Approximation Theory
• Best least squares approximation in an inner product space
• Polynomial interpolation and error analysis

Quadrature
• Interpolatory quadrature and error analysis
• Gaussian quadrature and error analysis

ODE Initial Value Problems
• Single step methods (e.g., Euler, Runge-Kutta)
• Multistep methods (e.g., Adams-Bashforth, Adams-Moulton)
• Order and convergence

Elliptic Boundary Value Problems
• Finite difference discretization
• Variational formulation of Boundary Value Problems
  • Natural and Essential Boundary Conditions
  • Euler-Lagrange Differential Equation
• Weak Formulation and Galerkin-finite elements
• Order, Stability, and convergence analysis

Initial Boundary Value Problems
• Methods for the diffusion equation
  • Method of lines discretization
  • Semi-Discrete and Fully Discrete Methods
  • Order, Stability, and convergence analysis
  • Stability analysis: Fourier techniques
  • Stability analysis: Eigenvalue techniques
  • Galerkin-finite element methods
• Methods for hyperbolic equations
  • Finite differences
  • Stability analysis: Fourier Techniques
  • Stability analysis: The Energy Method
  • Method of characteristics
  • Burgers' equation

Numerical Linear Algebra
• Standard matrix decompositions (LU, Choleski)
• Classical iterative methods (Jacobi, Gauss-Siedel, Conjugate Gradient)
• Convergence analysis for iterative methods

References:

• The Mathematical Theory of Finite Element Methods, S. Brenner & L. R. Scott, Springer.
• Finite Element Solution of Boundary-Value Problems, O. Axelsson & V.A. Barker, SIAM, Classics in Applied Mathematics.
  
• A First Course in the Numerical Analysis of Differential Equations, A. Iserles, Cambridge Texts in Applied Mathematics.
• Finite Difference Methods for Ordinary and Partial Differential Equations: Steady State and Time-Dependent Problems Randall J. LeVeque. SIAM