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Multigrid methods for parameter dependent problems. (English) Zbl 0848.73062
Summary: Multigrid methods for parameter dependent problems are discussed. The contraction numbers of the algorithms are proved within a unifying framework to be bounded away from one, independent of the parameter and the mesh levels. Examples include the pure displacement and pure traction boundary value problems in planar linear elasticity, the Timoshenko beam problem and the Reissner-Mindlin plate problem.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74B05 Classical linear elasticity
74Kxx Thin bodies, structures
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[1] D. N. ARNOLD, 1981, Discretization by finite elements of a parameter dependent problem, Numer. Math., 37, pp. 405-421. Zbl0446.73066 MR627113 · Zbl 0446.73066
[2] D. N. ARNOLD, F. BREZZI and M. FORTIN, 1984, A stable finite element for the Stokes equations, Calcolo, 21, pp. 337-344. Zbl0593.76039 MR799997 · Zbl 0593.76039
[3] D. N. ARNOLD and R. S. FALK, 1989, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal, 26, pp. 1276-1290. Zbl0696.73040 MR1025088 · Zbl 0696.73040
[4] D. N. ARNOLD and R. S. FALK, 1990, The boundary layer for the Reissner-Mindlin plate model, SIAM J. Math. Anal, 21, pp. 281-312. Zbl0698.73042 MR1038893 · Zbl 0698.73042
[5] R. E. BANK and T. DUPONT, 1981, An optimal order process for solving finite element equations, Math, Comp., 36, pp. 35-51. Zbl0466.65059 MR595040 · Zbl 0466.65059
[6] D. BRAESS and C. BLÖMER, 1990, A multigrid method for a parameter dependent problem in solid mechanics, Numer. Math., 57, pp. 747-761. Zbl0665.65077 MR1065522 · Zbl 0665.65077
[7] D. BRAESS and R. VERFÜRTH, 1990, Multigrid methods for nonconforming finite element methods, SIAM J. Numer. Anal, 27, pp. 979-986. Zbl0703.65067 MR1051117 · Zbl 0703.65067
[8] J. H. BRAMBLE, 1993, Multigrid Methods, Longman Scientific & Technical, Essex. Zbl0786.65094 MR1247694 · Zbl 0786.65094
[9] S. C. BRENNER, 1989, An optimal order multigrid method for PI nonconforming finite elements, Math. Comp., 52, pp. 1-15. Zbl0664.65103 MR946598 · Zbl 0664.65103
[10] S. C. BRENNER, 1989, Multigrid methods for nonconforming finite elements, in Proceedings of the Fourth Copper Mountain Conference on Multigrid Methods, J. Mandel et al, ed., Society for Industrial and Applied Mathematics, Philadelphia, pp. 54-65. MR1065626
[11] S. C. BRENNER, 1990, A nonconforming multigrid method for the stationary Stokes equations, Math. Comp., 55, 1993, pp. 411-437. Zbl0705.76027 MR1035927 · Zbl 0705.76027
[12] S. C. BRENNER, 1993, A nonconforming mixed multigrid method for the pure displacement problem in planar linear elasticity, SIAM J. Numer. Anal, 30,pp. 116-135. Zbl0767.73068 MR1202659 · Zbl 0767.73068
[13] S. C. BRENNER, 1994, A nonconforming mixed multigrid method for the pure traction problem in planar linear elasticity, Math. Comp., 63, pp. 435-460, S1-S5. Zbl0809.73064 MR1257574 · Zbl 0809.73064
[14] S. C. BRENNER and L. R. SCOTT, 1994, The Mathematical Theory of Finite Element Methods, Springer-Verlag. Zbl0804.65101 MR1278258 · Zbl 0804.65101
[15] S. C. BRENNER and L.-Y. SUNG, 1992, Linear finite element methods for planar linear elasticity, Math. Comp., 59, pp. 321-338. Zbl0766.73060 MR1140646 · Zbl 0766.73060
[16] F. BREZZI, K. J. BATHE and M. FORTIN, 1989, Mixed-interpolated elements for Reissner-Mindlin plates, Int. J. Numer. Math, Eng., 28, pp. 1787-1801. Zbl0705.73238 MR1008138 · Zbl 0705.73238
