Top

Journal of Scientific Computing

Published in:

09-10-2017

Guaranteed A Posteriori Error Estimates for a Staggered Discontinuous Galerkin Method

Authors: Eric T. Chung, Eun-Jae Park, Lina Zhao

Published in: Journal of Scientific Computing | Issue 2/2018

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

search-config

AI-assisted search

Off

Abstract

In this paper, we present for the first time guaranteed upper bounds for the staggered discontinuous Galerkin method for diffusion problems. Two error estimators are proposed for arbitrary polynomial degrees and provide an upper bound on the energy error of the scalar unknown and \(L^2\)-error of the flux, respectively. Both error estimators are based on the potential and flux reconstructions. The potential reconstruction is given by a simple averaging operator. The equilibrated flux reconstruction can be found by solving local Neumann problems on elements sharing an edge with a Raviart–Thomas mixed method. Reliability and efficiency of the two a posteriori error estimators are proved. Numerical results are presented to validate the theoretical results.

previous article Strong Stability Preserving Explicit Peer Methods for Discontinuous Galerkin Discretizations

next article Stability and Convergence Analysis of Finite Difference Schemes for Time-Dependent Space-Fractional Diffusion Equations with Variable Diffusion Coefficients

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

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

Alonso, A.: Error estimators for a mixed method. Numer. Math. 74, 385–395 (1996)MathSciNetCrossRefMATH

Arbogast, T., Chen, Z.: On the implementation of mixed methods as nonconforming methods for second-order elliptic problems. Math. Comput. 64, 943–972 (1995)MathSciNetMATH

Arnold, A.N., Brezzi, F.: Mixed and nonconforming finite element methods: imlementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. Anal. 19, 7–32 (1985)CrossRefMATH

Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736–754 (1978)MathSciNetCrossRefMATH

Babuška, I., Rheinboldt, W.C.: A posteriori error estimates for the fintie element method. Int. J. Numer. Methods Eng. 12, 1597–1615 (1978)CrossRefMATH

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

Braess, D., Schöberl, J.: Equilibrated residual error estimator for edge elements. Math. Comput. 77, 651–672 (2008)MathSciNetCrossRefMATH

Braess, D., Verfürth, R.: A posteriori error estimators for the Raviart–Thomas element. SIAM J. Numer. Anal. 33, 2431–2444 (1996)MathSciNetCrossRefMATH

10.

Carstensen, C.: A posteriori error estimates for the mixed finite element method. Math. Comput. 66, 465–476 (1997)MathSciNetCrossRefMATH

11.

Carstensen, C., Feischl, M., Page, M., Praetorius, D.: Axioms of adaptivity. Comput. Math. Appl. 67, 1195–1253 (2014)MathSciNetCrossRefMATH

12.

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

13.

Carstensen, C., Park, E.-J.: Convergence and optimality of adaptive least squares finite element methods. SIAM J. Numer. Anal. 53(1), 43–62 (2015)MathSciNetCrossRefMATH

14.

Chan, H.N., Chung, E.T., Cohen, G.: Stability and dispersion analysis of staggered discontinuous Galerkin method for wave propagation. Int. J. Numer. Model. 10, 233–256 (2013)MathSciNetMATH

15.

Chen, Z., Dai, S.: On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients. SIAM J. Sci. Comput. 24, 443–462 (2002)MathSciNetCrossRefMATH

16.

Chung, E.T., Ciarlet, P., Yu, T.F.: Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwell’s equations on Cartesian grids. J. Comput. Phys. 235, 14–31 (2013)MathSciNetCrossRefMATH

17.

Chung, E., Cockburn, B., Fu, G.: The staggered DG method is the limit of a hybridizable DG method. SIAM J. Numer. Anal. 52, 915–932 (2014)MathSciNetCrossRefMATH

18.

Chung, E.T., Engquist, B.: Optimal discontinuous galerkin methods for wave propagation. SIAM J. Numer. Anal. 44, 2131–2158 (2006)MathSciNetCrossRefMATH

19.

Chung, E.T., Engquist, B.: Optimal discontinuous galerkin methods for the acoustic wave equation in higher dimensions. SIAM J. Numer. Anal. 47, 3820–3848 (2009)MathSciNetCrossRefMATH

20.

Chung, E.T., Kim, H.: A deluxe FETI-DP algorithm for a hybrid staggered discontinuous Galerkin method for H(curl)-elliptic problems. Int. J. Numer. Method Eng. 98, 1–23 (2014)MathSciNetCrossRefMATH

21.

Chung, E.T., Kim, H.H., Widlund, O.: Two-level overlapping schwarz algorithms for a staggered discontinuous Galerkin method. SIAM J. Numer. Anal. 51, 47–67 (2013)MathSciNetCrossRefMATH

22.

Chung, E.T., Lee, C.S.: A staggered discontinuous galerkin method for the curl–curl operator. IMA J. Numer. Anal. 32, 1241–1265 (2012)MathSciNetCrossRefMATH

23.

