Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Journal of Scientific Computing
Published in: Journal of Scientific Computing 3/2018

21-08-2017

Space-Time Adaptive Methods for the Mixed Formulation of a Linear Parabolic Problem

Published in: Journal of Scientific Computing | Issue 3/2018

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this paper, we are concerned with space-time a posteriori error estimators for fully discrete solutions of linear parabolic problems. The mixed formulation with Raviart–Thomas finite element spaces is considered. A new second-order method in time is proposed so that mixed finite element spaces are permitted to change at different time levels. The new method can be viewed as a variant Crank–Nicolson (CN) scheme. Introducing a CN reconstruction appropriate for the mixed setting, we construct an a posteriori error estimator of second order in time for the variant CN mixed scheme. Various numerical examples are given to test our space-time adaptive algorithm and validate the theory proved in the paper. In addition, numerical results for backward Euler and CN schemes are presented to compare their performance in the time adaptivity setting over uniform/adaptive spatial meshes.
previous article Approximations on SO(3) by Wigner D-matrix and Applications
next article An Analysis of Stability of the Flux Reconstruction Formulation on Quadrilateral Elements for the Linear Advection–Diffusion Equation

Dont have a licence yet? Then find out more about our products and how to get one now:

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!

inform now

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!

inform now

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!

inform now
Literature
1.
Akrivis, G., Makridakis, C., Nochetto, R.: A posteriori error estimates for the CN method for parabolic equations. Math. Comput. 75(254), 511–531 (2006)CrossRefMATH
2.
3.
4.
Babuška, I., Rheinboldt, W.C.: Aposteriori error estimates for the finite element method. Numer. Math. 12, 1597–1615 (1978)
5.
Bansch, E., Karakatsani, F., Makridakis, Ch.: A posteriori error control for fully discrete CN schemes. SIAM J. Numer. Anal. 50(6), 2845–2872 (2012)MathSciNetCrossRefMATH
6.
Braess, D., Verfürth, R.: A posteriori error estimators for the Raviart–Thomas element. SIAM J. Numer. Anal. 33(6), 2431–2444 (1996)MathSciNetCrossRefMATH
7.
8.
Carstensen, C., Kim, D., Park, E.-J.: A priori and a posteriori pseudostress–velocity mixed finite element error analysis for the Stokes problem. SIAM J. Numer. Anal. 49, 2501–2523 (2011)MathSciNetCrossRefMATH
9.
Cascon, J.M., Ferragut, L., Asensio, M.I.: Space-time adaptive algorithm for the mixed parabolic problem. Numer. Math. 103(3), 367–392 (2006)MathSciNetCrossRefMATH
10.
Clement, P.: Approximation by finite element functions using local regularization. RAIRO R–2, 77–84 (1975)MathSciNetMATH
11.
Douglas, J., Roberts, J.E.: Global estimates for mixed methods for second order elliptic equations. Math. Comput. 44(169), 39–52 (1985)MathSciNetCrossRefMATH
12.
Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems I: a linear model problem. SIAM J. Numer. Anal. 28(1), 43–77 (1991)MathSciNetCrossRefMATH
13.
Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems IV: nonlinear problems. SIAM J. Numer. Anal. 2(6), 1729–1749 (1995)MathSciNetCrossRefMATH
14.
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1997)MATH
15.
Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations, pp. 27–28. Springer, New York (1986)CrossRefMATH
16.
Johnson, C.: Numerical Solution of Partial Differential Equationns by the Finite Element Method. Cambridge University Press, Cambridge (1987)
17.
Johnson, C., Thomée, V.: Error estimates for some mixed finite element methods for parabolic type problems. RAIRO Anal. Numér. 15, 41–78 (1981)MathSciNetCrossRefMATH
18.
Kim, D., Park, E.-J.: A posteriori error estimators for the upstream weighting mixed methods for convection diffusion problems. Comput. Methods Appl. Mech. Eng. 197(6–8), 806–820 (2008)MathSciNetCrossRefMATH
