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Mathematics of Computation

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Hierarchical bases for elliptic problems

Author: W. Dörfler
Journal: Math. Comp. 58 (1992), 513-529, S29
MSC: Primary 65N30; Secondary 65F35
MathSciNet review: 1122064
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Abstract: Linear systems of equations with positive and symmetric matrices often occur in the numerical treatment of linear and nonlinear elliptic boundary value problems. If the CG algorithm is used to solve these equations, one is able to speed up the convergence by "preconditioning." The method of preconditioning with hierarchical basis has already been considered for the Laplace equation in two space dimensions and for linear conforming elements. In the present work this method is generalized to a large class of conforming and nonconforming elements.

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  • [1] O. Axelsson and V. A. Barker, Finite element solutions of boundary value problems, Academic Press, London, 1984. MR 758437 (85m:65116)
  • [2] E. Bänsch, Local mesh refinement in 2 and 3 dimensions, SFB 256, report no. 6, Universität Bonn, 1989.
  • [3] J. H. Bramble, J. E. Pasciak, and J. Xu, Parallel multilevel preconditioners, Math. Comp. 55 (1990), 1-22. MR 1023042 (90k:65170)
  • [4] P. G. Ciarlet, The finite element method for elliptic problems, North-Holland, New York, 1978. MR 0520174 (58:25001)
  • [5] W. Dörfler, The conditioning of the stiffness matrix for certain elements approximating the incompressibility condition in fluid dynamics, Numer. Math. 58 (1990), 203-214. MR 1069279 (91k:65142)
  • [6] R. Glowinski, Numerical methods for nonlinear variational problems, Springer, New York, 1984. MR 737005 (86c:65004)
  • [7] W. Hackbusch, Theorie und Numerik elliptischer Differentialgleichungen, Teubner, Stuttgart, 1986. MR 1600003 (98j:35002)
  • [8] M. C. Rivara, Algorithms for refining triangular grids suitable for adaptive and multigrid techniques, Internat. J. Numer. Methods Engrg. 20 (1984), 745-756. MR 739618 (85h:65258)
  • [9] H. Yserentant, On the multi-level splitting of finite element spaces, Numer. Math. 49 (1986), 379-412. MR 853662 (88d:65068a)
  • [10] -, Two preconditioners based on the multi-level splitting of finite element spaces, manuscript, Universität Dortmund, 1990.

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Article copyright: © Copyright 1992 American Mathematical Society

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