A posteriori local discontinuous Galerkin error estimates for the one-dimensional sine-Gordon equation

ABSTRACT

In this paper, we present a posteriori error analysis of the local discontinuous Galerkin (LDG) method for the sine-Gordon nonlinear hyperbolic equations with smooth solutions. We show that the dominant components of the local LDG errors on each element are proportional to right and left Radau polynomials of degree p+1. Thus, the discretization errors for the p-degree LDG solution and its spatial derivative are $\mathcal{O}\left(h^{p+2}\right)$ superconvergent at the roots of $(p+1)$-degree right and left Radau polynomials, respectively. Numerical experiments indicate that our superconvergence results hold globally. We use the superconvergence results to construct simple, efficient, and asymptotically exact a posteriori LDG error estimates. The proposed error estimates are computationally simple and are obtained by solving local steady problems with no boundary conditions on each element. Numerical computations suggest that these a posteriori LDG error estimates for the solution and its spatial derivative, at any fixed time, converge to the true errors at $\mathcal{O}\left(h^{p+2}\right)$ rate, respectively. We also demonstrate that the global effectivity indices for the solution and its derivative in the $L^2$-norm converge to unity. We present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement. Finally, we present a local adaptive procedure that makes use of our local a posteriori error estimates.

Keywords:

• Local discontinuous Galerkin method
• sine-Gordon equation
• superconvergence
• Radau polynomials
• a posteriori error estimates
• adaptive mesh method

2010 AMS SUBJECT CLASSIFICATIONS:

• 65M12
• 65M15
• 65M60
• 65N12
• 65N30
• 35Q51

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Mahboub Baccouch http://orcid.org/0000-0002-6721-309X

Additional information

Funding

The author thank the anonymous reviewers for the valuable comments and suggestions which improve the quality of the paper. This research was partially supported by the NASA Nebraska Space Grant Program and UCRCA at the University of Nebraska at Omaha (UCRCA Proposal 2016-01-F).