Finite Elements

The aim of the overarching course AP3001 is to acquaint the student with the results and methods of the three disciplines to show how these are used in Physics.

For the Finite Elements part this translates to the introduction to the numerical discretization of partial differential equations (PDEs) with the Finite Element Method (FEM) in 1D and 2D.

Concepts introduced are:

1. Minimization problems

   • Euler-Lagrange equations

   • Ritz method

2. Finite Element Method (FEM)

   • Function spaces

   • Weak formulation

   • Boundary conditions

     • Essential boundary conditions

     • Natural boundary conditions

   • Interpolation and approximation of functions

     • Nodal (Lagrange) interpolation: Linear and high order polynomials (nodal basis)

     • Modal interpolation: Fourier basis, Hermite polynomials (modal basis)

   • Weighted residual methods (overview)

   • Galerkin method

3. Implementation aspects of FEM

   • Numerical integration (Gaussian quadrature)

   • Verification and validation of FEM (and general) numerical solvers

     • Method of manufactured solutions

     • Error convergence (h and p)