Numerical Analysis

Syllabus


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