Superconvergence of discontinuous Galerkin methods for nonlinear delay differential equations with vanishing delay

In this paper, we investigate the local superconvergence of the discontinuous Galerkin (DG) solutions on quasi-graded meshes for nonlinear delay differential equations with vanishing delay. It is shown that the optimal order of the DG solution at the mesh points is O(h2m+1). By analyzing the supercl...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2019-03, Vol.348, p.314-327
Main Authors: Xu, Xiuxiu, Huang, Qiumei
Format: Article
Language:English
Subjects:
Discontinuous Galerkin methods
Nonlinear delay differential equations
State dependent delay
Superconvergence
Vanishing delay
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Summary:In this paper, we investigate the local superconvergence of the discontinuous Galerkin (DG) solutions on quasi-graded meshes for nonlinear delay differential equations with vanishing delay. It is shown that the optimal order of the DG solution at the mesh points is O(h2m+1). By analyzing the supercloseness between the DG solution and the interpolation Πhu of the exact solution, we get the optimal order O(hm+2) of the DG solution at characteristic points. We then extend the convergence results of DG solutions to state dependent delay differential equations. Numerical examples are provided to illustrate the theoretical results.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2018.08.029