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Bücher Englische Bücher Wissen & Bildung Naturwissenschaft, Medizin, Informatik, Technik Mathematik Analysis Guaranteed Computational Methods for Self-Adjoint Differential Eigenvalue Problems
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Guaranteed Computational Methods for Self-Adjoint Differential Eigenvalue Problems
Xuefeng Liu (Broschiert, Englisch)
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Beschreibung
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. 64,19 €
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Technische Daten
Erscheinungsdatum
26.06.2024
Sprache
Englisch
EAN
9789819735761
Herausgeber
Springer Singapore
Serien- oder Bandtitel
SpringerBriefs in Mathematics
Sonderedition
Nein
Autor
Xuefeng Liu
Seitenanzahl
131
Einbandart
Broschiert
Schlagwörter
Lower eigenvalue bounds, Finite element method, Eigenvalue problems of differential operators, Interpolation error constants, Verified numerical computation
Thema-Inhalt
PBK - Mathematische Analysis, allgemein
PBKF - Funktionalanalysis und Abwandlungen
PBKS - Numerische Mathematik
Höhe
235 mm
Breite
15.5 cm
Transparenz & Sicherheit
Hersteller: Springer, Europaplatz 3, Heidelberg, Deutschland, 69115, ProductSafety@springernature.com, Springer Nature Customer Service Center GmbH
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