Numerical Methods for Elliptic Partial Differential Equations

last update: 2021-10-03

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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

- Examination questions: up
as   pdf-file  

- The super question: up
as   pdf-file  

- Examination dates: up
Link to examination dates


Time and room:

Tue, March 3, 200910:15 - 11:45 Room: T 911Introduction
Wed, March 4, 200910:15 - 11:45 Room: T 111Lecture 1
Thu, March 5, 200910:15 - 11:45 Room: T 911Lecture 2
Tue, March 10, 200910:15 - 11:45 Room: T 911Lecture 3
Wed, March 11, 200910:15 - 11:45 Room: T 111Lecture 4
Wed, March 18, 200910:15 - 11:45 Room: T 111Lecture 5
Thu, March 19, 200910:15 - 11:45 Room: T 911Lecture 6
Wed, March 25, 200910:15 - 11:45 Room: T 111Lecture 7
Thu, March 26, 200910:15 - 11:45 Room: T 911Lecture 8
Tue, March 31, 200910:15 - 11:45 Room: T 911Lecture 9
Wed, April 1, 200910:15 - 11:45 Room: T 111Lecture 10
Wed, April 22, 200910:15 - 11:45 Room: T 111Lecture 11
Thu, April 23, 200910:15 - 11:45 Room: T 911Lecture 12
Tue, April 28, 200910:15 - 11:45 Room: T 911Lecture 13
Wed, April 29, 200910:15 - 11:45 Room: T 111Lecture 14
Thu, April 30, 200910:15 - 11:45 Room: T 911Lecture 15
Wed, May 6, 200910:15 - 11:45 Room: T 111Lecture 16
Thu, May 7, 200910:15 - 11:45 Room: T 911Lecture 17
Thu, May 14, 200910:15 - 11:45 Room: T 911Lecture 18
Wed, May 20, 200910:15 - 11:45 Room: T 111Lecture 19
Thu, May 21, 2009Christi HimmelfahrtLecture is canceled
Tue, May 26, 200910:15 - 11:45 Room: T 911Lecture 20
Wed, May 27, 200910:15 - 11:45 Room: T 111Lecture 21
Thu, May 28, 200910:15 - 11:45 Room: T 911Lecture 22
Thu, June 4, 200910:15 - 11:45 Room: T 911Lecture 23
Wed, June 10, 200910:15 - 11:45 Room: T 111Lecture 24
Thu, June 11, 2009FronleichnamLecture is canceled
Wed, June 17, 200910:15 - 11:45 Room: T 111Lecture 25
Thu, June 18, 200910:15 - 11:45 Room: T 911Lecture 26
Wed, June 24, 200910:15 - 11:45 Room: T 111Lecture 27

Lecturer: O.Univ.-Prof. Dr. Ulrich Langer



Tutorial      up

Numerical Methods for Elliptic Partial Differential Equations - Tutorials

(CourseId 327.004, 2 hours per week, Semester 6)

Tutorials held by: Dr. Clemens Pechstein

Time and room:

Thu, March 12, 200910:15 - 11:45 Room: T 911Tutorial 01
Tue, March 17, 200910:15 - 11:45 Room: T 911Tutorial 02
Tue, March 24, 200910:15 - 11:45 Room: T 911Tutorial 03
Thu, April 2, 200910:15 - 11:45 Room: T 911Tutorial 04
Tue, April 21, 200910:15 - 11:45 Room: T 911Tutorial 05
Tue, May 5, 200910:15 - 11:45 Room: T 911Tutorial 06
Tue, May 12, 200910:15 - 11:45 Room: T 911Tutorial 07
Wed, May 13, 200910:15 - 11:45 Room: T 111Consultation LTTP
Tue, May 19, 200910:15 - 11:45 Room: T 911Tutorial 08
Tue, June 2, 2009Pfingstdienstagcanceled
Wed, June 3, 200910:15 - 11:45 Room: T 111Tutorial 09
Tue, June 9, 200910:15 - 11:45 Room: T 911Individual Consultation LTTP
Tue, June 16, 200910:15 - 11:45 Room: T 911Tutorial 10
Tue, June 23, 200910:15 - 11:45 Room: T 911Tutorial 11
Thu, June 25, 200910:15 - 11:45 Room: T 911Presentation LTTP

