Mathematics of Computation

Author(s): Susanne C. Brenner.
Journal: Math. Comp. 68 (1999), 25-53.
MSC (1991): Primary 65N55, 65N30
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Abstract: We consider nonconforming multigrid methods for symmetric positive definite second and fourth order elliptic boundary value problems which do not have full elliptic regularity. We prove that there is a bound (

) for the contraction number of the

-cycle algorithm which is independent of mesh level, provided that the number of smoothing steps is sufficiently large. We also show that the symmetric variable

-cycle algorithm is an optimal preconditioner.

1.: A. Adini and R.W. Clough, Analysis of plate bending by the finite element method, NSF Report G. 7337 (1961).
2.: T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for second order elliptic problems, Math. Comp. 64 (1995), 943-972.
3.: J.H. Argyris, I. Fried and D.W. Scharpf, The TUBA family of plate elements for the matrix displacement method, Aero. J. Roy. Aero. Soc. 72 (1968), 701-709.
4.: D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, R.A.I.R.O Modél. Math. Anal. Numér. 19 (1985), 7-32. MR 87g:65126
5.: D.N. Arnold and R.S. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal. 21 (1990), 281-312. MR 91c:65068
6.: R.E. Bank and T.F. Dupont, An optimal order process for solving finite element equations, Math. Comp. 36 (1981), 35-51. MR 82b:65113
7.: J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, Berlin, 1976. MR 58:2349
8.: F.K. Bogner, R.L. Fox and L.A. Schmit, The generation of interelement compatible stiffness and mass matrices by the use of interpolation formulas, Proc. Conf. Matrix Methods in Structural Mechanics (1965).
9.: D. Braess and R. Verfürth, Multigrid methods for nonconforming finite element methods, SIAM J. Numer. Anal. 27 (1990), 979-986. MR 95j:65164
10.: J.H. Bramble, Multigrid Methods, Longman Scientific & Technical, Essex, 1993. MR 95b:65002
11.: J.H. Bramble and S.R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal. 7 (1970), 113-124. MR 41:7819
12.: J.H. Bramble and J.E. Pasciak, New estimates for multigrid algorithms including the $V$ -cycle, Math. Comp. 60 (1993), 447-471. MR 94a:65064
13.: 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 91h:65159
14.: S.C. Brenner, Multigrid Methods for Nonconforming Finite Elements, Dissertation, The University of Michigan, 1988.
15.: -, An optimal-order multigrid method for nonconforming finite elements, Math. Comp. 52 (1989), 1-15. MR 89f:65119
16.: -, An optimal-order nonconforming multigrid method for the biharmonic equation, SIAM J. Numer. Anal. 26 (1989), 1124-1138. MR 90i:65189
17.: -, Multigrid methods for nonconforming finite elements, Copper Mountain Conference on Multigrid methods (J. Mandel, et. al., eds.), SIAM, Philadelphia, 1989, pp. 54-65. MR 91h:65189
18.: -, A nonconforming multigrid method for the stationary Stokes equations, Math. Comp. 52 (1990), 411-437. MR 91d:65167
19.: -, A multigrid algorithm for the lowest-order Raviart-Thomas mixed triangular finite element method, SIAM J. Numer. Anal. 29 (1992), 647-678. MR 93j:65175
20.: -, A nonconforming mixed multigrid method for the pure displacement problem in planar linear elasticity, SIAM J. Numer. Anal. 30 (1993), 116-135. MR 93m:65166
21.: -, A nonconforming mixed multigrid method for the pure traction problem in planar linear elasticity, Math. Comp. 63 (1994), 435-460, S1-S5. MR 95c:73076
