Introduction to the Finite Element Method in optics

Very frequently, equations in physics are too complicated to find solutions in closed form or by purely analytical means (e.g. by Laplace and Fourier transform methods, or in the form of a power series). Numerical approximations to the unknown analytical solution are therefore necessary.

The Finite Element Method (FEM) represents a powerful and general method for the approximate solution of partial differential equations. Although more complex to formulate and to implement than the popular Finite Difference Method (FDM), the FEM offers several important advantages such as the possibility of accurately following material interfaces, of imposing boundary continuity requirements for the approximated electromagnetic vector fields, etc.

The following topics will be covered:
Variational (weak) formulation of Partial Differential Equation (PDE) problems
Lagrange Finite element space
Finite element discretization of elliptic PDE
Edge/Vector Finite Elements for Maxwell equations

Course leader

Fabrice Mahé

Target group

Master level students in mathematics or physics. Students from third year undergraduate as well as PhD students are also be welcome.

Course aim

The aim of this course is to provide an introduction to both the mathematical theory and the numerical implementation of the FEM, with a special emphasis on applications in optics. Lectures will alternate with practical exercises using FEM software packages such as COMSOL Multi-Physics or Freefem++ and project work in groups of 3 or 4 students.