Lecture 07/02 (Section 2.1): Derivation of Poisson's equation in one dimension as the stationary solutions of the heat or wave equations. Derivation of the solution expressed by the Green's function. (Section 2.1.)

Lecture 12/02 (Section 2.1-2.2): Properties of the solution of Poisson's equation: Existence, uniqueness, regularity, maximum principles and stability. Started on finite difference methods (Sec 2.2).

Lecture 19/02 (Section 2.2-2.3): Tridiagonal and diagonally matrices, and their invertibility. Writing the PDE and the numerical method in the operator form $L u = f$ and $L_{h} v = f$ . Started on properties of $L$ and $L_{h}$ , in particular symmetry.

Lecture 21/02 (Section 2.2-2.3): Positive definiteness of $L$ and $L_{h}$ , and the fact that this implies uniqueness of solutions. Green's function for the discrete problem and the discrete maximum principle. Convergence of the approximate solution to the exact one: Linearity + Consistency + Stability = Convergence.

Lecture 26/02 (Section 2.4): The eigenvalue problem for the continuous operator $L$ .

Lecture 28/02 (Section 2.4): The eigenvalue problem for the discrete operator $L_{h}$ . Discrete Fourier transforms.