Institut für Numerische Mathematik
Lecture in the Summer Term 2021
Numerics and simulation/ Elective subject mathematics (Gemischte Finite Elemente Methoden & Anwendungen)
Content
We will deal with a more detailed analysis and application of finite element methods for solving practical problems in natural science and engineering.
Mixed methods denote a class of finite element methods which have more than one approximation space. Typical examples are saddle point problems with Lagrangian multipliers to satisfy constraints. The unique solvability of these problems follows from coercivity and the inf-sup condition. However, not all choices of finite element spaces lead to stable and convergent approximations. In particular, the discrete inf-sup condition, also known as BBL condition, must be satisfied. This can be guaranteed by an appropriate, problem depending choice of the finite element spaces.
Examples arise in electromagnetics, elasticity, and fluid mechanics. In addition, these methods play an important role in the coupling of different discretization methods, different trial spaces (non-matching grids), and different fields.
Expected prior knowledge
analysis and computational methods of partial differential equations, in particular finite element methods
Scheduled dates
• on Tuesdays and Wednesdays 8:15--9:45 (alternately with the practical) in seminar room STEG006 (if applicable)
• It is to be expected that the lecture will not be possible in the classroom at the start of the semester. Alternatively, the lecture will be recorded and hopefully available in TUBE (and the teach center) within 1 or 2 days.
• Scheduled dates: TUG Online
• first lecture: March 2, 2021
Exam
• oral exam by arrangement
• Due to study directives the language of the exam is English.
Practical
• Exam method and evaluation of the exercise course:
• There will probably be 7 exercise sheets with calculation examples and coding examples.
• The students have to mark (votieren) the calculation examples they were able to solve at the beginning of each class. The students can be asked to present any example marked as solved on the blackboard. Due to new study directives the language of presentation is English.
• The percentage of marked examples will account for 2/3 of the total number of points.
• The quality of the presentations will account for 1/3 of the total number of points.
• 50 % of the total number of points are sufficient to complete the practical succesfully.
• After voting or submission for two exam dates the course will be assessed negatively if applicable.
• on Tuesdays and Wednesdays 8:15--9:45 (alternately with the lecture) in a Webex meeting or in seminar room STEG006
• Scheduled dates: TUG Online
• first practical: March 16, 2021
• Exercise sheets and submission in the teach center
Selected References
• Daniele Boffi, Franco Brezzi, Michel Fortin. Mixed Finite Element Methods and Applications. Springer, Berlin, Heidelberg, 2013.
• Franco Brezzi, Michel Fortin. Mixed and hybrid Finite Element Methods. Springer-Verlag, New York, 1991.
• Dietrich Braess. Finite Elemente. Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie. Springer, Berlin Heidelberg, 2013.
• Susanne C. Brenner, L. Ridgway Scott. The Mathematical Theory of Finite Element Methods. Springer, New York, 2008.
• Carsten Carstensen, Peter Wriggers. Mixed finite element technologies. CISM Courses and Lectures, Springer, Wien New York, 2009.
• Alexandre Ern, Jean-Luc Guermond. Theory and Practice of Finite Elements. Springer-Verlag New York, 2004.
• Gabriel N. Gatica. A Simple Introduction to the Mixed Finite Element Method: Theory and Applications. Springer, 2014.
• Vivette Girault, Pierre-Arnaud Raviart. Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer-Verlag, Berlin Heidelberg, 1986.
• J.E. Roberts, J.-M- Thomas. Mixed and hybrid Methods. in: Handbook of Numerical Analysis, Vol II, edited by P.G. Ciralet, J.L Lions, Elseviert Science Publisher B.V. (North-Holland), 1991.
• Olaf Steinbach. Numerical Approximation Methods for Elliptic Boundary Value Problems. Finite and Boundary Elements. Springer, New York, 2008.
Contact
Contact and office hours Günther Of