MA5251 —– Spectral Methods and Applications （Outline）

(Semester 2 2015/16)

Detailed Course Outline

Introduction and Preliminaries ( Lecture 1)
- Weighted residual methods (WRM)
- Two examples
- Spectral collocation method
- Spectral methods of Galerkin type
- Tools for error analysis
- Comparisons of different spatial discretizations
  - Finite difference vs Spectral-collocation
  - Finite element vs Spectral-Galerkin
  - Spectral-Galerkin vs Spectral-collocation
- Review of iterative solvers and preconditioning
- Review of time discretization methods
Fourier Spectral Methods for Periodic Problems
- Introduction
- Continuous & Discrete Fourier transforms
  - Continuous Fourier series
  - Discrete Fourier transformation and FFT
  - Differentiation in physical space
  - Differentiation in frequency (or phase or Fourier) space
  - Differentiation matrices
- Fourier approximation & Fourier spectral methods
  - Inverse inequalities
  - Orthogonal projection & Interpolation
  - Fourier spectral methods and error estimates
- Applications
Orthogonal polynomials & Spectral methods
- Introduction
- Orthogonal polynomials
  - Chebyshev polynomials
  - Legendre polynomials
  - Jacobi polynomials
  - Error estimates for polynomials approximations
- Spectral methods for problems on bounded domains
  - Galerkin methods
  - Galerkin method with numerical integration
  - Collocation methods
- Error estimates
Spectral Methods for Problems in Unbounded Domains
- Introduction
- Orthogonal functions
  - Laguerre polynomials/functions
  - Hermite polynomials/functions
- Approximation results
  - Inverse inequalities
  - Orthogonal projections
  - Interpolation
- Spectral method via Laguerre or Hermite functions
- Error estimates
- Mapped spectral methods and rational approximation
Extensions and Applications in Applied Sciences and Engineering
- Extensions
  - Integral equations
  - Higher-order PDEs
  - Multi-dimensional problems
  - Spectral-element methods and applications
- Applications
  - In fluid dynamics
  - In heat transfer
  - In materials sciences
  - In quantum physics and nonlinear optics
  - In plasma and particle physics
  - In biology