This book discusses numerical methods for solving time-fractional evolution equations. The approach is based on first discretizing in the spatial variables by the Galerkin finite element method, using piecewise linear trial functions, and then applying suitable time stepping schemes, of the type either convolution quadrature or finite difference. The main concern is on stability and error analysis of approximate solutions, efficient implementation and qualitative properties, under various regularity assumptions on the problem data, using tools from semigroup theory and Laplace transform. The book provides a comprehensive survey on the present ideas and methods of analysis, and it covers most important topics in this active area of research. It is recommended for graduate students and researchers in applied and computational mathematics, particularly numerical analysis.
Author(s): Bangti Jin, Zhi Zhou
Series: Applied Mathematical Sciences, 214
Publisher: Springer
Year: 2023
Language: English
Pages: 427
City: Cham
1
Preface
Contents
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1 Existence, Uniqueness, and Regularity of Solutions
1.1 Basics of Fractional Calculus
1.2 Mittag–Leffler Function
1.3 Existence, Uniqueness, and Sobolev Regularity
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2 Spatially Semidiscrete Discretization
2.1 Galerkin Finite Element Method
2.2 Error Analysis via Mittag–Leffler Functions
2.3 Error Analysis via Laplace Transform
2.4 Lumped Mass FEM
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3 Convolution Quadrature
3.1 Convolution Quadrature Generated by BDF
3.2 BDFk CQ with Initial Correction
3.3 Fractional Crank–Nicolson Scheme
3.4 Parallel in Time Algorithm
3.5 Fast Convolution
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4 Finite Difference Methods: Construction and Implementation
4.1 Construction of Time-Stepping Schemes
4.2 Sum of Exponential Approximation
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5 Finite Difference Methods on Uniform Meshes
5.1 Error Analysis of L1 Scheme
5.2 Corrected L1 Scheme
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6 Finite Difference Methods on Graded Meshes
6.1 Error Analysis via Nonuniform Gronwall's Inequality
6.2 Error Analysis of the L1 Scheme via Barrier Functions
6.3 Error Analysis of Alikhanov's Scheme via Barrier Functions
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7 Nonnegativity Preservation
7.1 Nonnegativity Preservation
7.2 Spatially Semidiscrete Methods
7.3 Fully Discrete Scheme
7.4 Maximum-Norm Contractivity
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8 Discrete Maximal Regularity
8.1 R-Boundedness, UMD Spaces, and Fourier Multiplier Theorems
8.2 Convolution Quadrature Generated by BDF
8.3 L1 Scheme
8.4 Explicit Euler Method
8.5 Fractional Crank–Nicolson Method
8.6 Inhomogeneous Initial Condition
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9 Subdiffusion with Time-Dependent Coefficients
9.1 Regularity Theory
9.2 Semidiscrete Galerkin FEM
9.3 Time Discretization by Backward Euler CQ
9.4 Time Discretization by Corrected BDF2 CQ
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10 Semilinear Subdiffusion
10.1 Discrete Gronwall's Inequality
10.2 Error Estimates for the Linearized Scheme
10.3 High-Order Time-Stepping Schemes
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11 Time-Space Finite Element Approximation
11.1 Time-Space Petrov–Galerkin Formulation
11.2 Petrov–Galerkin FEM on Tensor-Product Meshes
11.3 Error Estimates
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12 Spectral Galerkin Approximation
12.1 Time-Space Galerkin Formulation
12.2 Log Orthogonal Functions
12.3 Spectral Galerkin Method
12.4 Fully Discrete Scheme
12.5 Fast Linear Solver
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13 Incomplete Iterative Solution at Time Levels
13.1 Incomplete Iterative Scheme
13.2 Error Analysis for Smooth Initial Data
13.3 Error Analysis for Nonsmooth Initial Data
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14 Optimal Control with Subdiffusion Constraint
14.1 Regularity Theory
14.2 Numerical Approximation of the Forward Problem
14.3 Numerical Approximation of the Optimal Control Problem
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15 Backward Subdiffusion
15.1 Stability and Regularization
15.2 Spatially Semidiscrete Scheme
15.3 Fully Discrete Scheme
1 (1)
Appendix A Mathematical Preliminaries
A.1 Gamma Function
A.2 Polylogarithmic Function
A.3 Integral Transforms
Appendix References
Index
