Superconvergence of discontinuous Galerkin method for neutral delay differential equations

References

• S. Adjerid, K. Devine, J. Flaherty, and L. Krivodonova, A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems, Comput. Methods Appl. Mech. Eng. 191 (2002), pp. 1097–1112.
   Web of Science ®Google Scholar
• M. Baccouch, Analysis of a posteriorierror estimates of the discontinuous Galerkin method for nonlinear ordinary differential equations, Appl. Numer. Math. 106 (2016), pp. 129–153.
   Web of Science ®Google Scholar
• C. Baker and C. Paul, Discontinuous solutions of neutral delay differential equations, Appl. Numer. Math. 56 (2006), pp. 284–304.
   Web of Science ®Google Scholar
• M.V. Barbarossa, K.P. Hadeler, and C. Kuttler, State-dependent neutral delay equations from population dynamics, J. Math. Biol. 69 (2014), pp. 1027–1056.
   PubMed Web of Science ®Google Scholar
• A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 2003.
   Google Scholar
• H. Brunner, Q. Huang, and H. Xie, Discontinuous Galerkin methods for delay differential equations of pantograph type, SIAM J. Numer. Anal. 48(5) (2010), pp. 1944–1967.
   Web of Science ®Google Scholar
• C. Chen, Structure Theory of Superconvergence of Finite Elements, Hunan Press of Science and Technology, Changsha, 2001.
   Google Scholar
• B. Cockburn, Discontinuous Galerkin methods, Z. Angew. Math. Mech. 83(11) (2003), pp. 731–754.
   Web of Science ®Google Scholar
• B. Cockburn and C.W. Shu, The local discontinuous Galerkin method for time-dependent convection diffusion systems, SIAM J. Numer. Anal. 35 (1998), pp. 2440–2463.
   Web of Science ®Google Scholar
• M. Delfour, W. Hager, and F. Trochu, Discontinuous Galerkin methods for ordinary diferential equations, Math. Comput. 36 (1981), pp. 455–473.
   Web of Science ®Google Scholar
• N. Guglielmi and E. Hairer, Computing breaking points in implicit delay differential equations, Adv. Comput. Math. 29(3) (2008), pp. 229–247.
   Web of Science ®Google Scholar
• K.P. Hadeler, Neutral delay equations from and for population dynamics, Electron. J. Qual. Theo 11 (2008), pp. 1–18.
   Google Scholar
• J.K. Hale and S.M.V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
   Google Scholar
• P. Hövel, Control of Complex Nonlinear Systems with Delay, Springer-Verlag, Berlin, 2010.
   Google Scholar
• G. Hu and G. Hu, Stabilization of an uncertain large-scale time-dependent bilinear neutral differential system by memory feedback control, IMA J. Math. Control I 18 (2001), pp. 1–18.
   Google Scholar
• Q. Huang, X. Xu, and H. Brunner, Continuous Galerkin methods on quasi-geometric meshes for delay differential equations of pantograph type, Discrete Contin. Dyn. Syst. A 36 (2016), pp. 5423–5443.
   Web of Science ®Google Scholar
• D. Li and C. Zhang, Superconvergence of a discontinuous Galerkin method for first-order linear delay differential equations, J. Comput. Math. 29(5) (2011), pp. 574–588.
   Web of Science ®Google Scholar
• D. Li and C. Zhang, L∞ error estimates of discontinuous Galerkin methods for delay differential equations, Appl. Numer. Math. 82 (2014), pp. 1–10.
   Web of Science ®Google Scholar
• T. Meng and L. Yi, An h-p version of the continuous Petrov-Galerkin method for nonlinear delay difierential equations, J. Sci. Comput. 74 (2018), pp. 1091–1114.
   Web of Science ®Google Scholar
• K. Mustapha and J.K. Ryan, Post-processing discontinuous Galerkin solutions to Volterra integro-differential equations: Analysis and simulations, J. Comput. Appl. Math. 253 (2013), pp. 89–103.
   Web of Science ®Google Scholar
• H. Qin, Q. Zhang, and S. Wan, The continuous Galerkin finite element methods for linear neutral delay differential equations, Appl. Math. Comput. 346 (2019), pp. 76–85.
   Web of Science ®Google Scholar
• A. Romkes, S. Prudhomme, and J.T. Oden, A priori error analyses of a stabilized discontinuous Galerkin method, Comput. Math. Appl. 46 (2003), pp. 1289–1311.
   Web of Science ®Google Scholar
• 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.
   Web of Science ®Google Scholar
• G. Szegö, Orthogonal Polynomials, 4th ed., American Mathematical Society, Providence, RI, 1975.
   Google Scholar
• C.P. Vyasarayani, Galerkin approximations for neutral delay differential equations, J. Comput. Nonlinear Dyn. 8(2) (2013), 021014, 5pp.
   PubMed Web of Science ®Google Scholar
• W. Wang, Nonlinear stability of one-leg methods for neutral Volterra delay-integro-differential equations, Math. Comput. Simul. 97(1) (2014), pp. 147–161.
   Google Scholar
• W. Wang, On A-stable one-leg methods for solving nonlinear Volterra functional differential equations, Appl. Math. Comput. 314 (2017), pp. 380–390.
   Web of Science ®Google Scholar
• W. Wang and C. Zhang, Preserving stability implicit Euler method for nonlinear Volterra and neutral functional differential equations in Banach space, Numer. Math. 115(3) (2010), pp. 451–474.
   Web of Science ®Google Scholar
• Y. Wei, T. Sung, and L. Yi, An hp-version of the discontinuous Galerkin time-stepping method for nonlinear second-order delay difierential equations with vanishing delays, J. Comput. Appl. Math. 364 (2020), 112348, 21pp.
   Web of Science ®Google Scholar
• J. Wu, Theory and Application of Partial Functional Differential Equation, Springer-Verlag, New York, 1996.
   Google Scholar
• A. Xiao, G. Zhang, and J. Zhou, Implicit-explicit time discretization coupled with finite element methods for delayed predator-prey competition reaction-diffusion system, Comput. Math. Appl. 71(10) (2016), pp. 2106–2123.
   Web of Science ®Google Scholar
• Z. Xiong, Superconvergence of the continuous Galerkin finite element method for delay-differential equation with several terms, J. Comput. Appl. Math. 198 (2007), pp. 160–166.
   Web of Science ®Google Scholar
• X. Xu and Q. Huang, Superconvergence of discontinuous Galerkin methods for nonlinear delay differential equations with vanishing delay, J. Comput. Appl. Math. 348 (2019), pp. 314–327.
   Web of Science ®Google Scholar
• Y. Yu, L. Wen, and S. Li, Nonlinear stability of Runge-Kutta methods for neutral delay integro-differential equations, Appl. Math. Comput. 191 (2007), pp. 543–549.
   Web of Science ®Google Scholar
• G. Zhang and A. Xiao, Stability and convergence analysis of implicit-explicit one-leg methods for stiff delay differential equations, Int. J. Comput. Math. 93(11) (2016), pp. 1964–1983.
   Web of Science ®Google Scholar
• G. Zhang, G. He, and X. Dai, Stability analysis of discontinuous Galerkin method for stiff Volterra functional differential equations, J. Math. Anal. Appl. 475 (2019), pp. 1293–1303.
   Web of Science ®Google Scholar