Numerical Methods for Elliptic Partial Differential Equations | last update: 2021-02-13 |
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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
- Due to rector's CORONA directives, the classes will be offered online until end of June. Please study Lecture 04-13 at home.
- Lectures 14-29 at home and at ZOOM to which you will be invited !
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Examination questions: | up |
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The super question: | up |
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Examination dates: | up |
| Wed, March 3, 2020 | 10:15 - 11:45 Room: S2 416-1 | Lecture 01 |
| Wed, March 4, 2020 | 08:30 - 10:00 Room: HS 14 | Lecture 02 |
| Wed, March 5, 2020 | 08:30 - 10:00 Room: HS 13 | Lecture 03 |
| Wed, March 11, 2020 | 08:30 - 10:00 Room: -- -- | Lecture 04 |
| Thu, March 12, 2020 | 08:30 - 10:00 Room: -- -- | Lecture 05 |
| Wed, March 18, 2020 | 08:30 - 10:00 Room: -- -- | Lecture 06 |
| Thu, March 19, 2020 | 08:30 - 10:00 Room: -- -- | Lecture 07 |
| Wed, March 25, 2020 | 08:30 - 10:00 Room: -- -- | Lecture 08 |
| Thu, March 26, 2020 | 08:30 - 10:00 Room: -- -- | Lecture 09 |
| Wed, April 1, 2020 | 08:30 - 10:00 Room: -- -- | Lecture 10 |
| Thu, April 2, 2020 | 08:30 - 10:00 Room: -- -- | Lecture 11 |
| Easter Break | ||
| Wed, April 22, 2020 | 08:30 - 10:00 Room: -- -- | Lecture 12 |
| Thu, April 23, 2020 | 08:30 - 10:00 Room: -- -- | Lecture 13 |
| Wed, April 29, 2020 | 08:30 - 10:00 Room: ZOOM | Lecture 14 |
| Thu, April 30, 2020 | 08:30 - 10:00 Room: ZOOM | Lecture 15 |
| Wed, May 6, 2020 | 08:30 - 10:00 Room: ZOOM | Lecture 16 |
| Thu, May 7, 2020 | 08:30 - 10:00 Room: ZOOM | Lecture 17 |
| Wed, May 13, 2020 | 08:30 - 10:00 Room: ZOOM | Lecture 18 |
| Thu, May 14, 2020 | 08:30 - 10:00 Room: ZOOM | Lecture 19 |
| Wed, May 20, 2020 | 08:30 - 10:00 Room: ZOOM | Lecture 20 |
| Thu, May 21, 2020 | Christi Himmelfahrt | Lecture is canceled |
| Wed, May 27, 2020 | 08:30 - 10:00 Room: ZOOM | Lecture 21 |
| Thu, May 28, 2020 | 08:30 - 10:00 Room: ZOOM | Lecture 22 |
| Wed, June 3, 2020 | 08:30 - 10:00 Room: ZOOM | Lecture 23 |
| Thu, June 4, 2020 | 08:30 - 10:00 Room: ZOOM | Lecture 24 |
| Wed, June 10, 2020 | 08:30 - 10:00 Room: ZOOM | Lecture 25 |
| Thu, June 11, 2020 | Fronleichnam | Lecture is canceled |
| Wed, June 17, 2020 | 08:30 - 10:00 Room: ZOOM | Lecture 26 |
| Thu, June 18, 2020 | 08:30 - 10:00 Room: ZOOM | Lecture 27 |
| Wed, June 24, 2020 | 08:30 - 10:00 Room: ZOOM | Lecture 28 | Thu, June 25, 2020 | 08:30 - 10:00 Room: ZOOM | Lecture 29 |
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 Rainer Schneckenleitner
Time and room:
| Tutorial 01 | Tue, March 10, 2020 | 10:15 - 11:45 Room: KEP3 | Tutorial 01 |
| Tutorial 02 | Tue, March 17, 2020 | 10:15 - 11:45 Room: -- | Tutorial 02 |
| Tutorial 03 | Tue, March 24, 2020 | 10:15 - 11:45 Room: -- | Tutorial 03 |
| Tutorial 04 | Tue, March 31, 2020 | 10:15 - 11:45 Room: -- | Tutorial 04 |
| Easter Break | |||
| Tutorial 05 | Tue, April 21, 2020 | 10:15 - 11:45 Room: -- | Tutorial 05 |
| Tutorial 06 | Tue, April 28, 2020 | 10:15 - 11:45 Room: -- | Tutorial 06 |
| Tutorial 07 | Tue, May 5, 2020 | 10:15 - 11:45 Room: -- | Tutorial 07 |
| Tutorial 08 | Tue, May 12, 2020 | 10:15 - 11:45 Room: -- | Tutorial 08 |
| Tutorial 09 | Tue, May 19, 2020 | 10:15 - 11:45 Room: -- | Tutorial 09 |
| Tutorial 10 | Tue, May 26, 2020 | 10:15 - 11:45 Room: -- | Tutorial 10 |
| Tue, June 2, 2020 | Pfingstdienstag | canceled | |
| Tutorial 11 | Tue, June 9, 2020 | 10:15 - 11:45 Room: -- | Tutorial 11 |
| Tutorial 12 | Tue, June 16, 2020 | 10:15 - 11:45 Room: -- | Tutorial 12 |
| Tutorial 13 | Tue, June 23, 2020 | 10:15 - 11:45 Room: -- | Tutorial 13 |
| Tutorial 14 | Tue, June 30, 2020 | 10:15 - 11:45 Room: -- | Tutorial 14 |
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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 | 1.2.2 Linear elasticity I |
| Transparency 04b: colour | 1.2.2 Linear elasticity II |
| Transparency 04c: colour | 1.2.2 Linear elasticity III |
| Transparency 04d: colour | 1.2.2 Linear elasticity IV |
| Transparency 04e: colour | 1.2.2 Linear elasticity V |
| 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 06: colour | GALERKIN-RITZ-Scheme |
| 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 11a: b/w | Mesh Generation 1.-2. |
| Transparency 11b: b/w | Mesh Generation 3. |
| Transparency 11c: colour | Mesh Generation 4. |
| Transparency 11d: colour | Mesh Generation 5. |
| Transparency 12: 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 T4-01a: colour | 4.1.1 DG VF: model problem |
| Transparency T4-01b: colour | 4.1.1 DG VF: notations and formulation |
| Transparency T4-02: colour | 4.1.1 DG VF: DG bilinear form |
| Transparency T4-03: colour | Alternative Proof |
| 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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CISM Courses | up |
[Part 2]
[Part 3]
[Part 4]
[Part 5]
see also [9] in Basic Lecture Notes.
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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.
[9] Langer U. and Neumæller M.: Direct and iterative solvers. In M. Kaltenbacher, editor, Computational Acoustics, volume 579 of CISM International Centre for Mechanical Sciences: Courses and Lectures, pages 205-251. Springer-Verlag, 2017.
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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)
- 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 Rainer Schneckenleitner
- First Tutorial: Tuesday, March 10, 2020, 10:15 - 11:45, Room KEP3 !
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 LTTP (Long-Term Training Problem).
