Skip to main content
SpringerLink
Log in

Stability and Convergence of Spectral Mixed Discontinuous Galerkin Methods for 3D Linear Elasticity on Anisotropic Geometric Meshes

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

Abstract

We consider spectral mixed discontinuous Galerkin finite element discretizations of the Lamé system of linear elasticity in polyhedral domains in \({\mathbb {R}}^3\). In order to resolve possible corner, edge, and corner-edge singularities, anisotropic geometric edge meshes consisting of hexahedral elements are applied. We perform a computational study on the discrete inf-sup stability of these methods, and especially focus on the robustness with respect to the Poisson ratio close to the incompressible limit (i.e. the Stokes system). Furthermore, under certain realistic assumptions (for analytic data) on the regularity of the exact solution, we illustrate numerically that the proposed mixed DG schemes converge exponentially in a natural DG norm.

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. Ainsworth, M., Coggins, P.: The stability of mixed \(hp\)-finite element methods for Stokes flow on high aspect ratio elements. SIAM J. Numer. Anal. 38(5), 1721–1761 (2000)

    Article  MathSciNet  Google Scholar 

  2. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2001)

    Article  MathSciNet  Google Scholar 

  3. Babuška, I., Guo, B.Q.: Approximation properties of the \(h\)-\(p\) version of the finite element method. Comput. Methods Appl. Mech. Eng. 133(3–4), 319–346 (1996)

    Article  MathSciNet  Google Scholar 

  4. Bangerth, W., Hartmann, R., Kanschat, G.: deal.II—a general purpose object oriented finite element library. ACM Trans. Math. Softw. 33(4), 1–27 (2007)

    Article  MathSciNet  Google Scholar 

  5. Bangerth, W., Heister, T., Kanschat, G., et al.: \(\mathtt{deal.II}\) Differential Equations Analysis Library, Technical Reference. http://www.dealii.org

  6. Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. In: Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)

  7. Dauge, M., Costabel, M., Nicaise, S.: Analytic regularity for linear elliptic systems in polygons and polyhedra. Math. Models Methods Appl. Sci. 22(8), 1250015 (2012)

    Article  MathSciNet  Google Scholar 

  8. Georgoulis, E.H., Hall, E., Houston, P.: Discontinuous Galerkin methods on \(hp\)-anisotropic meshes. I. A priori error analysis. Int. J. Comput. Sci. Math. 1(2–4), 221–244 (2007)

    Article  MathSciNet  Google Scholar 

  9. Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations. Springer, New York (1986)

    Book  Google Scholar 

  10. Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences, 3rd edn. Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  11. Guo, B.Q.: The \(h\)-\(p\) version of the finite element method for solving boundary value problems in polyhedral domains. Boundary Value Problems and Integral Equations in Nonsmooth Domains. Lecture Notes in Pure and Applied Mathematics, vol. 167, pp. 101–120. Dekker, New York (1995)

  12. Guo, B.Q., Babuška, I.: Regularity of the solutions for elliptic problems on nonsmooth domains in \({\mathbb{R}}^3\). I. Countably normed spaces on polyhedral domains. Proc. Roy. Soc. Edinburgh Sect. A 127(1), 77–126 (1997)

    Article  MathSciNet  Google Scholar 

  13. Guo, B.Q., Babuška, I.: Regularity of the solutions for elliptic problems on nonsmooth domains in \({\mathbb{R}}^3\). II. Regularity in neighbourhoods of edges. Proc. R. Soc. Edinburgh Sect. A 127(3), 517–545 (1997)

    Article  Google Scholar 

  14. Houston, P., Schötzau, D., Wihler, T.P.: An \(hp\)-adaptive mixed discontinuous Galerkin FEM for nearly incompressible linear elasticity. Comput. Methods Appl. Mech. Eng. 195(25–28), 3224–3246 (2006)

    Article  MathSciNet  Google Scholar 

  15. Houston, P., Wihler, T.P.: An \(hp\)-adaptive Newton-discontinuous-Galerkin finite element approach for semilinear elliptic boundary value problems. Math. Comput. 87(314), 2641–2674 (2018)

    Article  MathSciNet  Google Scholar 

  16. Maday, Y., Bernardi, C.: Uniform inf-sup conditions for the spectral discretization of the Stokes problem. Math. Models Methods Appl. Sci. 09(03), 395–414 (1999)

    Article  MathSciNet  Google Scholar 

  17. Maday, Y., Marcati, C.: Regularity and \(hp\) discontinuous Galerkin finite element approximation of linear elliptic eigenvalue problems with singular potentials. Math. Models Methods Appl. Sci. 29(8), 1585–1617 (2019)

