Error analysis of Nitsche's and discontinuous Galerkin methods of a reduced Landau-de Gennes problem

TL;DR

This paper provides error analysis and a posteriori estimates for Nitsche's and discontinuous Galerkin methods applied to a reduced Landau-de Gennes model for liquid crystal devices, with validation through numerical examples.

Contribution

It offers the first a priori and a posteriori error estimates for these methods applied to this nonlinear PDE system under relaxed regularity assumptions.

Findings

Error bounds in energy norm established for both methods.

Reliable a posteriori error estimators developed and validated.

Numerical experiments confirm theoretical predictions.

Abstract

We study a system of semi-linear elliptic partial differential equations with a lower order cubic nonlinear term, and inhomogeneous Dirichlet boundary conditions, relevant for two-dimensional bistable liquid crystal devices, within a reduced Landau-de Gennes framework. The main results are (i) a priori error estimates for the energy norm, within the Nitsche's and discontinuous Galerkin frameworks under milder regularity assumptions on the exact solution and (ii) a reliable and efficient {\it a posteriori} analysis for a sufficiently large penalization parameter and a sufficiently fine triangulation in both cases. Numerical examples that validate the theoretical results, are presented separately.