I gave a student seminar series on the Finite Element Method (FEM).

Time and Locations:

• 28 May: 13:40-15:30; Room M215
• 29 May: 13:40-15:30; Room M106
• 30 May: 13:40-15:30; Room M231
• 31 May: 13:40-15:30; Room M231

I used the following sources:

• S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Springer, 2008. https://doi.org/10.1007/978-0-387-75934-0.
• Süli, E. (2020). Lecture Notes on Finite Element Methods for Partial Differential Equations. [Lecture notes]. Mathematical Institute, University of Oxford.

For the implementation of the Finite Element Method for ODEs, I used Python.

• Sobolev spaces
  • Review of Lebesgue Integration Theory
  • Generalized (Weak) Derivatives
  • Sobolev Norms and Associated Spaces
  • Inclusion Relations and Sobolev’s Inequality
  • Trace Theorems
  • Negative Norms and Duality
• Variational Formulation of Elliptic Boundary Value Problems
  • Inner-Product Spaces
  • Hilbert Spaces
  • Riesz Representation Theorem
  • Formulation of Symmetric Variational Problems
  • Formulation of Nonsymmetric Variational Problems
  • The Lax-Milgram Theorem
  • Estimates for General Finite Element Approximation
  • Higher-dimensional Examples
• Finite Element Method for ODEs
  • Variational Formulation
  • Python Implementation
• The Construction of a Finite Element Space
  • The Finite Element
  • Triangular Finite Elements
  • The Interpolant
  • Equivalence of Elements
  • Finite Element Method for PDEs