Abstract
For the purpose of bounding eigenvalues of the Laplacian over a bounded polygonal domain, we propose an algorithm to give high-precision bound even in the case that the eigenfunction has singularities around reentrant corners. The algorithm is a combination of the finite element method and the Lehmann–Goerisch theorem. The interval arithmetic is adopted in floating point number computation. Since all the error in the computation, e.g., the function approximation error, the floating point number rounding error, are exactly estimated, the result can be mathematically correct. In the end of the chapter, there are computational examples over an L-shaped domain and a square-minus-square domain that demonstrate the efficiency of our proposed algorithm.
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References
Behnke, H.: The calculation of guaranteed bounds for eigenvalues using complementary variational principles. Computing 47, 11–27 (1991)
Behnke, H., Goerisch, F.: Inclusions for eigenvalues of selfadjoint problems. Topics in Validated Computations, pp. 277–322. Elsevier, Amsterdam (1994)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Heidelberg (1991)
Chen, C.-Y.: On the properties of sard kernels and multiple error estimates for bounded linear functionals of bivariate functions with application to non-product cubature. Numerische Mathematik 122, 603–643 (2012)
Dautray, R., Lions, J., Artola, M., Cessenat, M.: Mathematical Analysis and Numerical Methods for Science and Technology: Spectral Theory and Applications. Springer, Heidelberg (2000)
Goerisch, F.: Ein stufenverfahren zur berechnung von eigenwertschranken, in Numerical Treatment of Eigenvalue Problems, Vol. 4. In: Albrecht, J., Collatz, L., Velte, W., Wunderlich, W. (eds.) International Series of Numerical Mathematics, vol. 83, pp. 104–114. Birkhäuser, Basel (1987)
Goerisch, F., He, Z.: The determination of guaranteed bounds to eigenvalues with the use of variational methods I. Computer Arithmetic and Self-validating Numerical Methods, pp. 137–153. Academic Press Professional Inc, San Diego (1990)
Lehmann, N.: Optimale Eigenwerteinschließungen. Numer. Math. 5, 246–272 (1963)
Liu, X., Kikuchi, F.: Analysis and estimation of error constants for interpolations over triangular finite elements. J. Math. Sci. Univ. Tokyo 17, 27–78 (2010)
Liu, X., Oishi, S.: Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape. SIAM J. NUMER. ANAL. 51, 1634–1654 (2013)
Mason, J.: Chebyshev polynomial approximations for the l-membrane eigenvalue problem. SIAM J. Appl. Math. 15, 172–186 (1967)
Mathematica, Ver. 9.0, Wolfram Research Inc, Champaign, IL (2013)
Plum, M.: Eigenvalue inclusions for second-order ordinary differential operators by a numerical homotopy method. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 41, 205–226 (1990)
Plum, M.: Guaranteed numerical bounds for eigenvalues. In: Hinton, D., Schaefer, P. (eds.) Spectral Theory and Computational Methods of Sturm, pp. 313–332. Marcel Dekker, New York (1997)
Plum, M., Wieners, C.: New solutions of the Gelfand problem. J. Math. Anal. Appl. 269, 588–606 (2002)
Rump, S.: INTLAB - INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer Academic Publishers, Dordrecht (1999). http://www.ti3.tu-harburg.de/rump/
Still, G.: Approximation theory methods for solving elliptic eigenvalue problems. ZAMM Z. Angew. Math. Mech. 83, 468–478 (2003)
Yuan, Q., He, Z.: Bounds to eigenvalues of the Laplacian on L-shaped domain by variational methods. J. Comput. Appl. Math. 233, 1083–1090 (2009)
Acknowledgments
The first author of this chapter is supported by Japan Society for the Promotion of Science, Grand-in-Aid for Young Scientist (B) 23740092.
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Liu, X., Okayama, T., Oishi, S. (2014). High-Precision Eigenvalue Bound for the Laplacian with Singularities. In: Feng, R., Lee, Ws., Sato, Y. (eds) Computer Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43799-5_23
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DOI: https://doi.org/10.1007/978-3-662-43799-5_23
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