LTTP = Long-Term Training Problems

Tutorials      up
Tutorial 01March 12, 2009pdf
Tutorial 02March 17, 2009pdf
Tutorial 03March 24, 2009pdf
Tutorial 04April 2, 2009pdf
Tutorial 05April 21, 2009pdf
Tutorial 06May 5, 2009pdf
Tutorial 07May 12, 2009pdf
Tutorial 08May 19, 2009pdf
Tutorial 09June 3, 2009pdf
Tutorial 10June 16, 2009pdf
Tutorial 11June 23, 2009pdf

Practical exercises      up
PA 01Toothpdf
PA 02CourantpdfR. Schmid, C. Stadlmayr
PA 03PistonpdfX. Deng, A.G. Weldeyesus, A.A. Arara
PA 04MagnetpdfX. Wang, Y. Zhou
PA 05ChamberpdfN. Banagaaya, I.I. Wangwe
PA 06Schellbachpdf
PA 01-06Allzip

Transparencies      up
Transparency 00a: b/wMath. Models
Transparency 00b: b/wRemark 1.2
Transparency 01: b/wEx 1.1 - 1.2
Transparency 02: b/wEx 1.3 - 1.4
Transparency 03: b/wEx 1.5 - 1.6
Transparency 04: b/wEx 1.7 - 1.9
Transparency 05: b/wEx 1.10 - 1.11
Transparency 05a: b/w1.3.1. Mixed VF I: General
Transparency 05b: b/w1.3.1. Mixed VF II: Navier-Stokes
Transparency 05c: b/w1.3.1. Mixed VF III: Oseen/Stokes
Transparency 05d: b/w1.3.1. Mixed VF IV: Poisson equ.
Transparency 05e: b/w1.3.1. Mixed VF V: 1st bih. BVP
Transparency 05f: b/w1.3.2. Dual VF I: General
Transparency 05g: b/w1.3.2. Dual VF II: Cont.
Transparency 05h: b/w1.3.2. Dual VF III: Example
Transparency 06a: b/wCourant's idea
Transparency 06b: colourIllustration
Transparency 07a: colourRemark 2.1.1-2
Transparency 07b: b/wRemark 2.1.3-4
Transparency 08a: colourModel Problem
Transparency 08b: colourCHIP
Transparency 09: colourMesh for CHIP
Transparency 10a: b/wCHIP.NET
Transparency 10b: colourMeshing
Transparency 10c: colourTables
Transparency 10d: b/wFiner Mesh
Transparency 11: b/wMesh Generation 1.-2.
Transparency 12a: b/wMesh Generation 3.
Transparency 12b: colourMesh Generation 4.
Transparency 13a: colourstiffness matrix (1)
Transparency 13b: b/wstiffness matrix (2)
Transparency 13c: b/wstiffness matrix (3)
Transparency 14a: b/w2nd kind BC
Transparency 14b: b/w3rd kind BC
Transparency 14c: b/w1st kind BC
Transparency 15: colourIllustration
Transparency 16: b/wExercises 2.5 - 2.8
Transparency 17a: colourRoad Map I
Transparency 17b: b/wRoad Map II
Transparency 17c: colourTheorem 2.6
Transparency 18a: colourRemark 2.7.1
Transparency 18b: b/wRemark 2.7.2-5, E 2.9, E 2.10
Transparency 19: b/wTheorem 2.8 (H1-Convergence)
Transparency 20: b/wRemark 2.9.1-4
Transparency 21: b/wRemark 2.9.5
Transparency 22: b/wRemark 2.14
Transparency 23: colourVar.Crimes I
Transparency 24: colourVar.Crimes II
Transparency 25: colourVar.Crimes III
Transparency 26: b/wRemark 2.20
Transparency 27a: b/wDWR I
Transparency 27b: b/wDWR II
Transparency 27c: colourAFEM
Transparency 28: colourRemark 3.1
Transparency 29: colourExample, Remark 3.2
Transparency 30: b/wSecondary Grids I
Transparency 31: b/wSecondary Grids II
Transparency 32: colourRemark 3.3 + E 3.1
Transparency 33: b/wRemark 3.4
Transparency 34: colourBoundary boxes
Transparency 35: colourRemark 3.5 + E 3.2
Transparency 36a: b/wGalerkin-Petrov I
Transparency 36b: b/wGalerkin-Petrov II
Transparency 37a: b/wRemark 3.6.1-3.6.4
Transparency 37b: b/wRemark 3.6.5-3.6.6
Transparency 38: colourRef + Remark 3.7
Transparency 39: colourDiscrete Convergence I
Transparency 40: b/wDiscrete Convergence II
Transparency 41: b/wDiscrete Convergence III
Transparency 42: b/wDiscrete Convergence IV (E 3.3)
Transparency 43: b/wDiscrete Convergence V
Transparency 44: colourDiscrete Convergence VI
Transparency 39-44: b/wSummary
Transparency 45: b/w4. BEM 4.1 Introduction I
Transparency 46: b/w4.1 Introduction II
Transparency 47: b/w4.1 Introduction III
Transparency 48: b/w4.1 Introduction IV
Transparency 49a: b/wSubsection 4.2.1
Transparency 50a: colourSection 4.3: CM I
Transparency 50b: b/wSection 4.3: CM II
Transparency 51a: colourSection 4.3: CM III
Transparency 51b: b/wSection 4.3: CM IV
Transparency 52a: b/wSection 4.3: CM V
Transparency 52b: colourSection 4.3: CM VI
Transparency 53: b/wSection 4.3: CM VII
Transparency 54: b/wSection 4.3: CM VIII
Transparency 55: b/wSection 4.3: CM IV
Transparency 56: b/wSection 4.3: CM X
Transparency 57: b/wSection 4.3: CM XI
Transparency 58a: b/wBIO: Def.
Transparency 58b: b/wBIO: Calderon
Transparency 58c: b/wBIO: D2N
Transparency 59a: b/w4.4.2 Properties I
Transparency 59b: b/w4.4.2 Properties II
Transparency 60: b/wGalerkin I
Transparency 61: b/wGalerkin II
Transparency 62: b/wGalerkin III
Transparency 63: b/wGalerkin IV
Transparency 64: b/wGalerkin V

