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Guaranteed Computational Methods for Self-Adjoint Differential Eigenvalue Problems

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  • © 2024

Overview

  • Focuses uniquely on guaranteed computation for eigenvalue problems, providing rigorous eigenvalue bounds
  • Provides a comprehensive exploration of the hypercircle method for explicit error estimation
  • Unveils innovative error estimation algorithms for eigenfunction approximation in the case of clustered eigenvalues

Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)

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About this book

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.

 

Keywords

Table of contents (6 chapters)

Authors and Affiliations

  • Department of Information and Sciences, Tokyo Woman's Christian University, Suginami-ku, Tokyo-167-8585, Japan

    Xuefeng Liu

About the author

Xuefeng Liu is a Professor at Tokyo Woman's Christian University since 2023. He earned his bachelor's degree from the University of Science & Technology of China (1998-2003) and obtained his Master's and PhD degrees at the University of Tokyo by 2009. Liu was an Assistant Professor at Waseda University until 2014, then an Associate Professor at Niigata University until 2023. His research focuses on numerical error analysis for differential equations, especially verified computation for eigenvalue problems, and developing computer-assisted mathematical proofs for the Navier-Stokes equations in 3D domains.

Bibliographic Information

  • Book Title: Guaranteed Computational Methods for Self-Adjoint Differential Eigenvalue Problems

  • Authors: Xuefeng Liu

  • Series Title: SpringerBriefs in Mathematics

  • DOI: https://doi.org/10.1007/978-981-97-3577-8

  • Publisher: Springer Singapore

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024

  • Softcover ISBN: 978-981-97-3576-1Published: 26 June 2024

  • eBook ISBN: 978-981-97-3577-8Published: 25 June 2024

  • Series ISSN: 2191-8198

  • Series E-ISSN: 2191-8201

  • Edition Number: 1

  • Number of Pages: IX, 131

  • Number of Illustrations: 17 b/w illustrations, 2 illustrations in colour

  • Topics: Analysis, Functional Analysis, Computational Mathematics and Numerical Analysis

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