Abstract
In the error analysis of finite element method, the inf-sup condition or the uniform lifting property plays an important role. In this paper, we discuss the relationship between the uniform inf-sup condition and the essential spectrum of the operator that appears in the problem. In general, one can not expect the convergence of the finite element approximation due to the spectral pollution that stems from the inappropriate mixing of the eigen-subspaces that correspond to two distinct components of the essential spectrum. As examples of our consideration, we treat the Stokes problem, mixed approximations of elliptic problems and a structural-acoustic coupling problem. In these problems, two distinct components might appear in the essential spectrum of the corresponding operators.
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References
Babuška, I., Error-bounds for finite element method, Numer. Math., 16, 322–333 (1971).
Babuška, I., The finite element method with Lagrangian multipliers, Numer. Math., 20, 179–192 (1973).
Braess, D., Finite Elements, Theory, Fast Solvers and Applications in Solid Mechanics, Cambridge University Press, 1997.
Bramble, J.H. and Hilbert, S.R., Bounds for a class of linear functionals with applications to Hermite interpolation, Numer. Math., 16, 362–369 (1971).
Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, RAIRO Anal. Numér., 8, 129–151 (1974).
Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, Springer-Verlag (1991).
Ciarlet, P. G. and Lions, J.L., Handbook of Numerical Analysis, Vol II, Finite element methods, North Holland, 1991.
Deng, L. and Kako, T., Finite element approximation of eigenvalue problem for a coupled vibration between acoustic field and plate. J. Compu. Math., 15, no. 3, 265–278 (1997).
Descloux, J., Nassif, N. and Rappaz, J., On spectral approximation. Part 1. The problem of convergence, RAIRO Anal. Numér., 12, 87–112 (1978).
Griffiths, D.F., Discrete eigenvalue problems, LBB constants and stabilization, Numerical Analysis 1995, D F Griffiths and G A Watson eds., 57–75 (1996).
Heikkola, E., Kusnetsov, Y. A., Neittaanmaki, P. and Toivanen J., Fictitious domain methods for the numerical solution of two-dimensional scattering problems, J. Comput. Phys., 145, pp. 89–109 (1998).
Kako, T., Approximation of scattering state by means of the radiation boundary condition, Math. Meth. in the Appl. Sci., 3, pp. 506–515 (1981).
Kako, T., MHD numerical simulation and numerical analysis, Journal of the Japan Society for Simulation Technology, Vol.12, 91–98 (1993) (in Japanese).
Kako, T., Remark on the relation between spectral pollution and inf-sup condition, G AKUT O International Series Math. Sci. and Appli., 11, Recent Developments in Domain Decomposition Methods and Flow Problems, 252–258 (1998).
Kako, T. and Descloux, J., Spectral approximation for the linearized MHD operator in cylindrical region, Japan J. Indust. Appl. Math., Vol.8, 221–244 (1991).
Kikuchi, F., Mathematical Foundation of Finite Element Method, Baifukan (1994) (in Japanese).
Kuznetsov, Yu. A. and Lipnikov, K. N. 3D Helmholtz wave equation by fictitious domain method, Russ. J. Numer. Anal. Math. Modeling, 13 No. 5, pp. 371–387 (1998).
Liu, X., Study on Approximation method for the Helmholtz equation in unbounded region, PhD. thesis, The University of Electro-Communications, Japan, 1999.
Liu, X. and Kako, T., Higher order radiation boundary condition and finite element method for scattering problem, Advances in Mathematical Sciences and Applications, 8, No. 2, pp. 801–819(1998).
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Kako, T., Nasir, H.M. (2002). Essential Spectrum and Mixed Type Finite Element Method. In: Babuška, I., Ciarlet, P.G., Miyoshi, T. (eds) Mathematical Modeling and Numerical Simulation in Continuum Mechanics. Lecture Notes in Computational Science and Engineering, vol 19. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56288-4_11
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DOI: https://doi.org/10.1007/978-3-642-56288-4_11
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