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

Ruma Rani Maity; Apala Majumdar; Neela Nataraj

doi:10.1515/cmam-2020-0185

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

Ruma Rani Maity, Apala Majumdar, Neela Nataraj

Mathematics And Statistics

Research output: Contribution to journal › Article › peer-review

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 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.

Original language	English
Pages (from-to)	179–209
Number of pages	31
Journal	Computational Methods in Applied Mathematics
Volume	21
Issue number	1
Early online date	16 Dec 2020
DOIs	https://doi.org/10.1515/cmam-2020-0185
Publication status	E-pub ahead of print - 16 Dec 2020

Keywords

non-linear elliptic PDE
non-homogeneous Dirichlet boundary data
lower regularity
Nitsche's methodq

Access to Document

10.1515/cmam-2020-0185

Embargoed Document

Maity-etal-CMAM-2020-Error-analysis-of-Nitsches-and-discontinuous-Galerkin-methods
Accepted author manuscript, 1.98 MB
Embargo ends: 16/12/21

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

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{\textquoteright}s and discontinuous Galerkin frameworks under milder regularity assumptions on the exact solution and (ii) a reliable and efficient 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.",

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AU - Maity, Ruma Rani

AU - Majumdar, Apala

AU - Nataraj, Neela

PY - 2020/12/16

Y1 - 2020/12/16

N2 - 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 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.

AB - 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 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.

KW - non-linear elliptic PDE

KW - non-homogeneous Dirichlet boundary data

KW - lower regularity

KW - Nitsche's methodq

U2 - 10.1515/cmam-2020-0185

DO - 10.1515/cmam-2020-0185

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EP - 209

JO - Computational Methods in Applied Mathematics

JF - Computational Methods in Applied Mathematics

SN - 1609-9389

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