Numerical Methods for Elliptic Partial Differential Equations | last update: |
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Lecture | up |
Numerical Methods for Elliptic Partial Differential Equations - Lectures
(CourseId 327.003, 4 hours per week, Semester 6)Lecturer: O.Univ.-Prof. Dr. Ulrich Langer
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Examination questions: | up |
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The super question: | up |
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Examination dates: | up |
Time and room:
| Wed, March 7, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 01 |
| Thu, March 8, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 02 |
| Tue, March 14, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 03 |
| Wed, March 15, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 04 |
| Thu, March 21, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 05 |
| Thu, March 22, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 06 |
| Easter Break | ||
| Wed, April 11, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 07 |
| Thu, April 12, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 08 |
| Wed, April 18, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 09 |
| Thu, April 19, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 10 |
| Wed, April 25, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 11 |
| Thu, April 26, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 12 |
| Wed, May 2, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 13 |
| Thu, May 3, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 14 |
| Wed, May 9, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 15 |
| Thu, May 10, 2018 | Christi Himmelfahrt | Lecture is canceled |
| Wed, May 16, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 16 |
| Thu, May 17, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 17 |
| Wed, May 23, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 18 |
| Thu, May 24, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 19 |
| Wed, May 30, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 20 |
| Thu, May 31, 2018 | Fronleichnam | Lecture is canceled |
| Wed, June 06, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 21 |
| Thu, June 07, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 22 |
| Wed, June 13, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 23 |
| Thu, June 14, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 24 |
| Wed, June 20, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 25 |
| Thu, June 21, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 26 |
| Wed, June 27, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 27 |
| Thu, June 28, 2018 | 08:30 - 10:00 Room: HS 13 | Lecture 28 |
Lecturer: O.Univ.-Prof. Dr. Ulrich Langer
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Tutorial | up |
Numerical Methods for Elliptic Partial Differential Equations - Tutorials
(CourseId 327.004, 2 hours per week, Semester 6)Tutorials held by: DI Christoph Hofer
Time and room:
| Tue, March 13, 2018 | 10:15 - 11:45 Room: S2 120 | Tutorial 01 |
| Tue, March 20, 2018 | 10:15 - 11:45 Room: S2 120 | Tutorial 02 |
| Easter Break | ||
| Tue, April 10, 2018 | 10:15 - 11:45 Room: S2 048 | Tutorial 03 |
| Tue, April 17, 2018 | 10:15 - 11:45 Room: S2 346 | Tutorial 04 |
| Tue, April 24, 2018 | 10:15 - 11:45 Room: S2 346 | Tutorial 05 |
| Tue, May 1, 2018 | Staatsfeiertag | Tutorial is canceled |
| Tue, May 8, 2018 | 10:15 - 11:45 Room: S2 346 | Tutorial 06 |
| Tue, May 15, 2018 | 10:15 - 11:45 Room: S2 346 | Tutorial 07 |
| Tue, May 22, 2018 | lehrveranstaltungsfrei | |
| Tue, May 29, 2018 | 10:15 - 11:45 Room: S2 346 | Tutorial 08 |
| Tue, June 5, 2018 | 10:15 - 11:45 Room: S2 346 | Tutorial 09 |
| Tue, June 12, 2018 | 10:15 - 11:45 Room: S2 346 | Tutorial 10 |
| Tue, June 19, 2018 | 10:15 - 11:45 Room: S2 346 | Tutorial 11 |
| Tue, June 26, 2018 | 10:15 - 11:45 Room: S2 346 | Tutorial 12 |
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Tutorials | up |
| Tutorial 01 | March 13, 2018 | ||
| Tutorial 02 | March 20, 2018 | ||