19.
Kim, D., Park, E.-J.: Adaptive CN methods with dynamic finite-element spaces for parabolic problems. Discrete Contin. Dyn. Syst. Ser. B 10(4), 873–886 (2008)MathSciNetCrossRefMATH
20.
Kim, D., Park, E.-J.: A priori and a posteriori analysis of mixed finite element methods for nonlinear elliptic equations. SIAM J. Numer. Anal. 48(3), 1186–1207 (2010)MathSciNetCrossRefMATH
21.
Kim, M.-Y., Milner, F.A., Park, E.-J.: Some observations on mixed methods for fully nonlinear parabolic problems in divergence form. Appl. Math. Lett. 9, 75–81 (1996)MathSciNetCrossRefMATH
22.
Kim, M.-Y., Park, E.-J., Thomas, S.G., Wheeler, M.F.: A multiscale mortar mixed finite element method for slightly compressible flows in porous media. J. Korean Math. Soc. 44(5), 1103–1119 (2007)MathSciNetCrossRefMATH
23.
Larson, M.G., Malqvist, A.: A posteriori error estimates for mixed finite element approximations of parabolic problems. Numer. Math. 118, 33–48 (2011)MathSciNetCrossRef
24.
Makridakis, C., Nochetto, R.H.: Elliptic reconstruction and a posteriori error estimates for parabolic problems. SIAM J. Numer. Anal. 41(4), 1585–1594 (2003)MathSciNetCrossRefMATH
25.
Memon, S., Nataraj, N., Pani, A.K.: An a posteriori error analysis of mixed finite element Galerkin approximations to second order linear parabolic problems. SIAM J. Numer. Anal. 50(3), 1367–1393 (2012)MathSciNetCrossRefMATH
26.
Milner, F.A., Park, E.-J.: A mixed finite element method for a strongly nonlinear second-order elliptic problem. Math. Comput. 64, 973–988 (1995)MathSciNetCrossRefMATH
27.
Nochetto, R.H., Schmidt, A., Verdi, C.: A posteriori error estimation and adaptivity for degenerate parabolic problems. Math. Comput. 69, 1–24 (2000)MathSciNetCrossRefMATH
28.
29.
Park, E.-J.: Mixed finite element methods for generalized Forchheimer flow in porous media. Numer. Methods Partial Differ. Equ. 21(2), 213–228 (2005)MathSciNetCrossRefMATH
30.
Park, E.-J.: Mixed finite element methods for nonlinear second-order elliptic problems. SIAM J. Numer. Anal. 32(3), 865–885 (1995)MathSciNetCrossRefMATH
31.
Raviart, P.A., Thomas, J.: A mixed finite element method for 2-nd order elliptic problems. In: Mathematical Aspects of the Finite Elements Method. Lectures Notes in Math., vol. 606, pp. 292–315. Springer, Berlin (1977)
32.
Wheeler, M.F., Yotov, I.: A posteriori error estimates for the mortar mixed finite element method. SIAM J. Numer. Anal. 43, 1021–1042 (2005)MathSciNetCrossRefMATH
33.
Wohlmuth, B.I., Hoppe, R.H.W.: A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart–Thomas elements. Math. Comput. 68, 1347–1378 (1999)MathSciNetCrossRefMATH
34.
Yang, D.Q.: Mixed methods with dynamic finite-element spaces for miscible displacement in porous media. J. Comput. Appl. Math. 30(3), 313–328 (1990)MathSciNetCrossRefMATH
Metadata
Title
Space-Time Adaptive Methods for the Mixed Formulation of a Linear Parabolic Problem
Publication date
21-08-2017
Published in
Journal of Scientific Computing / Issue 3/2018
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-017-0514-8

Other articles of this Issue 3/2018

Journal of Scientific Computing 3/2018 Go to the issue

OriginalPaper

A Third Order Hierarchical Basis WENO Interpolation for Sparse Grids with Application to Conservation Laws with Uncertain Data

OriginalPaper

An Analysis of Stability of the Flux Reconstruction Formulation on Quadrilateral Elements for the Linear Advection–Diffusion Equation

OriginalPaper

Field-Aligned Interpolation for Semi-Lagrangian Gyrokinetic Simulations

OriginalPaper

Laguerre Functions and Their Applications to Tempered Fractional Differential Equations on Infinite Intervals

OriginalPaper

Enriched Spectral Method for Stiff Convection-Dominated Equations

OriginalPaper

Manifold Based Low-Rank Regularization for Image Restoration and Semi-Supervised Learning

Premium Partner

    Image Credits