# Approximation of the arch problem by residual-free bubbles

A. Agouzal; M. El Alami El Ferricha

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 35, Issue: 2, page 271-293
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topAgouzal, A., and M. El Alami El Ferricha. "Approximation of the arch problem by residual-free bubbles." ESAIM: Mathematical Modelling and Numerical Analysis 35.2 (2010): 271-293. <http://eudml.org/doc/197385>.

@article{Agouzal2010,

abstract = {
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.
},

author = {Agouzal, A., M. El Alami El Ferricha},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Mixed method; Lagrange multipliers; conforming approximations; residual-free-bubble.; loaded arch; conforming Galerkin mixed finite element method; residual-free bubble functions; small thickness; approximation of middle line},

language = {eng},

month = {3},

number = {2},

pages = {271-293},

publisher = {EDP Sciences},

title = {Approximation of the arch problem by residual-free bubbles},

url = {http://eudml.org/doc/197385},

volume = {35},

year = {2010},

}

TY - JOUR

AU - Agouzal, A.

AU - M. El Alami El Ferricha

TI - Approximation of the arch problem by residual-free bubbles

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 35

IS - 2

SP - 271

EP - 293

AB -
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.

LA - eng

KW - Mixed method; Lagrange multipliers; conforming approximations; residual-free-bubble.; loaded arch; conforming Galerkin mixed finite element method; residual-free bubble functions; small thickness; approximation of middle line

UR - http://eudml.org/doc/197385

ER -

## References

top- D.N. Arnold and R.S. Falk, A uniformly accurate finite element method for the Reissner Mindlin plate. SIAM J. Numer. Anal26 (1989) 1276-1250. Zbl0696.73040
- I. Babuska, The finite element method with Lagrangian multipliers. Numer. Math20 (1973) 179-192. Zbl0258.65108
- I. Babuska and M. Suri, On the locking and robustness in the finite element method. SIAM J. Numer. Anal.29 (1992) 1276-1290. Zbl0763.65085
- C. Baiocchi, F. Brezzi and L. Franca, Virtual bubbles and the Galerkin-Least-squares method. Comput. Methods Appl. Mech. Engrg.105 (1993) 125-141. Zbl0772.76033
- M. Bernadou and Y. Ducatel, Approximation of a general arch problems by straight beam elements. Numer. Math.40 (1982) 1-29. Zbl0508.73069
- F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO-Anal. Numér. (1974) 129-151. Zbl0338.90047
- F. Brezzi and I. Douglas, Stabilized mixed methods for the stokes problem. Numér. Math.53 (1988) 225-236. Zbl0669.76052
- F. Brezzi and M. Fortin, Mixed and hybrid finite Element Methods. Springer-Verlag, Berlin, New-York, Springer Ser. Comput. Math.15 (1991). Zbl0788.73002
- F. Brezzi and A. Russo, Choosing bubbles for advection-diffusion problems. Math. Models Methods Appl. Sci.4 (1994) 571-578. Zbl0819.65128
- B. Budiansky and J.L. Sanders, On the best first order linear shell theory. Progr. Appl. Mech., Mac Millan, New-York, 129-140.
- 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. Zbl0631.49015
- D. Chenais and J.-C. Paumier, On the locking phenomenon for a class of elliptic problems. Numer. Math.67 (1994) 427-440 Zbl0798.73054
- P.G. Ciarlet, The finite element method for elliptic problems. North Holland, Amsterdam (1978). Zbl0383.65058
- Ph. Destuyender, Some numerical aspects of mixed finite elements for bending plates. Comput. Methods. Appl. Mech. Engrg.78 (1990) 73-87.
- L.P. Franca and T.J.R. Hughes, Two classes of mixed finite element methods. Comput. Methods Appl. Mech. Engrg.69 (1986) 89-129. Zbl0629.73053
- L.P. Franca and A. Russo, Unlocking with residual-free bubbles. Comput. Methods Appl. Mech. Engrg.142 (1997) 361-364 Zbl0890.73064
- 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. Zbl0756.73088
- V. Lods, A new formulation for arch structures. Application to optimization problems. RAIRO-Modél. Math. Anal. Numér.28 (1994) 873-902. Zbl0817.73041
- 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. Zbl0607.73077
- Z. Ould Zeidane, Contributions théoriques en Optimisation et Modélisation des structures. Thèse Université de Nice Sophia-Antipolis, Nice (1995).
- 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.
- A. Russo, Bubble Stabilization of finite element methods for the linearized incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg.132 (1996) 333-343. Zbl0887.76038

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.