Advertisement

Adaptive Discontinuous Galerkin Methods on Polytopic Meshes

  • Joe Collis
  • Paul HoustonEmail author
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 12)

Abstract

In this article we consider the application of discontinuous Galerkin finite element methods, defined on agglomerated meshes consisting of general polytopic elements, to the numerical approximation of partial differential equation problems posed on complicated geometries. Here, we assume that the underlying computational domain may be accurately represented by a geometry-conforming fine mesh \(\mathcal{T}_{\text{fine}}\); the resulting coarse mesh is then constructed based on employing standard graph partitioning algorithms. To improve the accuracy of the computed numerical approximation, we consider the development of goal-oriented adaptation techniques within an automatic mesh refinement strategy. In this setting, elements marked for refinement are subdivided by locally constructing finer agglomerates; should further resolution of the underlying fine mesh \(\mathcal{T}_{\text{fine}}\) be required, then adaptive refinement of \(\mathcal{T}_{\text{fine}}\) will also be undertaken. As an example of the application of these techniques, we consider the numerical approximation of the linear elasticity equations for a homogeneous isotropic material. In particular, the performance of the proposed adaptive refinement algorithm is studied for the computation of the (scaled) effective Young’s modulus of a section of trabecular bone.

Keywords

Fine Mesh Discontinuous Galerkin Discontinuous Galerkin Method Adaptive Mesh Refinement Tree Data Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

Joe Collis acknowledges the financial support of the EPSRC under the grant EP/K039342/1.

