Abstract
We consider the C 0 interior penalty Galerkin method for biharmonic eigenvalue problems with the boundary conditions of the clamped plate, the simply supported plate and the Cahn-Hilliard type. We establish the convergence of the method and present numerical results to illustrate its performance. We also compare it with the Argyris C 1 finite element method, the Ciarlet-Raviart mixed finite element method, and the Morley nonconforming finite element method.
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References
A. Adini, R.W. Clough, Analysis of plate bending by the finite element method. NSF Report G.7337 (1961)
J.H. Argyris, I. Fried, D.W. Scharpf, The TUBA family of plate elements for the matrix displacement method. Aeronaut. J. R. Aeronaut. Soc. 72, 701–709 (1968)
I. Babuška, J. Osborn, Eigenvalue problems, in Handbook of Numerical Analysis II, ed. by P.G. Ciarlet, J.L. Lions (North-Holland, Amsterdam, 1991), pp. 641–787
I. Babuška, J. Osborn, J. Pitkäranta, Analysis of mixed methods using mesh dependent norms. Math. Comput. 35, 1039–1062 (1980)
P.E. Bjørstad, B.P. Tjøstheim, High precision solution of two fourth order eigenvalue problems. Computing 63, 97–107 (1999)
H. Blum, R. Rannacher, On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2, 556–581 (1980)
D. Boffi, F. Brezzi, L. Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comput. 69, 121–140 (2000)
S.C. Brenner, C 0 interior penalty methods, in Frontiers in Numerical Analysis - Durham 2010. Lecture Notes in Computational Science and Engineering, vol. 85 (Springer, Berlin, 2012), pp. 79–147
S.C. Brenner, M. Neilan, A C0 interior penalty method for a fourth order elliptic singular perturbation problem. SIAM J. Numer. Anal. 49, 869–892 (2011)
S.C. Brenner, L. Sung, C 0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22/23, 83–118 (2005)
S.C. Brenner, K. Wang, J. Zhao, Poincaré-Friedrichs inequalities for piecewise H 2 functions. Numer. Funct. Anal. Optim. 25, 463–478 (2004)
S.C. Brenner, S. Gu, T. Gudi, L.-Y. Sung, A quadratic C 0 interior penalty method for linear fourth order boundary value problems with boundary conditions of the Cahn-Hilliard type. SIAM J. Numer. Anal. 50, 2088–2110 (2012)
S.C. Brenner, M. Neilan, L.-Y. Sung, Isoparametric C 0 interior penalty methods. Calcolo 49, 35–67 (2013)
J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system-I: interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)
P.G. Ciarlet, P.-A. Raviart, A mixed finite element method for the biharmonic equation, in Mathematical Aspects of Finite Elements in Partial Differential Equations (Proc. Sympos., Math. Res. Center, University of Wisconsin, Madison, WI, 1974). (Academic Press, New York, 1974), pp. 125–145
G. Engel, K. Garikipati, T.J.R. Hughes, M.G. Larson, L. Mazzei, R.L. Taylor, Continuous/discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Methods Appl. Mech. Eng. 191, 3669–3750 (2002)
R. Falk, J. Osborn, Error estimates for mixed methods. RAIRO Anal. Numér. 14, 249–277 (1980)
J. Hu, Y. Huang, Q. Shen, The lower/upper bound property of approximate eigenvalues by nonconforming finite element methods for elliptic operators. J. Sci. Comput. 58, 574–591 (2014)
B. Mercier, J. Osborn, J. Rappaz, P.-A. Raviart, Eigenvalue approximation by mixed and hybrid methods. Math. Comput. 36, 427–453 (1981)
P. Monk, A mixed finite element method for the biharmonic equation. SIAM J. Numer. Anal. 24, 737–749 (1987)
L. Morley, The triangular equilibrium problem in the solution of plate bending problems. Aeronaut. Q. 19, 149–169 (1968)
R. Rannacher, Nonconforming finite element methods for eigenvalue problems in linear plate theory. Numer. Math. 33, 23–42 (1979)
J. Sun, A new family of high regularity elements. Numer. Methods Partial Differ. Equ. 28, 1–16 (2012)
G.N. Wells, N.T. Dung, A C 0 discontinuous Galerkin formulation for Kirchhoff plates. Comput. Methods Appl. Mech. Eng. 196, 3370–3380 (2007)
C. Wieners, Bounds for the N lowest eigenvalues of fourth-order boundary value problems. Computing 59, 29–41 (1997)
Acknowledgements
The work of the first author was supported in part by the NSF Grant DMS-1319172. The work of the second author is supported in part by the Air Force Office of Scientific Research Grant FA9550-13-1-0199 and by NSF Grant DMS-1216620. The work of the third author is supported in part by NSF Grants DMS-1521555 and DMS-132139.
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Brenner, S.C., Monk, P., Sun, J. (2015). C 0 Interior Penalty Galerkin Method for Biharmonic Eigenvalue Problems. In: Kirby, R., Berzins, M., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014. Lecture Notes in Computational Science and Engineering, vol 106. Springer, Cham. https://doi.org/10.1007/978-3-319-19800-2_1
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DOI: https://doi.org/10.1007/978-3-319-19800-2_1
Publisher Name: Springer, Cham
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