Department Mathematik
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Numerics II (Summer 2024)

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

News: The final grades are available. You can review your solutions on Friday July 19, 14:00-15:00, room B251. If you cannot make it, please write to Dr. Giao Ky Duong to get an appointment to see your solutions on Monday July 29.

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. Error estimates. Finite element method.

22.04. Relationship with Difference Method. Adaptive Approximation.

24.04. 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. Sobolev spaces. Completeness. Approximation by smooth functions. Sobolev inequalities in R^d. 1D case.

29.04. Fourier transform and weak derivatives. Sobolev spaces and Sobolev inequalities for p=2.

6.5. Extension operator. Sobolev inequalities for Lipschitz domains. Review of Chapter 0.

8.5. Trace theorems. Duality and negative Sobolev spaces.

13.5. Chapter 2: Variational Formulation of Elliptic Boundary Value Problems. Hilbert spaces. Projections onto Subspaces. Riesz Representation Theorem. Symmetric Variational Problems.

15.5. Nonsymmetric Variational Problems. The Lax-Milgram Theorem.

22.5. Higher-dimensional Examples. Chapter 3: Construction of Finite Element Spaces. The Finite Element.

27.5. Factorization lemma with hyperplanes. The triangular Lagrange finite element.

29.5. Midterm exam.

3.6. Hermite element. Argyris Element. Computation of the dual basis.

5.6. The Interpolant. Local and global interpolants.Triangulation. Regularity of Lagrange, Hermite and Argyris elements.

10.6. Equivalence of elements. Rectangular elements. Higher-dimensional elements.

12.6. Chapter 4: Polynomial Approximation in Sobolev Spaces. Averaged Taylor Polynomials. Star-shaped domains and the chunkiness parameter.

17.6. Error representation. Bounds for Riesz Potentials. Sobolev's Inequality. Bramble-Hilbert inequality.

19.6. Poincare-Friedrichs inequality. Bounds for the Interpolation Error. Non-degeneracy condition.

24.6. Chapter 5: Variational Problems in higher dimensions. Poisson’s equation with Dirichlet and Neumann boundary conditions. The Dirichlet problem: variational formulation and coercivity.

26.6. The Pure Neumann Problem. Elliptic Regularity Estimates.

8.7. Error estimates in H^1 and L^2 norms. Chapter 6: Finite Element Multigrid Methods. The model. Mesh-Dependent Norms. The Multigrid Algorithm.

10.7. Comparison of V- and W-cycle methods. The convergence rate for the W-cycle method.

15.7. The work estimate for the W-cycle method. Remarks on the V-cycle method.

17.7. Final exam.