(CourseId 327.003, 4 hours per week, Semester 6)
Lecturer: O.Univ.-Prof. Dr. Ulrich Langer
Files: The material is collected in the following zip files: NuEPDELecureA2020s.zip, NuEPDELecureB2020s.zip, NuEPDESlides2020s.zip, NuEPDEOther2020s.zip
Examination questions: Exam2020.pdf
The super question: Super2020.pdf
Time and room:
| Wed, March 3, 2020 | 10:15 - 11:45 Room: S2 416-1 | lecture01.pdf |
| Wed, March 4, 2020 | 08:30 - 10:00 Room: HS 14 | lecture02.pdf |
| Wed, March 5, 2020 | 08:30 - 10:00 Room: HS 13 | lecture03.pdf |
| Wed, March 11, 2020 | 08:30 - 10:00 Room: -- -- | lecture04.pdf |
| Thu, March 12, 2020 | 08:30 - 10:00 Room: -- -- | lecture05.pdf |
| Wed, March 18, 2020 | 08:30 - 10:00 Room: -- -- | lecture06.pdf |
| Thu, March 19, 2020 | 08:30 - 10:00 Room: -- -- | lecture07.pdf |
| Wed, March 25, 2020 | 08:30 - 10:00 Room: -- -- | lecture08.pdf |
| Thu, March 26, 2020 | 08:30 - 10:00 Room: -- -- | lecture09.pdf |
| Wed, April 1, 2020 | 08:30 - 10:00 Room: -- -- | lecture10.pdf |
| Thu, April 2, 2020 | 08:30 - 10:00 Room: -- -- | lecture11.pdf |
| Easter Break | ||
| Wed, April 22, 2020 | 08:30 - 10:00 Room: -- -- | lecture12.pdf |
| Thu, April 23, 2020 | 08:30 - 10:00 Room: -- -- | lecture13.pdf |
| Wed, April 29, 2020 | 08:30 - 10:00 Room: ZOOM | lecture14.pdf |
| Thu, April 30, 2020 | 08:30 - 10:00 Room: ZOOM | lecture15.pdf |
| Wed, May 6, 2020 | 08:30 - 10:00 Room: ZOOM | lecture16.pdf |
| Thu, May 7, 2020 | 08:30 - 10:00 Room: ZOOM | lecture17.pdf |
| Wed, May 13, 2020 | 08:30 - 10:00 Room: ZOOM | lecture18.pdf |
| Thu, May 14, 2020 | 08:30 - 10:00 Room: ZOOM | lecture19.pdf |
| Wed, May 20, 2020 | 08:30 - 10:00 Room: ZOOM | lecture20.pdf |
| Wed, May 27, 2020 | 08:30 - 10:00 Room: ZOOM | lecture21.pdf |
| Thu, May 28, 2020 | 08:30 - 10:00 Room: ZOOM | lecture22.pdf |
| Wed, June 3, 2020 | 08:30 - 10:00 Room: ZOOM | lecture23.pdf |
| Thu, June 4, 2020 | 08:30 - 10:00 Room: ZOOM | lecture24.pdf |
| Wed, June 10, 2020 | 08:30 - 10:00 Room: ZOOM | lecture25.pdf |
| Wed, June 17, 2020 | 08:30 - 10:00 Room: ZOOM | lecture26.pdf |
| Thu, June 18, 2020 | 08:30 - 10:00 Room: ZOOM | lecture27.pdf |
| Wed, June 24, 2020 | 08:30 - 10:00 Room: ZOOM | lecture28.pdf |
| Thu, June 25, 2020 | 08:30 - 10:00 Room: ZOOM | lecture29.pdf |
(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 | tutorial01.pdf |
| Tutorial 02 | Tue, March 17, 2020 | 10:15 - 11:45 Room: -- | tutorial02.pdf |
| Tutorial 03 | Tue, March 24, 2020 | 10:15 - 11:45 Room: -- | tutorial03.pdf |
| Tutorial 04 | Tue, March 31, 2020 | 10:15 - 11:45 Room: -- | tutorial04.pdf |
| Tutorial 05 | Tue, April 21, 2020 | 10:15 - 11:45 Room: -- | tutorial05.pdf |
