October  2016, 36(10): 5423-5443. doi: 10.3934/dcds.2016039

Continuous Galerkin methods on quasi-geometric meshes for delay differential equations of pantograph type

1. 

Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing 100124, China, China

2. 

Department of Mathematics, Hong Kong Baptist University, Hong Kong

Received  September 2015 Revised  December 2015 Published  July 2016

We analyze the optimal global and local convergence properties of continuous Galerkin (CG) solutions on quasi-geometric meshes for delay differential equations with proportional delay. It is shown that with this type of meshes the attainable order of nodal superconvergence of CG solutions is higher than of the one for uniform meshes. The theoretical results are illustrated by a broad range of numerical examples.
Citation: Qiumei Huang, Xiuxiu Xu, Hermann Brunner. Continuous Galerkin methods on quasi-geometric meshes for delay differential equations of pantograph type. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5423-5443. doi: 10.3934/dcds.2016039
References:
[1]

A. Bellen, Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay, IMA J. Numer. Anal., 22 (2002), 529-536. doi: 10.1093/imanum/22.4.529.

[2]

A. Bellen, H. Brunner, S. Maset and L. Torelli, Superconvergence in collocation methods on quasi-graded meshes for functional differential equations with vanishing delays, BIT Numer. Math., 46 (2006), 229-247. doi: 10.1007/s10543-006-0055-2.

[3]

A. Bellen, N. Guglielmi and L. Torelli, Asymptotic stability properties of $\theta$-methods for the pantograph equation, Appl. Numer. Math., 24 (1997), 279-293. doi: 10.1016/S0168-9274(97)00026-3.

[4]

A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198506546.001.0001.

[5]

H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234.

[6]

H. Brunner, Q. Hu and Q. Lin, Geometric meshes in collocation methods for Volterra integral equations with proportional delays, IMA J. Numer. Anal., 21 (2001), 783-798. doi: 10.1093/imanum/21.4.783.

[7]

H. Brunner, Q. Huang and H. Xie, Discontinuous Galerkin methods for delay differential equations of pantograph type, SIAM. J. Numer. Anal., 48 (2010), 1944-1967. doi: 10.1137/090771922.

[8]

H. Brunner and H. Liang, Stability of collocation methods for delay differential equations with vanishing delays, BIT Numer. Math., 50 (2010), 693-711. doi: 10.1007/s10543-010-0285-1.

[9]

C. Chen, Structure Theory of Superconvergence of Finite Elements, Hunan Press of Science and Technology, Changsha, 2001 (in Chinese).

[10]

M. Delfour, W. Hager and F. Trochu, Discontinuous Galerkin methods for ordinary differential equations, Math. Comp., 36 (1981), 455-473. doi: 10.1090/S0025-5718-1981-0606506-0.

[11]

K. Deng, Z. Xiong and Y. Huang, The Galerkin continuous finite element method for delay-differential equation with a variable term, Appl. Math. Comput., 186 (2007), 1488-1496. doi: 10.1016/j.amc.2006.07.147.

[12]

Q. Huang, H. Xie and H. Brunner, Superconvergence of discontinuous Galerkin solutions for delay differential equation of pantograph type, SIAM. J. Sci. Comput., 33 (2011), 2664-2684. doi: 10.1137/110824632.

[13]

A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl. Math., 4 (1993), 1-38. doi: 10.1017/S0956792500000966.

[14]

T. Koto, Stability of Runge-Kutta methods for the generalized pantograph equation, Numer. Math., 84 (1999), 233-247. doi: 10.1007/s002110050470.

[15]

D. Li and C. Zhang, Superconvergence of a discontinuous Galerkin Method for first-order linear delay differential equations, J. Comput. Math., 29 (2011), 574-588. doi: 10.4208/jcm.1107-m3433.

[16]

Y. Liu, On the $\theta$ -method for delay differential equations with infinite lag, J. Comput. Appl. Math., 71 (1996), 177-190. doi: 10.1016/0377-0427(95)00222-7.

[17]

N. Takama, Y. Muroya and E. Ishiwata, On the attainable order of collocation methods for delay differential equations with proportional delay, BIT Numer. Math., 40 (2000), 374-394. doi: 10.1023/A:1022351309662.

