Numerics II (Summer 2019)

Prof. Phan Thành Nam, Dr. Julien Ricaud

If you are interested in taking a retake exam (on October 4) please send an email to "nam@math.lmu.de".

Grades are available in the pin board in front of room B329.

You can review your solutions of the final exam on August 6 (Tuesday) at 10:00 AM, room B 046.

General Information

Goal: The course will focus on finite element methods and the applications in partial differential equations.

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, B047.
• Exercises: Tuesday, 16:15-18:00, B047.
• Tutorials: Tuesday, 14:15-16:00, B047.

References:

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

Exercises and Tutorials: There will be a homework sheet every week. Doing the homework is the best way to learn the course's materials and to prepare for the final exam.

The tutorial section is provided to help you in reviewing the lectures. You can bring up your questions, discuss complementary materials, and try some extra exercises.

Grade:  Final grade is determined by your total performance:

• You can get up to 100 points in the final exam.
• You can get 1 point for every homework sheet if you solve at least 50% problems in the sheet.
  

You need 50 points to pass the course and 85 points to get the final grade 1.0.

Exam: Please bring your student ID card with you. You can bring your notes (lecture notes, homework sheets and tutorial materials). Electronic devices are not allowed. The final exam takes place on July 30, 10:00-12:00, room B005.

Homework Sheets

Contents of the lectures

24.04.2019. Chapter 0: Basic Concepts. Example of an elliptic boundary value problem. Variational formulation of weak solutions. Ritz-Galerkin approximation method. Piecewise polynomial spaces. Error estimates.

29.04.2019. Computer implementation. Domain decomposition and parallel computing. Chapter 1: Sobolev spaces. Basic properties of L^p spaces. Weak-derivatives of locally integrable functions. Sobolev spaces W^{k,p}.

06.05.2019. Sobolev inequality for W^{k,p}(R^d). Domains with Lipschitz boundary. Extension theorem. Sobolev inequality for W^{k,p}(U).

08.05.2019. Trace theorem. Duality and negative Sobolev spaces. Chapter 2: Variational formulation of Elliptic boundary value problems. Review of Hilbert spaces: bilinear form and inner product, completeness, orthogonal projections on subspaces, Riesz representation theorem. Variational problem with symmetric bilinear form.

13.05.2019. Examples in 1D. Contraction mapping theorem. Nonsymmetric bilinear forms. Lax-Milgram theorem for continuous coercive bilinear forms.

15.05.2019. Finite dimensional approximation. Céa Theorem on error estimate. Examples in higher dimensions. Chapter 3: Construction of Finite Element Spaces. Finite element (K,P,N). Lagrange finite element for 1D intervals. Lagrange finite element for 2D triangles.

22.05.2019. Additional discussion on boundary value problems: inhomogeneous and higher dimensional cases.

27.05.2019. 2D triangular elements (cont.): Hermite element, Argyris element. The Interpolant.

29.05.2019. Examples of dual basis. The global interpolant. Continuity problem. Triangulation. C^r property. Affine equivalence of elements.

03.06.2019. Interpolation equivalence and affine-interpolation equivalence. Rectangle elements: Tensor product elements and Serendipity elements.

12.06.2019. Finite elements in higher dimensions. Chapter 4: Polynomial approximation in Sobolev spaces. Recall of Taylor expansion for smooth functions. Average Taylor expansion for locally integrable functions. A preliminary error estimate involving Riesz potentials.

17.06.2019. Riesz potential estimates (I and II). Bounds for the remainder u - Q^m u in L^infty and in Sobolev spaces.

19.06.2019. Non-degenerate family of subdivisions. Bounds for the interpolation error.

24.06.2019. Quasi-uniform family of subdivisions. Inverse estimates in finite dimensional spaces. Comparisons of different Sobolev norms. Log Sobolev (discrete Sobolev) inequality in 2D.

26.06.2019. Chapter 5: N-dimensional variational problems. Variational formulation of Poisson's equation for Neumann/Dirichlet/mixed boundary conditions.

1.07.2019. Continuity and coercivity of the bilinear form. Existence and uniqueness of the variational formulation. Elliptic regularity. Examples of failure of the elliptic regularity. Proof of elliptic regularity for R^N.

3.07.2019. Error estimates for the finite dimensional approximation.

8.07.2019. General second-order elliptic equation: variational formulation, finite dimensional reduction, error estimates.

10.07.2019. Chapter 6: Finite element multigrid methods. General multigrid algorithm. Work estimate.

15.07.2019. Convergence of W-cycle method.

17.07.2019. Convergence of V-cycle method.

22.07.2019. Chapter 7: Additive Schwarz Domain decomposition methods. Abstract additive Schwarz framework. Schwarz Lemma for preconditioners.

24.07.2019. Example of Poisson's equation: overlapping domain decompositions and upper bound on the condition number. The relation between the conditon number and the convergence rate.