A Parallel Adaptive Cartesian PDE Solver Using Space–Filling Curves

  • Hans-Joachim Bungartz
  • Miriam Mehl
  • Tobias Weinzierl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4128)


In this paper, we present a parallel multigrid PDE solver working on adaptive hierarchical cartesian grids. The presentation is restricted to the linear elliptic operator of second order, but extensions are possible and have already been realised as prototypes. Within the solver the handling of the vertices and the degrees of freedom associated to them is implemented solely using stacks and iterates of a Peano space–filling curve. Thus, due to the structuredness of the grid, two administrative bits per vertex are sufficient to store both geometry and grid refinement information. The implementation and parallel extension, using a space–filling curve to obtain a load balanced domain decomposition, will be formalised. In view of the fact that we are using a multigrid solver of linear complexity \(\mathcal{O}(n)\), it has to be ensured that communication cost and, hence, the parallel algorithm’s overall complexity do not exceed this linear behaviour.


Domain Decomposition Cartesian Grid Diploma Thesis Output Stream Geometric Element 
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  1. 1.
    Bader, M., Frank, A.C., Zenger, C.: An Octree-Based Approach for Fast Elliptic Solvers. High Performance Scientific and Engineering Computing. Springer, Heidelberg (2001)Google Scholar
  2. 2.
    Brenk, M., Bungartz, H.J., Mehl, M., Neckel, T.: Fluid–Structure Interaction on Cartesian Grids: Flow Simulation and Coupling Environment. Fluid-Structure Interaction, LNCS (to appear)Google Scholar
  3. 3.
    Gotsman, C., Lindenbaum, M.: On the Metric Properties of Discrete Space–Filling Curves. IEEE Transactions on Image Processing 5(5), 794–797 (1996)CrossRefGoogle Scholar
  4. 4.
    Griebel, M.: Multilevel algorithms considered as iterative methods on indefinite systems. SFB-Bericht 342/29/91 (1991)Google Scholar
  5. 5.
    Griebel, M., Zumbusch, G.: Hash–Storage Techniques for Adaptive Multilevel Solvers and Their Domain Decomposition Parallelization. Proceedings of Domain Decomposition Methods 10, DD10 218, 279–286 (1998)MathSciNetGoogle Scholar
  6. 6.
    Günther, F., Krahnke, A., Langlotz, M., Mehl, M., Pögl, M., Zenger, C.: On the Parallelization of a Cache-Optimal Iterative Solver for PDEs Based on Hierarchical Data Structures and Space-Filling Curves. In: Kranzlmüller, D., Kacsuk, P., Dongarra, J. (eds.) EuroPVM/MPI 2004. LNCS, vol. 3241, pp. 425–429. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. 7.
    Günther, F., Mehl, M., Pögl, M., Zenger, C.: A cache-aware algorithm for PDEs on hierarchical data structures based on space-filling curves. SIAM Journal on Scientific Computing (to appear)Google Scholar
  8. 8.
    Hungershöfer, J., Wierum, J.M.: On the Quality of Partitions based on Space–Filling Curves. In: Sloot, P.M.A., Tan, C.J.K., Dongarra, J., Hoekstra, A.G. (eds.) ICCS-ComputSci 2002. LNCS, vol. 2331, pp. 31–45. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Hartmann, J.: Entwicklung eines cache–optimalen Finite–Element–Verfahrens zur Lösung d-dimensionaler Probleme. Diploma thesis, Technical University Munich (2004)Google Scholar
  10. 10.
    Herder, W.: Entwicklung eines cache–optimalen Finite–Element–Verfahrens zur Lösung d-dimensionaler Probleme. Diploma thesis, Technical University Munich (2005)Google Scholar
  11. 11.
    Kowarschik, M., Weiß, C.: An Overview of Cache Optimization Techniques and Cache-Aware Numerical Algorithms. In: Meyer, U., Sanders, P., Sibeyn, J.F. (eds.) Algorithms for Memory Hierarchies. LNCS, vol. 2625, pp. 213–232. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  12. 12.
    Langlotz, M.: Parallelisierung eines Cache–optimalen 3D Finite–Element–Verfahrens. Diploma thesis, Technical University Munich (2005)Google Scholar
  13. 13.
    Mehl, M., Weinzierl, T., Zenger, C.: A cache–oblivious self–adaptive full multigrid method. Numerical Linear Algebra With Applications (to appear)Google Scholar
  14. 14.
    Neckel, T.: Einfache 2D–Fluid–Struktur–Wechselwirkungen mit einer cache–optimalen Finite–Element–Methode. Diploma thesis, Technical University Munich (2005)Google Scholar
  15. 15.
    Sagan, H.: Space-Filling Curves. Springer, Heidelberg (1994)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Hans-Joachim Bungartz
    • 1
  • Miriam Mehl
    • 1
  • Tobias Weinzierl
    • 1
  1. 1.Technical University MunichGarchingGermany

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