:08
    Last modified:04 Sep 2023 02:12
    URI:https://strathprints.strath.ac.uk/id/eprint/86371
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