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

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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.

**[1]**Ivo Babuška and A. K. Aziz,*Survey lectures on the mathematical foundations of the finite element method*, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR**0421106****[2]**Randolph E. Bank and Todd Dupont,*An optimal order process for solving finite element equations*, Math. Comp.**36**(1981), no. 153, 35–51. MR**595040**, https://doi.org/10.1090/S0025-5718-1981-0595040-2**[3]**Randolph E. Bank,*A comparison of two multilevel iterative methods for nonsymmetric and indefinite elliptic finite element equations*, SIAM J. Numer. Anal.**18**(1981), no. 4, 724–743. MR**622706**, https://doi.org/10.1137/0718048**[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 and A. H. Sherman,*Algorithmic aspects of the multilevel solution of finite element equations*, Sparse Matrix Proceedings 1978 (Sympos. Sparse Matrix Comput., Knoxville, Tenn., 1978) SIAM, Philadelphia, Pa., 1979, pp. 62–89. MR**566371****[7]**Achi Brandt,*Multi-level adaptive solutions to boundary-value problems*, Math. Comp.**31**(1977), no. 138, 333–390. MR**0431719**, https://doi.org/10.1090/S0025-5718-1977-0431719-X**[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 solutions of nonlinear elliptic equations*, Numer. Math.**32**(1979), no. 1, 83–95. MR**525639**, https://doi.org/10.1007/BF01397652**[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.**17**(1980), no. 6, 752–765. MR**595441**, https://doi.org/10.1137/0717063**[13]**R. A. Nicolaides,*On the 𝑙² convergence of an algorithm for solving finite element equations*, Math. Comp.**31**(1977), no. 140, 892–906. MR**0488722**, https://doi.org/10.1090/S0025-5718-1977-0488722-3**[14]**Alfred H. Schatz,*An observation concerning Ritz-Galerkin methods with indefinite bilinear forms*, Math. Comp.**28**(1974), 959–962. MR**0373326**, https://doi.org/10.1090/S0025-5718-1974-0373326-0

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DOI:
https://doi.org/10.1090/S0025-5718-1982-0669639-X

Article copyright:
© Copyright 1982
American Mathematical Society