    Article  MathSciNet  Google Scholar 

  18. Maday, Y., Meiron, D., Patera, A.T., Rønquist, E.M.: Analysis of iterative methods for the steady and unsteady Stokes problem: application to spectral element discretizations. SIAM J. Sci. Comput. 14(2), 310–337 (1993)

    Article  MathSciNet  Google Scholar 

  19. Maz’ya, V., Rossmann, J.: Elliptic Equations in Polyhedral Domains, Mathematical Surveys and Monographs, vol. 162. American Mathematical Society, Providence, RI (2010)

    Book  Google Scholar 

  20. Mazzucato, A.L., Nistor, V.: Well-posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks. Arch. Ration. Mech. Anal. 195(1), 25–73 (2010)

    Article  MathSciNet  Google Scholar 

  21. Schötzau, D., Schwab, C.: Mixed \(hp\)-FEM on anisotropic meshes. Math. Models Methods Appl. Sci. 8(5), 787–820 (1998)

    Article  MathSciNet  Google Scholar 

  22. Schötzau, D., Schwab, C., Stenberg, R.: Mixed \(hp\)-\(\rm FEM\) on anisotropic meshes. II. Hanging nodes and tensor products of boundary layer meshes. Numer. Math. 83(4), 667–697 (1999)

    Article  MathSciNet  Google Scholar 

  23. Schötzau, D., Schwab, C., Toselli, A.: Mixed \(hp\)-DGFEM for incompressible flows. SIAM J. Numer. Anal. 40, 2171–2194 (2003)

    Article  MathSciNet  Google Scholar 

  24. Schötzau, D., Schwab, C., Toselli, A.: Mixed \(hp\)-DGFEM for incompressible flows. II. Geometric edge meshes. IMA J. Numer. Anal. 24(2), 273–308 (2004)

    Article  MathSciNet  Google Scholar 

  25. Schötzau, D., Schwab, C., Wihler, T.P.: \(hp\)-dGFEM for second-order elliptic problems in polyhedra I: stability on geometric meshes. SIAM J. Numer. Anal. 51(3), 1610–1633 (2013). https://doi.org/10.1137/090772034

    Article  MathSciNet  MATH  Google Scholar 

  26. Schötzau, D., Schwab, C., Wihler, T.P.: \(hp\)-dGFEM for second order elliptic problems in polyhedra II: exponential convergence. SIAM J. Numer. Anal. 51(4), 2005–2035 (2013). https://doi.org/10.1137/090774276

    Article  MathSciNet  MATH  Google Scholar 

  27. Schötzau, D., Schwab, C., Wihler, T.P.: \(hp\)-DGFEM for second-order mixed elliptic problems in polyhedra. Math. Comput. 85(299), 1051–1083 (2016)

    Article  MathSciNet  Google Scholar 

  28. Schötzau, D., Wihler, T.P.: Exponential convergence of mixed \(hp\)-DGFEM for Stokes flow in polygons. Numer. Math. 96, 339–361 (2003)

    Article  MathSciNet  Google Scholar 

  29. Stenberg, R., Suri, M.: Mixed \(hp\) finite element methods for problems in elasticity and Stokes flow. Numer. Math. 72(3), 367–389 (1996)

    Article  MathSciNet  Google Scholar 

  30. Toselli, A., Schwab, C.: Mixed \(hp\)-finite element approximations on geometric edge and boundary layer meshes in three dimensions. Numer. Math. 94(4), 771–801 (2003)

    Article  MathSciNet  Google Scholar 

  31. Wihler, T.P., Wirz, M.: Mixed hp-discontinuous Galerkin FEM for linear elasticity and Stokes flow in three dimensions. Math. Models Methods Appl. Sci. 22(8), 1250016 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Institute, University of Bern, 3012, Bern, Switzerland

    Thomas P. Wihler & Marcel Wirz

Authors
  1. Thomas P. Wihler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcel Wirz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas P. Wihler.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Thomas P. Wihler acknowledges the financial support of the Swiss National Science Foundation under Grant no. 200021–182524.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wihler, T.P., Wirz, M. Stability and Convergence of Spectral Mixed Discontinuous Galerkin Methods for 3D Linear Elasticity on Anisotropic Geometric Meshes. J Sci Comput 82, 49 (2020). https://doi.org/10.1007/s10915-020-01153-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01153-9

Keywords

Mathematics Subject Classification

Search

Navigation