| Easter Break | |||
| Tutorial 03 | April 10, 2018 | ||
| Tutorial 04 | April 17, 2018 | ||
| Tutorial 05 | April 24, 2018 | ||
| Tutorial 06 | May 8, 2018 | ||
| Tutorial 07 | May 15, 2018 | zip | |
| Tutorial 08 | May 29, 2018 | ||
| Tutorial 09 | June 5, 2018 | ||
| Tutorial 10 | June 12, 2018 | ||
| Tutorial 11 | June 19, 2018 | ||
| Tutorial 12 | June 26, 2018 |
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Transparencies | up |
| Transparency 00a: colour | Math. Models |
| Transparency 00b: colour | Remark 1.2 |
| Transparency 01: colour | Ex 1.1 - 1.2 |
| Transparency 02: colour | Ex 1.3 - 1.4 |
| Transparency 03: colour | Ex 1.5 - 1.6 |
| Transparency 04a: colour | Ex 1.7 - 1.9 |
| Transparency 04b: colour | Remark 1.5 |
| Transparency 05: colour | Ex 1.10 - 1.11 |
| Transparency 05a: b/w | 1.3.1. Mixed VF I: General |
| Transparency 05b: b/w | 1.3.1. Mixed VF II: Navier-Stokes |
| Transparency 05c: b/w | 1.3.1. Mixed VF III: Oseen/Stokes |
| Transparency 05d: b/w | 1.3.1. Mixed VF IV: Poisson equ. |
| Transparency 05e: b/w | 1.3.1. Mixed VF V: 1st bih. BVP |
| Transparency 05f: b/w | 1.3.2. Dual VF I: General |
| Transparency 05g: b/w | 1.3.2. Dual VF II: Cont. |
| Transparency 05h: b/w | 1.3.2. Dual VF III: Example |
| Transparency 2-01: colour | D(/Omega) |
| Transparency 2-02: colour | Week derivatives |
| Transparency 2-03: colour | Distributions |
| Transparency 2-04: colour | Distributive derivatives |
| Transparency 2-05: colour | Lebesgue spaces Lp |
| Transparency 2-06: colour | Sobolev spaces W_p^k |
| Transparency 2-07: colour | Traces |
| Transparency 2-08: colour | Negative-order Sobolev spaces |
| Transparency 2-09: colour | H(div), H(curl), H^s |
| Transparency 2-10: colour | H^{1/2}(\Gamma) ~ \gamma_oH^1(\Omega) |
| Transparency 2-11: colour | Th. 2.13 Norm equivalence theorem |
| Transparency 2-12: colour | Exercise 2.14 |
| Transparency 2-13: colour | Friedrichs' inequalities I |
| Transparency 2-14: colour | Friedrichs' inequalities II |
| Transparency 2-15: colour | 2.4. Poincaré |
| Transparency 2-16: colour | 2.5. Main Formula of DIC |
| Transparency 2-17: colour | 2.5. Gauss' Theorem |
| Transparency 2-18: colour | 2.5. Further Integration Formulas |
| Transparency 2-19: colour | 2.5. H(div) - Trace Theorem |
| Transparency 2-20: colour | 2.5. H(div) Inverse Trace Theorem |
| Transparency 2-21: colour | 2.6. Extension Problem |
| Transparency 2-22: colour | 2.6. Extension Problem (cont) |
| Transparency 2-23: colour | 2.7. Embedding |
| Transparency 2-24: colour | 2.7. Embedding (cont) |
| Transparency 06a: b/w | Courant's idea |
| Transparency 06b: colour | Illustration |
| Transparency 07a: colour | Remark 2.1.1-2 |
| Transparency 07b: b/w | Remark 2.1.3-4 |
| Transparency 08a: colour | Model Problem |
| Transparency 08b: colour | CHIP |
| Transparency 09: colour | Mesh for CHIP |
| Transparency 10a: b/w | CHIP.NET |
| Transparency 10b: colour | Meshing |
| Transparency 10c: colour | Tables |
| Transparency 10d: b/w | Finer Mesh |
| Transparency 11: b/w | Mesh Generation 1.-2. |
| Transparency 12a: b/w | Mesh Generation 3. |
| Transparency 12b: colour | Mesh Generation 4. |
| Transparency 12c: colour | Mapping principle |
| Transparency 13a: colour | stiffness matrix (1) |
| Transparency 13b: b/w | stiffness matrix (2) |
| Transparency 13c: b/w | stiffness matrix (3) |
| Transparency 14a: b/w | 2nd kind BC |
| Transparency 14b: b/w | 3rd kind BC |
| Transparency 14c: b/w | 1st kind BC |
| Transparency 15: colour | Illustration |
| Transparency 16: b/w | Exercises 2.5 - 2.8 |
| Transparency 17a: colour | Road Map I |
| Transparency 17b: b/w | Road Map II |
| Transparency 17c: colour | Theorem 2.6 = Approximation Theorem |
| Transparency 17d: colour | Sketch of the Proof |
| Transparency 18a: colour | Remark 2.7.1 |
| Transparency 18b: b/w | Remark 2.7.2-5, E 2.9, E 2.10 |
| Transparency 19: b/w | Theorem 2.8 (H1-Convergence) |
| Transparency 20: b/w | Remark 2.9.1-4 |
| Transparency 21: b/w | Remark 2.9.5 |
| Transparency 22: b/w | Remark 2.14 |
| Transparency 23: colour | Var.Crimes I |
| Transparency 24: colour | Var.Crimes II |
