Delay-dependent elliptic reconstruction and optimal 𝐿^{∞}(𝐿²) a posteriori error estimates for fully discrete delay parabolic problems

Wang, Wansheng; Yi, Lijun

doi:10.1090/mcom/3761

Delay-dependent elliptic reconstruction and optimal $L^\infty (L^2)$ a posteriori error estimates for fully discrete delay parabolic problems
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Math. Comp. 91 (2022), 2609-2643 Request permission

Abstract:

We derive optimal order a posteriori error estimates for fully discrete approximations of linear parabolic delay differential equations (PDDEs), in the $L^\infty (L^2)$-norm. For the discretization in time we use Backward Euler and Crank-Nicolson methods, while for the space discretization we use standard conforming finite element methods. A novel space-time reconstruction operator is introduced, which is a generalization of the elliptic reconstruction operator, and we call it as delay-dependent elliptic reconstruction operator. The related a posteriori error estimates for the delay-dependent elliptic reconstruction play key roles in deriving optimal order a posteriori error estimates in the $L^\infty (L^2)$-norm. Numerical experiments verify and complement our theoretical results.

References

Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308, DOI 10.1002/9781118032824
Georgios Akrivis, Charalambos Makridakis, and Ricardo H. Nochetto, A posteriori error estimates for the Crank-Nicolson method for parabolic equations, Math. Comp. 75 (2006), no. 254, 511–531. MR 2196979, DOI 10.1090/S0025-5718-05-01800-4
C. T. H. Baker, G. A. Bocharov, and F. A. Rihan, A report on the use of delay differential equations in numerical modelling in the biosciences, MCCM Technical Report, vol. 343, Manchester, 1999, ISSN 1360-1725.
Christopher T. H. Baker and Christopher A. H. Paul, Discontinuous solutions of neutral delay differential equations, Appl. Numer. Math. 56 (2006), no. 3-4, 284–304. MR 2207590, DOI 10.1016/j.apnum.2005.04.009
E. Bänsch, F. Karakatsani, and Ch. Makridakis, A posteriori error control for fully discrete Crank-Nicolson schemes, SIAM J. Numer. Anal. 50 (2012), no. 6, 2845–2872. MR 3022245, DOI 10.1137/110839424
E. Bänsch, F. Karakatsani, and Ch. Makridakis, The effect of mesh modification in time on the error control of fully discrete approximations for parabolic equations, Appl. Numer. Math. 67 (2013), 35–63. MR 3031653, DOI 10.1016/j.apnum.2011.08.008
Sören Bartels and Rüdiger Müller, Quasi-optimal and robust a posteriori error estimates in $L^\infty (L^2)$ for the approximation of Allen-Cahn equations past singularities, Math. Comp. 80 (2011), no. 274, 761–780. MR 2772095, DOI 10.1090/S0025-5718-2010-02444-5
Alfredo Bellen and Marino Zennaro, Numerical methods for delay differential equations, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 2003. MR 1997488, DOI 10.1093/acprof:oso/9780198506546.001.0001
Luis Blanco-Cocom and Eric Ávila-Vales, Convergence and stability analysis of the $\theta$-method for delayed diffusion mathematical models, Appl. Math. Comput. 231 (2014), 16–25. MR 3174007, DOI 10.1016/j.amc.2013.12.188
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
Hermann Brunner, Collocation methods for Volterra integral and related functional differential equations, Cambridge Monographs on Applied and Computational Mathematics, vol. 15, Cambridge University Press, Cambridge, 2004. MR 2128285, DOI 10.1017/CBO9780511543234
Yu Chen, Jin Cheng, Yu Jiang, and Keji Liu, A time delay dynamical model for outbreak of 2019-nCoV and the parameter identification, J. Inverse Ill-Posed Probl. 28 (2020), no. 2, 243–250. MR 4080262, DOI 10.1515/jiip-2020-0010
Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739
J. R. Branco, J. A. Ferreira, and P. da Silva, Non-Fickian delay reaction-diffusion equations: theoretical and numerical study, Appl. Numer. Math. 60 (2010), no. 5, 531–549. MR 2610802, DOI 10.1016/j.apnum.2010.01.003
W. H. Enright and H. Hayashi, A delay differential equation solver based on a continuous Runge-Kutta method with defect control, Numer. Algorithms 16 (1997), no. 3-4, 349–364 (1998). MR 1617169, DOI 10.1023/A:1019107718128
