Summary

Proceedings of the 2012 International Symposium on Nonlinear Theory and its Applications

2012

Session Number:D1L-D

Session:

Number:812

On guaranteed eigenvalue estimation of compact differential operator with singularity

Xuefeng LIU,  Shin'ich OISHI,  

pp.812-815

Publication Date:

Online ISSN:2188-5079

DOI:10.15248/proc.1.812

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In this paper, we consider the eigenvalue problem of elliptic operator L := -Δu + σu over 2D domain Ω:
Find u ∈ H¹₀ (Ω) and λ ∈ R , Lu = λu   (1)
where σ ∈ L_∞(Ω). In case of domain being polygonal one with re-entrant corner, the eigenfunction of the problem above has singularity, which brings difficulty in bounding the eigenvalues. For this problem, we develop new method to deal with the singularity. Moreover, the Lehmann-Goerisch theorem is applied to produce high precision eigenvalue bounds.

[1] M. Plum, Bounds for eigenvalues of second-order elliptic differential operators, The Journal of Applied Mathematics and Physics(ZAMP), 42(6):848-863, 1991.

[2] X. Liu, S. Oishi, Verified eigenvalue evaluation for Laplacian over polygonal domain of arbitrary shape, submitted to SIAM Journal on Numerical Analysis, 2012

[3] N.J. Lehmann, Optimale eigenwerteinschlieBungen, Numerische Mathematik , 5(1):246-272, 1963.

[4] H. Behnke, F. Goerisch, Inclusions for eigenvalues of selfadjoint problems, Topics in Validated Computations (ed.J. Herzberger), North-Holland, Amsterdam, pp.277-322, 1994.

[5] P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, 1985.

[6] F. Kikuchi and H. Saito, Remarks on a posteriori error estimation for finite element solutions, Journal of Computational and Applied Mathematics, 199:329-336, 2007.

[7] Dietrich Braess, Finite elements: theory, fast solvers, and applications in elasticity theory. Cambridge University Press, 2007.

[8] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer, 1991.

[9] W. Prager and J.L. Synge, Approximations in elasticity based on the concept of function spaces. 5:241, 1947.

[10] F. Kikuchi and X. Liu, Estimation of interpolation error constants for the P0 and P1 triangular finite element. Computer Methods in Applied Mechanics and Engineering, 196:3750-3758, 2007.

[11] M. Plum, Bounds for eigenvalues of second-order elliptic differential operators, ZAMP 42 (1991), 848-863.

[12] F. Goerisch , Ein Stufenverfahren zur Berechnung von Eigenwertschranken, in: J. Albrecht, L. Collatz, W. Velte, W. Wunderlich (eds.): Numerical Treatment of Eigenvalue Problems, Vol. 4, pp. 104-114, ISNM 83, Birkhäuser, Basel, 1987.

[13] F. Goerisch and Z. He, The determination of guaranteed bounds to eigenvalues with the use of variational methods I, pages 137-153. Academic Press Professional, Inc., San Diego, CA, USA, 1990.

[14] N.J. Lehmann Optimale eigenwerteinschließungen. Numerische Mathematik, 5(1):246-272, 1963.