General Information

Description and Course Evaluation

The solutions of partial differential equations cannot usually be expressed with closed formulas, so we rely on numerical approximations. The lecture covers linear equations of both elliptic and parabolic type. The analytical solution theory (existence and interpretation) and the finite element method (FEM) for numerical approximation are the main objectives.

Recommended Knowledge:

• Basic analysis and linear algebra
• Linear functional analysis (will be supplemented during lectures)
• Programming knowledge (Python, Matlab, etc.) Python is preferred.

Exercise Sheets:

Exercise sheets will be uploaded on the course webpage and on Moodle, one week before the submission deadline. Each exercise sheet will contain two marked exercises (out of four or more). To qualify for the exam, students must submit at least one complete solution from the marked exercises on at least three of the six exercise sheets.

* Note: Submitted exercises do not contribute to the final grade; they are required solely for exam eligibility (Zulassung).

Course Evaluation:

The course evaluation consists of an oral exam at the end of the semester, lasting 30 minutes. Exact dates will be announced by the end of April.

Course Materials

* A login to Moodle with your university account is required. Students enrolled in Numerik Partieller Differentialgleichungen I should click on “Numerik PDE I,” and students enrolled in Theory and Numerical Analysis of Partial Differential Equations I should click on “Theory & Analysis I.” This will redirect to the respective Moodle page. Both PDFs are identical.

Theoretical Exercises

* TBA = To Be Announced

Programming Exercises

Here you will find some numerical exercises that are optional, but interesting.

Lecture Calendar

Literature

• [1] Susanne C. Brenner; L. Ridgeway Scott, The Mathematical Theory of Finite Element Methods, Third Edition, 2008.
• [2] D. Gallistl, Theorie und Numerik partieller Differentialgleichungen I (Computational PDEs I), Lecture notes.
• [3] Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Pub. Co, 1978.
• [4] Alexandre Ern, Jean-Luc Guermond, Theory and Practice of Finite Elements, Springer, 2010.
• [5] Kendall E. Atkinson, Weimin Han, Theoretical Numerical Analysis: A Functional Analysis Framework, Springer, 2007.
• [6] Mark S. Gockenbach, Understanding and Implementing the Finite Element Method, SIAM, 2006.
• [7] Mats G. Larson, Fredrik Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Springer Berlin, Heidelberg, 2015.
• [8] Lawrence C. Evans, Partial Differential Equations, AMS, 2015.
• [9] Vidar Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Berlin Heidelberg, 1984.