P. García, M. A. Castro, J. A. Martín, and A. Sirvent, Numerical solutions of diffusion mathematical models with delay, Math. Comput. Modelling 50 (2009), no. 5-6, 860–868. MR 2569247, DOI 10.1016/j.mcm.2009.05.015
P. García, M. A. Castro, J. A. Martín, and A. Sirvent, Convergence of two implicit numerical schemes for diffusion mathematical models with delay, Math. Comput. Modelling 52 (2010), no. 7-8, 1279–1287. MR 2718446, DOI 10.1016/j.mcm.2010.02.016
David Green Jr. and Harlan W. Stech, Diffusion and hereditary effects in a class of population models, Differential equations and applications in ecology, epidemics, and population problems (Claremont, Calif., 1981) Academic Press, New York-London, 1981, pp. 19–28. MR 645186
Nicola Guglielmi, Open issues in devising software for the numerical solution of implicit delay differential equations, J. Comput. Appl. Math. 185 (2006), no. 2, 261–277. MR 2169065, DOI 10.1016/j.cam.2005.03.010
Nicola Guglielmi and Ernst Hairer, Computing breaking points in implicit delay differential equations, Adv. Comput. Math. 29 (2008), no. 3, 229–247. MR 2438343, DOI 10.1007/s10444-007-9044-5
Nicola Guglielmi and Ernst Hairer, Asymptotic expansions for regularized state-dependent neutral delay equations, SIAM J. Math. Anal. 44 (2012), no. 4, 2428–2458. MR 3023382, DOI 10.1137/100801238
Eskil Hansen and Tony Stillfjord, Implicit Euler and Lie splitting discretizations of nonlinear parabolic equations with delay, BIT 54 (2014), no. 3, 673–689. MR 3259820, DOI 10.1007/s10543-014-0480-6
Desmond J. Higham and Tasneem Sardar, Existence and stability of fixed points for a discretised nonlinear reaction-diffusion equation with delay, Appl. Numer. Math. 18 (1995), no. 1-3, 155–173. Seventh Conference on the Numerical Treatment of Differential Equations (Halle, 1994). MR 1357914, DOI 10.1016/0168-9274(95)00051-U
P. J. van der Houwen, B. P. Sommeijer, and Christopher T. H. Baker, On the stability of predictor-corrector methods for parabolic equations with delay, IMA J. Numer. Anal. 6 (1986), no. 1, 1–23. MR 967678, DOI 10.1093/imanum/6.1.1
Z. Jackiewicz and B. Zubik-Kowal, Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations, Appl. Numer. Math. 56 (2006), no. 3-4, 433–443. MR 2207601, DOI 10.1016/j.apnum.2005.04.021
V. Kolmanovskii and A. Myshkis, Introduction to the theory and applications of functional-differential equations, Mathematics and its Applications, vol. 463, Kluwer Academic Publishers, Dordrecht, 1999. MR 1680144, DOI 10.1007/978-94-017-1965-0
J. X. Kuang and Y. H. Cong, Stability of Numerical Methods for Delay Differential Equations, Science Press, Beijing, 2005.
Omar Lakkis and Charalambos Makridakis, Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems, Math. Comp. 75 (2006), no. 256, 1627–1658. MR 2240628, DOI 10.1090/S0025-5718-06-01858-8
Omar Lakkis, Charalambos Makridakis, and Tristan Pryer, A comparison of duality and energy a posteriori estimates for $\mathrm {L}_\infty (0,T;\mathrm {L}_2(\Omega ))$ in parabolic problems, Math. Comp. 84 (2015), no. 294, 1537–1569. MR 3335883, DOI 10.1090/S0025-5718-2014-02912-8
Dongfang Li, Chengjian Zhang, and Wansheng Wang, Long time behavior of non-Fickian delay reaction-diffusion equations, Nonlinear Anal. Real World Appl. 13 (2012), no. 3, 1401–1415. MR 2863967, DOI 10.1016/j.nonrwa.2011.11.005
Shoufu Li, High order contractive Runge-Kutta methods for Volterra functional differential equations, SIAM J. Numer. Anal. 47 (2010), no. 6, 4290–4325. MR 2585188, DOI 10.1137/080741148
Shoufu Li and Yunfei Li, $B$-convergence theory of Runge-Kutta methods for stiff Volterra functional differential equations with infinite integration interval, SIAM J. Numer. Anal. 53 (2015), no. 6, 2570–2583. MR 3422441, DOI 10.1137/130944837
S. F. Li, Numerical Analysis for Stiff Ordinary and Functional Differential Equations, Xiangtan University Press, Xiangtan, 2010.
Hui Liang, Convergence and asymptotic stability of Galerkin methods for linear parabolic equations with delays, Appl. Math. Comput. 264 (2015), 160–178. MR 3351600, DOI 10.1016/j.amc.2015.04.104
