nach oben

Journal of Scientific Computing

Erschienen in:

01.07.2020

A Posteriori Error Estimates for Fully Discrete Finite Element Method for Generalized Diffusion Equation with Delay

verfasst von: Wansheng Wang, Lijun Yi, Aiguo Xiao

Erschienen in: Journal of Scientific Computing | Ausgabe 1/2020

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

In this paper, we derive several a posteriori error estimators for generalized diffusion equation with delay in a convex polygonal domain. The Crank–Nicolson method for time discretization is used and a continuous, piecewise linear finite element space is employed for the space discretization. The a posteriori error estimators corresponding to space discretization are derived by using the interpolation estimates. Two different continuous, piecewise quadratic reconstructions are used to obtain the error due to the time discretization. To estimate the error in the approximation of the delay term, linear approximations of the delay term are used in a crucial way. As a consequence, a posteriori upper and lower error bounds for fully discrete approximation are derived for the first time. In particular, long-time a posteriori error estimates are obtained for stable systems. Numerical experiments are presented which confirm our theoretical results.

Vorheriger Artikel The Fine Error Estimation of Collocation Methods on Uniform Meshes for Weakly Singular Volterra Integral Equations

Nächster Artikel Two-Level Schwarz Methods for a Discontinuous Galerkin Approximation of Elliptic Problems with Jump Coefficients

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Akrivis, G., Makridakis, Ch., Nochetto, R.H.: A posteriori error estimates for the Crank–Nicolson method for parabolic equations. Math. Comput. 75, 511–531 (2006)MathSciNetMATH

Baker, C.T.H., Bocharov, G.A., Rihan, F.A.: A report on the use of delay differential equations in numerical modelling in the biosciences, MCCM Technical Report, vol. 343, pp. 1360–1725. Manchester, ISSN (1999)

Baker, C.T.H., Paul, C.A.H.: Discontinuous solutions of neutral delay differential equations. Appl. Numer. Math. 56, 284–304 (2006)MathSciNetMATH

Bänsch, E., Karakatsani, F., Makridakis, Ch.: A posteriori error control for fully discrete Crank–Nicolson schemes. SIAM J. Numer. Anal. 50, 2845–2872 (2012)MathSciNetMATH

Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford (2003)MATH

Bellen, A., Maset, S., Zennaro, M., Guglielmi, N.: Recent trends in the numerical solution of retarded functional differential equations. Acta Numer. 18, 1–110 (2009)MathSciNetMATH

Blanco-Cocom, L., Ávila-Vales, E.: Convergence and stability analysis of the \(\theta \)-method for delayed diffusion mathematical models. Appl. Math. Comput. 231, 16–26 (2014)MathSciNetMATH

Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (2002)MATH

Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge (2004)MATH

10.

Chen, Y., Cheng, J., Jiang, J., Liu, K. J.: A time delay dynamical model for outbreak of 2019-nCoV and the parameter identification (2020). ArXiv: 2002.00418

11.

Enright, W.H., Hayashi, H.: A delay differential equation solver based on a continuous Runge–Kutta method with defect control. Numer. Algorithms 16, 349–364 (1998)MathSciNetMATH

12.

Gan, S.Q.: Dissipativity of \(\theta \)-methods for nonlinear delay differential equations of neutral type. Appl. Numer. Math. 59, 1354–1365 (2009)MathSciNetMATH

13.

García, P., Castro, M.A., Martín, J.A., Sirvent, A.: Numerical solutions of diffusion methematical models with delay. Math. Comput. Model. 50, 860–868 (2009)MATH

14.

García, P., Castro, M.A., Martín, J.A., Sirvent, A.: Convergence of two implicit numerical schemes for diffusion mathematical models with delay. Math. Comput. Model. 52, 1279–1287 (2010)MathSciNetMATH

15.

Green, D., Stech, H.W.: Diffusion and hereditary effects in a class of population models. In: Busenberg, S.N., Cooke, K.L. (eds.) Differential Equations and Applications in Ecology, Epidemics, and Population Problems. Academic Press, New York (1981)

16.

