1D model problems using finite differences.
Formulation, accuracy, spectra.
Equivalent differential equation.
A tour of timesteppers: accuracy, efficiency, stability.
1D problems using spectral Galkerkin and FEM.
Formulation, accuracy, spectra.
Comparison with finite differences.
Neumann and Robin boundary conditions.
Extension to two space dimensions.
Tensor-product forms.
Fast operator evaluation / fast solvers.
Poisson equation with constant and variable coefficients.
General geometries: isoparametric mappings.
Boundary condition choices.
Extension to three space dimensions.
Comparison of different bases: FEM/spectral/Fourier.
Iterative solvers:
Krylov-subspace projection: CG / Lanczos / GMRES.
Eigenvalue estimates, projection for unsteady problems.
Exponential integrators.
Multi-domain spectral methods: 1D, 2D; matrix assembly.
Preconditioning: overlapping Schwarz, multigrid.
Stabilization in higher space dimensions:
Dealiasing / filtering / bubble functions / SUPG.
Other systems:
anisotropic diffusion
Stokes & Navier-Stokes
Maxwell's equations
Characteristics-based timesteppers.
Spectral-element/discontinuous-Galerkin (SEDG) methods.