# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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.  doi: 10.1093/imanum/22.4.529.  Google Scholar [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.  doi: 10.1007/s10543-006-0055-2.  Google Scholar [3] A. Bellen, N. Guglielmi and L. Torelli, Asymptotic stability properties of $\theta$-methods for the pantograph equation,, Appl. Numer. Math., 24 (1997), 279.  doi: 10.1016/S0168-9274(97)00026-3.  Google Scholar [4] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations,, Oxford University Press, (2003).  doi: 10.1093/acprof:oso/9780198506546.001.0001.  Google Scholar [5] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations,, Cambridge University Press, (2004).  doi: 10.1017/CBO9780511543234.  Google Scholar [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.  doi: 10.1093/imanum/21.4.783.  Google Scholar [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.  doi: 10.1137/090771922.  Google Scholar [8] H. Brunner and H. Liang, Stability of collocation methods for delay differential equations with vanishing delays,, BIT Numer. Math., 50 (2010), 693.  doi: 10.1007/s10543-010-0285-1.  Google Scholar [9] C. Chen, Structure Theory of Superconvergence of Finite Elements,, Hunan Press of Science and Technology, (2001).   Google Scholar [10] M. Delfour, W. Hager and F. Trochu, Discontinuous Galerkin methods for ordinary differential equations,, Math. Comp., 36 (1981), 455.  doi: 10.1090/S0025-5718-1981-0606506-0.  Google Scholar [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.  doi: 10.1016/j.amc.2006.07.147.  Google Scholar [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.  doi: 10.1137/110824632.  Google Scholar [13] A. Iserles, On the generalized pantograph functional-differential equation,, European J. Appl. Math., 4 (1993), 1.  doi: 10.1017/S0956792500000966.  Google Scholar [14] T. Koto, Stability of Runge-Kutta methods for the generalized pantograph equation,, Numer. Math., 84 (1999), 233.  doi: 10.1007/s002110050470.  Google Scholar [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.  doi: 10.4208/jcm.1107-m3433.  Google Scholar [16] Y. Liu, On the $\theta$ -method for delay differential equations with infinite lag,, J. Comput. Appl. Math., 71 (1996), 177.  doi: 10.1016/0377-0427(95)00222-7.  Google Scholar [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.  doi: 10.1023/A:1022351309662.  Google Scholar [18] X. Xu and Q. Huang, Continuous Galerkin methods for delay differential equations of pantograph type,, Math. Pract. Theory, 24 (2014), 280.   Google Scholar [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.   Google Scholar

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##### 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.  doi: 10.1093/imanum/22.4.529.  Google Scholar [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.  doi: 10.1007/s10543-006-0055-2.  Google Scholar [3] A. Bellen, N. Guglielmi and L. Torelli, Asymptotic stability properties of $\theta$-methods for the pantograph equation,, Appl. Numer. Math., 24 (1997), 279.  doi: 10.1016/S0168-9274(97)00026-3.  Google Scholar [4] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations,, Oxford University Press, (2003).  doi: 10.1093/acprof:oso/9780198506546.001.0001.  Google Scholar [5] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations,, Cambridge University Press, (2004).  doi: 10.1017/CBO9780511543234.  Google Scholar [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.  doi: 10.1093/imanum/21.4.783.  Google Scholar [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.  doi: 10.1137/090771922.  Google Scholar [8] H. Brunner and H. Liang, Stability of collocation methods for delay differential equations with vanishing delays,, BIT Numer. Math., 50 (2010), 693.  doi: 10.1007/s10543-010-0285-1.  Google Scholar [9] C. Chen, Structure Theory of Superconvergence of Finite Elements,, Hunan Press of Science and Technology, (2001).   Google Scholar [10] M. Delfour, W. Hager and F. Trochu, Discontinuous Galerkin methods for ordinary differential equations,, Math. Comp., 36 (1981), 455.  doi: 10.1090/S0025-5718-1981-0606506-0.  Google Scholar [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.  doi: 10.1016/j.amc.2006.07.147.  Google Scholar [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.  doi: 10.1137/110824632.  Google Scholar [13] A. Iserles, On the generalized pantograph functional-differential equation,, European J. Appl. Math., 4 (1993), 1.  doi: 10.1017/S0956792500000966.  Google Scholar [14] T. Koto, Stability of Runge-Kutta methods for the generalized pantograph equation,, Numer. Math., 84 (1999), 233.  doi: 10.1007/s002110050470.  Google Scholar [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.  doi: 10.4208/jcm.1107-m3433.  Google Scholar [16] Y. Liu, On the $\theta$ -method for delay differential equations with infinite lag,, J. Comput. Appl. Math., 71 (1996), 177.  doi: 10.1016/0377-0427(95)00222-7.  Google Scholar [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.  doi: 10.1023/A:1022351309662.  Google Scholar [18] X. Xu and Q. Huang, Continuous Galerkin methods for delay differential equations of pantograph type,, Math. Pract. Theory, 24 (2014), 280.   Google Scholar [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.   Google Scholar
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