Chung, E.T., Lee, C.S.: A staggered discontinuous galerkin method for the convection–diffusion equation. J. Numer. Math. 20, 1–31 (2012)MathSciNetCrossRefMATH

24.

Chung, E.T., Yuen, M., Zhong, L.: A-posteriori error analysis for a staggered discontinuous Galerkin discretization of the time-harmonic Maxwell’s equations. Appl. Math. Comput. 237, 613–631 (2014)MathSciNetMATH

25.

Dörfler, W., Wilderotter, O.: An adaptive finite element method for a linear elliptic equation with variable coefficients. Z. Angew. Math. Mech. 80, 481–491 (2000)MathSciNetCrossRefMATH

26.

Ern, A., Nicaise, S., Vohralík, M.: An accurate H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems. C.R. Math. Acad. Sci. Paris Ser. I(345), 709–712 (2007)MathSciNetCrossRefMATH

27.

Ern, A., Vohralík, M.: Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations. SIAM J. Numer. Anal. 53, 1058–1081 (2015)MathSciNetCrossRefMATH

28.

Ern, A., Vohralík, M.: Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids. C.R. Math. Acad. Sci. Paris Ser. I(347), 441–444 (2009)MathSciNetCrossRefMATH

29.

Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41, 2374–2399 (2003)MathSciNetCrossRefMATH

30.

Kim, D., Park, E.-J.: A posteriori error estimator for expanded mixed hybrid methods. Numer. Methods Partial Differ. Equ. 23, 973–988 (2007)MathSciNetCrossRefMATH

31.

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, 806–820 (2008)MathSciNetCrossRefMATH

32.

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, 1186–1207 (2010)MathSciNetCrossRefMATH

33.

Kim, H.H., Chung, E.T., Lee, C.S.: A staggered discontinuous Galerkin method for the Stokes system. SIAM J. Numer. Anal. 51, 3327–3350 (2013)MathSciNetCrossRefMATH

34.

Kim, H.H., Chung, E.T., Lee, C.S.: A BDDC algorithm for a class of staggered discontinuous Galerkin methods. Comput. Math. Appl. 67, 1373–1389 (2014)MathSciNetCrossRefMATH

35.

Kim, K.Y.: A posteriori error analysis for locally conservative mixed methods. Math. Comput. 76, 43–66 (2007)MathSciNetCrossRefMATH

36.

Kim, K.Y.: A posteriori error estimators for locally conservative methods of nonlinear elliptic problem. Appl. Numer. Math. 57, 1065–1080 (2007)MathSciNetCrossRefMATH

37.

Ladevèze, P.: Comparaison de modèles de milieux continus. Ph.D. thesis, Université Pierre et Marie Curie, Paris6 (1975)

38.

Larson, M.G., Målqvist, A.: A posteriori error estimates for mixed finite element approximations of elliptic problems. Numer. Math. 108, 487–500 (2008)MathSciNetCrossRef

39.

Luce, E., Wohlmuth, B.I.: A local a posteriori error estimator based on equilibrated fluxes. SIAM J. Numer. Anal. 42, 1394–1414 (2004)MathSciNetCrossRefMATH

40.

Prager, W., Synge, J.L.: Approximations in elasticity based on the concept of function space. Q. Appl. Math. 6, 241–269 (1947)MathSciNetCrossRefMATH

41.

Rivère, B., Wheeler, M.F.: A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems. Comput. Math. Appl. 46, 141–163 (2003)MathSciNetCrossRef

42.

Synge, J.L.: The Hypercircle in Mathematical Physics: A Method for the Approximate Solution of Boundary Value Problems. Cambridge University Press, New York (1957)MATH

43.

Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Teubner-Wiley, Stuttgart (1996)MATH

44.

Vohralík, M.: A posteriori error estimates for lowest-order mixed finite element discretization of convection–diffusion–reaction equations. SIAM J. Numer. Anal. 45, 1570–1599 (2007)MathSciNetCrossRefMATH

45.

Vohralík, M.: Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods. Math. Comput. 79, 2001–2032 (2010)MathSciNetCrossRefMATH

46.

Vohralík, M.: Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients. J. Sci. Comput. 46, 397–438 (2011)MathSciNetCrossRefMATH

Title: Guaranteed A Posteriori Error Estimates for a Staggered Discontinuous Galerkin Method
Authors: Eric T. Chung
Eun-Jae Park
Lina Zhao
Publication date: 09-10-2017
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 2/2018
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-017-0575-8

Springer Professional

Abstract

Please log in to get access to your license.

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

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 2/2018

On the Operator Splitting and Integral Equation Preconditioned Deferred Correction Methods for the “Good” Boussinesq Equation

A Dual Consistent Finite Difference Method with Narrow Stencil Second Derivative Operators

Unconditionally Energy Stable Linear Schemes for the Diffuse Interface Model with Peng–Robinson Equation of State

A Dimension Reduction Shannon-Wavelet Based Method for Option Pricing

A Weak Galerkin Method for the Reissner–Mindlin Plate in Primary Form

Conservative and Stable Degree Preserving SBP Operators for Non-conforming Meshes

Premium Partner