Skip to main content
SpringerLink
Log in

Introduction

  • Chapter
  • First Online:
hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

  • 1031 Accesses

Abstract

In this chapter we present a historical overview of discontinuous Galerkin finite element methods; in particular, we discuss their key properties and relative advantages compared to other schemes employed for the numerical approximation of partial differential equations. In addition, we highlight the key computational advantages of exploiting general computational meshes consisting of polygonal/ polyhedral elements, in terms of both meshing complicated geometries in an affordable manner, as well as providing sequences of coarse geometry-conforming meshes needed for the design of efficient multi-level solvers. Finally, we outline some standard notation used throughout this volume.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 29.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 39.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. R.A. Adams, J.J.F. Fournier, Sobolev Spaces. Pure and Applied Mathematics (Amsterdam), vol. 140, 2nd edn. (Elsevier/Academic, Amsterdam, 2003)

    Google Scholar 

  2. B. Ahmad, A. Alsaedi, F. Brezzi, L.D. Marini, A. Russo, Equivalent projectors for virtual element methods. Comput. Math. Appl. 66(3), 376–391 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. P.F. Antonietti, B. Ayuso, Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case. ESAIM Math. Model. Numer. Anal. 41(1), 21–54 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. P.F. Antonietti, P. Houston, Preconditioning high-order discontinuous Galerkin discretizations of elliptic problems, in Domain Decomposition Methods in Science and Engineering XX, ed. by R. Bank, M. Holst, O. Widlund, J. Xu. Lecture Notes in Computational Science and Engineering, vol. 91 (Springer, Berlin, 2013)

    Google Scholar 

  5. P.F. Antonietti, S. Giani, P. Houston, hp-Version composite discontinuous Galerkin methods for elliptic problems on complicated domains. SIAM J. Sci. Comput. 35(3), A1417–A1439 (2013)

    Google Scholar 

  6. P.F. Antonietti, S. Giani, P. Houston, Domain decomposition preconditioners for discontinuous Galerkin methods for elliptic problems on complicated domains. J. Sci. Comput. 60(1), 203–227 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. P.F. Antonietti, M. Sarti, M. Verani, Multigrid algorithms for hp-discontinuous Galerkin discretizations of elliptic problems. SIAM J. Numer. Anal. 53(1), 598–618 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. P.F. Antonietti, P. Houston, I. Smears, A note on optimal spectral bounds for nonoverlapping domain decomposition preconditioners for hp-version discontinuous Galerkin methods. Int. J. Numer. Anal. Model. 13(4), 513–524 (2016)

    MathSciNet  MATH  Google Scholar 

  9. P.F. Antonietti, A. Cangiani, J. Collis, Z. Dong, E.H. Georgoulis, S. Giani, P. Houston, Review of discontinuous Galerkin finite element methods for partial differential equations on complicated domains, in Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, ed. by G.R. Barrenechea, F. Brezzi, A. Cangiani, E.H. Georgoulis (Springer International Publishing, Cham, 2016), pp. 281–310

    Chapter  Google Scholar 

  10. P.F. Antonietti, M. Sarti, M. Verani, Multigrid algorithms for high order discontinuous Galerkin methods. Lect. Notes Comput. Sci. Eng. 104, 3–13 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. P.F. Antonietti, P. Houston, X. Hu, M. Sarti, M. Verani, Multigrid algorithms for hp-version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes, in CALCOLO (2017). https://doi.org/10.1007/s10092-017-0223-6

    Google Scholar 

  12. D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  13. D.N. Arnold, D. Boffi, R.S. Falk, L. Gastaldi, Finite element approximation on quadrilateral meshes. Commun. Numer. Meth. Eng. 17(11), 805–812 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. E. Baas, J.H. Kuiper, A numerical model of heterogeneous surface strains in polymer scaffolds. J. Biomech. 41, 1374–1378 (2008)

    Article  Google Scholar 

  17. E. Baas, J.H. Kuiper, Y. Yang, M.A. Wood, A.J. El Haj, In vitro bone growth responds to local mechanical strain in three-dimensional polymer scaffolds. J. Biomech. 43, 733–739 (2010)

    Article  Google Scholar 

  18. I. Babuška, The finite element method with penalty. Math. Comput. 27(122), 221–228 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  19. I. Babuška, J.E. Osborn, Generalized finite element methods: their performance and their relation to mixed methods. SIAM J. Numer. Anal. 20(3), 510–536 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  20. G.A. Baker, Finite element methods for elliptic equations using nonconforming elements. Math. Comput. 31(137), 45–59 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  21. G.A. Baker, W.N. Jureidini, O.A. Karakashian, Piecewise solenoidal vector fields and the Stokes problem. SIAM J. Numer. Anal. 27(6), 1466–1485 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  22. G.R. Barrenechea, F. Brezzi, A. Cangiani, E.H. Georgoulis (eds.), Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol. 114 (Springer International Publishing, Cham, 2016)

