Seminar on Numerical Analysis

letzte Änderung: 2021-09-30

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Seminar on Numerical Analysis

Adaptive Finite Element Methods (AFEM)

(Course Id 327.014, 2 hours per week, semester 5)

Supervisors: A.Univ.-Prof. Dr. Helmut Gfrerer

                       O.Univ.-Prof. Dr. Ulrich Langer

                       A.Univ.-Prof. Dr. Walter Zulehner

Start of the seminar: Tue, 2019-10-08, 13:45 - 15:15 Uhr, Room: S2 054

Time, room, seminar, student, supervisor:

Tue, 2019-10-08 13:45 - 15:15 Room: S2 054 Seminar 01: Introduction Langer
Tue, 2019-10-15 13:45 - 15:15 Room: S2 054 Consultation
Tue, 2019-10-22 13:45 - 15:15 Room: S2 054 Seminar 02 Endtmayer Gfrerer
Tue, 2019-10-29 13:45 - 15:15 Room: S2 054 Consultation/WS2 RICAM
Tue, 2019-11-05 13:45 - 15:15 Room: S2 054 Seminar 03 Mitter Zulehner
Tue, 2019-11-12 13:45 - 15:15 Room: S2 054 Consultation
Tue, 2019-11-19 13:45 - 15:15 Room: S2 054 Consultation
Tue, 2019-11-26 13:45 - 15:15 Room: S2 054 Seminar 04 Gobrial Zulehner
Tue, 2019-12-03 13:45 - 15:15 Room: S2 054 cont.
Tue, 2019-12-10 13:45 - 15:15 Room: S2 054 Seminar 05 Jodlbauer Zulehner
Tue, 2019-12-17 13:45 - 15:15 Room: S2 054 cont.
Tue, 2020-01-07 13:45 - 15:15 Room: S2 054 Seminar 06 Schneckenleitner Langer
Tue, 2020-01-14 13:45 - 15:15 Room: S2 054 cont.
Tue, 2020-01-21 13:45 - 15:15 Room: S2 054 Seminar 07 Schafelner Langer
Tue, 2020-01-28 13:45 - 15:15 Room: S2 054 cont.


- Seminar Topics up

The seminar deals with the theoretical aspects of AFEM for elliptic BVP such as

> Reliability of a posteriori error estimates
> Efficiency of a posteriori error estimates
> Constants
> Computable bounds
> Convergence
> Convergence rates

but also with a posteriori error estimates and AFEM for

> parabolic initial-boundary value problems
> hyperbolic problems
> optimal control problems


Seminar 01: 2019-10-08
Introduction by Langer
Seminar 02: 2019-10-22
Title: Convergence of AFEM
Literature:   [1]
Supervisor: Gfrerer
Student:     Endtmayer
Seminar 03: 2019-11-05
Title: Data oscillation and convergence of adaptive FEM
Literature:   [2]
Supervisor: Zulehner
Student:     Mitter
Seminar 04: 2019-11-26
Title: Newest vertex bisection method
Literature:   [3]
Supervisor: Zulehner
Student:     Gobrial
Seminar 05: 2019-12-10
Title: Optimality of a standard AFEM
Literature:   [4]
Supervisor: Zulehner
Student:     Jodlbauer
Seminar 06: 2020-01-07
Title: Axioms of Adaptivity I
Literature:   [5], Sections 1 - 3, i.e. pp. 1195 - 1207,Subsection 5.1
Supervisor: Langer
Student:     Schneckenleitner
Seminar 07: 2020-01-21
Title: Axioms of Adaptivity II
Literature:   [5], Sections 4 + Subsection 5.1, i.e. pp. 1207 -1217
Supervisor: Langer
Student:     Schafelner


- Literature up

[1]   W. Dörfler, Convergent Adaptive Algorithm for Poisson’s Equation. SIAM J. Numer. Anal., 33(3), 1106–1124.
[2]   P. Morin, R. Nochetto, and K. Siebert. Data oscillation andconvergence of adaptive FEM. SIAM J. Numer. Anal., 38(2):466–488,2000.
[3]   P. Binev, W. Dahmen, and R. DeVore. Adaptive finite elementmethods withconvergence rates. Numer. Math., 97(2):219 – 268, 2004.
[4]   R. Stevenson, Optimality of a standard adaptive finiteelement method, Found. Comput. Math. 7 (2) (2007) 245–269.
[5]   C. Carstensen, M. Feischl, M. Page, and C. Praetorius. Axiomsof adaptivity. Comput. Methods Appl. Math. 2014; 67:1195-1253.

- Additional Literature: Books on AFEM up

[A]   R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, 1996 (150 pp).
[B]   R. Verfürth, A Posteriori Error Estimation Techniques for Finite Element Methods. Oxford University Press, 2013 (416 pp).
[C]   M. Ainsworth, J.T. Oden, A posteriori error estimation infinite element analysis, in: Pure and Applied Mathematics (New York),Wiley-Interscience, New York, 2000.
[D]   I. Babuska, T. Strouboulis, The finite element method andits reliability, in: Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2001.
[E]   P. Neittaanmäki, S. Repin, Reliable Methods for Computer Simulation, in: Studies in Mathematics and its Applications, vol. 33,Elsevier Science B.V, Amsterdam, 2004.
[F]   S. Repin, A Posteriori Estimates for Partial Differential Equations, in: Radon Series on Computational and Applied Mathematics,vol. 4, Walter de Gruyter GmbH & Co. KG, Berlin, 2008.
[G]   W. Han, A POSTERIORI ERROR ANALYSIS VIA DUALITY THEORY. With Applications in Modeling and Numerical Approximations. Springer, New York, 2005.

- Participants up

Endtmayer Bernhard
Gobrial Mario
Jodlbauer Daniel
Mitter Ludwig
Pistrich Mario
Schafelner Andreas
Schneckenleitner Rainer

- General Remarks up