**Numerical methods for Elliptic PDEs**

- 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

**Codes:** elpot.m, MyFEMCat.m, cat.png

**Refs:**

[2] K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Second Edition, Cambridge University Press, 2005, Chapter 6

[4] S. Larsson and V. Thomee, Partial Differential Equations with Numerical Methods

**Numerical Linear Algebra for Sparse Matrices**

- Basic iterative methods: Jacobi, Gauss-Seidel, SOR.

- Multigrid.

- Krylov subspace methods: the conjugate gradient and generalizations

Codes: basic_iterative_methods.m, Smoothers4multigrid.m, spectra.m, multigrid.m

**Refs:**

[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 Conjugate Gradient methods)

**Numerical Methods for Time-Dependent PDEs**

- 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.

Codes: MyFEMheat.m, cat.png, advection.m

**Refs:**

[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

[3] Fourier Transform links: Trefethen, P. Cheung

[4] Method of stationary phase

[5] Cameron's notes on Burgers's equation

**Fourier and Wavelet Transform Methods**

- 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

Codes: linear_dispersion.m, linear_dispersion0

**Codes:**

[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

**Refs:**

[2] Cameron's notes on Fourier spectral methods

Cameron’s note on the Korteweg - de Vries equation

[3] 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)