Syllabus for the Preliminary Examination in Numerical Analysis

The Prelim sequence is M387C and M387D. The first part of the Prelim examination will cover algebra and approximation and the second part of the Prelim examination will cover diferential equations.

Principles of discretization of differential equations:

  • ODEs: Stability and convergence theory, Stiff problems,Symplectic integrators
  • FEM (finite element method) and FDM (finite difference method) for boundary value problems
  • FEM for PDEs (main focus on elliptic problems): Basic theory, weak formulations, Lax-Milgram theorem, finite element spaces, approximation theory, a priori and a posteriori error estimates, practical algorithms, extensions, mixed methods etc.
  • FDM for PDEs (main focus on hyperbolic and parabolic problems): Lax equivalence theorem, Von Neumann and other stability analysis, nonlinear conservation laws, shocks, entropy, practical algorithms

Brief survey of other methods for PDEs:

  • FVM, DG, Spectral and particle methods
  • Applications: Elasticity (FEM), Fluids (FVM), and Waves (FDM)
  • Solution of linear and nonlinear equations
  • Solution of integral equations
  • Eigenvalues
  • Optimization
  • Monte Carlo methods
  • Fast Fourier, wavelet transforms, approximation theory
  • Basic undergraduate numerical methods
    • Interpolation, fixed point iterations, Newton's method for root finding
    • Direct and iterative methods for solving linear equations
    • Quadratures

Recommended texts:

  • Dahlquist and Bjorck, Numerical methods. Dover
  • Lambert, Numerical methods for ordinary differential systems. Wiley
  • Gustafsson, Kreiss, and Oliger, Time dependent problems and difference methods
  • Iserles, A first course in the numerical analysis of differential equations, Cambridge
  • Claes Johnson, Numerical solution of partial differential equations by the finite element method. Cambridge University Press