Superconvergence of discontinuous Galerkin method for neutral delay differential equations

ABSTRACT

In this paper, we investigate how many convergence orders of discontinuous Galerkin (DG) method for numerically solving neutral delay differential equations (NDDEs). Although discontinuous behaviour may occur in the derivatives of the exact solution at every breaking point, it is shown that the convergence order of the p-degree DG solution at the mesh points and characteristic points can achieve $\mathcal{O}(h^{2p+1})$ and $\mathcal{O}(h^{p+2})$, respectively. Numerical examples are also reported to confirm the theoretical results.

Keywords:

• Superconvergence
• discontinuous Galerkin method
• neutral delay differential equations
• breaking point
• error estimate

2010 AMS Subject Classifications:

• 65L60
• 65L20

Acknowledgments

The authors wish to thank the anonymous referees for their valuable comments and suggestions. The work is supported by the National Natural Science Foundation of China (Grant Nos. 11701110, 11671343), and the Postgraduate Innovation Fund of Hunan Province in China (No. CX20190420).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work is supported by the National Natural Science Foundation of China (Grant Nos. 11701110, 11671343), and the Postgraduate Innovation Fund of Hunan Province in China (No. CX20190420).