Extending and automating Fourier analysis for multigrid methods / Hannah Rittich. Wuppertal, June 2017
Inhalt
- Preface
- Multigrid Essentials
- Wire under Tension
- Numerical Approximation
- Multigrid Methods
- A Relaxation Scheme
- The Error Iteration
- The Jacobi Method and the Poisson Equation
- Coarse Approximations
- Two Grids
- Multiple Grids
- Weighted Restriction
- The General Multigrid Method
- The Error Propagator
- Galerkin Coarse Approximation
- Formal Eigenfunction Analysis
- Higher Dimensions
- Multi-Index Notation
- The Poisson Equation in 2D
- Formal Eigenfunctions
- Wave Functions
- Low and High Frequencies
- Harmonic Frequencies
- A 2D Fourier Symbol
- Operator Norm and Spectral Radius
- Further Applications
- Literature and Contributions
- Local Fourier Analysis
- Operators on Infinite Grids
- Fourier Representation
- Multiplication Operators
- Operators with Fourier Symbols
- Smoothing Factor
- Literature and Contributions
- Matrix Symbols
- The Two-Grid Method
- Matrix Symbols
- Fourier Matrix Symbols
- Interpretation and Visualization of Matrix Symbols
- Spectral Properties of Matrix Multiplication Operators
- Periodic Stencils
- Block Shift Invariance
- Expansion
- Smoothing Factor
- Three- and n-Grid Analysis
- Literature and Contributions
- Applications
- PDEs with Jumping Coefficients
- Finite Volume Method
- Fourier Analysis I
- Adaptive Interpolation
- Fourier Analysis II
- Numerical Evaluation
- Block Smoothers
- Literature and Contributions
- Automating Fourier Analysis
- Constant Stencils
- Periodic Stencils
- Approximating Fourier Symbols
- Approximating Fourier Matrix Symbols
- Frequency Splitting
- Matrix Symbols
- Fourier Matrix Symbols
- Periodic Stencil Operators
- Restriction and Interpolation
- Expansion
- Smoothing Factor
- Matrix Representation
- A Language for LFA
- Evaluating Formulas
- Building Expression Trees
- Matching Resolutions
- Evaluating Fourier Matrix Symbols
- Literature and Contributions
- Conclusions
- Exponential Basis
- Nomenclature