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


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.

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  1. 1.

    Babuška, I., Guo, B.: Problems of unbounded lattices, mathematical framework. in preparation.

  2. 2.

    Babuška, I, Sauter, S.: Algebraic Algorithms for the Analysis of Mechanical Trusses. Technical Report 16-2002, Universität Zürich, http://www.math.unizh.ch/fileadmin/math/preprints/16-02.pdf, 2002. to appear in MathComp.

  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 1995

  4. 4.

    Chan, T., Xu, J., Zikatanov, L.: An agglomeration multigrid method for unstructured grids. Contemp. Math. 218, 67–81 (1998)

    Article  Google Scholar 

  5. 5.

    Chew, L.P.: Guaranteed-quality Delaunay meshing in 3D. In Proc. 13th Symp. Comp. Geom., pp. 391–393, ACM, 1997

  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)

    Article  Google 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)

    MathSciNet  Article  Google 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 1996

  10. 10.

    Gibson, L., Ashby, M.: Cellular Solids, Structures and Properties. Exeter: Pergamon Press 1989

  11. 11.

    Glasser, M.L., Boersma, J.: Exact Values for the Cubic Lattice Green Functions. J. Phys., 133, 5017–5023 (2000)

    MathSciNet  Google 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. 59

  13. 13.

    Hackbusch, W.: A sparse matrix arithmetic based on ℋ-Matrices. Part I: Introduction to ℋ-Matrices. Computing 62, 89–108 (1999)

    MathSciNet  Google 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)

    MathSciNet  Google 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 2000

  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)

    Article  Google Scholar 

  18. 18.

    Joyce, G.S.: On the Simple Cubic Lattice Green Functions. Proc. Roy. Soc., London Ser. A 445, 463–477 (1994)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Mandel, J., Brezina, M., Vaněk, P.: Energy optimization of algebraic multigrid bases. Computing 62, 205–228 (1999)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Martinson, P.G.: Fast Multiscale Methods for Lattice Equations. PhD thesis, University of Texas at Austin, 2002

  22. 22.

    Ostoja-Starzewski, M.: Lattice Models in Micromechanics. App. Mech. Review 55(1), 35–60 (2002)

    Article  Google 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 1993

  25. 25.

    Ruge, J., Stüben, K.: Algebraic multigrid. In: McCormick, S. (ed.), Multigrid Methods, pp. 73–130, Philadelphia, Pennsylvania: SIAM 1987

  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 1996

  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 1998

  28. 28.

    Xu, J.: The auxiliary space method and optimal multigrid precondition-ing techniques for unstructured grids. Computing 56, 215–235 (1996)

    MathSciNet  Article  Google Scholar 

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G. Wittum

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Babuška, I., Sauter, S. Efficient solution of lattice equations by the recovery method Part I: Scalar elliptic problems. Comput. Visual Sci. 7, 113–119 (2004). https://doi.org/10.1007/s00791-004-0146-z

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  • Bilinear Form
  • Element Discretisation
  • Lattice Equation
  • Multigrid Method
  • Recovery Method