    Google Scholar 

  23. F. Bassi, L. Botti, A. Colombo, S. Rebay, Agglomeration based discontinuous Galerkin discretization of the Euler and Navier-Stokes equations. Comput. Fluids 61, 77–85 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. F. Bassi, L. Botti, A. Colombo, Agglomeration-based physical frame dG discretizations: an attempt to be mesh free. Math. Models Methods Appl. Sci. 24(8), 1495–1539 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. P. Bastian, C. Engwer, An unfitted finite element method using discontinuous Galerkin. Int. J. Numer. Methods Eng. 79(12), 1557–1576 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini, A. Russo, Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199–214 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. L. Beirão da Veiga, K. Lipnikov, G. Manzini, The Mimetic Finite Difference Method for Elliptic Problems. MS&A. Modeling, Simulation and Applications, vol. 11 (Springer, Cham, 2014)

    Google Scholar 

  28. L. Beirão da Veiga, F. Brezzi, L.D. Marini, A. Russo, Serendipity nodal VEM spaces. Comput. Fluids 141, 2–12 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. K.S. Bey, T. Oden, hp-Version discontinuous Galerkin methods for hyperbolic conservation laws. Comput. Methods Appl. Mech. Eng. 133, 259–286 (1996)

    Google Scholar 

  30. S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15, 3rd edn. (Springer, New York, 2008)

    Google Scholar 

  31. F. Brezzi, A. Russo, Choosing bubbles for advection-diffusion problems. Math. Models Methods Appl. Sci. 4, 571–587 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  32. F. Brezzi, P. Houston, D. Marini, E. Süli, Modeling subgrid viscosity for advection-diffusion problems. Comput. Methods Appl. Mech. Eng. 190(13), 1601–1610 (2000)

    Article  MATH  Google Scholar 

  33. A.N. Brooks, T.J.R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32(1), 199–259 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  34. E. Burman, P. Hansbo, An interior-penalty-stabilized Lagrange multiplier method for the finite-element solution of elliptic interface problems. IMA J. Numer. Anal. 30, 870–885 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. E. Burman, P. Hansbo, Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 62, 328–341 (2012)

    MATH  Google Scholar 

  36. A. Cangiani, The residual-free bubble method for problems with multiple scales, Ph.D. thesis, University of Oxford, 2004

    Google Scholar 

  37. A. Cangiani, E.H. Georgoulis, P. Houston, hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 24(10), 2009–2041 (2014)

    Google Scholar 

  38. A. Cangiani, G. Manzini, O.J. Sutton, Conforming and nonconforming virtual element methods for elliptic problems. IMA J. Numer. Anal. 37(3), 1317–1354 (2017)

    MathSciNet  Google Scholar 

  39. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications (North-Holland, Amsterdam, 1978)

    Google Scholar 

  40. B. Cockburn, An introduction to the discontinuous Galerkin method for convection-dominated problems, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations (Cetraro, 1997) (Springer, Berlin, 1998), pp. 151–268

    Google Scholar 

  41. B. Cockburn, Discontinuous Galerkin methods for convection-dominated problems, in High-Order Methods for Computational Physics (Springer, Berlin, 1999), pp. 69–224

    Book  MATH  Google Scholar 

  42. B. Cockburn, Static condensation, hybridization, and the devising of the HDG methods, in Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, ed. by G.R. Barrenechea, F. Brezzi, A. Cangiani, E.H. Georgoulis (Springer International Publishing, Cham, 2016), pp. 129–177

    Chapter  Google Scholar 

  43. B. Cockburn, C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput. 52(186), 411–435 (1989)

    MATH  Google Scholar 

  44. B. Cockburn, C.-W. Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems. J. Comput. Phys. 141(2), 199–224 (1998)

    MathSciNet  MATH  Google Scholar 

  45. B. Cockburn, S.Y. Lin, C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems. J. Comput. Phys. 84(1), 90–113 (1989)

    MATH  Google Scholar 

  46. B. Cockburn, S. Hou, C.-W. Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math. Comput. 54(190), 545–581 (1990)

    MATH  Google Scholar 

  47. B. Cockburn, G.E. Karniadakis, C.-W. Shu (eds.), Discontinuous Galerkin Methods (Springer, Berlin, 2000). Theory, computation and applications, Papers from the 1st International Symposium held in Newport, RI, May 24–26, 1999

    Google Scholar 

  48. B. Cockburn, J. Gopalakrishnan, R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  49. B. Cockburn, D.A. Di Pietro, A. Ern, Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM Math. Model. Numer. Anal. 50(3), 635–650 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  50. D.A. Di Pietro, A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 69 (Springer, Heidelberg, 2012)

    Google Scholar 

  51. D.A. Di Pietro, A. Ern, A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Methods Appl. Mech. Eng. 283, 1–21 (2015)

