TY - JOUR
T1 - The hp Discontinuous Galerkin Method for Delay Differential Equations with Nonlinear Vanishing Delay
AU - Huang, Qiumei
AU - Xie, Hehu
AU - Brunner, Hermann
N1 - Funding information:
College of Applied Sciences, Beijing University of Technology, Beijing 100124, China ([email protected]). The work of this author was supported by Beijing Natural Science Foundation (1112002), the National Natural Science Foundation of China (NSFC 11101018), Key Lab of Virtual Geographic Environment (Nanjing Normal University), and Ministry of Education.
^ LSEC, ICMSEC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China ([email protected]). The work of this author was supported by the National Natural Science Foundation of China (NSFC 11001259, 2010DFR00700), the Croucher Foundation of Hong Kong Baptist University, the National Center for Mathematics and Interdisciplinary Science, CAS, and the President Foundation of AMSS-CAS.
§ Department of Mathematics, Hong Kong Baptist University, Hong Kong SAR, China and Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7, Canada ([email protected]). The work of this a uthor was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Grant 9406), and the Hong Kong Research Grants Council (HKBU 200207).
Publisher copyright:
© 2013, Society for Industrial and Applied Mathematics
PY - 2013/6/19
Y1 - 2013/6/19
N2 - We present the hp-version of the discontinuous Galerkin method for the numerical solution of delay differential equations with nonlinear vanishing delays and derive error bounds that are explicit in the time steps, the degrees of the approximating polynomials, and the regularity properties of the exact solutions. It is shown that the hp discontinuous Galerkin method exhibits exponential rates of convergence for smooth solutions on uniform meshes, and for nonsmooth solutions on geometrically graded meshes. The theoretical results are illustrated by various numerical examples.
AB - We present the hp-version of the discontinuous Galerkin method for the numerical solution of delay differential equations with nonlinear vanishing delays and derive error bounds that are explicit in the time steps, the degrees of the approximating polynomials, and the regularity properties of the exact solutions. It is shown that the hp discontinuous Galerkin method exhibits exponential rates of convergence for smooth solutions on uniform meshes, and for nonsmooth solutions on geometrically graded meshes. The theoretical results are illustrated by various numerical examples.
KW - Discontinuous Galerkin method
KW - Hp-version
KW - Nonlinear vanishing delay
KW - Pantograph delay differential equations
KW - Spectral and exponential accuracy
UR - http://www.scopus.com/inward/record.url?scp=84884915099&partnerID=8YFLogxK
U2 - 10.1137/120901416
DO - 10.1137/120901416
M3 - Journal article
AN - SCOPUS:84884915099
SN - 1064-8275
VL - 35
SP - A1604-A1620
JO - SIAM Journal on Scientific Computing
JF - SIAM Journal on Scientific Computing
IS - 3
ER -