Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Analysis of a multilevel iterative method for nonlinear finite element equations


Authors: Randolph E. Bank and Donald J. Rose
Journal: Math. Comp. 39 (1982), 453-465
MSC: Primary 65N30; Secondary 65H10
DOI: https://doi.org/10.1090/S0025-5718-1982-0669639-X
MathSciNet review: 669639
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The multilevel iterative technique is a powerful technique for solving the systems of equations associated with discretized partial differential equations. We describe how this technique can be combined with a globally convergent approximate Newton method to solve nonlinear partial differential equations. We show that asymptotically only one Newton iteration per level is required; thus the complexity for linear and nonlinear problems is essentially equal.


References [Enhancements On Off] (What's this?)

  • [1] Ivo Babuška & A. K. Aziz, "Survey lectures on the mathematical foundations of the finite element method," in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A. K. Aziz, Ed., Academic Press, New York, 1972, pp. 111-184. MR 0421106 (54:9111)
  • [2] Randolph E. Bank & Todd F. Dupont, "An optimal order process for solving finite element equations," Math. Comp., v. 36, 1981, pp. 35-51. MR 595040 (82b:65113)
  • [3] Randolph E. Bank, "A comparison of two multi-level iterative methods for non-symmetric and indefinite elliptic finite element equations," SIAM J. Numer. Anal., v. 18, 1981, pp. 724-743. MR 622706 (82f:65110)
  • [4] Randolph E. Bank & Donald J. Rose, "Parameter selection for Newton-like methods applicable to nonlinear partial differential equations," SIAM J. Numer. Anal., v. 17, 1980, 806-822.
  • [5] Randolph E. Bank & Donald J. Rose, "Global approximate Newton methods," Numer. Math., v. 37, 1981, pp. 279-295.
  • [6] Randolph E. Bank & Andrew H. Sherman, "Algorithmic aspects of the multi-level solution of finite element equations," in Sparse Matrix Proceedings--1978, (I. S. Duff and G. W. Stewart, Eds.), SIAM, Philadelphia, Pa., 1979, pp. 62-89. MR 566371 (81g:65144)
  • [7] Achi Brandt, "Multi-level adaptive solutions to boundary value problems," Math. Comp., v. 31, 1977, pp. 333-390. MR 0431719 (55:4714)
  • [8] Achi Brandt & Steve McCormick, Private communication, 1980.
  • [9] Wolfgang Hackbusch, On the Convergence of a Multi-Grid Iteration Applied to Finite Element Equations, Technical Report 77-8, Mathematisches Institut, Universität zu Köln, 1977.
  • [10] Wolfgang Hackbusch, "On the fast solution of nonlinear elliptic equations," Numer. Math., v. 32, 1979, pp. 83-95. MR 525639 (80b:65128)
  • [11] A. R. Hutson, "Role of dislocations in the electrical conductivity of cds," Phys. Rev. Lett., v. 46, 1981, pp. 1159-1162.
  • [12] Lois Mansfield, "On the solution of nonlinear finite element systems," SIAM J. Numer. Anal., v. 17, 1980, pp. 752-765. MR 595441 (82a:65090)
  • [13] R. A. Nicolaides, "On the $ {l^2}$ convergence of an algorithm for solving finite element systems," Math. Comp., v. 31, 1977, pp. 892-906. MR 0488722 (58:8239)
  • [14] Alfred H. Schatz, "An observation concerning Ritz-Galerkin methods with indefinite bilinear forms," Math. Comp., v. 28, 1974, pp. 959-962. MR 0373326 (51:9526)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N30, 65H10

Retrieve articles in all journals with MSC: 65N30, 65H10


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1982-0669639-X
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society