A nonconforming mixed multigrid method for the pure traction problem in planar linear elasticity. (English) Zbl 0809.73064

Brenner, Susanne C.

Math. Comput. 63, No. 208, 435-460, S1-S5 (1994).

Summary: A robust optimal-order multigrid method for the pure traction problem in two-dimensional linear elasticity is studied. The finite element discretization is a variant of a mixed method proposed by R. S. Falk [Math. Comput. 57, 529-550 (1991; Zbl 0747.73044)] where the displacement is approximated by nonconforming piecewise linear functions and the “pressure” is approximated by piecewise constant functions on a coarser grid. Full multigrid convergence is obtained. The performance of this multigrid algorithm does not deteriorate as the material becomes nearly incompressible.

MSC:

74S05	Finite element methods applied to problems in solid mechanics
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74B05	Classical linear elasticity
65N55	Multigrid methods; domain decomposition for boundary value problems involving PDEs

Keywords:

pressure; displacement; nonconforming piecewise linear functions; piecewise constant functions; convergence

Citations:

Zbl 0747.73044

References:

[1]	Randolph E. Bank and Todd Dupont, An optimal order process for solving finite element equations, Math. Comp. 36 (1981), no. 153, 35 – 51. · Zbl 0466.65059
[2]	Susanne C. Brenner, A nonconforming multigrid method for the stationary Stokes equations, Math. Comp. 55 (1990), no. 192, 411 – 437. · Zbl 0705.76027
[3]	Susanne C. Brenner and Li-Yeng Sung, Linear finite element methods for planar linear elasticity, Math. Comp. 59 (1992), no. 200, 321 – 338. · Zbl 0766.73060
[4]	Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. · Zbl 0383.65058
[5]	M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 33 – 75. · Zbl 0302.65087
[6]	Richard S. Falk, Nonconforming finite element methods for the equations of linear elasticity, Math. Comp. 57 (1991), no. 196, 529 – 550. · Zbl 0747.73044
[7]	P. Grisvard, Problèmes aux limites dans les polygones. Mode d’emploi, EDF Bull. Direction Études Rech. Sér. C Math. Inform. 1 (1986), 3, 21 – 59 (French). · Zbl 0623.35031
[8]	P. Grisvard, Singularités en elasticité, Arch. Rational Mech. Anal. 107 (1989), no. 2, 157 – 180 (French, with English summary). · Zbl 0706.73013 · doi:10.1007/BF00286498
[9]	J. Nečas, Les méthodes directes en théorie des équations elliptiques, Masson, Paris, 1967. · Zbl 1225.35003
[10]	J. A. Nitsche, On Korn’s second inequality, RAIRO Anal. Numér. 15 (1981), no. 3, 237 – 248 (English, with French summary). · Zbl 0467.35019
[11]	I.D. Parsons and J.F. Hall, The multigrid method in solid mechanics: part I–algorithm description and behaviour, Internat. J. Numer. Methods Engrg. 29 (1990), 719-738. · Zbl 0724.73269
[12]	Ridgway Scott, Interpolated boundary conditions in the finite element method, SIAM J. Numer. Anal. 12 (1975), 404 – 427. · Zbl 0357.65082 · doi:10.1137/0712032
[13]	Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation. · Zbl 0356.65096
[14]	J. M. Thomas, Sur l’analyse numérique des méthodes d’éléments finis hybrides et mixtes, Thèse d’Etat, Université Pierre et Marie Curie, Paris, 1977.