Skip to main content
Springer Nature Link
Log in

A generic interface for parallel and adaptive discretization schemes: abstraction principles and the Dune-Fem module

  • Published:
Computing Aims and scope Submit manuscript

Abstract

Starting from an abstract mathematical notion of discrete function spaces and operators, we derive a general abstraction for a large class of grid-based discretization schemes for stationary and instationary partial differential equations. Special emphasis is put on concepts for local adaptivity and parallelization with dynamic load balancing. The concepts are based on a corresponding abstract definition of a parallel and hierarchical adaptive grid given in Bastian et al. (Computing 82(2–3):103–119, 2008). Based on the abstract framework, we describe an efficient object oriented implementation of a generic interface for grid-based discretization schemes that is realized in the Dune-Fem library (http://dune.mathematik.uni-freiburg.de). By using interface classes we manage to separate functionality from data structures. Efficiency is obtained by using modern template based generic programming techniques, including static polymorphism, the engine concept, and template metaprogramming. We present numerical results for several benchmark problems and some advanced applications.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Abdelaziz Y, Hamouine A (2008) A survey of the extended finite element. Comput Struct 86(11–12): 1141–1151

    Article  Google Scholar 

  2. Arnold DN, Brezzi F, Cockburn B, Marini LD (2002) Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J Numer Anal 39(5): 1749–1779

    Article  MATH  MathSciNet  Google Scholar 

  3. Bangerth W, Hartmann R, Kanschat G (2007) deal.II—a general-purpose object-oriented finite element library. ACM Trans Math Softw 33(4):24, 27

    Google Scholar 

  4. Bastian P, Birken K, Johannsen K, Lang S, Neuss N, Rentz-Reichert H, Wieners C (1997) UG—a flexible software toolbox for solving partial differential equations. Comput Visual Sci 1: 27–40

    Article  MATH  Google Scholar 

  5. Bastian P, Blatt M, Dedner A, Engwer C, Klöfkorn R, Kornhuber R, Ohlberger M, Sander O (2008) A generic grid interface for parallel and adaptive scientific computing. II: Implementation and tests in dune. Computing 82(2–3): 121–138

    Article  MATH  MathSciNet  Google Scholar 

  6. Bastian P, Blatt M, Dedner A, Engwer C, Klöfkorn R, Ohlberger M, Sander O (2008) A generic grid interface for parallel and adaptive scientific computing. I: Abstract framework. Computing 82 (2–3): 103–119

    Article  MATH  MathSciNet  Google Scholar 

  7. Blatt M, Bastian P (2007) The iterative solver template library. In: Kagström B, Elmroth E, Dongarra J, Wasniewski J (eds) Applied parallel computing—state of the art in scientific computing. Springer, Berlin, pp 666–675

    Chapter  Google Scholar 

  8. Brenner SC, Ridgway Scott L (2008) The mathematical theory of finite element methods. In: Texts in applied mathematics, 3rd edn, vol 15. Springer, New York

  9. Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. Springer, Berlin

    MATH  Google Scholar 

  10. Burri A, Dedner A, Diehl D, Klöfkorn R, Ohlberger M (2006) A general object oriented framework for discretizing nonlinear evolution equations. In: Shokin YI, Resch M, Danaev N, Orunkhanov M, Shokina N (eds) Advances in high performance computing and computational sciences. Notes on numerical fluid mechanics and multidiciplinary design (NNFM), vol 93. Springer, Berlin

  11. Burri A, Dedner A, Klöfkorn R, Ohlberger M (2006) An efficient implementation of an adaptive and parallel grid in dune. In: Krause E, Shokin YI, Resch M, Shokina N (eds) Advances in high performance computing and computational sciences. Notes on numerical fluid mechanics adn multidisciplinary design (NNFM), vol 91. Springer, Berlin

  12. Castillo P, Rieben R, White D (2005) FEMSTER: an object-oriented class library of high-order discrete differential forms. ACM Trans Math Softw 31(4): 425–457

    Article  MATH  MathSciNet  Google Scholar 

  13. Ciarlet PG (1987) The finite element methods for elliptic problems. North-Holland, Amsterdam

    Google Scholar 

  14. Cockburn B, Shu C-W (2001) Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J Sci Comput 16(3): 173–261

    Article  MATH  MathSciNet  Google Scholar 

  15. Davis TA (2004) Algorithm 832: UMFPACK v4.3—an unsymmetric-pattern multifrontal method. ACM Trans Math Softw 30(2): 196–199

