Skip to main content
SpringerLink
Log in

A Superconvergent Discontinuous Galerkin Method for Hyperbolic Problems on Tetrahedral Meshes

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this manuscript we present a superconvergent discontinuous Galerkin method equipped with an element residual error estimator applied to scalar first-order hyperbolic problems using tetrahedral meshes. We present a local error analysis to derive a discontinuous Galerkin orthogonality condition for the leading term of the discretization error and establish new superconvergence points, lines and surfaces. We also derive new basis functions spanning the error and propose an implicit error estimation procedure by solving a local problem on each tetrahedron. The DG method combined with the a posteriori error estimation procedure yields both accurate error estimates and \(O(h^{p+2})\) superconvergent solutions. Computations validate our theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Adjerid, S., Baccouch, M.: The discontinuous Galerkin method for two-dimensional hyperbolic problems. Part I: superconvergence error analysis. J. Sci. Comput. 33, 75–113 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Adjerid, S., Baccouch, M.: The discontinuous Galerkin method for two-dimensional hyperbolic problems. Part II: a posteriori error estimation. J. Sci. Comput. 38, 15–49 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. Adjerid, S., Baccouch, M.: A Posteriori error analysis of the discontinuous Galerkin method for two-dimensional hyperbolic problems on unstructured meshes. Comput. Methods Appl. Mech. Eng. 200, 162–177 (2011)

    Article  MathSciNet  Google Scholar 

  4. Adjerid, S., Baccouch, M.: A superconvergent local discontinuous Galerkin method for elliptic problems. J. Sci. Comput. 52, 113–152 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Adjerid, S., Devine, K.D., Flaherty, J.E., Krivodonova, L.: A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Comput. Methods Appl. Mech. Eng. 191, 1097–1112 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Adjerid, S., Klauser, A.: Superconvergence of discontinuous finite element solutions for convection-diffusion problems. J. Sci. Comput. 21–22, 5–24 (2005)

    Article  MathSciNet  Google Scholar 

  7. Adjerid, S., Massey, T.C.: A posteriori discontinuous finite element error estimation for two-dimensional hyperbolic problems. Comput. Methods Appl. Mech. Eng. 191, 5877–5897 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Adjerid, S., Massey, T.C.: Superconvergence of discontinuous finite element solutions for nonlinear hyperbolic problems. Comput. Methods Appl. Mech. Eng. 195, 3331–3346 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Adjerid, S., Mechai, I.: A posteriori discontinuous Galerkin error estimation on tetrahedral meshes. Comput. Methods Appl. Mech. Eng. 201–204, 157–178 (2012)

    Article  MathSciNet  Google Scholar 

  10. Adjerid, S., Weinhart, T.: Discontinuous Galerkin error estimation for linear symmetric hyperbolic systems. Comput. Methods Appl. Mech. Eng. 198, 3113–3129 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Adjerid, S., Weinhart, T.: Asymptotically exact discontinuous Galerkin error estimates for linear symmetric hyperbolic systems. submitted (2011)

  12. Adjerid, S., Weinhart, T.: Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems. Math. Comput. 80, 1335–1367 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  13. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, D.: Discontinuous Galerkin methods for elliptic problems. In: Cockburn, B., Karniadakis, G., Shu, C.-W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications, volume 11 of Lecture Notes in Computational Science and Engineering, pp. 89–102. Springer, Berlin (2000)

    Chapter  Google Scholar 

  14. Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible navier-stokes equations. J. Comput. Phys. 131, 267–279 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Baumann, C.E., Oden, J.T.: A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Eng. 175, 311–341 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  16. Biswas, R., Devine, K., Flaherty, J.E.: Parallel adaptive finite element methods for conservation laws. Appl. Numer. Math. 14, 255–284 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  17. Bottcher, K., Rannacher, R.: Adaptive Error Control in Solving Ordinary Differential Equations by the Discontinuous Galerkin Method. University of Heidelberg, Tech. report, (1996)

  18. Brezzi, F., Manzini, M., Marini, D., Pietra, P., Russo, A.: Discontinuous finite elements for diffusion problems. Atti Convegno in onore Di F. Brioschi (Milano 1997), Instituto Lombardo, Accademia di Scienze e Lettere (1999)