References

  1. 1.
    Antonietti, P.F., Giani, S., Houston, P.: hp-Version composite discontinuous Galerkin methods for elliptic problems on complicated domains. SIAM J. Sci. Comput. 35 (3), A1417–A1439 (2013)Google Scholar
  2. 2.
    Antonietti, P.F., Giani, S., Houston, P.: Domain decomposition preconditioners for discontinuous Galerkin methods for elliptic problems on complicated domains. J. Sci. Comput. 60 (1), 203–227 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Antonietti, P.F., Cangiani, A., Collis, J., Dong, Z., Georgoulis, E.H., Giani, S., Houston, P.: Review of discontinuous Galerkin finite element methods for partial differential equations on complicated domains. In: Barrenechea, G.R., Brezzi, F., Cangiani, A., Georgoulis, E.H. (eds.) Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2015)Google Scholar
  4. 4.
    Antonietti, P.F., Houston, P., Sarti, M., Verani, M.: Multigrid algorithms for hp-version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes (2015, submitted for publication)Google Scholar
  5. 5.
    Bassi, F., Botti, L., Colombo, A., Rebay, S.: Agglomeration based discontinuous Galerkin discretization of the Euler and Navier-Stokes equations. Comput. Fluids 61, 77–85 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    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. 231 (1), 45–65 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bassi, F., Botti, L., Colombo, A.: Agglomeration-based physical frame dG discretizations: an attempt to be mesh free. Math. Models Methods Appl. Sci. 24 (8), 1495–1539 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bassi, F., Botti, L., Colombo, A.: h-Multigrid agglomeration based solver strategies for BR2 discontinuous Galerkin discretizations of elliptic problems (2015, Submitted for publication)Google Scholar
  9. 9.
    Becker, R., Rannacher, R.: An optimal control approach to a-posteriori error estimation in finite element methods. In: Iserles, A. (ed.) Acta Numerica, pp. 1–102. Cambridge University Press, Cambridge (2001)Google Scholar
  10. 10.
    Beirão da Veiga, L., Droniou, J., Manzini, G.: A unified approach for handling convection terms in finite volumes and mimetic discretization methods for elliptic problems. IMA J. Numer. Anal. 31 (4), 1357–1401 (2011)Google Scholar
  11. 11.
    Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (1), 199–214 (2013)Google Scholar
  12. 12.
    Brezzi, F., Buffa, A., Lipnikov, K.: Mimetic finite differences for elliptic problems. ESAIM Math. Modell. Numer. Anal. 43 (2), 277–295 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Burman, E., Hansbo, P.: An interior-penalty-stabilized Lagrange multiplier method for the finite-element solution of elliptic interface problems. IMA J. Numer. Anal. 30, 870–885 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Burman, P., Hansbo, P.: Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method. Comput. Methods Appl. Mech. Eng. 199, 2680–2686 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Burman, E., Hansbo, P.: Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 62, 328–341 (2012)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Cangiani, A., Manzini, G., Russo, A.: Convergence analysis of the mimetic finite difference method for elliptic problems. SIAM J. Numer. Anal. 47 (4), 2612–2637 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cangiani, A., Georgoulis, E.H., Houston, P.: hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 24 (10), 2009–2041 (2014)Google Scholar
  18. 18.
    Cangiani, A., Dong, Z., Georgoulis, E.H., Houston, P.: hp-Version discontinuous Galerkin methods for advection–diffusion–reaction problems on polytopic meshes. ESAIM Math. Modell. Numer. Anal. 50 (3), 699–725 (2016)Google Scholar
  19. 19.
    Di Pietro, D.A., Ern, A.: Hybrid High-Order methods for variable diffusion problems on general meshes. C.R. Math. 353, 31–34 (2014)Google Scholar
  20. 20.
    Di Pietro, D.A., Ern, A., Lemaire, S.: A review of Hybrid High-Order methods: formulations, computational aspects, comparison with other methods. In: Barrenechea, G.R., Brezzi, F., Cangiani, A., Georgoulis, E.H. (eds.) Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2015)Google Scholar
  21. 21.
    Fries, T.-P., Belytschko, T.: The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Methods Eng. 84 (3), 253–304 (2010)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Giani, S., Houston, P.: Domain decomposition preconditioners for discontinuous Galerkin discretizations of compressible fluid flows. Numer. Math. Theory Methods Appl. 7 (2), 123–128 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Giani, S., Houston, P.: Goal-oriented adaptive composite discontinuous Galerkin methods for incompressible flows. J. Comput. Appl. Math. 270, 32–42 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Giani, S., Houston, P.: hp-Adaptive composite discontinuous Galerkin methods for elliptic problems on complicated domains. Numer. Methods Part. Diff. Equ. 30 (4), 1342–1367 (2014)Google Scholar
  25. 25.
    Hackbusch, W., Sauter, S.A.: Composite finite elements for problems containing small geometric details. Part II: implementation and numerical results. Comput. Visual Sci. 1, 15–25 (1997)zbMATHGoogle Scholar
  26. 26.
    Hackbusch, W., Sauter, S.A.: Composite finite elements for the approximation of PDEs on domains with complicated micro-structures. Numer. Math. 75, 447–472 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hansbo, P., Larson, M.G.: Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method. Comput. Methods Appl. Mech. Eng. 191 (17–18), 1895–1908 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Johansson, A., Larson, M.G.: A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary. Numer. Math. 123 (4), 607–628 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Karypis, G., Kumar, V.: A fast and highly quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20 (1), 359–392 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Massing, A.: Analysis and implementation of finite element methods on overlapping and fictitious domains. Ph.D. thesis, University of Oslo (2012)Google Scholar
  31. 31.
    Perilli, E., Baruffaldi, F.: Proposal for shared collections of X-ray microCT datasets of bone specimens. In: ICCB03, Zaragoza, Spain, 24–26 September 2003Google Scholar
  32. 32.
    Shewchuk, J.R.: Triangle: engineering a 2D quality mesh generator and Delaunay triangulator. In: Lin, M.C., Manocha, D. (eds). Applied Computational Geometry: Towards Geometric Engineering. Lecture Notes in Computer Science, vol. 1148, pp. 203–222. Springer, Berlin (1996); From the First ACM Workshop on Applied Computational GeometryGoogle Scholar
  33. 33.
    Si, H.: TetGen, a Delaunay-based quality tetrahedral mesh generator. ACM Trans. Math. Softw. 41 (2), 11:1–11:36 (2015)Google Scholar
  34. 34.
    Sukumar, N., Tabarraei, A.: Conforming polygonal finite elements. Int. J. Numer. Methods Eng. 61 (12), 2045–2066 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Verhoosel, C.V., van Zwieten, G.J., van Rietbergen, B., de Borst, R.: Image-based goal-oriented adaptive isogeometric analysis with application to the micro-mechanical modeling of trabecular bone. Comput. Methods Appl. Mech. Eng. 284, 138–164 (2015)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Wihler, T.P.: Locking-free adaptive discontinuous Galerkin FEM for linear elasticity problems. Math. Comput. 75 (255), 1087–1102 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of NottinghamNottinghamUK

Personalised recommendations