Basic Lecture Notes:      up
[1]   Langer U.: Numerik I (Operatorgleichungen), JKU, Linz 1996 (Sobolev-Spaces and Tools).
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. Teubner-Verlag, Stuttgart, Leipzig, Wiesbaden 2001 (practical aspects of the FEM).
Methode der Finiten Elemente für Ingenieure
[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).

Additional Literature:      up
[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 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]   Heinrich B.: Finite Difference Methods on Irregular Networks. Akademie-Verlag, Berlin 1987.
[6]   Knaber P., Angermann L.: Numerik partieller Differentialgleichungen. Eine anwendungsorientierte Einführung. Springer-Verlag, Berlin-Heidelberg 2000.
[7]   Monk P.: Finite Element Methods for Maxwell's Equations. Oxford Science Publications, Oxford 2003.
[8]   Schwarz H.R.: FORTRAN-Programme zur Methode der finiten Elemente. B.G. Teubner, Stuttgart, 1991.
[9]   Schwarz H.R.: Methode der finiten Elemente. B.G. Teubner, Stuttgart, 1991.
[10]   Verfürth R.: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley - Teubner, 1996.


Software:      up
FEM1D FEM2D NETREFINER FEM EP Mesh Generation


Links:      up

      NETGEN
      NGSolve
      SPIDER

General Information      up
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.