Guglielmi, N.: On the asymptotic stability properties of Runge-Kutta methods for delay differential equations. Numer. Math. 77, 467–485 (1997)MathSciNetMATH

17.

Guglielmi, N.: Asymptotic stability barriers for natural Runge–Kutta processes for delay equations. SIAM J. Numer. Anal. 39, 763–783 (2001)MathSciNetMATH

18.

Guglielmi, N.: Open issues in devising software for the numerical solution of implicit delay differential equations. J. Comput. Appl. Math. 185(2), 261–277 (2006)MathSciNetMATH

19.

Guglielmi, N., Hairer, E.: Computing breaking points in implicit delay differential equations. Adv. Comput. Math. 29, 229–247 (2008)MathSciNetMATH

20.

Guglielmi, N., Hairer, E.: Asymptotic expansions for regularized state-dependent neutral delay equations. SIAM J. Math. Anal. 44, 2428–2458 (2012)MathSciNetMATH

21.

Houwen, P.J.V., Sommeijer, B.P., Baker, C.T.H.: On the stability of predictor–corrector methods for parabolic equations with delay. IMA J. Numer. Anal. 6, 1–23 (1986)MathSciNetMATH

22.

Hu, G.D., Mitsui, T.: Stability analysis of numerical methods for systems of neutral delay-differential equations. BIT 35, 504–515 (1995)MathSciNetMATH

23.

Huang, C.M., Li, S.F., Fu, H.Y., Chen, G.N.: Stability and error analysis of one-leg methods for nonlinear delay differential equations. J. Comput. Appl. Math. 103, 263–279 (1999)MathSciNetMATH

24.

In’t Hout, K.J., Spijker, M.N.: Stability analysis of numerical methods for delay differential equations. Numer. Math. 59, 807–814 (1991)MathSciNetMATH

25.

In’t Hout, K.J.: The stability of a class of Runge–Kutta methods for delay differential equations. Appl. Numer. Math. 9, 347–355 (1992)MathSciNetMATH

26.

Jackiewicz, Z.: Asymptotic stability analysis of \(\theta \)-methods for functional differential equations. Numer. Math. 43, 389–396 (1984)MathSciNetMATH

27.

Jackiewicz, Z., Lo, E.: Numerical solution of neutral functional differential equations by Adams methods in divided difference form. J. Comput. Appl. Math. 189, 592–605 (2006)MathSciNetMATH

28.

Kolmanovskii, V.B., Myshkis, A.: Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers, Dordrecht (1999)MATH

29.

Kuang, J.X., Cong, Y.H.: Stability of Numerical Methods for Delay Differential Equations. Science Press, Beijing (2005)

30.

Li, S.F.: High order contractive Runge–Kutta methods for Volterra functional differential equations. SIAM J. Numer. Anal. 47, 4290–4325 (2010)MathSciNetMATH

31.

Li, S.F., Li, Y.F.: B-convergence theory of Runge–Kutta methods for stiff Volterra functional differential equations with infinite integration interval. SIAM J. Numer. Anal. 53, 2570–2583 (2015)MathSciNetMATH

32.

Li, S.F.: Numerical Analysis for Stiff Ordinary and Functional Differential Equations. Xiangtan University Press, Xiangtan (2010)

33.

Liang, H.: Convergence and asymptotic stability of Galerkin methods for linear parabolic equations with delays. Appl. Math. Comput. 264, 160–178 (2015)MathSciNetMATH

34.

Liu, M.Z., Spijker, M.N.: The stability of \(\theta \)-methods in the numerical solution of delay differential equations. IMA J. Numer. Anal. 10, 31–48 (1990)MathSciNetMATH

35.

Lozinski, A., Picasso, M., Prachittham, V.: An anisotropic error estimator for the Crank–Nicolson method: application to a parabolic problem. SIAM J. Sci. Comput. 31, 2757–2783 (2009)MathSciNetMATH

36.

Maset, S., Zennaro, M.: Good behavior with respect to the stiffness in the numerical integration of retarded functional differential equations. SIAM J. Numer. Anal. 52, 1843–1866 (2014)MathSciNetMATH

37.

