×

Explicit eigenvalue bounds of differential operators defined by symmetric positive semi-definite bilinear forms. (English) Zbl 1433.65054

Summary: Recently, the eigenvalue problems formulated with symmetric positive definite bilinear forms have been well investigated with the aim of explicit bounds for the eigenvalues. In this paper, the existing theorems for bounding eigenvalues are further extended to deal with the case of eigenvalue problems defined by positive semi-definite bilinear forms. As an application, the eigenvalue estimation theorems are applied to the error constant estimation for polynomial projections.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15A18 Eigenvalues, singular values, and eigenvectors
15A63 Quadratic and bilinear forms, inner products
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Kikuchi, Fumio; Liu, Xuefeng, Estimation of interpolation error constants for the \(P_0\) and \(P_1\) triangular finite element, Comput. Methods Appl. Mech. Engrg., 196, 3750-3758 (2007) · Zbl 1173.65346
[2] Liu, Xuefeng; Kikuchi, Fumio, Analysis and estimation of error constants for \(P_0\) and \(P_1\) interpolations over triangular finite elements, J. Math. Sci. Univ. Tokyo, 17, 27-78 (2010) · Zbl 1248.65118
[3] Kobayashi, Kenta, On the interpolation constants over triangular elements (in Japanese), Kyoto Univ. Res. Inf. Repos., 1733, 58-77 (2011)
[4] Kobayashi, Kenta, On the interpolation constants over triangular elements, (Brandts, J.; Korotov, S.; Krızek, M.; Segeth, K.; Sıstek, J.; Vejchodsky, T., Proc. Internatonal Conf. Appl. Math. Vol. 2015 (2015), Institue of Mathematics, Czech Academy of Sciences), 110-124 · Zbl 1363.65014
[5] Liu, Xuefeng; Oishi, Shin’ichi, Verified eigenvalue evaluation for Laplace operator on arbitrary polygonal domain, RIMS Kokyuroku, 1733, 31-39 (2011)
[6] Liu, Xuefeng; Oishi, Shin’ichi, Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape, SIAM J. Numer. Anal., 51, 3, 1634-1654 (2013) · Zbl 1273.65179
[7] Liu, Xuefeng, A framework of verified eigenvalue bounds for self-adjoint differential operators, Appl. Math. Comput., 267, 341-355 (2015) · Zbl 1410.35088
[8] Carstensen, Carsten; Gedicke, Joscha, Guaranteed lower bounds for eigenvalues, Math. Comp., 83, 290, 2605-2629 (2014) · Zbl 1320.65162
[9] Carstensen, Carsten; Gallistl, Dietmar, Guaranteed lower eigenvalue bounds for the biharmonic equation, Numer. Math., 126, 1, 33-51 (2014) · Zbl 1298.65165
[10] Liu, Xuefeng; You, Chun’guang, Explicit bound for quadratic Lagrange interpolation constant on triangular finite elements, Appl. Math. Comput., 319, 693-701 (2018) · Zbl 1426.65013
[11] Liao, Shih-Kang; Shu, Yu-Chen; Liu, Xuefeng, Ooptimal estimation for the Fujino-Morley interpolation error constants, Jpn. J. Ind. Appl. Math. (2019) · Zbl 1418.65178
[12] Xie, Manting; Xie, Hehu; Liu, Xuefeng, Explicit lower bounds for Stokes eigenvalue problems by using nonconforming finite elements, Jpn. J. Ind. Appl. Math., 35, 1, 335-354 (2018) · Zbl 1450.65143
[13] You, Chun’guang; Xie, Hehu; Liu, Xuefeng, Guaranteed eigenvalue bounds for the Steklov eigenvalue problem, SIAM J. Numer. Anal., 57, 3, 1395-1410 (2019) · Zbl 1427.65384
[14] Babuška, Ivo; Osborn, John E., Eigenvalue problems, (Handbook of Numerical Analysis, Vol. II (1991), North-Holland: North-Holland Amsterdam), 641-787 · Zbl 0875.65087
[15] Behnke, Henning, The calculation of guaranteed bounds for eigenvalues using complementary variational principles, Computing, 47, 1, 11-27 (1991) · Zbl 0753.65032
[16] Rump, S. M., INTLAB - INTerval LABoratory, (Csendes, Tibor, Dev. Reliab. Comput. (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 77-104 · Zbl 0949.65046
[17] Zhang, Shangyou, On the divergence-free finite element method for the Stokes equations and the P1 Powell-Sabin divergence-free element, Math. Comp., 74, 250, 543-554 (2004) · Zbl 1085.76042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.