| Tutorial 06 | Tue, April 28, 2020 | 10:15 - 11:45 Room: -- | tutorial06.pdf |
| Tutorial 07 | Tue, May 5, 2020 | 10:15 - 11:45 Room: -- | tutorial07.pdf |
| Tutorial 08 | Tue, May 12, 2020 | 10:15 - 11:45 Room: -- | tutorial08.pdf |
| Tutorial 09 | Tue, May 19, 2020 | 10:15 - 11:45 Room: -- | tutorial09.pdf |
| Tutorial 10 | Tue, May 26, 2020 | 10:15 - 11:45 Room: -- | tutorial10.pdf |
| Tutorial 11 | Tue, June 9, 2020 | 10:15 - 11:45 Room: -- | tutorial11.pdf |
| Tutorial 12 | Tue, June 16, 2020 | 10:15 - 11:45 Room: -- | tutorial12.pdf |
| Tutorial 13 | Tue, June 23, 2020 | 10:15 - 11:45 Room: -- | tutorial13.pdf |
| Tutorial 14 | Tue, June 30, 2020 | 10:15 - 11:45 Room: -- | No new sheet |
| Transparency 00a: nuepde00aco.pdf | Math. Models |
| Transparency 00b: nuepde00bco.pdf | Remark 1.2 |
| Transparency 01: nuepde01co.pdf | Ex 1.1 - 1.2 |
| Transparency 02: nuepde02co.pdf | Ex 1.3 - 1.4 |
| Transparency 03: nuepde03co.pdf | Ex 1.5 - 1.6 |
| Transparency 04a: nuepde04a.pdf | 1.2.2 Linear elasticity I |
| Transparency 04b: nuepde04b.pdf | 1.2.2 Linear elasticity II |
| Transparency 04c: nuepde04c.pdf | 1.2.2 Linear elasticity III |
| Transparency 04d: nuepde04d.pdf | 1.2.2 Linear elasticity IV |
| Transparency 04e: nuepde04e.pdf | 1.2.2 Linear elasticity V |
| Transparency 05: nuepde05co.pdf | Ex 1.10 - 1.11 |
| Transparency 05a: nuepde05asw.pdf | 1.3.1. Mixed VF I: General |
| Transparency 05b: nuepde05bsw.pdf | 1.3.1. Mixed VF II: Navier-Stokes |
| Transparency 05c: nuepde05csw.pdf | 1.3.1. Mixed VF III: Oseen/Stokes |
| Transparency 05d: nuepde05dsw.pdf | 1.3.1. Mixed VF IV: Poisson equ. |
| Transparency 05e: nuepde05esw.pdf | 1.3.1. Mixed VF V: 1st bih. BVP |
| Transparency 05f: nuepde05fsw.pdf | 1.3.2. Dual VF I: General |
| Transparency 05g: nuepde05gsw.pdf | 1.3.2. Dual VF II: Cont. |
| Transparency 05h: nuepde05hsw.pdf | 1.3.2. Dual VF III: Example |
| Transparency 2-01: nuepde2-01co.pdf | D(/Omega) |
| Transparency 2-02: nuepde2-02co.pdf | Week derivatives |
| Transparency 2-03: nuepde2-03co.pdf | Distributions |
| Transparency 2-04: nuepde2-04co.pdf | Distributive derivatives |
| Transparency 2-05: nuepde2-05co.pdf | Lebesgue spaces Lp |
| Transparency 2-06: nuepde2-06co.pdf | Sobolev spaces W_p^k |
| Transparency 2-07: nuepde2-07co.pdf | Traces |
| Transparency 2-08: nuepde2-08co.pdf | Negative-order Sobolev spaces |
| Transparency 2-09: nuepde2-09co.pdf | H(div), H(curl), H^s |
| Transparency 2-10: nuepde2-10co.pdf | H^{1/2}(\Gamma) ~ \gamma_oH^1(\Omega) |
| Transparency 2-11: nuepde2-11co.pdf | Th. 2.13 Norm equivalence theorem |
| Transparency 2-12: nuepde2-12co.pdf | Exercise 2.14 |
| Transparency 2-13: nuepde2-13co.pdf | Friedrichs' inequalities I |
