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Higher-dimensional nonnested multigrid methods

Authors: L. Ridgway Scott and Shangyou Zhang
Journal: Math. Comp. 58 (1992), 457-466
MSC: Primary 65N55; Secondary 65N30
MathSciNet review: 1122077
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Abstract: Nonnested multigrid methods are shown to be optimal-order solvers for systems of finite element equations arising from elliptic boundary problems in any space dimension. Results are derived for Lagrange-type elements of arbitrary degree.

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  • [1] R. A. Adams Sobolev spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
  • [2] R. Bank and T. Dupont, An optimal order process for solving finite element equations, Math. Comp. 36 (1981), pp. 35-51. MR 595040 (82b:65113)
  • [3] J. H. Bramble, J. E. Pasciak, and J. Xu, The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms, Math. Comp. 56 (1991), 1-34. MR 1052086 (91h:65159)
  • [4] S. C. Brenner, An optimal-order multigrid method for $ {P_1}$ nonconforming finite elements, Math. Comp. 52 (1989), pp. 1-15. MR 946598 (89f:65119)
  • [5] P. G. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, New York, Oxford, 1978. MR 0520174 (58:25001)
  • [6] M. Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Math., vol. 1341, Springer, Berlin and New York, 1988. MR 961439 (91a:35078)
  • [7] P. Grisvard, Elliptic problems in nonsmooth domains, Pitman, New York and London, 1985. MR 775683 (86m:35044)
  • [8] W. Hackbusch, Multigrid methods and applications, Springer, Berlin and New York, 1985. MR 814495 (87e:65082)
  • [9] S. F. McCormick, ed., Multigrid Methods, Frontiers in Applied Mathematics, SIAM, Philadelphia, PA, 1987. MR 972752 (89m:65004)
  • [10] L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), 483-493. MR 1011446 (90j:65021)
  • [11] L. R. Scott and S. Zhang, A nonnested multigrid method for three dimensional boundary value problems: An introduction to the NMGTM code, in preparation.
  • [12] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ, 1970. MR 0290095 (44:7280)
  • [13] R. Verfürth, A multilevel algorithm for mixed problems, SIAM J. Numer. Anal. 21 (1984), pp. 264-271. MR 736330 (85f:65112)
  • [14] J. Wloka, Partial differential equations, Cambridge Univ. Press, London and New York, 1987. MR 895589 (88d:35004)
  • [15] S. Zhang, Multi-level iterative techniques, Ph.D. thesis, Pennsylvania State University, 1988.
  • [16] -, Successive subdivisions of tetrahedra and multigrid methods on tetrahedral meshes, (submitted), also in [15].
  • [17] -, Optimal-order nonnested multigrid methods for solving finite element equations I: On quasiuniform meshes, Math. Comp. 55 (1990), 23-36. MR 1023054 (91g:65268)
  • [18] -, Optimal-order nonnested multigrid methods for solving finite element equations II: On non-quasi-uniform meshes, Math. Comp. 55 (1990), 439-450. MR 1035947 (91g:65269)
  • [19] -, Optimal-order nonnested multigrid methods for solving finite element equations III: On degenerate meshes, (submitted).

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Article copyright: © Copyright 1992 American Mathematical Society

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