- Linear elliptic equations. Modeling using elliptic PDEs. Existence and Uniqueness theorems, weak and strong maximum principles.
- Finite difference methods in 2D: different types of boundary conditions, convergence.
- Variational and weak formulations for elliptic PDEs.
- Finite element method in 2D

- [1] Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM 2007, ( freely available online via the UMD library), Chapter 3
- [2] K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Second Edition, Cambridge University Press, 2005, Chapter 6
- [3] Cameron's lecture notes.
- [3] S. Larsson and V. Thomee, Partial Differential Equations with Numerical Methods
- [5] Jochen Alberty, Carsten Carstensen and Stefan A. Funken, "Remarks around 50 lines of Matlab: short finite element implementation", Numerical Algorithms 20 (1999) 117–137

- Basic iterative methods: Jacobi, Gauss-Seidel, SOR.
- Multigrid.
- Krylov subspace methods: the conjugate gradient and generalizations

- [1] Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM 2007, ( freely available online via the UMD library), Chapter 4
- [2] K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Second Edition, Cambridge University Press, 2005, Chapter 7
- [3] Yousef Saad, Iterative Methods for Sparse Linear System, SIAM 2003 (see chapter 13 for Multigrid)
- [4] J. Nocedal and S. Wrigth, Numerical Optimization, 2nd edition, Springer (see Chapter 5 for Conjucate Gradient methods)

- Parabolic equations:
- Heat equation. Finite difference methods: explicit and implicit. Basic facts about stability and convergence.
- Solving heat equation in 2D using finite element method.
- Method of lines. An example of a nonlinear equation (the Boussinesq equation).

- Linear advection equation:
- Finite difference methods. Basic facts about stability and convergence. The CFL condition.
- Fourier transform. Dispersion analysis. Phase and group velocities.

- Hyperbolic conservation laws:
- Shock speed and the Rankine-Hugoriot condition, weak solutions, entropy condition and vanishing viscosity solution
- Numerical methods for conservation laws: conservative form, consistency, Godunov's and Glimm's methods.

- [1] Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM 2007, ( freely available online via the UMD library), Chapter 9, Chapter 10, Appendix E
- [2] Jochen Alberty, Carsten Carstensen and Stefan A. Funken, "Remarks around 50 lines of Matlab: short finite element implementation", Numerical Algorithms 20 (1999) 117–137
- [3] Fourier Transform links: Trefethen, P. Cheung
- [4] Method of stationary phase
- [5] Cameron's notes on Burger's equation

- Continuous and Discrete Fourier transforms
- Spectral methods for solving linear and nonlinear PDEs
- The fast Fourier transform
- Nyquist frequency, sampling theorem
- Continuous and discrete wavelet transforms
- Haar and Daubechies wavelets, approximation properties, fast wavelet transforms
- Application of wavelets to image processing

- [1] linear_dispersion.m: solution of u_t +u_{xxx} = 0 for x \in [0, 2pi] using DFT and exact time integration
- [2] linear_dispersion01.m: solution of u_t +u_{xxx} = 0 for x \in [0, 1] using DFT and exact time integration
- [3] KdVrkm1.m: solution of the Korteweg - de Vries equation using DFT and a method involving DFT and exact time integration of the linear part of the RHS
- [4] WLena.m: image compression using wavelets. Input image: Lenna.png available at https://en.wikipedia.org/wiki/File:Lenna.png (Links to an external site)

- [1] John P. Boyd, Chebyshev and Fourier Spectral methods, 2nd edition, Dover Publication, Inc., Mineola, New York, 2001
- [2] Cameron's notes on Fourier spectral methods
- [3] Cameron’s note on the Korteweg - de Vries equation
- [4] Ingrid Daubechies, Ten lectures on wavelets, 1992
- [4] Stephane Mallet, A wavelet tour of signal processing. The sparse way. 3rd edition. Academic Press, Elsevier, 2009
- [5] Lecture notes on wavelets and multi resolution analysis:
- Phillip K. Poon (U. of Arizona),
- Brani Vidakovich (GATech),
- Vlad Balan and Cosmin Condea (USC)
- [6] Sonja Grgic, Kresimir Kers, Mislav Grgic, Image Compression Using Wavelets, IEEE, 0-7803-5662-4/99 (1999)