MA5240 —– Finite Element Method (FEM)

May 5, 2019

(Semester 2 2020/21)

Detailed Course Outline

  • Introduction and Preliminaries ( Lecture 1)
    • Brief history
    • A framework of FEM
    • Various types of FEM
    • Different FEM software packages
    • A simple example in one dimension (1D)
    • Comparison to the finite difference method
    • Link with the gradient discretization method
    • Review of Sobolov spaces
  • Variational Formulation of Elliptic Boundary Value Problems
    • Introduction
    • Variational problems
    • Minimization problems
    • Ritz-Galerkin Approximation
    • The Lax-Milgram theorem
    • Error estimates framework
    • An example in higher dimensions
  • The Construction of a Finite Element Space
    • The finite element
    • Triangular finite elements
      • The Lagrange elements
      • The Hermite elements
      • The Argyris elements
    • Rectangular elements
      • Tensor product elements
      • The serendipity element
    • Higher-dimensional elements
    • Exotic elements
  • FEM for Elliptic Problems
    • A model problem in two dimensions (2D)
    • FEM and its corresponding linear system
    • Error estimates
    • Aubin-Nitsche technique
    • Max-norm estimates
    • Extensuion to problems on unbounded domains
    • Extension to linear elasticity
    • Nonconforming finite elements
  • Mixed FEM
    • Examples of mixed variational formulations
    • Abstract mixed formulation
    • Discrete mixed formulation
    • Convergence results
    • The discrete inf-sup condition
    • Applications for the Stokes or Navier-Stokes problems
  • Advanced topics
    • Finite element multigrid methods
    • Adatpive FEM
    • Discontinuous Galerkin (DG) methods
  • Applications
    • For eigenvalue problems
    • For time-dependent problems
    • For Maxwell equations