Institut für Angewandte Mathematik
Lecture in winter term 2024/2025
Numerical Mathematics 4
Content
More detailed analysis and application of finite element methods in particular mixed methods. 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 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 ansatz spaces (non-matching grids), and different fields.
Previous knowledge expected
Computational Mathematics 3, Partial Differential Equations
Scheduled dates
  • Mondays, 8:15-9:45, seminar room STEG006
  • Thursdays, 8:00-9:30, seminar room STEG006
  • Scheduled dates: TUG Online
  • first lecture: October 3, 2024
Exam
  • oral exam by arrangement
  • Due to study directives the language of presentation is English.
Exercise course
  • Exam method and evaluation of the exercise course:
    • There will probably be 6 or 7 exercise sheets with calculation examples (including coding examples).
    • The students have to mark (votieren) the 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 study directives the language of presentation is English. Each marked example will account for 2/3/(number of examples) of the total number of points.
    • The particpants have to present at least 4 examples succesfully. 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.
  • Thursdays, 8:00-9:30, seminar room STEG006
  • Scheduled dates: TUG Online
  • October 17, 2024: exercise sheet 1
  • October 31, 2024: exercise sheet 2
  • November 14, 2024: exercise sheet 3
  • November 28, 2024: exercise sheet 4
  • December 12, 2024: exercise sheet 5
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, Elsevier 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