Mathematics of Computation

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ISSN 1088-6842(e) ISSN 0025-5718(p)

Multigrid and multilevel methods for nonconforming $Q_1$ elements

Author(s): Zhangxin Chen; Peter Oswald.
Journal: Math. Comp. 67 (1998), 667-693.
MSC (1991): Primary 65N30, 65N22, 65F10
MathSciNet review: 1451319
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Abstract: In this paper we study theoretical properties of multigrid algorithms and multilevel preconditioners for discretizations of second-order elliptic problems using nonconforming rotated $Q_1$ finite elements in two space dimensions. In particular, for the case of square partitions and the Laplacian we derive properties of the associated intergrid transfer operators which allow us to prove convergence of the $\mathcal{W}$ -cycle with any number of smoothing steps and close-to-optimal condition number estimates for $\mathcal{V}$ -cycle preconditioners. This is in contrast to most of the other nonconforming finite element discretizations where only results for $\mathcal{W}$ -cycles with a sufficiently large number of smoothing steps and variable $\mathcal{V}$ -cycle multigrid preconditioners are available. Some numerical tests, including also a comparison with a preconditioner obtained by switching from the nonconforming rotated $Q_1$ discretization to a discretization by conforming bilinear elements on the same partition, illustrate the theory.

References:

1.: T. Arbogast and Zhangxin Chen, On the implementation of mixed methods as nonconforming methods for second order elliptic problems, Math. Comp. 64 (1995), 943-972. MR 95k:65102
2.: R. Bank and T. Dupont, An optimal order process for solving finite element equations, Math. Comp. 36 (1981), 35-51. MR 82b:65113
3.: D. Braess and R. Verfürth, Multigrid methods for nonconforming finite element methods, SIAM J. Numer. Anal. 27 (1990), 979-986. MR 91j:65164
4.: J. Bramble, Multigrid Methods, Pitman Research Notes in Math., vol. 294, Longman, London, 1993. MR 95b:65002
5.: J. Bramble, J. Pasciak, and J. Xu, Parallel multilevel preconditioners, Math. Comp. 55 (1991), 1-22. MR 90k:65170
6.: J. Bramble, J. Pasciak, and J. Xu, The analysis of multigrid algorithms with non-nested spaces or non-inherited quadratic forms, Math. Comp. 56 (1991), 1-34. MR 91h:65159
7.: S. Brenner, An optimal-order multigrid method for P1 nonconforming finite elements, Math. Comp. 52 (1989), 1-15. MR 89f:65119
8.: S. Brenner, Multigrid methods for nonconforming finite elements, Proceedings of Fourth Copper Mountain Conference on Multigrid Methods, J. Mandel, et al., eds., SIAM, Philadelphia, 1989, pp. 54-65. MR 91h:65189
9.: S. Brenner, Convergence of nonconforming multigrid methods without full elliptic regularity, Preprint, 1995, submitted.
10.: Zhangxin Chen, Analysis of mixed methods using conforming and nonconforming finite element methods, RAIRO Modèl. Math. Anal. Numer. 27 (1993), 9-34. MR 94c:65132
11.: Zhangxin Chen, Projection finite element methods for semiconductor device equations, Computers Math. Applic. 25 (1993), 81-88. MR 93k:65092
12.: Zhangxin Chen, Equivalence between and multigrid algorithms for nonconforming and mixed methods for second order elliptic problems, East-West J. Numer. Math. 4 (1996), 1-33. CMP 96:13
13.: Zhangxin Chen, R. E. Ewing, Y. Kuznetsov, R. Lazarov, and S. Maliassov, Multilevel preconditioners for mixed methods for second order elliptic problems, J. Numer. Lin. Alg. Appl. 30 (1996), 427-453. CMP 97:04
14.: Zhangxin Chen, R. Ewing, and R. Lazarov, Domain decomposition algorithms for mixed methods for second order elliptic problems, Math. Comp. 65 (1996), 467-490. MR 96g:65117
15.: Zhangxin Chen, D. Y. Kwak, and Y. J. Yon, Multigrid algorithms for nonconforming and mixed methods for nonsymmetric and indefinite problems, IMA Preprint Series #1277, 1994, SIAM J. Scientific Computing, 1998 to appear.
16.: M. Griebel and P. Oswald, On the abstract theory of additive and multiplicative Schwarz algorithms, Numer. Math. 70 (1995), 163-180. MR 96a:65164
17.: P. Kloucek, B. Li, and M. Luskin, Analysis of a class of nonconforming finite elements for crystalline microstructures, Math. Comp. 65 (1996), 1111-1135. MR 97a:73076
18.: P. Kloucek and M. Luskin, The computation of the dynamics of martensitic microstructure, Continuum Mech. Thermodyn. 6 (1994), 209-240. MR 95d:73009
19.: C. Lee, A nonconforming multigrid method using conforming subspaces, Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, N. Melson et al., eds., NASA Conference Publication, vol. 3224, 1993, pp.317-330.
20.: P. Oswald, On a hierarchical basis multilevel method with nonconforming P1 elements, Numer. Math. 62 (1992), 189-212. MR 93b:65059
21.: P. Oswald, Multilevel Finite Element Approximation : Theory and Application, Teubner Skripten zur Numerik, Teubner, Stuttgart, 1994. MR 95k:65110
22.: P. Oswald, Preconditioners for nonconforming discritizations, Math. Comp. 65 (1996), 923-941. MR 96j:65056
23.: P. Oswald, Intergrid transfer operators and multilevel preconditioners for nonconforming discretizations, Appl. Numer. Math. 23 (1997), 139-158. CMP 97:09
24.: R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numer. Meth. Partial Diff. Equations 8 (1992), 97-111. MR 92i:65170
25.: P. Raviart and J. Thomas, A mixed finite element method for second order elliptic problems, Mathematical aspects of the FEM, Lecture Notes in Mathematics, 606, Springer-Verlag, Berlin & New York (1977), pp.292-315. MR 58:3547
26.: S. Turek, Multigrid techniques for a divergence-free finite element discretization, East-West J. Numer. Math. 2 (1994), 229-255. MR 96c:65195
27.: M. Wang, The W-cycle multigrid method for finite elements with nonnested spaces, Adv. in Math. 23 (1994), 238-250. MR 95e:65118
28.: H. Yserentant, Old and new convergence proofs for multigrid methods, Acta Numerica, Cambr. Univ. Press, Cambridge, 1993, pp.285-236. MR 94i:65128
29.: J. Xu, Convergence estimates for some multigrid algorithms, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, T. F. Chan et al., eds., SIAM, Philadelphia, 1990, pp. 174-187. MR 91f:65194
30.: J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Review 34 (1992), 581-613. MR 93k:65029

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