Approximation of the arch problem by residual-free bubbles
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 2, pp. 271-293.

We consider a general loaded arch problem with a small thickness. To approximate the solution of this problem, a conforming mixed finite element method which takes into account an approximation of the middle line of the arch is given. But for a very small thickness such a method gives poor error bounds. the conforming Galerkin method is then enriched with residual-free bubble functions.

On considère un problème de déplacement d'une arche chargée et de faible épaisseur. Pour approcher la solution de ce problème, on donne une méthode d'éléments finis Galerkin mixte conforme qui tient compte d'une approximation de la forme de l'arche. Cependant une application directe d'une telle méthode ne donne pas de résultat de convergence satisfaisant pour une faible épaisseur. On propose d'enrichir cette méthode par des fonctions bulles résiduelles.

Classification: 65N12
Keywords: mixed method, Lagrange multipliers, conforming approximations, residual-free-bubble
@article{M2AN_2001__35_2_271_0,
     author = {Agouzal, A. and El Alami El Ferricha, M.},
     title = {Approximation of the arch problem by residual-free bubbles},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     pages = {271--293},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {2},
     year = {2001},
     zbl = {1007.74070},
     mrnumber = {1825699},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2001__35_2_271_0/}
}
TY  - JOUR
AU  - Agouzal, A.
AU  - El Alami El Ferricha, M.
TI  - Approximation of the arch problem by residual-free bubbles
JO  - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY  - 2001
DA  - 2001///
SP  - 271
EP  - 293
VL  - 35
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/item/M2AN_2001__35_2_271_0/
UR  - https://zbmath.org/?q=an%3A1007.74070
UR  - https://www.ams.org/mathscinet-getitem?mr=1825699
LA  - en
ID  - M2AN_2001__35_2_271_0
ER  - 
%0 Journal Article
%A Agouzal, A.
%A El Alami El Ferricha, M.
%T Approximation of the arch problem by residual-free bubbles
%J ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
%D 2001
%P 271-293
%V 35
%N 2
%I EDP-Sciences
%G en
%F M2AN_2001__35_2_271_0
Agouzal, A.; El Alami El Ferricha, M. Approximation of the arch problem by residual-free bubbles. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 2, pp. 271-293. http://www.numdam.org/item/M2AN_2001__35_2_271_0/

[1] D.N. Arnold and R.S. Falk, A uniformly accurate finite element method for the Reissner Mindlin plate. SIAM J. Numer. Anal 26 (1989) 1276-1250. | Zbl

[2] I. Babuska, The finite element method with Lagrangian multipliers. Numer. Math 20 (1973) 179-192. | Zbl

[3] I. Babuska and M. Suri, On the locking and robustness in the finite element method. SIAM J. Numer. Anal. 29 (1992) 1276-1290. | Zbl

[4] C. Baiocchi, F. Brezzi and L. Franca, Virtual bubbles and the Galerkin-Least-squares method. Comput. Methods Appl. Mech. Engrg. 105 (1993) 125-141. | Zbl

[5] M. Bernadou and Y. Ducatel, Approximation of a general arch problems by straight beam elements. Numer. Math. 40 (1982) 1-29. | Zbl

[6] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO-Anal. Numér. (1974) 129-151. | Numdam | Zbl

[7] F. Brezzi and I. Douglas, Stabilized mixed methods for the stokes problem. Numér. Math. 53 (1988) 225-236. | Zbl

[8] F. Brezzi and M. Fortin, Mixed and hybrid finite Element Methods. Springer-Verlag, Berlin, New-York, Springer Ser. Comput. Math. 15 (1991). | MR | Zbl

[9] F. Brezzi and A. Russo, Choosing bubbles for advection-diffusion problems. Math. Models Methods Appl. Sci. 4 (1994) 571-578. | Zbl

[10] B. Budiansky and J.L. Sanders, On the best first order linear shell theory. Progr. Appl. Mech., Mac Millan, New-York, 129-140.

[11] D. Chenais, Rousselet and B. Benedict, Design sensibivity for arch structures with respect to midsurface shape under static loading. J. Optim. Theory Appl. 58 (1988) 225-239. | Zbl

[12] D. Chenais and J.-C. Paumier, On the locking phenomenon for a class of elliptic problems. Numer. Math. 67 (1994) 427-440 | Zbl

[13] P.G. Ciarlet, The finite element method for elliptic problems. North Holland, Amsterdam (1978). | MR | Zbl

[14] Ph. Destuyender, Some numerical aspects of mixed finite elements for bending plates. Comput. Methods. Appl. Mech. Engrg. 78 (1990) 73-87. | Zbl

[15] L.P. Franca and T.J.R. Hughes, Two classes of mixed finite element methods. Comput. Methods Appl. Mech. Engrg. 69 (1986) 89-129. | Zbl

[16] L.P. Franca and A. Russo, Unlocking with residual-free bubbles. Comput. Methods Appl. Mech. Engrg. 142 (1997) 361-364 | Zbl

[17] A. Habbal and D. Chenais, Deterioration of a finite element method for arch structures when thickness goes to zero. Numer. Math. 62 (1992) 321-341. | Zbl

[18] V. Lods, A new formulation for arch structures. Application to optimization problems. RAIRO-Modél. Math. Anal. Numér. 28 (1994) 873-902. | Numdam | Zbl

[19] A.F.D. Loula, L.P. Franca, T.J.R. Hughes and I. Miranda, Stability Convergence and accuracy of a New finite element method for the circular arch problem. Comput. Methods Appl. Mech. Engrg. 63 (1987) 281-303. | Zbl

[20] Z. Ould Zeidane, Contributions théoriques en Optimisation et Modélisation des structures. Thèse Université de Nice Sophia-Antipolis, Nice (1995).

[21] A. Russo, Residual-free bubbles and Stabilized methods, in Proc. of the ninth International Conference on finite Elements in Fluids-New Trends and Applications, M.M. Cacchi, K. Morgan, J. Pariaux, B.A. Schreffer, O.C. Zienkiewicz, Eds., Venice (1995) 377-386.

[22] A. Russo, Bubble Stabilization of finite element methods for the linearized incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 132 (1996) 333-343. | Zbl