nach oben

Journal of Scientific Computing

Erschienen in:

18.04.2016

A Posteriori Error Analysis of Two-Step Backward Differentiation Formula Finite Element Approximation for Parabolic Interface Problems

verfasst von: Jhuma Sen Gupta, Rajen K. Sinha, G. M. M. Reddy, Jinank Jain

Erschienen in: Journal of Scientific Computing | Ausgabe 1/2016

Einloggen

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

search-config

KI-gestützte Suche

Aus

Abstract

This paper studies a residual-based a posteriori error estimates for linear parabolic interface problems in a bounded convex polygonal domain in \(\mathbb {R}^2\). We use the standard linear finite element spaces in space which are allowed to change in time and the two-step backward differentiation formula (BDF-2) approximation at equidistant time step is used for the time discretizations. The essential ingredients in the error analysis are the continuous piecewise quadratic space–time BDF-2 reconstruction and Scott–Zhang interpolation estimates. Optimal order in time and an almost optimal order in space error estimates are derived in the \(L^{\infty }(L^{2})\)-norm using only energy method. The interfaces are assumed to be of arbitrary shape but are smooth for our purpose. Numerical experiments are performed to validate the asymptotic behaviour of the derived error estimators.

Vorheriger Artikel On Solving the Singular System Arisen from Poisson Equation with Neumann Boundary Condition

Nächster Artikel Error Analysis of a B-Spline Based Finite-Element Method for Modeling Wind-Driven Ocean Circulation

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

Adams, R.A., Fournier, J.J.F.: Sobolev Spaces vol. 140 of Pure and Applied Mathematics, 2nd edn. Elsevier, Amsterdam (2003)

Akrivis, G., Chatzipantelidis, P.: A posteriori error estimates for the two-step backward differentiation formula method for parabolic equations. SIAM J. Numer. Anal. 48, 109–132 (2010)MathSciNetCrossRefMATH

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

Becker, J.: A second order backward difference method with variable steps for a parabolic problem. BIT 38, 644–662 (1998)MathSciNetCrossRefMATH

Bernardi, C., Verfürth, R.: Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85, 579–608 (2000)MathSciNetCrossRefMATH

Berrone, S.: Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients. M2AN Math. Model. Numer. Anal. 40, 991–1021 (2006)MathSciNetCrossRefMATH

Bramble, J.H., Pasciak, J.E., Sammon, P.H., Thomée, V.: Incomplete iterations in multistep backward difference methods for parabolic problems with smooth and nonsmooth data. Math. Comput. 52, 339–367 (1989)MathSciNetCrossRefMATH

Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, vol. 15 of Texts in Applied Mathematics, 3rd edn. Springer, New York (2008)CrossRef

10.

Cai, Z., Ye, X., Zhang, S.: Discontinuous Galerkin finite element methods for interface problems: a priori and a posteriori error estimations. SIAM J. Numer. Anal. 49, 1761–1787 (2011)MathSciNetCrossRefMATH

11.

Cai, Z., Zhang, S.: Recovery-based error estimator for interface problems: conforming linear elements. SIAM J. Numer. Anal. 47, 2132–2156 (2009)MathSciNetCrossRefMATH

12.

Chen, Z., Zou, J.: Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79, 175–202 (1998)MathSciNetCrossRefMATH

13.

Ladyženskaja, O.A., Solonnikov, V.A., Ural\(^{\prime }\)ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, vol. 23. American Mathematical Society (1968)

14.

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

15.

Le Roux, M.-N.: Semidiscretization in time for parabolic problems. Math. Comput. 33, 919–931 (1979)MathSciNetCrossRefMATH

16.

Lumer, G., Weis, L.: Evolution Equations and Their Applications in Physical and Life sciences, vol. 215 of Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, Inc., New York (2001)

17.

Luskin, M., Rannacher, R.: On the smoothing property of the Crank–Nicolson scheme. Appl. Anal. 14, 117–135 (1983)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.

McLean, W., Thomée, V.: Numerical solution of an evolution equation with a positive-type memory term. J. Aust. Math. Soc. Ser. B 35, 23–70 (1993)MathSciNetCrossRefMATH

20.

Reusken, A., Nguyen, T.H.: Nitsche’s method for a transport problem in two-phase incompressible flows. J. Fourier Anal. Appl. 15, 663–683 (2009)MathSciNetCrossRefMATH

21.

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

22.

Sen Gupta, J.: A Posteriori Error Analysis of Finite Elemet Method For Parabolic InterfaceProblems. PhD Thesis, Indian Institute of Technology, Guwahati, India (2015)

23.

Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems, vol. 25 of Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (2006)

24.

Tu, C., Peskin, C.S.: Stability and instability in the computation of flows with moving immersed boundaries: a comparison of three methods. SIAM J. Sci. Stat. Comput. 13, 1361–1376 (1992)MathSciNetCrossRefMATH

25.

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

Titel: A Posteriori Error Analysis of Two-Step Backward Differentiation Formula Finite Element Approximation for Parabolic Interface Problems
verfasst von: Jhuma Sen Gupta
Rajen K. Sinha
G. M. M. Reddy
Jinank Jain
Publikationsdatum: 18.04.2016
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 1/2016
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-016-0203-z

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 1/2016

Stabilized Finite Element Methods for the Oberbeck–Boussinesq Model

Source Term Discretization Effects on the Steady-State Accuracy of Finite Volume Schemes

Optimal Error Estimates of Linearized Crank–Nicolson Galerkin Method for Landau–Lifshitz Equation

Application of a Multi-dimensional Limiting Process to Central-Upwind Schemes for Solving Hyperbolic Systems of Conservation Laws

$$L^p$$ L p Error Estimates of Two-Grid Method for Miscible Displacement Problem

Numerical Discretization of Coupling Conditions by High-Order Schemes