| Transparency 2-14: nuepde2-14co.pdf | Friedrichs' inequalities II |
| Transparency 2-15: nuepde2-15co.pdf | 2.4. Poincaré |
| Transparency 2-16: nuepde2-16co.pdf | 2.5. Main Formula of DIC |
| Transparency 2-17: nuepde2-17co.pdf | 2.5. Gauss' Theorem |
| Transparency 2-18: nuepde2-18co.pdf | 2.5. Further Integration Formulas |
| Transparency 2-19: nuepde2-19co.pdf | 2.5. H(div) - Trace Theorem |
| Transparency 2-20: nuepde2-20co.pdf | 2.5. H(div) Inverse Trace Theorem |
| Transparency 2-21: nuepde2-21co.pdf | 2.6. Extension Problem |
| Transparency 2-22: nuepde2-22co.pdf | 2.6. Extension Problem (cont) |
| Transparency 2-23: nuepde2-23co.pdf | 2.7. Embedding |
| Transparency 2-24: nuepde2-24co.pdf | 2.7. Embedding (cont) |
| Transparency 06: nuepde06.pdf | GALERKIN-RITZ-Scheme |
| Transparency 06a: nuepde06asw.pdf | Courant's idea |
| Transparency 06b: nuepde06b.pdf | Illustration |
| Transparency 07a: nuepde07.pdf | Remark 2.1.1-2 |
| Transparency 07b: nuepde08sw.pdf | Remark 2.1.3-4 |
| Transparency 08a: nuepde08a.pdf | Model Problem |
| Transparency 08b: nuepde08b.pdf | CHIP |
| Transparency 09: nuepde09.pdf | Mesh for CHIP |
| Transparency 10a: nuepde10sw.pdf | CHIP.NET |
| Transparency 10b: nuepde10b.pdf | Meshing |
| Transparency 10c: nuepde10c.pdf | Tables |
| Transparency 10d: nuepde10dsw.pdf | Finer Mesh |
| Transparency 11a: nuepde11asw.pdf | Mesh Generation 1.-2. |
| Transparency 11b: nuepde11bsw.pdf | Mesh Generation 3. |
| Transparency 11c: nuepde11c.pdf | Mesh Generation 4. |
| Transparency 11d: nuepde11d.pdf | Mesh Generation 5. |
| Transparency 12: nuepde12.pdf | Mapping principle |
| Transparency 13a: nuepde13a.pdf | stiffness matrix (1) |
| Transparency 13b: nuepde13bsw.pdf | stiffness matrix (2) |
| Transparency 13c: nuepde13csw.pdf | stiffness matrix (3) |
| Transparency 14a: nuepde14asw.pdf | 2nd kind BC |
| Transparency 14b: nuepde14bsw.pdf | 3rd kind BC |
| Transparency 14c: nuepde14csw.pdf | 1st kind BC |
| Transparency 15: nuepde15.pdf | Illustration |
| Transparency 16: nuepde16sw.pdf | Exercises 2.5 - 2.8 |
| Transparency 17a: nuepde17a.pdf | Road Map I |
| Transparency 17b: nuepde17bsw.pdf | Road Map II |
| Transparency 17c: nuepde17c.pdf | Theorem 2.6 = Approximation Theorem |
| Transparency 17d: nuepde17d.pdf | Sketch of the Proof |
| Transparency 18a: nuepde18a.pdf | Remark 2.7.1 |
| Transparency 18b: nuepde18bsw.pdf | Remark 2.7.2-5, E 2.9, E 2.10 |
| Transparency 19: nuepde19sw.pdf | Theorem 2.8 (H1-Convergence) |
| Transparency 20: nuepde20sw.pdf | Remark 2.9.1-4 |
| Transparency 21: nuepde21sw.pdf | Remark 2.9.5 |
| Transparency 22: nuepde22sw.pdf | Remark 2.14 |
| Transparency 23: nuepde23.pdf | Var.Crimes I |
| Transparency 24: nuepde24.pdf | Var.Crimes II |