22.: -, Two-level additive Schwarz preconditioners for nonconforming finite elements, Domain Decomposition Methods in Scientific and Engineering Computing (D. Keyes and J. Xu, eds.), Proceedings of the Seventh International Conference on Domain Decomposition, Contemp. Math. 180, 1994, pp. 9-14. MR 95j:65134
23.: -, Multigrid methods for parameter dependent problems, RAIRO Modél. Math. Anal. Numér. 30 (1996), 265-297. MR 97c:73076
24.: -, Two-level additive Schwarz preconditioners for nonconforming finite elements, Math. Comp. 65 (1996), 897-921. MR 96j:65117
25.: -, A two-level additive Schwarz preconditioner for nonconforming plate elements, Numer. Math. 72 (1996), 419-447. MR 97h:65147
26.: -, Preconditioning complicated FEMs by simple FEMs, SIAM J. Sci. Comput. 17 (1996), 1269-1274. MR 97g:65226
27.: S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, Berlin, 1994. MR 95f:65001
28.: S.C. Brenner and L.-Y. Sung, Linear finite element methods for planar linear elasticity, Math. Comp. 59 (1992), 321-338. MR 93a:73078
29.: P.G. Ciarlet, Conforming and nonconforming finite element methods for solving the plate problem, Conference on the Numerical Solution of Differential Equations (G.A. Watson, ed.), Lecture Notes in Mathematics, vol. 363, Springer-Verlag, Berlin, 1974, pp. 21-31. MR 54:11806
30.: -, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. MR 58:25001
31.: -, Interpolation error estimates for the reduced Hsieh-Clough-Tocher triangle, Math. Comp. 32 (1978), 335-344. MR 80i:65114
32.: J.F. Ciavaldini and J.C. Nedelec, Sur l'élément de Fraeijs de Veubeke et Sander, RAIRO R-2 (1974), 29-45. MR 52:2247
33.: P. Clément, Approximation by finite element functions using local regularization, R.A.I.R.O. R-2 (1975), 77-84. MR 53:4569
34.: R.W. Clough and J.L. Tocher, Finite element stiffness matrices for analysis of plates in bending, Proceedings of the Conference on Matrix Methods in Structural Mechanics, Wright Patterson A.F.B., Ohio, 1965.
35.: M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations I, R.A.I.R.O. R-3 (1973), 33-75. MR 49:8401
36.: M. Dauge, Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Math., vol. 1341, Springer-Verlag, Berlin, 1988. MR 91a:35078
37.: R.S. Falk, Nonconforming finite element methods for the equations of linear elasticity, Math. Comp. 57 (1991), 529-550. MR 92a:65290
38.: B. Fraeijs de Veubeke, A conforming finite element for plate bending, Int. J. Solids and Structure 4 (1968), 95-108.
39.: P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985. MR 86m:35044
40.: -, Problèmes aux limites dans les polygones, Mode d'emploi, EDF Bull. Direction Études Rech. Sér. C. Math. Inform. 1 (1986), 21-59. MR 87g:35073
41.: -, Singularities in Boundary Value Problems, Springer-Verlag, Berlin, 1992. MR 93h:35004
42.: W. Hackbusch, Multigrid Methods and Applications, Springer-Verlag, Berlin, 1985. MR 87e:65082
43.: M.R. Hanisch, Multigrid preconditioning for the biharmonic Dirichlet problem, SIAM J. Numer. Anal. 30 (1993), 184-214. MR 94e:65131
44.: S.G. Kre[??]in, Ju. I. Petunin and E.M. Semenov, Interpolation of Linear Operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, 1982.
45.: P. Lascaux and P. Lesaint, Some nonconforming finite elements for the plate bending problem, R.A.I.R.O. R-1 (1975), 9-53. MR 54:11941
46.: S. McCormick (ed.), Multigrid Methods, SIAM Frontiers in Applied Mathematics 3, SIAM, Philadelphia, 1987. MR 89m:65004
47.: L.S.D. Morley, The triangular equilibrium problem in the solution of plate bending problems, Aero. Quart. 19 (1968), 149-169.
48.: J. Ne\v{c}as, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, 1967. MR 37:3168