[18]

X. Xu and Q. Huang, Continuous Galerkin methods for delay differential equations of pantograph type, Math. Pract. Theory, 24 (2014), 280-288 (in Chinese).

[19]

X. Xu, Q. Huang and H. Chen, Local superconvergence of continuous Galerkin solutions for delay differential equations of pantograph type, J. Comput. Math., 34 (2016), 186-199.

show all references

References:
[1]

A. Bellen, Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay, IMA J. Numer. Anal., 22 (2002), 529-536. doi: 10.1093/imanum/22.4.529.

[2]

A. Bellen, H. Brunner, S. Maset and L. Torelli, Superconvergence in collocation methods on quasi-graded meshes for functional differential equations with vanishing delays, BIT Numer. Math., 46 (2006), 229-247. doi: 10.1007/s10543-006-0055-2.

[3]

A. Bellen, N. Guglielmi and L. Torelli, Asymptotic stability properties of $\theta$-methods for the pantograph equation, Appl. Numer. Math., 24 (1997), 279-293. doi: 10.1016/S0168-9274(97)00026-3.

[4]

A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198506546.001.0001.

[5]

H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511543234.

[6]

H. Brunner, Q. Hu and Q. Lin, Geometric meshes in collocation methods for Volterra integral equations with proportional delays, IMA J. Numer. Anal., 21 (2001), 783-798. doi: 10.1093/imanum/21.4.783.

[7]

H. Brunner, Q. Huang and H. Xie, Discontinuous Galerkin methods for delay differential equations of pantograph type, SIAM. J. Numer. Anal., 48 (2010), 1944-1967. doi: 10.1137/090771922.

[8]

H. Brunner and H. Liang, Stability of collocation methods for delay differential equations with vanishing delays, BIT Numer. Math., 50 (2010), 693-711. doi: 10.1007/s10543-010-0285-1.

[9]

C. Chen, Structure Theory of Superconvergence of Finite Elements, Hunan Press of Science and Technology, Changsha, 2001 (in Chinese).

[10]

M. Delfour, W. Hager and F. Trochu, Discontinuous Galerkin methods for ordinary differential equations, Math. Comp., 36 (1981), 455-473. doi: 10.1090/S0025-5718-1981-0606506-0.

[11]

K. Deng, Z. Xiong and Y. Huang, The Galerkin continuous finite element method for delay-differential equation with a variable term, Appl. Math. Comput., 186 (2007), 1488-1496. doi: 10.1016/j.amc.2006.07.147.

[12]

Q. Huang, H. Xie and H. Brunner, Superconvergence of discontinuous Galerkin solutions for delay differential equation of pantograph type, SIAM. J. Sci. Comput., 33 (2011), 2664-2684. doi: 10.1137/110824632.

[13]

A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl. Math., 4 (1993), 1-38. doi: 10.1017/S0956792500000966.

[14]

T. Koto, Stability of Runge-Kutta methods for the generalized pantograph equation, Numer. Math., 84 (1999), 233-247. doi: 10.1007/s002110050470.

[15]

D. Li and C. Zhang, Superconvergence of a discontinuous Galerkin Method for first-order linear delay differential equations, J. Comput. Math., 29 (2011), 574-588. doi: 10.4208/jcm.1107-m3433.

[16]

Y. Liu, On the $\theta$ -method for delay differential equations with infinite lag, J. Comput. Appl. Math., 71 (1996), 177-190. doi: 10.1016/0377-0427(95)00222-7.

[17]

N. Takama, Y. Muroya and E. Ishiwata, On the attainable order of collocation methods for delay differential equations with proportional delay, BIT Numer. Math., 40 (2000), 374-394. doi: 10.1023/A:1022351309662.

[18]

X. Xu and Q. Huang, Continuous Galerkin methods for delay differential equations of pantograph type, Math. Pract. Theory, 24 (2014), 280-288 (in Chinese).

[19]

X. Xu, Q. Huang and H. Chen, Local superconvergence of continuous Galerkin solutions for delay differential equations of pantograph type, J. Comput. Math., 34 (2016), 186-199.

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