Skip to main content
SpringerLink
Log in

Influence of Reference-to-Physical Frame Mappings on Approximation Properties of Discontinuous Piecewise Polynomial Spaces

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

Abstract

In this manuscript we compare physical and reference frame discontinuous Galerkin (dG) discretizations with emphasis on the influence of reference-to-physical frame mappings on the discrete space properties. We assess the excellence of physical frame discrete spaces in terms of approximation capabilities as well as the increased flexibility compared to reference frame discretizations. As a matter of fact, whenever curved elements are considered, non-affine reference-to-physical frame mappings are able to spoil the convergence properties of reference frame discrete spaces. This poorly documented drawback does not affect basis functions defined directly in the physical frame.

The convergence degradation associated to reference frame discretizations is evaluated theoretically, providing error bounds for the approximation error of the L 2-orthogonal projection operator, and the findings are justified by means of numerical test cases. In particular we exemplify by means of quadrilateral elements grids challenging grid configurations characterized by non-affine mappings and demonstrate the ability to predict the convergence rates without stringent assumptions on the element shapes.

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

Similar content being viewed by others

References

  1. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arnold, D.N., Boffi, D., Falk, R.S.: Approximation by quadrilateral finite elements. Math. Comput. 71(239), 909–922 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bassi, F., Botti, L., Colombo, A., Di Pietro, D.A., Tesini, P.: On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations. J. Comput. Phys. (2011). doi:10.1016/j.jcp.2011.08.018

    Google Scholar 

  4. Bassi, F., Crivellini, A., Di Pietro, D.A., Rebay, S.: An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier-Stokes equations. J. Comput. Phys. 218, 794–815 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)

    Book  MATH  Google Scholar 

  6. Ciarlet, P.G., Raviart, P.-A.: Interpolation theory over curved elements with applications to finite element methods. Comput. Methods Appl. Mech. Eng. 1, 217–249 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  7. Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Maths & Applications, vol. 69. Springer, Berlin (2011)

    Google Scholar 

  8. Ern, A., Guermond, J.-L.: Eléments finís: théorie, applications, mise en œuvre. Mathématiques & Applications, vol. 36. Springer, Berlin (2002)

    MATH  Google Scholar 

  9. Gassner, G.J., Lörcher, F., Munz, C.-D., Hesthaven, J.S.: Polymorphic nodal elements and their application in discontinuous Galerkin methods. J. Comput. Phys. 228, 1573–1590 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Georgoulis, E.H.: hp-version interior penalty discontinuous Galerkin finite element methods on anisotropic meshes. Int. J. Numer. Anal. Model. 3, 52–79 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Karniadakis, G.E., Sherwin, S.: Spectral/hp Element Methods for Computational Fluid Dynamics. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2005)

    Book  Google Scholar 

  12. Kikuchi, F., Okabe, M., Fujio, H.: Modification of the 8-node serendipity element. Comput. Methods Appl. Mech. Eng. 179, 91–109 (1999)

    Article  MATH  Google Scholar 

  13. Kurtz, J., Xenophontos, C.: On the effects of using curved elements in the approximation of the Reissner–Mindlin plate by the p version of the finite element method. Appl. Numer. Math. 46, 231–246 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. McNeal, R.H., Harder, R.L.: Eight nodes or nine? Int. J. Numer. Methods Eng. 33, 1049–1058 (1992)

    Article  Google Scholar 

  15. Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs (1971)

    MATH  Google Scholar 

  16. Suri, M.: Analytical and computational assessment of locking in the hp finite element method. Comput. Methods Appl. Mech. Eng. 133, 347–371 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tesini, P.: An h-multigrid approach for high-order discontinuous Galerkin methods. Ph.D. thesis, Università degli Studi di Bergamo, January 2008

  18. Xiao, H., Gimbutas, Z.: A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions. Comput. Math. Appl. 59, 663–676 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhang, J., Kikuchi, F.: Interpolation error estimates of a modified 8-node serendipity finite element. Numer. Math. 85, 503–524 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria Industriale, Università degli Studi Di Bergamo, Via Marconi 5, 24044, Dalmine, BG, Italy

    Lorenzo Botti

  2. Biomedical Engineering Department, Mario Negri Institute for Pharmacological Research, Ranica, Italy

    Lorenzo Botti

Authors
  1. Lorenzo Botti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lorenzo Botti.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Botti, L. Influence of Reference-to-Physical Frame Mappings on Approximation Properties of Discontinuous Piecewise Polynomial Spaces. J Sci Comput 52, 675–703 (2012). https://doi.org/10.1007/s10915-011-9566-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-011-9566-3

Keywords

Search

Navigation