A tentative plan for the course:
1) Introduction
- Review: Green's theorem
- Classification of second-order linear PDEs
2) Boundary value problems for ordinary differential equations (Sections 2.1-2.4, 4.1, 5.1)
- The problem and boundary conditions
- The maximum principle
- Green's function
- Variational formulation
- Solution & error estimates using finite differences
- Solution & error estimates using finite elements
3) Elliptic PDEs (Chapters 3-5)
- Maximum principle
- Green's function
- Variational formulation
- Finite difference methods for elliptic equations
- Finite element methods for elliptic equations
4) The elliptic eigenvalue problem (Chapter 6)
- Eigenfunction expansions
- Numerical methods
5) Sparse matrix methods (Appendix B)
- Direct methods for sparse linear systems
- Iterative methods for sparse linear systems
- Multigrid methods
- Methods for sparse eigenproblems
6) Initial value problems for ODEs (Chapter 7)
- The problem
- Numerical solution
7) Parabolic PDEs (Chapters 8-10)
- The initial value problems
- Solution by eigenvalue expansion
- Variational formulation
- Maximum principle
- The finite difference method
- The finite element method
8) Hyperbolic PDEs (Chapters 11-13)
- First order equations
- Hyperbolic systems
- Finite difference methods for hyperbolic problems
- Finite element method for hyperbolic problems
9) Fourier and wavelet methods (not in the book)
- Spectral methods
- A very brief introduction to wavelets
The chapters correspond to the textbook.
This is a tentative plan and it is subject to changes.