| Transparency 25: colour | Var.Crimes III |
| Transparency 26: colour | Remark 3.21 |
| Transparency 27a: b/w | DWR I |
| Transparency 27b: b/w | DWR II |
| Transparency 27c: colour | AFEM |
| Transparency T4-04: colour | Consistency + DG-Scheme |
| Transparency T4-05: colour | Remark 4.3.: Pros & Cons |
| Transparency T4-06a: colour | Lemma 4.4. |
| Transparency T4-06b: colour | Alternative Proof |
| Transparency T4-07: colour | Lemma 4.5.: ellipticity |
| Transparency T4-08: colour | Proof |
| Transparency T4-09: colour | Lemma 4.7.: boundedness |
| Transparency T4-10: colour | Lemma 4.9.: Trace inequality |
| Transparency T4-11: colour | Theoerem 4.10.: Error estimate |
| Transparency T4-12: colour | Proof (cont.) |
| Transparency T4-13: colour | Proof (cont.) |
| Transparency T4-14: colour | Proof (cont.) + Remark 4.11 |
| Transparency T4-15: colour | 4.2.: FDM |
| Transparency T4-16: colour | 4.2.: FVM |
| Transparency T4-17: colour | 4.2.: Stability+Appr.=>discrete Conv. |
| Transparency T4-18: colour | Summary |
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Additional Transparencies | up |
| Transparency 28: colour | Remark 3.1 |
| Transparency 29: colour | Example, Remark 3.2 |
| Transparency 30: b/w | Secondary Grids I |
| Transparency 31: b/w | Secondary Grids II |
| Transparency 32: colour | Remark 3.3 + E 3.1 |
| Transparency 33: b/w | Remark 3.4 |
| Transparency 34: colour | Boundary boxes |
| Transparency 35: colour | Remark 3.5 + E 3.2 |
| Transparency 36a: b/w | Galerkin-Petrov I |
| Transparency 36b: b/w | Galerkin-Petrov II |
| Transparency 36c: colour | Galerkin-Petrov Approach |
| Transparency 36d: colour | Two Galerkin-Petrov Schemes |
| Transparency 36e: colour | System of FV-Equations |
| Transparency 37a: b/w | Remark 3.6.1-3.6.4 |
| Transparency 37b: b/w | Remark 3.6.5-3.6.6 |
| Transparency 38: colour | Ref + Remark 3.7 |
| Transparency 39: colour | Discrete Convergence I |
| Transparency 40: b/w | Discrete Convergence II |
| Transparency 41: b/w | Discrete Convergence III |
| Transparency 42: b/w | Discrete Convergence IV (E 3.3) |
| Transparency 43: b/w | Discrete Convergence V |
| Transparency 44: colour | Discrete Convergence VI |
| Transparency 39-44: b/w | Summary |
| Transparency 45: b/w | 4. BEM 4.1 Introduction I |
| Transparency 46: b/w | 4.1 Introduction II |
| Transparency 47: b/w | 4.1 Introduction III |
| Transparency 48: b/w | 4.1 Introduction IV |
| Transparency 49a: b/w | Subsection 4.2.1 |
| Transparency 50a: colour | Section 4.3: CM I |
| Transparency 50b: b/w | Section 4.3: CM II |
| Transparency 51a: colour | Section 4.3: CM III |
| Transparency 51b: b/w | Section 4.3: CM IV |
| Transparency 52a: b/w | Section 4.3: CM V |
| Transparency 52b: colour | Section 4.3: CM VI |
| Transparency 53: b/w | Section 4.3: CM VII |
| Transparency 54: b/w | Section 4.3: CM VIII |
| Transparency 55: b/w | Section 4.3: CM IV |
| Transparency 56: b/w | Section 4.3: CM X |
| Transparency 57: b/w | Section 4.3: CM XI |
| Transparency 58a: b/w | BIO: Def. |
| Transparency 58b: b/w | BIO: Calderon |
| Transparency 58c: b/w | BIO: D2N |
| Transparency 59a: b/w | 4.4.2 Properties I |
| Transparency 59b: b/w | 4.4.2 Properties II |
| Transparency 60: b/w | Galerkin I |
| Transparency 61: b/w | Galerkin II |
| Transparency 62: b/w | Galerkin III |
| Transparency 63: b/w | Galerkin IV |
| Transparency 64: b/w | Galerkin V |
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Basic Lecture Notes: | up |
Postscript-File
[2] Langer U.: Numerik II (Numerische Verfahren für Randwertaufgaben), JKU, Linz 1996 (FEM and FVM).
Postscript-File
[3] Jung M., Langer U.: Methode der finiten Elemente für Ingenieure: Eine Einführung in die numerischen Grundlagen und Computersimulation. Springer Fachmedien, Wiesbaden 2013, 2., überarb. u. erw. Aufl. 2013, XVI, 639 S. 172 Abb. (practical aspects of the FEM).
http://www.springer.com/springer+vieweg/maschinenbau/book/978-3-658-01100-0
[4] Steinbach O.: Numerische Näherungsverfahren für elliptische Randwertprobleme. Teubner-Verlag, Stuttgart, Leipzig, Wiesbaden 2003 (FEM and BEM).
