An Abridged History of Cell Discretization

  • John Greenstadt
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 11)


This brief account will call attention to a line of research which stretches over 40 years, and which appears now to be joining the mainstream of work on the discretization of linear partial differential equations. The first in a series of papers [1] describing the Method of Cells, now called the Cell Discretization Algorithm (CD or CDA), was published in 1959.


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  1. 1.
    J. Greenstadt, “On the reduction of continuous problems to discrete form”, IBM Jour. Res. Dev., vol. 3, pp. 355–363 (1959)MathSciNetCrossRefGoogle Scholar
  2. 2.
    J. Greenstadt, “Cell discretization I - variational basis”, IBM New York Sci. Ctr. Report (1967)Google Scholar
  3. 3.
    J. Greenstadt, “Cell discretization” in Conference on Applications of Numerical Analysis, Dundee, Scotland, ed. J. Ll. Morris, Springer-Verlag (1971)Google Scholar
  4. 4.
    J. Greenstadt, “Some numerical tests of cell discretization”, IBM Palo Alto Sci. Ctr. Report (1972)Google Scholar
  5. 5.
    P.A. Raviart, J.M. Thomas, “Primal hybrid finite element methods for second-order elliptic equations”, Math. Comp., 31 (138) Rpp. 391–413 (1977)Google Scholar
  6. 6.
    J. Greenstadt, “The cell discretization algorithm for elliptic partial differential equations”, Siam J. Sci. Stat. Comput., 3, pp. 261–288 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    M.R. Dorr, “On the discretization of interdomain coupling in elliptic boundary value problems”, in: Domain Decomposition Methods, Eds., T.F. Chan, R. Glowinski, J. Periaux, O.B. Widlund, SIAM, Philadelphia, PA 1989Google Scholar
  8. 8.
    C. Bernardi, Y. Maday & A.T. Patera, “A new nonconforming approach to domain decomposition; the Mortar Element Method”, College de France Seminar, (1990), Pitman, eds., H. Brezis, J.-L. Lions.Google Scholar
  9. 9.
    J. Greenstadt, “Cell discretization of nonselfadjoint linear elliptic PDE’s”, SIAM J. Sci. Stat. Comput., 12 (1991), pp. 1074–1108.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    J. Greenstadt, “ Solution of elliptic systems of partial differential equations by cell discretization”, SIAM J. Sci. Stat. Comput., 14, pp. 627–653 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
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    H. Swann, “ On the use of Lagrange multipliers in domain decomposition for solving elliptic problems”, Math. of Comp., vol. 60, no. 201, pp. 49–78 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    J. Greenstadt, “The removal of overshoot in P.D.E. solutions by the use of special basis functions”, Computer Meth. in Appl. Mech. and Eng., (1994)Google Scholar
  13. 13.
    J. Greenstadt, “The application of the cell discretization method to time-dependent problems”, Computer Meth. in Appl. Math. and Engineering, (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • John Greenstadt
    • 1
  1. 1.NASA-Ames Research CenterUSA

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