Discontinuous galerkin methods for delay differential equations of pantograph type

SIAM Journal on Numerical Analysis

Volumn 48, Issue 5, 2010, Pages 1944-1967

  Brunner, Hermann ^a,b   Huang, Qiumei ^c   Xie, Hehu ^d  

^a MEMORIAL UNIVERSITY OF NEWFOUNDLAND   (Canada)

^b HONG KONG BAPTIST UNIVERSITY   (Hong Kong)

^c BEIJING UNIVERSITY OF TECHNOLOGY   (China)

^d CHINESE ACADEMY OF SCIENCES   (China)

Author keywords

Discontinuous Galerkin method; Optimal order of superconvergence; Pantograph delay differential equations; Vanishing proportional delay

Indexed keywords

COLLOCATION METHOD; DELAY DIFFERENTIAL EQUATIONS; DISCONTINUOUS GALERKIN METHODS; NUMERICAL EXAMPLE; SUPER-CONVERGENCE; THEORETICAL RESULT; UNIFORM MESH; VANISHING PROPORTIONAL DELAY;

DIFFERENTIAL EQUATIONS; OPTIMIZATION; PANTOGRAPHS;

GALERKIN METHODS;

EID: 78649980140     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/090771922     Document Type: Article

References (24)

1. A. Bellen, Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay, IMA J. Numer. Anal., 22 (2002), pp. 529-536.
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, 46 (2006), pp. 229-247.
3. A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 2003.
4. H. Brunner, The numerical analysis of functional integral and integro-differential equations of Volterra type, Acta Numer., 13 (2004), pp. 55-145.
5. H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, 2004.
6. H. Brunner, Current work and open problems in the numerical analysis of Volterra functional equations with vanishing delays, Front. Math. China, 4 (2009), pp. 3-22.
7. H. Brunner and Q. Hu, Optimal superconvergence results for delay integro-differential equations of pantograph type, SIAM J. Numer. Anal., 45 (2007), pp. 986-1004.
8. H. Brunner, Q. Y. Hu, and Q. Lin, Geometric meshes in collocation methods for Volterra integral equations with proportional delay, IMA J. Numer. Anal., 21 (2001), pp. 783-798.
9. H. Brunner and D. Schötzau, hp-Discontinuous Galerkin time-stepping for Volterra integrodifferential equations, SIAM J. Numer. Anal., 44 (2006), pp. 224-245.
10. M. Buhmann and A. Iserles, Stability of the discretized pantograph differential equation, Math. Comp., 60 (1993), pp. 575-589.
11. B. Cockburn, Discontinuous Galerkin methods, Z. Angew. Math. Mech., 83 (2003), pp. 731-754.
12. B. Cockburn, G. Karniadakis, and C. Shu, Discontinuous Galerkin Methods: Theory, Computation, and Applications, Springer-Verlag, Berlin, 2000.
13. M. Delfour, W. Hager, and F. Trochu, Discontinuous Galerkin methods for ordinary differential equations, Math. Comp., 36 (1981), pp. 455-473.
14. L. Fox, D. F. Mayers, J. R. Ockendon, and A. B. Tayler, On a functional differential equation, J. Inst. Math. Appl., 8 (1971), pp. 271-307.
15. A. Iserles, On the generalized pantograph functional differential equation, European J. Appl. Math., 4 (1993), pp. 1-38.
16. A. Iserles, Numerical analysis of delay differential equations with variable delays, Ann. Numer. Math., 1 (1994), pp. 133-152.
17. A. Iserles, On nonlinear delay-differential equations, Trans. Amer. Math. Soc., 344 (1994), pp. 441-477.
18. T. Kato and J. B. McLeod, The functional-differential equation y′(x) = ay(λx)+by(x), Bull. Amer. Math. Soc., 77 (1971), pp. 891-937.
19. S. Larsson, V. Thomée and L. Wahlbin, Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method, Math. Comp., 67 (1998), pp. 45-71. (Pubitemid 128381508)
20. P. Lesaint and P. A. Raviart, On a finite element method for solving the neutron tranport equation, in Mathematical Aspects of Finite Elements in Partial Differential Equations, C. de Boor, ed., Academic Press, New York, 1974, pp. 89-145.
21. D. Schötzau and C. Schwab, An hp a priori error analysis of the DG time-stepping method for initial value problems, Calcolo, 37 (2000), pp. 207-232.
22. D. Schötzau and C. Schwab, Time discretization of parabolic problems by the hp-version of the discontinuous Galerkin finite element method, SIAM J. Numer. Anal., 38 (2000), pp. 837-875.
23. N. Takama, Y. Muroya, and E. Ishiwata, On the attainable order of collocation methods for delay differential equations with proportional delay, BIT, 40 (2000), pp. 374-394.
24. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin, 1997.