English version:
Steinbach O.: Numerical Approximation Methods for Elliptic Boundary Value Problem: Finite and Boundary Elements. Springer, New York 2008 (FEM and BEM):
FEBEBook
[5] Steinbach O.: Lösungsverfahren für lineare Gleichungssysteme: Algorithmen und Anwendungen. Teubner-Verlag, Stuttgart, Leipzig, Wiesbaden 2005 (solvers for systems of algebraic equations).
[6] Zulehner W.: Numerische Mathematik: Eine Einführung anhand von Differentialgleichungsproblemen. Band 1: Stationäre Probleme. Mathematik Kompakt. Birkhäuser Verlag, Basel-Bosten-Berlin 2008.
[7] Rivière B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM, Philadelphia 2008.
[8] Di Pietro D.A., Ern A.: Mathematical Aspects of Discontinuous Galerkin Method. Springer-Verlag, Berlin, Heidelberg, 2012.
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Additional Literature: | up |
English version: Braess D.: Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics. Cambridge University Press, Cambridge, 1997, 2001, 2007. - ISBN: 0 521 70518-9 Homepage: http://homepage.ruhr-uni-bochum.de/Dietrich.Braess/ftp.html#books
[2] Brenner S.C., Scott L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York 1994.
[3] Ciarlet P.G.: The finite element method for elliptic problems. Classics in Applied Mathematics (40), SIAM, Philadelphia PA, 2002. [4] Großmann C., Roos H.-G.: Numerik partieller Differentialgleichungen. Teubner-Verlag, Stuttgart 1992. (3. völlig überarbeitete und erweiterte Auflage, November 2005)
[5] Deuflhard P., Weiser M.: Numerische Mathematik: Band 3 "Adaptive Lösung partieller Differentialgleichungen. de Gruyter Verlag, Berlin 2011 (englische Version ist 2012 ebenfalls bei de Gruyter erschienen).
[6] Heinrich B.: Finite Difference Methods on Irregular Networks. Akademie-Verlag, Berlin 1987.
[7] Knaber P., Angermann L.: Numerik partieller Differentialgleichungen. Eine anwendungsorientierte Einführung. Springer-Verlag, Berlin-Heidelberg 2000.
[8] Monk P.: Finite Element Methods for Maxwell's Equations. Oxford Science Publications, Oxford 2003.
[9] Schwarz H.R.: FORTRAN-Programme zur Methode der finiten Elemente. B.G. Teubner, Stuttgart, 1991.
[10] Schwarz H.R.: Methode der finiten Elemente. B.G. Teubner, Stuttgart, 1991.
[11] Verfürth R.: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley - Teubner, 1996.
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History: | up |
Historical Papers
Gander's presentation
M.J. Gander and G. Wanner: From Euler, Ritz and Galerkin to Modern Computing , SIAM Review, 54(4)
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Software: | up |
| FEM1D | FEM2D | NETREFINER | FEM EP | Mesh Generation |
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Links: | up |
NETGEN
NGSolve
SPIDER
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General Information | up |
- Linear algebra and analytic geometry 1 and 2
- Analysis 1 - 3 (particularly Analysis 3)
- Knowledge of Computer Science and Programming
- Numerical analysis
- Partial differential equations and Integral equations
- Mathematical models in engineering
- Numerical methods for partial differential equations
- Numerical Methods for Non-Stationary Problems
- Special Topics in Computational Mathematics
- Special Seminars in Computational Mathematics
Get familiar with advanced numerical methods for the solution of multidimensional elliptic Boundary Value Problems (BVP) for Partial Differential Equations (PDE) and with tools for their analysis.
Contents:
- Variational formulation of multidimensional elliptic boundary value problems (BVP) and their analysis
- Useful Tools from the theory of Sobolev spaces
- The Galerkin Finite Element Methods (FEM) for BVP
- Other Discretization Methods (DG, FDM, FVM, BEM)
- Solvers
- There is a Tutorial accompanying the lectures. This tutorial provides practical skills for applying numerical methods to the solution of elliptic boundary value problems. The tutorial has 2 hours per week.
- Supervisor: DI Christoph Hofer
- First Tutorial: Tuesday, March 13, 2018, 10:15 - 11:45, Room S2 120 !
Lecture:
The lecture contains an oral examination.
Tutorial:
The mark of the tutorial consists of the assessment of the individual exercises, the presentations on the blackboard and a practical exercise on a LLTP (Long-Term Training Problem).
