Institut für Angewandte Mathematik

Lecture in the Summer Term 2022 
Numerics and simulation/ Elective subject mathematics (Domain Decomposition Methods) 
Content 
We will deal with a more detailed analysis and application of finite element methods for solving practical problems in natural science and engineering.
An introduction is given to domain decomposition methods, a class of efficient computational methods, and to the underlying theory. In domain decomposition methods, many small problems are solved a certain number of times instead of solving a huge problem once. Using suitable methods, the computational times are reduced. In particular, many variations of domain decomposition methods and the abstract theory of Schwarz methods are covered.
Further, some basics of the parallel implementation of domain decomposition methods are discussed using MPI.

Expected prior knowledge 
analysis and computational methods of partial differential equations, in particular finite element methods

Scheduled dates 
 on Tuesdays 8:159:45 in seminar room STEG006
 on Wednesday 12:1513:45 in seminar room STEG006 (alternately with the Practical)
 Scheduled dates: TUG Online
 first lecture: March 1, 2022

Practical 
 Exam method and evaluation of the exercise course:
 There will probably be some exercise sheets with 2025 calculation examples in total and 2 exercise sheets with 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 40 % of the total number of points.
 The particpants have to present at least ? calculation examples succesfully. The quality of the presentations will account for 20 % of the total number of points.
 The solutions of the coding examples have to be submitted. Each coding exercise sheet accounts for approximately 20 % of the total number of points.
 50 % of the total number of points are necessary to pass the practical.
 After voting or submission for two exam dates the course will be assessed negatively if applicable.
 on Wednesdays 12:1513:45 in seminar room STEG006 (alternately with the lecture)
 Scheduled dates: TUG Online
 first practical: March 8, 2022
 March 8, 2022: Exercise sheet 1
 March 23, 2022: Exercise sheet 2
 April 6, 2022: Exercise sheet 3
 May 3, 2022: Exercise sheet 4
 May 18, 2022: Exercise sheet 5, code, template
 June 1, 2022: Exercise sheet 6
 May 18, 2022: Exercise sheet 5, code

Selected References 
 Andrea Toselli and Olof Widlund:
Domain decomposition methods—algorithms and theory, volume 34 of
Springer Series in Computational Mathematics.
SpringerVerlag, Berlin, 2005.
 Alfio Quarteroni and Alberto Valli:
Domain decomposition methods for partial differential equations.
Numerical Mathematics and Scientific Computation. The Clarendon Press Oxford
University Press, New York, 1999.
Oxford Science Publications.
 Barry F. Smith,
Petter E. Bjørstad, and William D. Gropp:
Domain decomposition.
Cambridge University Press, Cambridge, 1996.
Parallel multilevel methods for elliptic partial differential equations.
 Clemens Pechstein: Finite and Boundary Element tearing and Interconnecting Solvers for Multiscale Problems. Volume 90 of Lecture Notes in Computational Science and Engineering. Springer, Berlin, Heidelberg, 2013.
 Tarek P. A. Mathew.
Domain
decomposition methods for the numerical solution of partial differential
equations, volume 61 of Lecture Notes in Computational
Science and Engineering.
SpringerVerlag, Berlin, 2008.
 Victorita Dolean, Pierre Jolivet, and Frédéric Nataf.
An Introduction to Domain Decomposition Methods: Algorithms, Theory, and Parallel Implementation. SIAM, Philadelphia, 2015.

Contact 
Contact and office hours Günther Of 