| Transparency 25: nuepde25.pdf | Var.Crimes III |
| Transparency 26: nuepde26.pdf | Remark 3.21 |
| Transparency 27a: nuepde27asw.pdf | DWR I |
| Transparency 27b: nuepde27bsw.pdf | DWR II |
| Transparency 27c: nuepde27c.pdf | AFEM |
| Transparency T4-01a: nuepdeT4-01a.pdf | 4.1.1 DG VF: model problem |
| Transparency T4-01b: nuepdeT4-01b.pdf | 4.1.1 DG VF: notations and formulation |
| Transparency T4-02: nuepdeT4-02.pdf | 4.1.1 DG VF: DG bilinear form |
| Transparency T4-03: nuepdeT4-03.pdf | Alternative Proof |
| Transparency T4-04: nuepdeT4-04.pdf | Consistency + DG-Scheme |
| Transparency T4-05: nuepdeT4-05.pdf | Remark 4.3.: Pros & Cons |
| Transparency T4-06a: nuepdeT4-06a.pdf | Lemma 4.4. |
| Transparency T4-06b: nuepdeT4-06b.pdf | Alternative Proof |
| Transparency T4-07: nuepdeT4-07.pdf | Lemma 4.5.: ellipticity |
| Transparency T4-08: nuepdeT4-08.pdf | Proof |
| Transparency T4-09: nuepdeT4-09.pdf | Lemma 4.7.: boundedness |
| Transparency T4-10: nuepdeT4-10.pdf | Lemma 4.9.: Trace inequality |
| Transparency T4-11: nuepdeT4-11.pdf | Theoerem 4.10.: Error estimate |
| Transparency T4-12: nuepdeT4-12.pdf | Proof (cont.) |
| Transparency T4-13: nuepdeT4-13.pdf | Proof (cont.) |
| Transparency T4-14: nuepdeT4-14.pdf | Proof (cont.) + Remark 4.11 |
| Transparency T4-15: nuepdeT4-15.pdf | 4.2.: FDM |
| Transparency T4-16: nuepdeT4-16.pdf | 4.2.: FVM |
| Transparency T4-17: nuepdeT4-17.pdf | 4.2.: Stability+Appr.=>discrete Conv. |
| Transparency T4-18: nuepdeT4-18.pdf | Summary |
| Transparency 28: nuepde28.pdf | Remark 3.1 |
| Transparency 29: nuepde29.pdf | Example, Remark 3.2 |
| Transparency 30: nuepde30sw.pdf | Secondary Grids I |
| Transparency 31: nuepde31sw.pdf | Secondary Grids II |
| Transparency 32: nuepde32.pdf | Remark 3.3 + E 3.1 |
| Transparency 33: nuepde33sw.pdf | Remark 3.4 |
| Transparency 34: nuepde34.pdf | Boundary boxes |
| Transparency 35: nuepde35.pdf | Remark 3.5 + E 3.2 |
| Transparency 36a: nuepde36asw.pdf | Galerkin-Petrov I |
| Transparency 36b: nuepde36bsw.pdf | Galerkin-Petrov II |
| Transparency 36c: nuepde36c.pdf | Galerkin-Petrov Approach |
| Transparency 36d: nuepde36d.pdf | Two Galerkin-Petrov Schemes |
| Transparency 36e: nuepde36e.pdf | System of FV-Equations |
| Transparency 37a: nuepde37asw.pdf | Remark 3.6.1-3.6.4 |
| Transparency 37b: nuepde37bsw.pdf | Remark 3.6.5-3.6.6 |
| Transparency 38: nuepde38.pdf | Ref + Remark 3.7 |
| Transparency 39: nuepde39.pdf | Discrete Convergence I |
| Transparency 40: nuepde40sw.pdf | Discrete Convergence II |
| Transparency 41: nuepde41sw.pdf | Discrete Convergence III |
| Transparency 42: nuepde42sw.pdf | Discrete Convergence IV (E 3.3) |
| Transparency 43: nuepde43sw.pdf | Discrete Convergence V |
