nach oben

Journal of Scientific Computing

Erschienen in:

30.10.2018

A Posteriori Error Analysis of the Crank–Nicolson Finite Element Method for Parabolic Integro-Differential Equations

verfasst von: G. Murali Mohan Reddy, Rajen Kumar Sinha, José Alberto Cuminato

Erschienen in: Journal of Scientific Computing | Ausgabe 1/2019

Einloggen

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

search-config

KI-gestützte Suche

Aus

Abstract

We study a posteriori error analysis for the space-time discretizations of linear parabolic integro-differential equation in a bounded convex polygonal or polyhedral domain. The piecewise linear finite element spaces are used for the space discretization, whereas the time discretization is based on the Crank–Nicolson method. The Ritz–Volterra reconstruction operator (IMA J Numer Anal 35:341–371, 2015), a generalization of elliptic reconstruction operator (SIAM J Numer Anal 41:1585–1594, 2003), is used in a crucial way to obtain optimal rate of convergence in space. Moreover, a quadratic (in time) space-time reconstruction operator is introduced to establish second order convergence in time. The proposed method uses nested finite element spaces and the standard energy technique to obtain optimal order error estimator in the \(L^{\infty }(L^2)\)-norm. Numerical experiments are performed to validate the optimality of the error estimators.

Vorheriger Artikel Partitioned Time Stepping Method for a Dual-Porosity-Stokes Model

Nächster Artikel A Nonconforming Immersed Finite Element Method for Elliptic Interface Problems

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

Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York (2000)CrossRefMATH

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

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

Bänsch, E., Karakatsani, F., Makridakis, C.: The effect of mesh modification in time on the error control of fully discrete approximations for parabolic equations. Appl. Numer. Math. 67, 35–63 (2013)MathSciNetCrossRefMATH

Bergam, A., Bernardi, C., Mghazli, Z.: A posteriori analysis of the finite element discretization of some parabolic equations. Math. Comput. 74, 1117–1138 (2004)MathSciNetCrossRefMATH

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

Capasso, V.: Asymptotic stability for an integro-differential reaction-diffusion system. J. Math. Anal. Appl. 103, 575–588 (1984)MathSciNetCrossRefMATH

Dolejsi, V., Ern, A., Vohralík, M.: A framework for robust a posteriori error control in unsteady nonlinear advection-diffusion problems. SIAM J. Numer. Anal. 51, 773–793 (2013)MathSciNetCrossRefMATH

Chen, C., Shih, T.: Finite Element Methods for Integro-Differential Equations. World Scientific, Singapore (1998)CrossRef

10.

Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems I: a linear model problem. SIAM J. Numer. Anal. 28, 43–77 (1991)MathSciNetCrossRefMATH

11.

Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems IV: nonlinear problems. SIAM J. Numer. Anal. 32, 1729–1749 (1995)MathSciNetCrossRefMATH

12.

Ern, A., Vohralík, M.: A posteriori error estimation based on potential and flux re-construction for the heat equation. SIAM J. Numer. Anal. 48, 198–223 (2010)MathSciNetCrossRefMATH

13.

Gurtin, M.E., Pipkin, A.C.: A general theory of heat conduction with finite wave speeds. Arch. Ration. Mech. Anal. 31, 113–126 (1968)MathSciNetCrossRefMATH

14.

Habetler, G.J., Schiffman, R.L.: A finite difference method for analysing the compression of poro-viscoelasticity media. Computing 6, 342–348 (1970)CrossRefMATH

15.

Lakkis, O., Makridakis, C.: Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems. Math. Comput. 75, 1627–1658 (2006)MathSciNetCrossRefMATH

16.

Lin, Y.P., Thomée, V., Wahlbin, L.B.: Ritz–Volterra projections to finite-element spaces and applications to integrodifferential and related equations. SIAM J. Numer. Anal. 28(4), 1047–1070 (1991)MathSciNetCrossRefMATH

17.

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)MathSciNetCrossRefMATH

18.

Makridakis, C., Nochetto, R.H.: Elliptic reconstruction and a posteriori error estimates for parabolic problems. SIAM J. Numer. Anal. 41, 1585–1594 (2003)MathSciNetCrossRefMATH

19.

Nochetto, R.H., Savaré, G., Verdi, C.: A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations. Commun. Pure Appl. Math. 53, 525–589 (2000)MathSciNetCrossRefMATH

20.

Pani, A.K., Peterson, T.E.: Finite element methods with numerical quadrature for parabolic integrodifferential equations. SIAM J. Numer. Anal. 33(3), 1084–1105 (1996)MathSciNetCrossRefMATH

21.

Pao, C.V.: Solution of a nonlinear integro-differential system arising in nuclear reactor dynamics. J. Math. Anal. Appl. 48, 470–492 (1974)MathSciNetCrossRefMATH

22.

Picasso, M.: Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Eng. 167, 223–237 (1998)MathSciNetCrossRefMATH

23.

Reddy, G.M.M., Sinha, R.K.: Ritz–Volterra reconstructions and a posteriori error analysis of finite element method for parabolic integro-differential equations. IMA J. Numer. Anal. 35, 341–371 (2015)MathSciNetCrossRefMATH

24.

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

25.

Reddy, G.M.M., Sinha, R.K.: The backward Euler anisotropic a posteriori error analysis for parabolic integro-differential equations. Numer. Methods Partial Differ. Equ. 2016(32), 1309–1330 (2016)MathSciNetCrossRefMATH

26.

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

27.

Thomée, V., Zhang, N.Y.: Error estimates for semidiscrete finite element methods for parabolic integro-differential equations. Math. Comput. 53(187), 121–139 (1989)MathSciNetCrossRefMATH

28.

Verfürth, R.: A posteriori error estimates for non linear problems: \(L^r(0, t; L^p(\Omega ))\)-error estimates for finite element discretizations of parabolic equations. Math. Comput. 67, 1335–1360 (1998)CrossRefMATH

29.

Verfürth, R.: A posteriori error estimates for non linear problems: \(L^r(0, t;W^ {1, p}(\Omega ))\)-error estimates for finite element discretizations of parabolic equations. Numer. Methods Partial Differ. Equ. 14, 487–518 (1998)MathSciNetCrossRefMATH

30.

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

31.

Yanik, E.G., Fairweather, G.: Finite element methods for parabolic and hyperbolic partial integro-differential equations. Nonlinear Anal. 12, 785–809 (1988)MathSciNetCrossRefMATH

Titel: A Posteriori Error Analysis of the Crank–Nicolson Finite Element Method for Parabolic Integro-Differential Equations
verfasst von: G. Murali Mohan Reddy
Rajen Kumar Sinha
José Alberto Cuminato
Publikationsdatum: 30.10.2018
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 1/2019
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-018-0860-1

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 1/2019

A Gauss–Jacobi Kernel Compression Scheme for Fractional Differential Equations

Dispersive Behavior of an Energy-Conserving Discontinuous Galerkin Method for the One-Way Wave Equation

Error Analysis of a Second-Order Method on Fitted Meshes for a Time-Fractional Diffusion Problem

A Finite Element/Operator-Splitting Method for the Numerical Solution of the Two Dimensional Elliptic Monge–Ampère Equation

Numerical Methods for Cauchy Bisingular Integral Equations of the First Kind on the Square

Fast Rank-One Alternating Minimization Algorithm for Phase Retrieval