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020 _a9789819735778
_9978-981-97-3577-8
024 7 _a10.1007/978-981-97-3577-8
_2doi
050 4 _aQA299.6-433
072 7 _aPBK
_2bicssc
072 7 _aMAT034000
_2bisacsh
072 7 _aPBK
_2thema
100 1 _aLiu, Xuefeng.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_979309
245 1 0 _aGuaranteed Computational Methods for Self-Adjoint Differential Eigenvalue Problems
_h[electronic resource] /
_cby Xuefeng Liu.
250 _a1st ed. 2024.
264 1 _aSingapore :
_bSpringer Nature Singapore :
_bImprint: Springer,
_c2024.
300 _aIX, 131 p. 19 illus., 2 illus. in color.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aSpringerBriefs in Mathematics,
_x2191-8201
520 _aThis monograph presents a study of newly developed guaranteed computational methodologies for eigenvalue problems of self-adjoint differential operators. It focuses on deriving explicit lower and upper bounds for eigenvalues, as well as explicit estimations for eigenfunction approximations. Such explicit error estimations rely on the finite element method (FEM) along with a new theory of explicit quantitative error estimation, diverging from traditional studies that primarily focus on qualitative results. To achieve quantitative error estimation, the monograph begins with an extensive analysis of the hypercircle method, that is, the Prager–Synge theorem. It introduces a novel a priori error estimation technique based on the hypercircle method. This facilitates the explicit estimation of Galerkin projection errors for equations such as Poisson's and Stokes', which are crucial for obtaining lower eigenvalue bounds via conforming FEMs. A thorough exploration of the fundamental theory of projection-based explicit lower eigenvalue bounds under a general setting of eigenvalue problems is also offered. This theory is extensively detailed when applied to model eigenvalue problems associated with the Laplace, biharmonic, Stokes, and Steklov differential operators, which are solved by either conforming or non-conforming FEMs. Moreover, there is a detailed discussion on the Lehmann–Goerisch theorem for the purpose of high-precision eigenvalue bounds, showing its relationship with previously established theorems, such as Lehmann–Maehly's method and Kato's bound. The implementation details of this theorem with FEMs, a topic rarely covered in existing literature, are also clarified. Lastly, the monograph introduces three new algorithms to estimate eigenfunction approximation errors, revealing the potency of classical theorems. Algorithm I extends Birkhoff’s result that works for simple eigenvalues to handle clustered eigenvalues, while Algorithm II generalizes the Davis–Kahan theorem, initially designed for strongly formulated eigenvalue problems, to address weakly formulated eigenvalue problems. Algorithm III utilizes the explicit Galerkin projection error estimation to efficiently handle Galerkin projection-based approximations.
650 0 _aMathematical analysis.
_964178
650 0 _aFunctional analysis.
650 0 _aMathematics
_xData processing.
_965715
650 1 4 _aAnalysis.
650 2 4 _aFunctional Analysis.
650 2 4 _aComputational Mathematics and Numerical Analysis.
_966073
710 2 _aSpringerLink (Online service)
773 0 _tSpringer Nature eBook
776 0 8 _iPrinted edition:
_z9789819735761
776 0 8 _iPrinted edition:
_z9789819735785
830 0 _aSpringerBriefs in Mathematics,
_x2191-8201
_970788
856 4 0 _uhttps://doi.org/10.1007/978-981-97-3577-8
912 _aZDB-2-SMA
912 _aZDB-2-SXMS
999 _c61016
_d61016