| Transparency 44: nuepde44.pdf | Discrete Convergence VI |
| Transparency 39-44: nuepde39-44sw.pdf | Summary |
| Transparency 45: nuepde45sw.pdf | 4. BEM 4.1 Introduction I |
| Transparency 46: nuepde46sw.pdf | 4.1 Introduction II |
| Transparency 47: nuepde47sw.pdf | 4.1 Introduction III |
| Transparency 48: nuepde48sw.pdf | 4.1 Introduction IV |
| Transparency 49a: nuepde49asw.pdf | Subsection 4.2.1 |
| Transparency 50a: nuepde50a.pdf | Section 4.3: CM I |
| Transparency 50b: nuepde50bsw.pdf | Section 4.3: CM II |
| Transparency 51a: nuepde51a.pdf | Section 4.3: CM III |
| Transparency 51b: nuepde51bsw.pdf | Section 4.3: CM IV |
| Transparency 52a: nuepde52asw.pdf | Section 4.3: CM V |
| Transparency 52b: nuepde52b.pdf | Section 4.3: CM VI |
| Transparency 53: nuepde53sw.pdf | Section 4.3: CM VII |
| Transparency 54: nuepde54sw.pdf | Section 4.3: CM VIII |
| Transparency 55: nuepde55sw.pdf | Section 4.3: CM IV |
| Transparency 56: nuepde56sw.pdf | Section 4.3: CM X |
| Transparency 57: nuepde57sw.pdf | Section 4.3: CM XI |
| Transparency 58a: nuepde58asw.pdf | BIO: Def. |
| Transparency 58b: nuepde58bsw.pdf | BIO: Calderon |
| Transparency 58c: nuepde58csw.pdf | BIO: D2N |
| Transparency 59a: nuepde59asw.pdf | 4.4.2 Properties I |
| Transparency 59b: nuepde59bsw.pdf | 4.4.2 Properties II |
| Transparency 60: nuepde60sw.pdf | Galerkin I |
| Transparency 61: nuepde61sw.pdf | Galerkin II |
| Transparency 62: nuepde62sw.pdf | Galerkin III |
| Transparency 63: nuepde63sw.pdf | Galerkin IV |
| Transparency 64: nuepde64sw.pdf | Galerkin V |
[Part 1] Part1DirectSolvers.pdf: Direct Solvers
[Part 2] Part2IterativeSolvers.pdf: Iterative Solvers
[Part 3] Part3Preconditioners.pdf: Preconditioners
[Part 4] Part4MultigridI.pdf: Multigrid I
[Part 5] Part5MultigridII.pdf: Multigrid II
see also [9] in Basic Lecture Notes.
[1] Langer U.: Numerik I (Operatorgleichungen), JKU, Linz 1996 (Sobolev-Spaces and Tools): PDF
[2] Langer U.: Numerik II (Numerische Verfahren für Randwertaufgaben), JKU, Linz 1996 (FEM and FVM): PDF
[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).
[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):
[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 Neumüller 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
[1] Braess D.: Finite Elemente. Springer Lehrbuch, Berlin, Heidelberg 1997.
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
[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.
Previous Knowledge:
These lectures are required for:
Objectives of the Lectures: 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:
Additional Information:
Examinations:
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).