Computing and Visualization in Science

, Volume 7, Issue 3–4, pp 113–119 | Cite as

Efficient solution of lattice equations by the recovery method Part I: Scalar elliptic problems

  • I. Babuška
  • S.A. SauterEmail author
Regular article


In this paper, an efficient solver for high dimensional lattice equations will be introduced. We will present a new concept, the recovery method, to define a bilinear form on the continuous level which has equivalent energy as the original lattice equation. The finite element discretisation of the continuous bilinear form will lead to a stiffness matrix which serves as an quasi-optimal preconditioner for the lattice equations. Since a large variety of efficient solvers are available for linear finite element problems the new recovery method allows to apply these solvers for unstructured lattice problems.


Bilinear Form Element Discretisation Lattice Equation Multigrid Method Recovery Method 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Babuška, I., Guo, B.: Problems of unbounded lattices, mathematical framework. in preparation.Google Scholar
  2. 2.
    Babuška, I, Sauter, S.: Algebraic Algorithms for the Analysis of Mechanical Trusses. Technical Report 16-2002, Universität Zürich,, 2002. to appear in MathComp.Google Scholar
  3. 3.
    Bank, R.E., Xu, J.: A Hierarchical Basis Multi-Grid Method for Unstructured Grids. In: Hackbusch, W., Wittum, G. (eds.), Fast Solvers for Flow Problems, Proceedings of the Tenth GAMM-Seminar, Kiel: Verlag Vieweg 1995Google Scholar
  4. 4.
    Chan, T., Xu, J., Zikatanov, L.: An agglomeration multigrid method for unstructured grids. Contemp. Math. 218, 67–81 (1998)CrossRefGoogle Scholar
  5. 5.
    Chew, L.P.: Guaranteed-quality Delaunay meshing in 3D. In Proc. 13th Symp. Comp. Geom., pp. 391–393, ACM, 1997Google Scholar
  6. 6.
    Constantinides, G., Payatakes, A.C.: A Three Dimensional Network Model for Consolidated Porous Media. Basic Studies. Chem. Eng. Comm. 81, 55–81 (1980)CrossRefGoogle Scholar
  7. 7.
    Fatt, T.: The Network Model of Porous Media. Trans. Am. Inst. Mem. Metall Pet. Eng. 207, 144–181 (1956)Google Scholar
  8. 8.
    Feuchter, D., Heppner, I., Sauter, S., Wittum, G.: Bridging the gap between geometric and algebraic multigrid methods. Comput. Visual. Sci. 6(1), 1–13 (2003)MathSciNetCrossRefGoogle Scholar
  9. 9.
    George, P.: Automatic Mesh Generation and Finite Element Computation, Vol. IV, chapter Finite Element Methods (Part 2), pp. 69–192. In: Ciarlet, P.G., Lions, J.L. (eds.), Handbook of Numerical Analysis, North-Holland 1996Google Scholar
  10. 10.
    Gibson, L., Ashby, M.: Cellular Solids, Structures and Properties. Exeter: Pergamon Press 1989Google Scholar
  11. 11.
    Glasser, M.L., Boersma, J.: Exact Values for the Cubic Lattice Green Functions. J. Phys., 133, 5017–5023 (2000)MathSciNetGoogle Scholar
  12. 12.
    Griebel, M., Knapek, S.: A multigrid-homogenization method. In: Hackbusch, W., Wittum, G. (eds.), Modeling and Computation in Environmental Sciences, pp. 187–202, Braunschweig: Vieweg 1997. Notes Numer. Fluid Mech. 59Google Scholar
  13. 13.
    Hackbusch, W.: A sparse matrix arithmetic based on ℋ-Matrices. Part I: Introduction to ℋ-Matrices. Computing 62, 89–108 (1999)MathSciNetGoogle Scholar
  14. 14.
    Hackbusch, W., Khoromskij, B.: A sparse ℋ-Matrix Arithmetic. Part II: Applications to Multi-Dimensional Problems. Computing 64(22), 21–47 (2000)MathSciNetGoogle Scholar
  15. 15.
    Hackbusch, W., Khoromskij, B., Sauter, S.: On ℋ2-matrices. In: Bungartz, H.-J., Hoppe, R., Zenger, C. (eds.), Lectures on Applied Mathematics, pp. 9–30, Heidelberg: Springer-Verlag 2000Google Scholar
  16. 16.
    Hackbusch, W., Sauter, S.: Composite Finite Elements for Problems Containing Small Geometric Details. Part II: Implementation and Numerical Results. Comput. Visual. Sci. 1(1), 15–25 (1997)Google Scholar
  17. 17.
    Hansen, J.C., Chien, S., Skalog, R., Hoger, A.: A Classic Network Model Based on the Structure of the Red Blood Cell Membrane. Biophy. J. 70, 146–166 (1996)CrossRefGoogle Scholar
  18. 18.
    Joyce, G.S.: On the Simple Cubic Lattice Green Functions. Proc. Roy. Soc., London Ser. A 445, 463–477 (1994)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kornhuber, R., Yserentant, H.: Multilevel Methods for Elliptic Problems on Domains not Resolved by the Coarse Grid. Contemp. Math. 180, 49–60 (1994)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mandel, J., Brezina, M., Vaněk, P.: Energy optimization of algebraic multigrid bases. Computing 62, 205–228 (1999)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Martinson, P.G.: Fast Multiscale Methods for Lattice Equations. PhD thesis, University of Texas at Austin, 2002Google Scholar
  22. 22.
    Ostoja-Starzewski, M.: Lattice Models in Micromechanics. App. Mech. Review 55(1), 35–60 (2002)CrossRefGoogle Scholar
  23. 23.
    Ostoja-Starzewski, M., Shang, P.Y., Alzebdoh, K.: Spring Network Models in Elasticity and Fractures of Composites and Polycristals. Comp. Math. Sci. 7, 89–93 (1996)Google Scholar
  24. 24.
    Pshenichnov, G.I.: Theory of Lattice Plates and Shells. Singapure: World Scientific 1993Google Scholar
  25. 25.
    Ruge, J., Stüben, K.: Algebraic multigrid. In: McCormick, S. (ed.), Multigrid Methods, pp. 73–130, Philadelphia, Pennsylvania: SIAM 1987Google Scholar
  26. 26.
    Shewchuk, J.: Triangle:Engineering a 2D Quality Mesh Generator and Delaunay Triangulator. In First Workshop on Appl. Geom., pp. 124–133 , Philadelphia, Pennsylvania, USA: Association for Computing Machinery 1996Google Scholar
  27. 27.
    Shewchuk, J.: Tetrahedral Mesh Generation by Delaunay Refinement. In Proc. Of the 14th Annual Symp. On Comp. Geom., pp. 86–95, Minneapolis, USA: Association of Computing Machinery 1998Google Scholar
  28. 28.
    Xu, J.: The auxiliary space method and optimal multigrid precondition-ing techniques for unstructured grids. Computing 56, 215–235 (1996)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.ICESUniversity of Texas at AustinAustinUSA
  2. 2.Institut für MathematikUniversität ZürichZürichSwitzerland

Personalised recommendations