An a priori error analysis of the hp-version of the C⁰-continuous Petrov-Galerkin method for nonlinear second-order delay differential equations

Abstract

We study an hp-version of the $C^0$-continuous Petrov-Galerkin time-stepping method for nonlinear second-order delay differential equations with vanishing delays. We derive a priori error bound in the $H^1$-norm that is fully explicit in the local time steps, in the local approximation degrees, and in the local regularity indexes of the exact solutions. Moreover, we prove that the $C^0$-continuous Petrov-Galerkin based on special hp-version discretization can yield exponential rates of convergence for analytic solutions with initial singularities. Numerical experiments are provided to illustrate the theoretical results.

Keywords:

• Second-order delay differential equations
• hp-version
• continuous Petrov-Galerkin method
• time-stepping
• exponential convergence

2000 AMS SUBJECT CLASSIFICATIONS:

• 65L60
• 65L05
• 65L70

Disclosure statement

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

Additional information

Funding

The work of this author is supported in part by the National Natural Science Foundation of China [grant numbers 12171322, 11771298 and 11871043], the Natural Science Foundation of Shanghai [grant numbers 21ZR1447200 and 20ZR1441200], and the Science and Technology Innovation Plan of Shanghai [grant number 20JC1414200].