[17] F. BREZZI and M. FORTIN, 1986, Numerical approximation of Mindlin-Reissner plates, Math. Comp., 47, pp. 151-158. Zbl0596.73058 MR842127 · Zbl 0596.73058
[18] F. BREZZI and M. FORTIN, 1991, Mixed and Hybrid Finite Element Methods, Springer-Verlag. Zbl0788.73002 MR1115205 · Zbl 0788.73002
[19] F. BREZZI, M. FORTIN and R. STENBERG, 1991, Error analysis of mixed-interpolated elements for Reissner-Mindlin plates, Math. Models and Methods in Appl. Sci., 1, pp. 125-151. Zbl0751.73053 MR1115287 · Zbl 0751.73053
[20] M. CROUZEIX and P.-A. RAVIART, 1973, Conforming and nonconforming finite element methods for solving the stationary Stokes equations I, R.A.I.R.O., R-3, pp. 33-75. Zbl0302.65087 MR343661 · Zbl 0302.65087
[21] P. G. CIARLET, 1978, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam-New York-Oxford. Zbl0383.65058 MR520174 · Zbl 0383.65058
[22] R. DURÁN and E. LIEBERMAN, 1992, On mixed finite element methods for the Reissner-Mindlin plate model, Math. Comp., 58, pp. 561-573. Zbl0763.73054 MR1106965 · Zbl 0763.73054
[23] R. S. FALK, 1991, Nonconforming finite element methods for the equations of linear elasticity, Math. Comp., 57, pp. 529-550. Zbl0747.73044 MR1094947 · Zbl 0747.73044
[24] W. HACKBUSCH, 1985, Multi-Grid Methods and Applications, Springer-Verlag, Heidelberg. Zbl0595.65106 MR833988 · Zbl 0595.65106
[25] W. HACKBUSCH, 1980, Analysis and multigrid solutions of mixed finite element and mixed difference equations, Report, Ruhr-Universität Bochum.
[26] H. HAN, 1986, An analysis of penalty-nonconforming finite element method for Stokes equations, J Comput. Math., 4, pp. 164-172. Zbl0623.76020 MR854393 · Zbl 0623.76020
[27] Z. HUANG, 1990, A multi-grid algorithm for mixed problems with penalty, Numer. Math., 57, pp. 227-247. Zbl0712.73106 MR1057122 · Zbl 0712.73106
[28] M. JUNG, 1987, Konvergenzfaktoren für Mehrgitterverfahren zur Lösung von Problemen der Ebenen, Linearen Elastizitatstheorie, ZAMM, 67, pp. 165-173. Zbl0619.73086 MR887522 · Zbl 0619.73086
[29] S. MCCORMlCK ed., 1987, Multigrid Methods, SIAM Frontiers in Applied Mathematics 3, Society for Industrial and Applied Mathematics, Philadelphia. Zbl0659.65094 MR972752 · Zbl 0659.65094
[30] I. D. PARSONS and J. F. HALL, 1990, The multigrid method in solid mechanics; part I - algorithm description and behaviour, Int. J. Numer. Meth. Engrg., 29, pp. 719-738. Zbl0724.73269 · Zbl 0724.73269
[31] P. PEISKER, 1991, A multigrid method for Reissner-Mindlin plates, Numer. Math., 59, pp. 511-528. Zbl0736.73071 MR1121656 · Zbl 0736.73071
[32] P. PEISKER and D. BRAESS, 1992, Uniform convergence of mixed interpolated elements for Reissner-Mindlin plates, M2AN, 26, pp. 557-574. Zbl0758.73050 MR1177387 · Zbl 0758.73050
[33] P. PEISKER, W. RUST and E. STEIN, 1990, Iterative solution methods for plate bending problems : multigrid and preconditioned cg algorithm, SIAM J. Numer. Anal, 27, pp. 1450-1465. Zbl0721.73040 MR1080331 · Zbl 0721.73040
[34] R. VERFÜRTH, 1984, A multilevel algorithm for mixed problems, SIAM J. Numer. Anal, 21, pp. 264-271. Zbl0534.65065 MR736330 · Zbl 0534.65065
[35] R. VERFÜRTH, 1988, Multi-level algorithms for mixed problems II, treatment of the mini-clement, SIAM J. Numer. Anal, 25, pp. 285-293. Zbl0669.65083 MR933725 · Zbl 0669.65083
[36] S. ZHANG and Z. ZHANG, 1991, Treatment of discontinuity and bublle functions in multigrid methods II, Technical Report 153, Center for Applied Mathematics, Purdue University.
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