  19. Cockburn, B.: Discontinuous Galerkin methods for convection dominated problems. In: Barth, T., Deconink, H. (eds.) High-order Methods for Computational Physics, volume 9 of Lecture Notes in Computational Science and Engineering, pp. 69–224. Springer, Berlin (1999)

    Google Scholar 

  20. Cockburn, B., Hou, S., Shu, C.-W.: The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comput. 54, 545–581 (1990)

    MATH  MathSciNet  Google Scholar 

  21. Cockburn, B., Lin, S.Y., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin methods of scalar conservation laws III: one dimensional systems. J. Comput. Phys. 84, 90–113 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  22. Cockburn, B., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin methods for scalar conservation laws II: general framework. Math. Comput. 52, 411–435 (1989)

    MATH  MathSciNet  Google Scholar 

  23. Cockburn, B., Shu, C.-W.: The Runge–Kutta local projection \(P^1\)-discontinuous-Galerkin finite element method for scalar conservation laws. Math. Model. Numer. Anal. 25, 337–361 (1991)

    MATH  MathSciNet  Google Scholar 

  24. Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin finite element method for convection-diffusion systems. SIAM J. Numer. Anal. 35, 2240–2463 (1998)

    MathSciNet  Google Scholar 

  25. Cockburn, B., Shu, C.-W.: The Runge–Kutta discontinuous Galerkin finite element method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  26. Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd edn. Springer, New York (2007)

    Book  Google Scholar 

  27. Dawson, C.N.: Godunov-mixed methods for advection-diffusion equations in one space dimension. SIAM J. Numer. Anal. 30, 1282–1309 (1991)

    Article  Google Scholar 

  28. Dawson, C.N.: Godunov-mixed methods for advection-diffusion equations in multidimensions. SIAM J. Numer. Anal. 30, 1315–1332 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  29. Devine, K.D., Flaherty, J.E.: Parallel adaptive hp-refinement techniques for conservation laws. Comput. Methods Appl. Mech. Eng. 20, 367–386 (1996)

    MATH  MathSciNet  Google Scholar 

  30. Ericksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems I: a linear model problem. SIAM J. Numer. Anal. 28, 12–23 (1991)

    Google Scholar 

  31. Ericksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems II: optimal error estimates in \( L_\infty L_2\) and \( L_\infty L_\infty \). SIAM J. Numer. Anal. 32, 706–740 (1995)

    Article  MathSciNet  Google Scholar 

  32. Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Springer, New York (2004)

    Book  MATH  Google Scholar 

  33. Estep, D.: A posteriori error bounds and global error control for approximation of ordinary differential equations. SIAM J. Numer. Anal. 32, 1–48 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  34. Grundmann, A., Moeller, M.: Invariant integration formulas for the n-simplex by combinatorial methods. SIAM J. Numer. Anal. 15, 282–290 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  35. Johnson, C.: Error estimates and adaptive time-step control for a class of one-step methods for stiff ordinary differential equations. SIAM J. Numer. Anal. 25, 908–926 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  36. Karniadakis, G.E., Sherwin, S.J.: Spectral/hp Element Methods for CFD. Oxford University Press, New York (1999)

    MATH  Google Scholar 

  37. Krivodonova, L., Flaherty, J.E.: Error estimation for discontinuous Galerkin solutions of multidimensional hyperbolic problems. Adv. Comput. Math. pp. 57–71 (2003)

  38. Lesaint, P., Raviart, P.: On a finite element method for solving the neutron transport equations. In: de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 89–145. Academic Press, New York (1974)

    Google Scholar 

  39. Loehr, N.: Personal Communication (2011)

  40. Mechai, I.: A posteriori Error Analysis of a Discontinuous Galerkin Method Applied to Hyperbolic Problems on Tetrahedral Meshes. PhD thesis, Virginia Tech (2012)

  41. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos (1973)

Download references

Acknowledgments

This research was partially supported by the National Science Foundation (Grant Number DMS 0809262).

Author information

Authors and Affiliations

  1. Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061, USA

    Slimane Adjerid & Idir Mechai

Authors
  1. Slimane Adjerid
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Idir Mechai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Slimane Adjerid.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Adjerid, S., Mechai, I. A Superconvergent Discontinuous Galerkin Method for Hyperbolic Problems on Tetrahedral Meshes. J Sci Comput 58, 203–248 (2014). https://doi.org/10.1007/s10915-013-9735-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-013-9735-7

Keywords

Search

Navigation