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Guaranteed Computational Methods for Self-Adjoint Differential Eigenvalue Problems / by Xuefeng Liu
(SpringerBriefs in Mathematics. ISSN:21918201)
版 | 1st ed. 2024. |
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出版者 | Singapore : Springer Nature Singapore : Imprint: Springer |
出版年 | 2024 |
本文言語 | 英語 |
大きさ | IX, 131 p. 19 illus., 2 illus. in color : online resource |
著者標目 | *Liu, Xuefeng author SpringerLink (Online service) |
件 名 | LCSH:Mathematical analysis LCSH:Functional analysis LCSH:Mathematics -- Data processing 全ての件名で検索 FREE:Analysis FREE:Functional Analysis FREE:Computational Mathematics and Numerical Analysis |
一般注記 | This 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 HTTP:URL=https://doi.org/10.1007/978-981-97-3577-8 |
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電子ブック | オンライン | 電子ブック |
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Springer eBooks | 9789819735778 |
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EB00239014 |
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データ種別 | 電子ブック |
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分 類 | LCC:QA299.6-433 DC23:515 |
書誌ID | 4001118545 |
ISBN | 9789819735778 |