Analysis of a multilevel iterative method for nonlinear finite element equations
Authors:
Randolph E. Bank and Donald J. Rose
Journal:
Math. Comp. 39 (1982), 453465
MSC:
Primary 65N30; Secondary 65H10
MathSciNet review:
669639
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Abstract 
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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.
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 [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. 111184. 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. 3551. MR 595040 (82b:65113)
 [3]
 Randolph E. Bank, "A comparison of two multilevel iterative methods for nonsymmetric and indefinite elliptic finite element equations," SIAM J. Numer. Anal., v. 18, 1981, pp. 724743. MR 622706 (82f:65110)
 [4]
 Randolph E. Bank & Donald J. Rose, "Parameter selection for Newtonlike methods applicable to nonlinear partial differential equations," SIAM J. Numer. Anal., v. 17, 1980, 806822.
 [5]
 Randolph E. Bank & Donald J. Rose, "Global approximate Newton methods," Numer. Math., v. 37, 1981, pp. 279295.
 [6]
 Randolph E. Bank & Andrew H. Sherman, "Algorithmic aspects of the multilevel solution of finite element equations," in Sparse Matrix Proceedings1978, (I. S. Duff and G. W. Stewart, Eds.), SIAM, Philadelphia, Pa., 1979, pp. 6289. MR 566371 (81g:65144)
 [7]
 Achi Brandt, "Multilevel adaptive solutions to boundary value problems," Math. Comp., v. 31, 1977, pp. 333390. MR 0431719 (55:4714)
 [8]
 Achi Brandt & Steve McCormick, Private communication, 1980.
 [9]
 Wolfgang Hackbusch, On the Convergence of a MultiGrid Iteration Applied to Finite Element Equations, Technical Report 778, Mathematisches Institut, Universität zu Köln, 1977.
 [10]
 Wolfgang Hackbusch, "On the fast solution of nonlinear elliptic equations," Numer. Math., v. 32, 1979, pp. 8395. MR 525639 (80b:65128)
 [11]
 A. R. Hutson, "Role of dislocations in the electrical conductivity of cds," Phys. Rev. Lett., v. 46, 1981, pp. 11591162.
 [12]
 Lois Mansfield, "On the solution of nonlinear finite element systems," SIAM J. Numer. Anal., v. 17, 1980, pp. 752765. MR 595441 (82a:65090)
 [13]
 R. A. Nicolaides, "On the convergence of an algorithm for solving finite element systems," Math. Comp., v. 31, 1977, pp. 892906. MR 0488722 (58:8239)
 [14]
 Alfred H. Schatz, "An observation concerning RitzGalerkin methods with indefinite bilinear forms," Math. Comp., v. 28, 1974, pp. 959962. MR 0373326 (51:9526)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0025571819820669639X
PII:
S 00255718(1982)0669639X
Article copyright:
© Copyright 1982
American Mathematical Society
