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Huang, Ziping

A multi-grid algorithm for mixed problems with penalty. (English) Zbl 0712.73106

Numer. Math. 57, No. 3, 227-247 (1990).
The author develops a multigrid algorithm for the finite element approximation of mixed problems with penalty by the MINI element. Previously the Mindlin-Reissner plate model was applied in a mixed method with penalty [the author, A multigrid algorithm for mixed problems with penalty. Bochum: Univ. Bochum, Abt. f. Math., Diss. (1989; Zbl 0696.73043)] giving an optimal convergence rate uniformly with respect to the plate thickness. It enabled to avoid spurious phenomena like locking. Here it is proved that the convergence rate in a given multigrid algorithm is bounded away from 1 independently of the mesh size and of the thickness parameter. This aim is obtained by using smoothing procedures with an appropriate t-dependence. The convergence also holds when another finite element used in fluid mechanics is considered.
This mathematical paper is addressed to scientists working in the theory of the finite element method and for theoreticians who develop numerical models for practical use.
Reviewer: Cz.I.Bajer

Cited in 7 Documents

MSC:

74S30 Other numerical methods in solid mechanics (MSC2010)
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs

Keywords:

Jacobi relaxation; mixed problems; penalty; MINI element; convergence rate; smoothing procedures; appropriate t-dependence

Citations:

Zbl 0696.73043
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Full Text: DOI EuDML

References:

[1] Adams, R.A.: Sobolev spaces. New York: Academic 1975 · Zbl 0314.46030
[2] Arnold, D.N., Brezzi, F., Fortin, M.: A stable finite element for the Stokes equations. Calcolo21, 337-344 (1984) · Zbl 0593.76039 · doi:10.1007/BF02576171
[3] Arnold, D.N., Falk, R.S.: A uniformly accurate finite element method for the Mindlin-Reissner plate. SIAM J. Numer. Anal. (to appear) · Zbl 0696.73040
[4] Babu?ka, I., Aziz, A.K.: Survey lectures on the mathematical fundations of the finite element method. In: Aziz, A.K. (ed.). The mathematical fundations of finite element method with applications to partial differential equations, pp. 3-359. New York: Academic 1972
[5] Bank, R.E.: A comparison of two multi-level iterative methods for nonsymmetric and indefinite elliptic finite element equations. SIAM J. Numer. Anal.18, 724-734 (1981) · Zbl 0471.65074 · doi:10.1137/0718048
[6] Bank, R.E., Dupont, T.: An optimal order process for solving finite element equations. Math. Comput.36, 35-51 (1981) · Zbl 0466.65059 · doi:10.1090/S0025-5718-1981-0595040-2
[7] Braess, D., Blömer, C.: A multigrid method for a parameter dependent problem in solid mechanics. Numer. Math.57 (to appear) · Zbl 0665.65077
[8] Braess, D., Verfürth, R.: Multi-grid methods for non-conforming finite element methods. (Submitted) · Zbl 0703.65067
[9] Brezzi, F.: On the existence, uniqueness and approximations of saddle point problems arising from Lagrangian multipliers. RAIRO8, 129-151 (1974) · Zbl 0338.90047
[10] Brezzi, F., Fortin, M.: Numerical approximation of Mindlin-Reisser plates. Math. Comput.47, 151-158 (1986) · Zbl 0596.73058 · doi:10.1090/S0025-5718-1986-0842127-7
[11] Ciarlet, P.G.: The finite element method for elliptic problems. Amsterdam: North Holland 1978 · Zbl 0383.65058
[12] Girault, V., Raviart, P.A.: Finite element methods for Navier-Stokes equations. Berlin Heidelberg New York: Springer 1986 · Zbl 0585.65077
[13] Hackbusch, W.: Multi-grid methods and Applications. Berlin Heidelberg New York: Springer 1985 · Zbl 0595.65106
[14] Verfürth, R.: Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO Anal. Numer.18, 175-182 (1984) · Zbl 0557.76037
[15] Verfürth, R.: A multilevel algorithm for mixed problems. SIAM J. Numer. Anal.21, 264-271 (1984) · Zbl 0534.65065 · doi:10.1137/0721019
[16] Verfürth, R.: Multi-level algorithms for mixed problem II, Treatment of the mini-element. SIAM J. Numer. Anal.25, 285-293 (1988) · Zbl 0669.65083 · doi:10.1137/0725020
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