Preconditioning and twolevel multigrid methods of arbitrary degree of approximation
Authors:
O. Axelsson and I. Gustafsson
Journal:
Math. Comp. 40 (1983), 219242
MSC:
Primary 65N50; Secondary 65N30
MathSciNet review:
679442
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Abstract: Let h be a mesh parameter corresponding to a finite element mesh for an elliptic problem. We describe preconditioning methods for twolevel meshes which, for most problems solved in practice, behave as methods of optimal order in both storage and computational complexity. Namely, per mesh point, these numbers are bounded above by relatively small constants for all , where is small enough to cover all but excessively fine meshes. We note that, in practice, multigrid methods are actually solved on a finite, often even a fixed number of grid levels, in which case also these methods are not asymptotically optimal as . Numerical tests indicate that the new methods are about as fast as the best implementations of multigrid methods applied on general problems (variable coefficients, general domains and boundary conditions) for all but excessively fine meshes. Furthermore, most of the latter methods have been implemented only for difference schemes of second order of accuracy, whereas our methods are applicable to higher order approximations. We claim that our scheme could be added fairly easily to many existing finite element codes.
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 O. Axelsson, "A class of iterative methods for finite element equations," Comput. Methods Appl. Mech. Engrg., v. 9, 1976, pp. 123137. MR 0433836 (55:6807)
 [2]
 O. Axelsson & I. Gustafsson, A Preconditioned Conjugate Gradient Method for Finite Element Equations, Which is Stable for Rounding Errors, Information Processing 80 (S. H. Lavington, ed.), NorthHolland, Amsterdam, 1980, pp. 723728.
 [3]
 R. Bank & T. Dupont, Analysis of a TwoLevel Scheme for Solving Finite Element Equations, Report CNA159, Center for Numerical Analysis, The University of Texas at Austin, 1980.
 [4]
 A. Brandt, "Multilevel adaptive solutions to boundary value problems," Math. Comp., v. 31, 1977, pp. 333390. MR 0431719 (55:4714)
 [5]
 I. Fried, "Bounds on the extremal eigenvalues of the finite element stiffness and mass matrices and their spectral condition numbers," J. Sound Vibration, v. 22, 1972, pp. 407418.
 [6]
 I. Gustafsson, Stability and Rate of Convergence of Modified Incomplete Cholesky Factorization Methods, Thesis, Report 79.02R, Department of Computer Sciences, Chalmers University of Technology, Göteborg, Sweden, 1979.
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 P. W. Hemker, "Introduction to multigrid methods," Colloquium Numerical Solution of Partial Differential Equations (J. G. Verwer, ed.), MC SYLLABUS 44, Mathematisch Centrum, Amsterdam, 1980, pp. 5967. MR 607518 (83d:65179a)
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 D. Kershaw, "The incomplete Cholesky conjugate gradient method for the iterative solution of systems of linear equations," J. Comput. Phys., v. 26, 1978, pp. 4365. MR 0488669 (58:8190)
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 J. A. Meijerink & H. A. van der Vorst, "An iterative solution method for linear systems of which the coefficient matrix is a symmetric Mmatrix," Math. Comp., v. 31, 1977, pp. 148162. MR 0438681 (55:11589)
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 U. Trottenberg, Private communication, 1981.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198306794423
PII:
S 00255718(1983)06794423
Article copyright:
© Copyright 1983
American Mathematical Society