    Article  MATH  Google Scholar 

  16. Dedner A, Klöfkorn R (2008) A generic stabilization approach for higher order Discontinuous Galerkin methods for convection dominated problems. Preprint no. 8. Submitted to SIAM Sci. Comput. Mathematisches Institut, Unversität Freiburg. http://www.mathematik.uni-freiburg.de/IAM/homepages/robertk/postscript/dedner_kloefkorn_limiter.pdf

  17. Dedner A, Luethi M, Albrecht T, Vetter T (2007) Curvature guided level set registration using adaptive finite elements. In: Hamprecht F, Schnorr C, Jahne B (eds) Proceedings of the 29th annual symposium of the German association for pattern recognition. Springer, Berlin

  18. Dedner A, Rohde C, Schupp B, Wesenberg M (2004) A parallel, load balanced mhd code on locally adapted, unstructured grids in 3d. Comput Visual Sci 7: 79–96

    Article  MATH  MathSciNet  Google Scholar 

  19. Diehl D (2007) Higher order schemes for simulation of compressible liquid-vapor flows with phase change. PhD thesis, Universität Freiburg. http://www.freidok.uni-freiburg.de/volltexte/3762/

  20. Dune Fem Dune-Fem—The FEM Module. http://dune.mathematik.uni-freiburg.de/

  21. Eymard R, Galluoët T, Herbin R (2000) Finite volume methods. In: Handbook of numerical analysis, vol VII. North-Holland, Amsterdam, pp 713–1020

  22. Gerbeau J-F, Perthame B (2001) Derivation of viscous Saint–Venant system for laminar shallow water; numerical validation. Discrete Contin Dyn Syst Ser B 1(1): 89–102

    Article  MATH  MathSciNet  Google Scholar 

  23. Gersbacher C (2008) Local Discontiunous Galerkin Verfahren zur Simulation flacher dreidimensionaler Strömungen mit freier Oberfläche. Diploma thesis, Universität Freiburg

  24. Haasdonk B, Ohlberger M (2008) Reduced basis method for finite volume approximations of parametrized linear evolution equations. M2AN Math Model Numer Anal 42(2): 277–302

    Article  MATH  MathSciNet  Google Scholar 

  25. Henning P, Ohlberger M (2009) Advection diffusion problems with rapidly oscillating coefficients and large expected drift. Part 2: The heterogeneous multiscale finite element method. Technical report, University of Münster (to be submitted)

  26. Karypis G, Kumar V (1999) A fast and highly quality multilevel scheme for partitioning irregular graphs. SIAM J Sci Comput 20(1): 359–392

    Article  MATH  MathSciNet  Google Scholar 

  27. Karypis G, Kumar V (1999) Multilevel k-way partitioning scheme for irregular graphs. SIAM Rev 41(2): 278–300

    Article  MATH  MathSciNet  Google Scholar 

  28. Kröner D (1997) Numerical schemes for conservation laws. Wiley, Stuttgart

    MATH  Google Scholar 

  29. Lehn ML (2008) FLENS A flexible library for efficient numerical solutions. PhD thesis, Fakultät für Mathematik und Wirtschaftswissenschaften, Universität Ulm. http://flens.sourceforge.net/

  30. Leveque RJ (2002) Finite volume methods for hyperbolic problems. In: Cambridge texts in applied mathematics. Cambridge University Press, Cambridge

  31. Patera AT, Rozza G (2007) Reduced Basis approximation and a posteriori error estimation for parametrized partial differential equations. MIT, Version 1.0, Copyright MIT 2006-2007. In: (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering (to appear)

  32. Schmidt A, Siebert KG (2005) Design of adaptive finite element software—the finite element toolbox ALBERTA. Springer, Berlin

    MATH  Google Scholar 

  33. Veldhuizen T The object-oriented numerics page. http://www.oonumerics.org/oon/

Download references

Author information

Authors and Affiliations

  1. University of Freiburg, Freiburg, Germany

    Andreas Dedner, Robert Klöfkorn & Martin Nolte

  2. University of Münster, Münster, Germany

    Mario Ohlberger

Authors
  1. Andreas Dedner
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Robert Klöfkorn
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Martin Nolte
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Mario Ohlberger
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Andreas Dedner.

Additional information

Communicated by C.C. Douglas.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dedner, A., Klöfkorn, R., Nolte, M. et al. A generic interface for parallel and adaptive discretization schemes: abstraction principles and the Dune-Fem module. Computing 90, 165–196 (2010). https://doi.org/10.1007/s00607-010-0110-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00607-010-0110-3

Keywords

Mathematics Subject Classification (2000)