49.: P. Peisker and D. Braess, A conjugate gradient method and a multigrid algorithm for Morley's finite element approximation of the biharmonic equation, Numer. Math. 50 (1987), 567-586. MR 88e:65147
50.: P. Peisker, W. Rust and E. Stein, Iterative solution methods for plate bending problems: multigrid and preconditioned cg algorithm, SIAM J. Numer. Anal. 27 (1990), 1450-1465. MR 91h:73055
51.: P. Percell, On cubic and quartic Clough-Tocher finite elements, SIAM J. Numer. Anal. 13 (1976), 100-103. MR 53:11963
52.: R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numer. Meth. PDE 8 (1992), 97-111. MR 92i:65170
53.: P.-A. Raviart and J.M. Thomas, A mixed finite element method for second order elliptic problems, Mathematical Aspects of the Finite Element Method (I.Galligani and E. Magenes, eds.), Lecture Notes in Mathematics, vol. 606, Springer-Verlag, Berlin, 1977, pp. 292-315. MR 58:3547
54.: G. Sander, Bornes supérieures et inférieures dans l'analyse matricielle des plaques en flexion-torsion, Bull. Soc. Roy. Sci. Liège 33 (1964), 456-494. MR 30:764
55.: P. Schreiber and S. Turek, Multigrid results for the nonconforming Morley element, preprint (1993).
56.: L.R. Scott and S. Zhang, Finite element interpolation of non-smooth functions satisfying boundary conditions, Math. Comp. 54 (1990), 483-493. MR 90j:65021
57.: R. Seeley, Interpolation in $L^{p}$ with boundary conditions, Studia Math. 44 (1972), 47-60. MR 47:3981
58.: Z. Shi, On the convergence of the incomplete biquadratic nonconforming plate element, Math. Numer. Sinica 8 (1986), 53-62 (Chinese). MR 87j:73084
59.: -, Error estimates of Morley element, Chinese J. Num. Math. & Appl. 12 (1990), 9-15. MR 91i:65182
60.: Z. Shi, X. Yu and Z. Xie, A multigrid method for Bergan's energy-orthogonal plate element, Advances in Numerical Mathematics: Proc. Second Japan-China Sem. (Tokyo, 1994), Lecture Notes Numer. Appl. Anal., vol. 14, Kinokuniya, Tokyo, 1995, pp. 159-169. MR 98g:73041
61.: H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978. MR 80i:46032
62.: S. Turek, Ein robustes und effizientes Mehrgitterverfahren zur Lösung der instationären, inkompressiblen 2-D Navier-Stokes-Gleichungen mit diskret divergenzfreien finiten Elementen, Dissertation, Universität Heidelberg, 1991.
63.: -, Multigrid techniques for a divergence-free finite element discretization, East-West J. Numer. Math. 2 (1994), 229-255. MR 96c:65195
64.: M. Wang, The multigrid method for TRUNC plate element, J. Comput. Math. 11 (1993), 178-187. MR 94m:65192
65.: -, The $W$ -cycle multigrid method for finite elements with nonnested spaces, Adv. in Math. 23 (1994), 238-250. MR 95e:65118
66.: S. Zhang, An optimal order multigrid method for biharmonic, $C^{1}$ finite element equations, Numer. Math. 56 (1989), 613-624. MR 90j:65135
67.: Z. Zhang and S. Zhang, Wilson's element for the Reissner-Mindlin plate, Comput. Methods Appl. Mech. Engrg. 113 (1994), 55-65. MR 94k:73078
68.: S. Zhou and G. Feng, Multigrid method for the Zienkiewicz element approximation of biharmonic equation, J. Hunan Univ. 20 (1993), 1-6 (Chinese). MR 94c:65155

Susanne C. Brenner
Affiliation: Department of Mathematics, University of South Carolina, Columbia, SC 29208
Email: brenner@math.sc.edu

DOI: 10.1090/S0025-5718-99-01035-2
PII: S 0025-5718(99)01035-2
Keywords: Multigrid methods, nonconforming finite elements, macro elements, preconditioner, $W$-cycle, variable $V$-cycle
Received by editor(s): April 13, 1995
Additional Notes: This work was supported in part by the National Science Foundation under Grant No. DMS-94-96275.
Copyright of article: Copyright 1999, American Mathematical Society

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