Efficient Parallel Simulation of Atherosclerotic Plaque Formation Using Higher Order Discontinuous Galerkin Schemes

  • Stefan GirkeEmail author
  • Robert Klöfkorn
  • Mario Ohlberger
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 78)


The compact Discontinuous Galerkin 2 (CDG2) method was successfully tested for elliptic problems, scalar convection-diffusion equations and compressible Navier-Stokes equations. In this paper we use the newly developed DG method to solve a mathematical model for early stages of atherosclerotic plaque formation. Atherosclerotic plaque is mainly formed by accumulation of lipid-laden cells in the arterial walls which leads to a heart attack in case the artery is occluded or a thrombus is built through a rupture of the plaque. After describing a mathematical model and the discretization scheme, we present some benchmark tests comparing the CDG2 method to other commonly used DG methods. Furthermore, we take parallelization and higher order discretization schemes into account.


Wall Shear Stress Arterial Wall Discontinuous Galerkin Discontinuous Galerkin Method Numerical Flux 
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This work was supported by the Deutsche Forschungsgemeinschaft, Collaborative Research Center SFB 656 “Cardiovascular Molecular Imaging”, project B07, Münster, Germany. Scaling results were produced using the super computer Yellowstone (ark:/85065/ d7wd3xhc) provided by NCAR’s Computational and Information Systems Laboratory, sponsored by the National Science Foundation. Robert Klöfkorn is partially funded by the DEO program BER under award DE-SC0006959.


  1. 1.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Kornhuber, R., Ohlberger, M., Sander, O.: A generic grid interface for parallel and adaptive scientific computing. part ii: implementation and tests in dune. Computing 82(2–3), 121–138 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Brdar, S., Dedner, A., Klöfkorn, R.: Compact and stable discontinuous Galerkin methods for convection-diffusion problems. SIAM J. Sci. Comput. 34(1), 263–282 (2012). doi:
  4. 4.
    Calvez, V., Houot, J., Meunier, N., Raoult, A., Rusnakova, G., et al.: Mathematical and numerical modeling of early atherosclerotic lesions. ESAIM Proc. 30, 1–14 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Dedner, A., Klöfkorn, R.: A generic stabilization approach for higher order discontinuous Galerkin methods for convection dominated problems. J. Sci. Comput. 47(3), 365–388 (2011). doi:
  6. 6.
    Dedner, A., Klöfkorn, R., Nolte, M., Ohlberger, M.: A generic interface for parallel and adaptive discretization schemes: abstraction principles and the dune-fem module. Computing 90(3–4), 165–196 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Ibragimov, A., McNeal, C., Ritter, L., Walton, J.: A mathematical model of atherogenesis as an inflammatory response. Math. Med. Biol. 22(4), 305–333 (2005)CrossRefzbMATHGoogle Scholar
  8. 8.
    Klöfkorn, R.: Benchmark 3d: the compact discontinuous Galerkin 2 scheme. In: Jaroslav (ed.) et al., Finite Volumes for Complex Applications VI Problems & Perspectives, pp. 1023–1033. Springer, Heidelberg (2011)Google Scholar
  9. 9.
    Klöfkorn, R.: Efficient matrix-free implementation of discontinuous Galerkin methods for compressible flow problems. In: Handlovičová, A. et al. (eds.) Algoritmy 2012. 19th Conference on Scientific Computing, Vysoké Tatry, Podbanské, Slovakia, 9–14 Sept 2012. Proceedings of Contributed Papers and Posters. Bratislava: Slovak University of Technology, Faculty of Civil Engineering, Department of Mathematics and Descriptive Geometry, 11–21 (2012). ISBN 978-80-227-3742-5/pbk.Google Scholar
  10. 10.
    Knoll, D.A., Keyes, D.E.: Jacobian-free newton-krylov methods: a survey of approaches and applications. J. Comput. Phys. 193(2), 357–397 (2004). doi:
  11. 11.
    Kuhlmann, M.T., Cuhlmann, S., Hoppe, I., Krams, R., Evans, P.C., Strijkers, G.J., et al.: Implantation of a carotid cuff for triggering shear-stress induced atherosclerosis in mice. J. Visualized Exp. 59(e3308), 1–6 (2012)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Stefan Girke
    • 1
    Email author
  • Robert Klöfkorn
    • 2
  • Mario Ohlberger
    • 1
  1. 1.Institute for Computational and Applied MathematicsUniversity of MünsterMünsterGermany
  2. 2.National Center for Atmospheric Research1850 Table Mesa DriveBoulderUSA

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