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Fully computable a posteriori error bounds for eigenfunctions
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SYSNO ASEP 0561025 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title Fully computable a posteriori error bounds for eigenfunctions Author(s) Liu, X. (JP)
Vejchodský, Tomáš (MU-W) RID, SAI, ORCIDSource Title Numerische Mathematik - ISSN 0029-599X
Roč. 152, č. 1 (2022), s. 183-221Number of pages 39 s. Language eng - English Country DE - Germany Keywords eigenvalue problems ; Laplace eigenvalues ; approximation Subject RIV BA - General Mathematics OECD category Pure mathematics R&D Projects GA20-01074S GA ČR - Czech Science Foundation (CSF) Method of publishing Limited access Institutional support MU-W - RVO:67985840 UT WOS 000824307900001 EID SCOPUS 85134293670 DOI 10.1007/s00211-022-01304-0 Annotation For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both algorithms apply well to the case of tight clusters and multiple eigenvalues, under the settings of target eigenvalue problems. Algorithm I is based on the Rayleigh quotient and the min-max principle that characterizes the eigenvalue problems. The formula for the error estimate provided by Algorithm I is easy to compute and applies to problems with limited information of Rayleigh quotients. Algorithm II, as an extension of the Davis–Kahan method, takes advantage of the dual formulation of differential operators along with the Prager–Synge technique and provides greatly improved accuracy of the estimate, especially for the finite element approximations of eigenfunctions. Numerical examples of eigenvalue problems of matrices and the Laplace operators over convex and non-convex domains illustrate the efficiency of the proposed algorithms. Workplace Mathematical Institute Contact Jarmila Štruncová, struncova@math.cas.cz, library@math.cas.cz, Tel.: 222 090 757 Year of Publishing 2023 Electronic address https://doi.org/10.1007/s00211-022-01304-0
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