Department Mathematik
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Numerics II (Summer 2024)

Phan Thành Nam, Giao Ky Duong, Homework , Moodle (ID: 33920, pass: Numerics&II)

General Information

Goal: The course will focus on finite element methods and the applications in partial differential equations. In particular, we will discuss the variational formulation of elliptic boundary value problem in Sobolev spaces, the construction of finite element spaces, finite element multigrid methods as well as Schwarz domain decomposition methods.

Audience : Master students of Mathematics and Physics, TMP-Master. Bachelor students will get certificates ("Schein") if pass the course.

Time and place:
  • Lectures: Monday and Wednesday, 12:15-14:00, A027.
  • Exercises: Tuesday, 14:15-16:00, A027.
  • Tutorials: Wednesday, 8:15-10:00, B004.

References:
  • Susanne Brenner and Ridgway Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, Springer, 2008.

Exercises and Tutorials: Every week, there will be a homework sheet that will be discussed during the exercise sessions. Please handle your solutions at the beginning of the exercise sessions before they are discussed. Your solutions will be graded. Additionally, tutorial sessions will be provided to further help in reviewing the lectures.

Exam and Grade:
  • You can get 1 point for each homework sheet if you solve more than 50% problems there.
  • You can get up to 20 points in the midterm exam.
  • You can get up to 100 points in the final exam.
For exams you can bring your notes (lecture notes, homework sheets and tutorial materials). Electronic devices are not allowed. You need totally 50 points to pass the course and 85 points to get the final grade 1.0.

Contents of the lectures

15.04.2024. Introduction. Chapter 0: Basic Concepts. 1D example of Poisson equation. Variational formulation of weak solutions. Ritz-Galerkin approximation.

17.04. 2024. Error estimates. Finite element method.

22.04. 2024. Relationship with Difference Method. Adaptive Approximation.

24.04. 2024. Chapter 1: Sobolev spaces. L^p spaces. Convolution and the mollification method. Fundamental lemma of the calculus of variations. Weak derivatives. Duality argument. Relation to distributions.

29.04. 2024. Sobolev spaces. Completeness. Approximation by smooth functions. Sobolev inequalities.