    Article  MathSciNet  Google Scholar 

  52. D.A. Di Pietro, A. Ern, Hybrid high-order methods for variable-diffusion problems on general meshes. C. R. Math. Acad. Sci. Paris 353(1), 31–34 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  53. D.A. Di Pietro, A. Ern, S. Lemaire, An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Comput. Methods Appl. Math. 14(4), 461–472 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  54. A. Ern, J.-L. Guermond, Theory and Practice of Finite Elements. Applied Mathematical Sciences (Springer, New York, 2004)

    Book  MATH  Google Scholar 

  55. R.S. Falk, G.R. Richter, Local error estimates for a finite element method for hyperbolic and convection-diffusion equations. SIAM J. Numer. Anal. 29(3), 730–754 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  56. X. Feng, O.A. Karakashian, Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal. 39(4), 1343–1365 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  57. B. Fischer, A. Ramage, D.J. Silvester, A.J. Wathen, On parameter choice and iterative convergence for stabilised discretisations of advection-diffusion problems. Comput. Methods Appl. Mech. Eng. 179(1), 179–195 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  58. T.-P. Fries, T. Belytschko, The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Methods Eng. 84(3), 253–304 (2010)

    MathSciNet  MATH  Google Scholar 

  59. S. Giani, P. Houston, Domain decomposition preconditioners for discontinuous Galerkin discretizations of compressible fluid flows. Numer. Math. Theory Methods Appl. 7(2), 123–148 (2014)

    MathSciNet  MATH  Google Scholar 

  60. S. Giani, P. Houston, hp-Adaptive composite discontinuous Galerkin methods for elliptic problems on complicated domains. Numer. Methods Partial Differ. Equ. 30(4), 1342–1367 (2014)

    Google Scholar 

  61. W. Hackbusch, S.A. Sauter, Composite finite elements for problems containing small geometric details. Part II: implementation and numerical results. Comput. Vis. Sci. 1, 15–25 (1997)

    MATH  Google Scholar 

  62. W. Hackbusch, S.A. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated micro-structures. Numer. Math. 75, 447–472 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  63. P. Hansbo, C. Johnson, Streamline Diffusion Finite Element Methods for Fluid Flow (Von Karman Institute for Fluid Dynamics, Rhode St. Genése, 1995)

    Google Scholar 

  64. J.S. Hesthaven, T. Warburton, Nodal Discontinuous Galerkin Methods. Texts in Applied Mathematics, vol. 54 (Springer, New York, 2008). Algorithms, analysis, and applications

    Google Scholar 

  65. P. Houston, N. Sime, Numerical modelling of MPA-CVD reactors with the discontinuous Galerkin finite element method. J. Phys. D Appl. Phys. 50(29), 295202 (2017)

    Google Scholar 

  66. A. Johansson, M.G. Larson, A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary. Numer. Math. 123(4), 607–628 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  67. C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method (Cambridge University Press, Cambridge, 1987)

    MATH  Google Scholar 

  68. C. Johnson, J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput. 46(173), 1–26 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  69. M.G. Larson, F. Bengzon, The Finite Element Method: Theory, Implementation, and Applications (Springer, Heidelberg, 2013)

    Book  MATH  Google Scholar 

  70. P. Lesaint, P.-A. Raviart, On a finite element method for solving the neutron transport equation, in Mathematical Aspects of Finite Elements in Partial Differential Equations. Proceedings of Symposia, Mathematics Research Center, University of Wisconsin, Madison, WI, 1974, vol. 33 (Mathematics Research Center, University of Wisconsin-Madison, Academic, New York, 1974), pp. 89–123

    Google Scholar 

  71. A. Massing, Analysis and implementation of finite element methods on overlapping and fictitious domains, Ph.D. thesis, University of Oslo, 2012

    Google Scholar 

  72. J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Uni. Hamburg 36, 9–15 (1971)

    Article  MATH  Google Scholar 

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

    Google Scholar 

  74. M. Renardy, R.C. Rogers, An Introduction to Partial Differential Equations. Texts in Applied Mathematics, vol. 13 (Springer, New York, 1993)

    Google Scholar 

  75. B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. Frontiers in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2008). Theory and implementation

    Google Scholar 

  76. N. Sime, Numerical modelling of chemical vapour deposition reactors, Ph.D. thesis, University of Nottingham, 2016

    Google Scholar 

  77. N. Sukumar, A. Tabarraei, Conforming polygonal finite elements. Int. J. Numer. Methods Eng. 61(12), 2045–2066 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  78. M.F. Wheeler, An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15(1), 152–161 (1978)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Leicester, Leicester, UK

    Andrea Cangiani, Zhaonan Dong & Emmanuil H. Georgoulis

  2. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Emmanuil H. Georgoulis

  3. School of Mathematical Sciences, University of Nottingham, Nottingham, UK

    Paul Houston

Authors
  1. Andrea Cangiani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhaonan Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Emmanuil H. Georgoulis
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Paul Houston
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Cangiani, A., Dong, Z., Georgoulis, E.H., Houston, P. (2017). Introduction. In: hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-67673-9_1

Download citation

Publish with us

Policies and ethics

Search

Navigation