In this paper, a nonconforming finite element method has been proposed and analyzed for the von Kármán equations that describe bending of thin elastic plates. Optimal order error estimates in broken energy and norms are derived under minimal regularity assumptions. Numerical results that justify the theoretical results are presented.
DOI : 10.1051/m2an/2015052
Mots-clés : Von Kármán equations, Morley element, plate bending, non-linear, error estimates
@article{M2AN_2016__50_2_433_0,
author = {Mallik, Gouranga and Nataraj, Neela},
title = {A {Nonconforming} {Finite} {Element} {Approximation} for the von {Karman} equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {433--454},
publisher = {EDP-Sciences},
volume = {50},
number = {2},
year = {2016},
doi = {10.1051/m2an/2015052},
mrnumber = {3482550},
zbl = {1375.74089},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2015052/}
}
TY - JOUR AU - Mallik, Gouranga AU - Nataraj, Neela TI - A Nonconforming Finite Element Approximation for the von Karman equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 433 EP - 454 VL - 50 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015052/ DO - 10.1051/m2an/2015052 LA - en ID - M2AN_2016__50_2_433_0 ER -
%0 Journal Article %A Mallik, Gouranga %A Nataraj, Neela %T A Nonconforming Finite Element Approximation for the von Karman equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 433-454 %V 50 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015052/ %R 10.1051/m2an/2015052 %G en %F M2AN_2016__50_2_433_0
Mallik, Gouranga; Nataraj, Neela. A Nonconforming Finite Element Approximation for the von Karman equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 433-454. doi : 10.1051/m2an/2015052. http://www.numdam.org/articles/10.1051/m2an/2015052/
, and , Remarks around 50 lines of Matlab: short finite element implementation. Numer. Algorithms 20 (1999) 117–137. | DOI | MR | Zbl
, On von Kármán equations and the buckling of a thin elastic plate, I the clamped plate. Commun. Pure Appl. Math. 20 (1967) 687–719. | DOI | MR | Zbl
M.S. Berger, Nonlinearity and functional analysis. Academic Press (1977). | MR | Zbl
and , On von Kármán equations and the buckling of a thin elastic plate. Bull. Amer. Math. Soc. 72 (1966) 1006–1011. | DOI | MR | Zbl
and , Von Kármán equations and the buckling of a thin elastic plate. II plate with general edge conditions. Commun. Pure Appl. Math. 21 (1968) 227–241. | DOI | MR | Zbl
and , On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2 (1980) 556–581. | DOI | MR | Zbl
D. Braess, Finite Elements, Theory, Fast Solvers, and Applications in Elasticity Theory, 3rd edition. Cambridge (2007). | MR | Zbl
, Forty years of the Crouzeix-Raviart element. Numer. Methods Partial Differ. Equations 31 (2015) 367–396. | DOI | MR | Zbl
, , and , penalty methods for the fully nonlinear Monge−Ampère equation. Math. Comput. 80 (2011) 1979–1995. | DOI | MR | Zbl
S.C. Brenner and L. R. Scott, The mathematical theory of finite element methods, 3rd edition. Springer (2007). | MR
, , and , A Morley finite element method for the displacement obstacle problem of clamped Kirchhoff plates. J. Comput. Appl. Math. 254 (2013) 31–42. | DOI | MR | Zbl
, Finite element approximations of the von Kármán equations. RAIRO Anal. Numér. 12 (1978) 303–312. | DOI | Numdam | MR | Zbl
P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978). | MR | Zbl
P. G. Ciarlet, Mathematical Elasticity: Theory of Plates. In vol. II. North-Holland, Amsterdam (1997). | MR | Zbl
L.C. Evans, Partial Differential Equations. In vol. 19. American Mathematical Society (1998). | MR | Zbl
P. Grisvard, Singularities in Boundary Value Problems. Vol. RMA 22. Masson & Springer-Verlag (1992). | MR | Zbl
and , The best norm error estimate of lower order finite element methods for the fourth order problem. J. Comput. Math. 30 (2012) 449–460. | DOI | MR | Zbl
S. Kesavan, Topics in Functional Analysis and Applications. New Age International Publishers (2008).
, An existence theorem for the von Kármán equations. Arch. Ration. Mech. Anal. 27 (1967) 233–242. | DOI | MR | Zbl
and , Some nonconforming finite elements for the plate bending problem. RAIRO Anal. Numér. 9 (1975) 9–53. | Numdam | MR | Zbl
and , The Morley element for fourth order elliptic equations in any dimensions. Numer. Math. 103 (2006) 155–169. | DOI | MR | Zbl
, A mixed finite element method for the solution of the von Kármán equations. Numer. Math. 26 (1976) 255–269. | DOI | MR | Zbl
, A nonconforming Morley finite element method for the fully nonlinear Monge−Ampère equation. Numer. Math. 115 (2010) 371–394. | DOI | MR | Zbl
, Hybrid finite element methods for the von Kármán equations. Calcolo 16 (1979) 271–288. | DOI | MR | Zbl
, On the numerical analysis of the von Kármán equations: mixed finite element approximation and continuation techniques. Numer. Math. 39 (1982) 371–404. | DOI | MR | Zbl
, and , A two level additive Schwarz method for the Morley nonconforming element approximation of a nonlinear biharmonic equation. IMA J. Numer. Anal. 24 (2004) 97–122. | DOI | MR | Zbl
Cité par Sources :
