The application of sparse matrix methods to the numerical solution of nonlinear elliptic partial differential equations

  • S. C. Eisenstat
  • M. H. Schultz
  • A. H. Sherman
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 430)


We present a new algorithm for solving general semilinear, elliptic partial differential equations. The algorithm is based on Newton's Method but uses an approximate iterative method to solve the linear systems that arise at each step of Newton's Method. We show that the algorithm can maintain the quadratic convergence of Newton's Method and that it may be substantially faster than other available methods for semilinear or nonlinear partial differential equations.


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  1. [1]
    R. Bartels and J. W. Daniel. A conjugate gradient approach to nonlinear elliptic boundary value problems in irregular regions. Report #CNA 63, Center for Numerical Analysis, The University of Texas at Austin, 1973.Google Scholar
  2. [2]
    G. Birkhoff and D. J. Rose. Elimination by nested dissection. Complexity of Sequential and Parallel Numerical Algorithms, edited by J. F. Traub, Academic Press, New York, 1973.Google Scholar
  3. [3]
    A. Chang. Application of sparse matrix methods in electric power system analysis. Sparse Matrix Proceedings, edited by R. A. Willoughby, IBM Research Report #RA1, Yorktown Heights, New York, 1968.Google Scholar
  4. [4]
    M. A. Diamond. An Economical Algorithm for the Solution of Finite Difference Equations, PhD dissertation, Department of Computer Science, University of Illinois, 1971.Google Scholar
  5. [5]
    E. G. D'Jakonov. On certain iterative methods for solving non-linear difference equations. Proceedings of the Conference on the Numerical Solution of Differential Equations, (Scotland, June 1969), Springer-Verlag, Heidelberg, 1969.Google Scholar
  6. [6]
    F. W. Dorr. The direct solution of the discrete Poisson equation on a rectangle. SIAM Review 12:248–263, 1970.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    A. J. Hoffman, M. S. Martin, and D. J. Rose. Complexity bounds for regular finite difference and finite element grids. SIAM Journal on Numerical Analysis 10:364–369, 1973.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    J. M. Ortega and W. C. Rheinboldt. Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.zbMATHGoogle Scholar
  9. [9]
    L. F. Richardson. The approximate arithmetical solution by finite differences of physical problems involving differential equations with an application to the stresses in a masonry dam. Philosophical Transactions of the Royal Society, London, Series A(210): 307–357, 1910.CrossRefzbMATHGoogle Scholar
  10. [10]
    A. H. Sherman. PhD dissertation, Department of Computer Science, Yale University. To appear.Google Scholar
  11. [11]
    H. L. Stone. Iterative solution of implicit approximations of multidimensional partial differential equations. SIAM Journal on Numerical Analysis 10:530–558, 1968.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    J. H. Wilkinson. The Algebraic Eigenvalue Problem, Clarendon Press, London, 1965.zbMATHGoogle Scholar
  13. [13]
    D. M. Young. Iterative Solutions of Large Linear Systems, Academic Press, New York, 1971.Google Scholar

Copyright information

© Springer-Verlag 1974

Authors and Affiliations

  • S. C. Eisenstat
  • M. H. Schultz
  • A. H. Sherman
    • 1
  1. 1.Department of Computer ScienceYale UniversityUSA

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