Alexei Lozinski, Marco Picasso, and Virabouth Prachittham, An anisotropic error estimator for the Crank-Nicolson method: application to a parabolic problem, SIAM J. Sci. Comput. 31 (2009), no. 4, 2757–2783. MR 2520298, DOI 10.1137/080715135
S. Maset and M. Zennaro, Good behavior with respect to the stiffness in the numerical integration of retarded functional differential equations, SIAM J. Numer. Anal. 52 (2014), no. 4, 1843–1866. MR 3240853, DOI 10.1137/130908543
Charalambos Makridakis and Ricardo H. Nochetto, Elliptic reconstruction and a posteriori error estimates for parabolic problems, SIAM J. Numer. Anal. 41 (2003), no. 4, 1585–1594. MR 2034895, DOI 10.1137/S0036142902406314
Elia Reyes, Francisco Rodríguez, and José Antonio Martín, Analytic-numerical solutions of diffusion mathematical models with delays, Comput. Math. Appl. 56 (2008), no. 3, 743–753. MR 2435581, DOI 10.1016/j.camwa.2008.02.011
L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
Oliver J. Sutton, Long-time $L^\infty (L^2)$ a posteriori error estimates for fully discrete parabolic problems, IMA J. Numer. Anal. 40 (2020), no. 1, 498–529. MR 4050548, DOI 10.1093/imanum/dry078
Rüdiger Verfürth, A posteriori error estimation techniques for finite element methods, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2013. MR 3059294, DOI 10.1093/acprof:oso/9780199679423.001.0001
Wansheng Wang and Shoufu Li, Stability analysis of $\Theta$-methods for nonlinear neutral functional differential equations, SIAM J. Sci. Comput. 30 (2008), no. 4, 2181–2205. MR 2407157, DOI 10.1137/060654116
Wan-Sheng Wang, Yuan Zhang, and Shou-Fu Li, Nonlinear stability of one-leg methods for delay differential equations of neutral type, Appl. Numer. Math. 58 (2008), no. 2, 122–130. MR 2382677, DOI 10.1016/j.apnum.2006.11.002
Wan-Sheng Wang, Shou-Fu Li, and Kai Su, Nonlinear stability of general linear methods for neutral delay differential equations, J. Comput. Appl. Math. 224 (2009), no. 2, 592–601. MR 2492892, DOI 10.1016/j.cam.2008.05.050
Wansheng Wang, Yuan Zhang, and Shoufu Li, Stability of continuous Runge-Kutta-type methods for nonlinear neutral delay-differential equations, Appl. Math. Model. 33 (2009), no. 8, 3319–3329. MR 2524122, DOI 10.1016/j.apm.2008.10.038
Wansheng Wang and Chengjian Zhang, Preserving stability implicit Euler method for nonlinear Volterra and neutral functional differential equations in Banach space, Numer. Math. 115 (2010), no. 3, 451–474. MR 2640054, DOI 10.1007/s00211-009-0281-z
Wansheng Wang, Ting Rao, Weiwei Shen, and Peng Zhong, A posteriori error analysis for Crank-Nicolson-Galerkin type methods for reaction-diffusion equations with delay, SIAM J. Sci. Comput. 40 (2018), no. 2, A1095–A1120. MR 3788199, DOI 10.1137/17M1143514
Wansheng Wang, Lijun Yi, and Aiguo Xiao, A posteriori error estimates for fully discrete finite element method for generalized diffusion equation with delay, J. Sci. Comput. 84 (2020), no. 1, Paper No. 13, 27. MR 4121184, DOI 10.1007/s10915-020-01262-5
David R. Willé and Christopher T. H. Baker, The tracking of derivative discontinuities in systems of delay-differential equations, Appl. Numer. Math. 9 (1992), no. 3-5, 209–222. International Conference on the Numerical Solution of Volterra and Delay Equations (Tempe, AZ, 1990). MR 1158484, DOI 10.1016/0168-9274(92)90016-7
Jianhong Wu, Theory and applications of partial functional-differential equations, Applied Mathematical Sciences, vol. 119, Springer-Verlag, New York, 1996. MR 1415838, DOI 10.1007/978-1-4612-4050-1
Y. Yan, Y. Chen, K. J. Liu, X. Y. Luo, B. X. Xu, Y. Jiang, and J. Cheng, Modeling and prediction for the trend of outbreak of NCP based on a time-delay dynamic system (Chinese), Sci. Sin. Math. 50 (2020), 385–392.
Qifeng Zhang, Chengjian Zhang, and Li Wang, The compact and Crank-Nicolson ADI schemes for two-dimensional semilinear multidelay parabolic equations, J. Comput. Appl. Math. 306 (2016), 217–230. MR 3505894, DOI 10.1016/j.cam.2016.04.016
Barbara Zubik-Kowal, Stability in the numerical solution of linear parabolic equations with a delay term, BIT 41 (2001), no. 1, 191–206. MR 1829669, DOI 10.1023/A:1021930104326