Guaranteed Computational Methods for Self-Adjoint Differential Eigenvalue Problems

Preface
Contents
1 Introduction to Eigenvalue Problems
	1.1 Overview of Research on Rigorous Eigenvalue Bounds
	1.2 Model Eigenvalue Problems
	1.3 Sobolev Space Settings and Weak Formulation of Eigenvalue Problem
	1.4 Min-Max Principle and Upper Eigenvalue Bounds
	1.5 Finite Element Method
		1.5.1 Mesh for Numerical Examples
		1.5.2 Approximate Eigenvalue Evaluation
2 Explicit Error Estimation for Boundary Value Problems
	2.1 Poisson\'s Equation and Its FE Solution
		2.1.1 Poisson\'s Equation
		2.1.2 Finite Element Solution
	2.2 Interpolation Error Estimation and Several Constants
		2.2.1 Interpolation Function and Error Estimation
		2.2.2 Constants in the Trace Theorem
	2.3 A Priori Error Estimate for Solutions with H2-Regularity
	2.4 Error Estimate for Solutions Without H2-Regularity
		2.4.1 Space Settings and Hypercircle in the Prager–Synge Theorem
		2.4.2 A Posteriori Error Estimation
		2.4.3 A Priori Error Estimation
		2.4.4 Numerical Examples
			2.4.4.1 Square Domain
			2.4.4.2 L-Shaped Domain
	2.5 Poisson\'s Equation with General Settings
		2.5.1 General Hypercircle Involving c(x)
		2.5.2 A Priori Error Estimation
		2.5.3 Computation of κh and Its Upper Bound
		2.5.4 Numerical Computation
	2.6 Error Estimation for Stokes Equations
		2.6.1 Problem Settings
		2.6.2 Finite Element Spaces
			2.6.2.1 Construction of Vh
			2.6.2.2 Projection Operators
		2.6.3 Explicit Error Estimation for FE Solutions
			2.6.3.1 A Posteriori Error Estimation
			2.6.3.2 A Priori Error Estimation
			2.6.3.3 Computation of κh
		2.6.4 Numerical Computation Results
			2.6.4.1 A Priori Error Estimation over 3D Domains
3 Fundamental Theorem for Explicit Eigenvalue Bounds
	3.1 Eigenvalue Problem with Positive Definite a(·, ·)
		3.1.1 Explicit Eigenvalue Bounds
	3.2 Eigenvalue Problems with Positive Semi-definite a(·,·)
		3.2.1 Problem Setting and Explicit Eigenvalue Bounds
	3.3 Evaluation of the Constant in Projection Error Estimation
4 Explicit Eigenvalue Bounds for Various Differential Operators
	4.1 Preparation: Non-conforming FEMs
		4.1.1 Crouzeix–Raviart FEM
			4.1.1.1 Interpolation Operator ΠhCR
			4.1.1.2 Interpolation Error Constant CCR(K)
		4.1.2 Enriched Crouzeix–Raviart FEM
		4.1.3 Composite Enriched Crouzeix–Raviart FEM
		4.1.4 Fujino–Morley FEM
	4.2 Laplacian Eigenvalue Problems
		4.2.1 Case of c=0 (-Δu = λu)
		4.2.2 Case of c> 0 (-Δu + cu = λu)
	4.3 Stokes Eigenvalue Problems
		4.3.1 Weak Formulation of the Stokes Eigenvalue Problem
		4.3.2 Lower Bounds Using Non-conforming FEMs
		4.3.3 Lower and Upper Bounds Using Conforming FEMs
		4.3.4 Numerical Results
	4.4 Steklov Eigenvalue Problems
		4.4.1 Lower Bound Using Conforming FEMs
		4.4.2 Lower Bound Using Non-conforming FEMs
		4.4.3 Computation Results
	4.5 Biharmonic Eigenvalue Problems
		4.5.1 Lower Bounds Using Fujino–Morley FEMs
		4.5.2 Computation Examples
5 Lehmann–Goerisch Method for High-Precision Eigenvalue Bounds
	5.1 Lehmann–Goerisch Method
	5.2 Application of the Lehmann–Goerisch Method
		5.2.1 Dirichlet Eigenvalue Problems
		5.2.2 Steklov Eigenvalue Problems
	5.3 Computational Results and Applications
		5.3.1 Eigenvalue Bounds for Dirichlet Eigenvalues
		5.3.2 Eigenvalue Bounds for Steklov Eigenvalues
6 Guaranteed Eigenfunction Computation
	6.1 Preliminaries
		6.1.1 Distance Between Subspaces
		6.1.2 Eigenspaces for Operators
	6.2 Algorithm I: Rayleigh Quotient-Based Error Estimation
	6.3 Algorithm II: Residual-Based Estimation
		6.3.1 Extension of the Davis–Kahan sinθ Theorem to Weakly Formulated Problems
		6.3.2 Weakly Formulated Residual Error Estimation
		6.3.3 Direct Estimate of δa: Another Application of the Davis–Kahan Theorem
	6.4 Algorithm III: Galerkin Projection-Based Estimation
		6.4.1 A Priori Error Estimation for FE Solutions of Boundary Value Problems
		6.4.2 Galerkin Projection-Based Estimate in L2 Norm
	6.5 Numerical Examples
		6.5.1 Unit Square Domain
		6.5.2 L-Shaped Domain
A Introduction to VFEM Library
References