Numerische Mathematik

, Volume 43, Issue 1, pp 59–82 | Cite as

A method of solving nonlinear variational problems by nonlinear transformation of the objective functional. Part I

  • Alexander Eydeland
Article

Summary

A general globally convergent iterative method for solving nonlinear variational problems is introduced. The method is applied to a temperature control problem and to the minimal surface problem. Several aspects of finite element implementation of the method are discussed.

Subject Classifications

AMS(MOS): 65K10 CR: 5.15 5.41 

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References

  1. 1.
    Axelsson, O.: Solution of linear systems of equations: Iterative methods. In: Sparse matrix techniques. Barker, V.A. (ed.). Berlin, Heidelberg, New York: Springer, pp. 1–52, 1977Google Scholar
  2. 2.
    Ciarlet, P.G.: Numerical analysis of finite element method. Montreal: Les Presses de l'Université de Montreal 1976Google Scholar
  3. 3.
    Concus, P.: Numerical solution of the minimal surface equation. Math. Comput.21, 340–350 (1967)Google Scholar
  4. 4.
    Concus, P.: Numerical solution of the minimal surface equation by block nonlinear successive over-relaxation. Information processing. Amsterdam: North-Holland Publishing Company,68, pp. 153–157, 1969Google Scholar
  5. 5.
    Concus, P., Golub, G.H., O'Leary, D.P.: Numerical solution of nonlinear elliptic partial differential equations by a generalized conjugate gradient method. Computing19, 321–339 (1978)Google Scholar
  6. 6.
    Duvant, G., Lions, J.L.: Les inequalities en mecanique et en physique. Paris: Dunod 1972Google Scholar
  7. 7.
    Eydeland, A.: Ph.D. thesis. New York: Courant Institute of Math. Sciences 1982Google Scholar
  8. 8.
    Eydeland, A.: Method of transformation of objective functions. In: Problems of mathematical simulation. Moscow: Radio and Electronics Inst. of Acad. of Sci. of the USSR, pp. 79–86, 1979 (in Russian)Google Scholar
  9. 9.
    George, J.A.: Solution of linear systems of equations: Direct methods for finite element problems. In: Sparse matrix techniques. Barker, V.A. (ed.). Berlin, Heidelberg, New York: Springer, pp. 52–101, 1977Google Scholar
  10. 10.
    Glowinsky, R., Lions, J.L., Tremolieres, R.: Analyse numerique des inequations variationelles. Paris: Dunod 1976Google Scholar
  11. 11.
    Korobochkin, B., Eydeland, A.: Application of the objective function transformation method to one class of optimization problems. In: Mathematical methods of solving economical problems. Moscow: Nauka, pp. 72–84, 1980 (in Russian)Google Scholar
  12. 12.
    Ladyzhenskaya, O.A., Uraltseva, N.N.: Linear and quasilinear elliptic equations, 2nd ed. Moscow: Nauka 1973Google Scholar
  13. 13.
    Young, D.M.: Iterative solution of large linear systems. New York: Academic Press 1971Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Alexander Eydeland
    • 1
  1. 1.Department of MathematicsUniversity of MassachusettsAmherstUSA

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