Murali Mohan Reddy, G., Sinha, R.K.: On the Crank–Nicolson anisotropic a posteriori error analysis for parabolic integro-differential equations. Math. Comput. 85, 2365–2390 (2016)MathSciNetMATH

38.

Picasso, M., Prachittham, V.: An adaptive algorithm for the Crank–Nicolson scheme applied to a time-dependent convection–diffusion problem. J. Comput. Appl. Math. 233, 1139–1154 (2009)MathSciNetMATH

39.

Reyes, E., Rodríguez, F., Martín, J.A.: Analytic-numerical solutions of diffusion mathematical models with delays. Comput. Math. Appl. 56, 743–753 (2008)MathSciNetMATH

40.

Scott, L.R., Zhang, S.: Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)MATH

41.

Tian, H.J.: Asymptotic stability of numerical methods for linear delay parabolic differential equations. Comput. Math. Appl. 56, 1758–1765 (2008)MathSciNetMATH

42.

Verfürth, R.: A posteriori error estimates for finite element discretizations of the heat equation. Calcolo 40, 195–212 (2003)MathSciNetMATH

43.

Wang, W.S., Li, S.F.: Stability analysis of \(\theta \)-methods for nonlinear neutral functional differential equations. SIAM J. Sci. Comput. 30, 2181–2205 (2008)MathSciNet

44.

Wang, W.S., Li, S.F., Su, K.: Nonlinear stability of general linear methods for neutral delay differential equations. J. Comput. Appl. Math. 224, 592–601 (2009)MathSciNetMATH

45.

Wang, W.S., Zhang, Y., Li, S.F.: Stability of continuous Runge–Kutta-type methods for nonlinear neutral delay-differential equations. Appl. Math. Model. 33, 3319–3329 (2009)MathSciNetMATH

46.

Wang, W.S., Zhang, C.J.: Preserving stability implicit Euler method for nonlinear Volterra and neutral functional differential equations in Banach space. Numer. Math. 115, 451–474 (2010)MathSciNetMATH

47.

Wang, W.S., Rao, T., Shen, W.W., Zhong, P.: A posteriori error analysis for the Crank–Nicolson–Galerkin method for the reaction–diffusion equations with delay. SIAM J. Sci. Comput. 40, A1095–A1120 (2018)MathSciNetMATH

48.

Willé, D.R., Baker, C.T.H.: The tracking of derivative discontinuities in systems of delay differential equations. Appl. Numer. Math. 9, 209–222 (1992)MathSciNetMATH

49.

Wu, J.H.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996)MATH

50.

Yan, Y., Chen, Y., Liu, K.J., Luo, X.Y., Xu, B.X., Jiang, Y., Cheng, J.: Modeling and prediction for the trend of outbreak of NCP based on a time-delay dynamic system. Sci. Sin. Math. 50, 1–8 (2020). https://doi.org/10.1360/SSM-2020-0026. (in Chinese) CrossRef

51.

Zhang, C.J., Zhou, S.Z.: Non-linear stability and D-convergence of Runge–Kutta methods for delay differential equations. J. Comput. Appl. Math. 85, 225–237 (1997)MathSciNetMATH

52.

Zhang, C.J., Li, S.F.: Dissipativity and exponentially asymptotic stability of the solutions for nonlinear neutral functional-differential equations. Appl. Math. Comput. 119, 109–115 (2001)MathSciNetMATH

Titel: A Posteriori Error Estimates for Fully Discrete Finite Element Method for Generalized Diffusion Equation with Delay
verfasst von: Wansheng Wang
Lijun Yi
Aiguo Xiao
Publikationsdatum: 01.07.2020
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 1/2020
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-020-01262-5

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 1/2020

Stochastic Primal Dual Fixed Point Method for Composite Optimization

Asymptotic Analysis for Overlap in Waveform Relaxation Methods for RC Type Circuits

EXtended HDG Methods for Second Order Elliptic Interface Problems

A Combined Offline–Online Algorithm for Hodgkin–Huxley Neural Networks

Numerical Approximation for Fractional Diffusion Equation Forced by a Tempered Fractional Gaussian Noise

Error Estimates for Backward Fractional Feynman–Kac Equation with Non-Smooth Initial Data

Premium Partner