Course Modules

1 Classification of PDEs

1 Classification of PDEs 1 Classification of PDEs

Context Module Sub Header

Lecture 1: Classification of PDEs 1

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Attachment

Skeleton-Notes-Lecture1.pdf Skeleton-Notes-Lecture1.pdf 2

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Attachment

Notes-Lecture1.pdf Notes-Lecture1.pdf 3

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Attachment

Mysterious PDE (video) Mysterious PDE (video) 4

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Assignment

HW 1 HW 1 2.5 5

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2 Second-Order Elliptic Equations

2 Second-Order Elliptic Equations 2 Second-Order Elliptic Equations

Context Module Sub Header

Lecture 2: Elliptic maximum principle 1

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Page

Pre-class video by Timm: An elliptic maximum principle Pre-class video by Timm: An elliptic maximum principle 2

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Attachment

Skeleton-Notes-Lecture2.pdf Skeleton-Notes-Lecture2.pdf 3

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Attachment

Notes-Lecture2.pdf Notes-Lecture2.pdf 4

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Context Module Sub Header

Lecture 3: From strong solutions to weak solutions 5

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Page

Pre-class video by Timm: Poisson's Equation in Strong and Weak Form Pre-class video by Timm: Poisson's Equation in Strong and Weak Form 6

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Attachment

Skeleton-Notes-Lecture3.pdf Skeleton-Notes-Lecture3.pdf 7

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Attachment

Notes-Lecture3.pdf Notes-Lecture3.pdf 8

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Context Module Sub Header

Lecture 4: Well posedness of weak formulations 9

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Page

Pre-class video by Timm: Poincare's inequality Pre-class video by Timm: Poincare's inequality 10

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Attachment

Skeleton-Notes-Lecture4.pdf Skeleton-Notes-Lecture4.pdf 11

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Notes-Lecture4.pdf Notes-Lecture4.pdf 12

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Assignment

HW 2 HW 2 2.5 13

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Context Module Sub Header

Lecture 5: Finite Differences for Poisson's equation 14

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Page

Pre-class video by Timm: Finite Differences for Poisson's Equation in 1D Pre-class video by Timm: Finite Differences for Poisson's Equation in 1D 15

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Attachment

Skeleton-Notes-Lecture5.pdf Skeleton-Notes-Lecture5.pdf 16

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Attachment

Notes-Lecture5.pdf Notes-Lecture5.pdf 17

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Assignment

HW 3 HW 3 2.5 18

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Context Module Sub Header

Lecture 6: Conditiong of linear systems from Finite-Difference methods 19

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Page

Pre-class video by Timm: Diagonal dominance Pre-class video by Timm: Diagonal dominance 20

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Attachment

Skeleton-Notes-Lecture6.pdf Skeleton-Notes-Lecture6.pdf 21

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Notes-Lecture6.pdf Notes-Lecture6.pdf 22

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Assignment

HW 4 HW 4 2.5 23

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Context Module Sub Header

Lecture 7: Finite Element for the Poisson's equation 24

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Page

Pre-class video by Timm: Best approximatiom theorem Pre-class video by Timm: Best approximatiom theorem 25

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Attachment

Skeleton-Notes-Lecture7.pdf Skeleton-Notes-Lecture7.pdf 26

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Attachment

Notes-Lecture7.pdf Notes-Lecture7.pdf 27

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Context Module Sub Header

Lecture 8: Linear Finite Element 28

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Skeleton-Notes-Lecture8.pdf Skeleton-Notes-Lecture8.pdf 29

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Notes-Lecture8.pdf Notes-Lecture8.pdf 30

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Context Module Sub Header

Lecture 9: Quadrature formulas 31

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Skeleton-Notes-Lecture9.pdf Skeleton-Notes-Lecture9.pdf 32

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Page

Post-class video with Timm: Data Structure for Triangulated Domains Post-class video with Timm: Data Structure for Triangulated Domains 33

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Post-class video by Timm: Unisolvence Post-class video by Timm: Unisolvence 34

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Attachment

Notes-Lecture9.pdf Notes-Lecture9.pdf 35

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Assignment

HW 5 HW 5 2.5 36

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Assignment

HW 6 HW 6 2.5 37

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Context Module Sub Header

Lecture 10: General A Priori Error Analysis of Galerkin Methods 38

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Attachment

Skeleton-Notes-Lecture10.pdf Skeleton-Notes-Lecture10.pdf 39

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Notes-Lecture10.pdf Notes-Lecture10.pdf 40

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Context Module Sub Header

Lecture 11: Interpolation with Finite Elements 41

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Skeleton-Notes-Lecture11.pdf Skeleton-Notes-Lecture11.pdf 42

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Notes-Lecture11.pdf Notes-Lecture11.pdf 43

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HW 7 HW 7 2.5 44

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Context Module Sub Header

Lecture 12: Error estimates: FE versus FD 45

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Context Module Sub Header

Lecture 13: Boundary Approximations on Non-Polygonal Domains 48

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Page

Pre-class video by Timm: Parametric and Isoparametric Elements Pre-class video by Timm: Parametric and Isoparametric Elements 49

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Context Module Sub Header

Lecture 14: A Posteriori Error Analysis and Adaptive Finite Elements 52

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Pre-class video by Timm: A Priori vs A Posteriori Error Estimates Pre-class video by Timm: A Priori vs A Posteriori Error Estimates 53

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aPosteriori.py aPosteriori.py 56

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adaptivePoisson.py adaptivePoisson.py 58

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Context Module Sub Header

Lecture 15: The Method of Manufactured Solutions 60

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transport.py transport.py 63

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transport.ipynb transport.ipynb 64

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convergence_tofill.xlsx convergence_tofill.xlsx 65

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convergence.xlsx convergence.xlsx 66

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Assignment

HW 8 HW 8 2.5 67

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3 Second-Order Parabolic Equations

3 Second-Order Parabolic Equations 3 Second-Order Parabolic Equations

Context Module Sub Header

Lecture 16: Characteristic features 1

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Skeleton-Notes-Lecture16.pdf Skeleton-Notes-Lecture16.pdf 2

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Notes-Lecture16.pdf Notes-Lecture16.pdf 3

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Context Module Sub Header

Lecture 17: Finite Element for the heat equation - Time-stepping methods Part I 4

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Assignment

HW 9 HW 9 2.5 7

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Context Module Sub Header

Lecture 18: Finite Element for the heat equation - Time-stepping methods Part II 8

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Context Module Sub Header

Lecture 19: Space-time discretization - additional comments 11

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4 Second-Order Hyperbolic Equations

4 Second-Order Hyperbolic Equations 4 Second-Order Hyperbolic Equations

Context Module Sub Header

Lecture 20: The wave equation 1

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Page

Pre-class video by Timm: Solving Hamiltonian Systems with Symplectic Runge-Kutta Methods Pre-class video by Timm: Solving Hamiltonian Systems with Symplectic Runge-Kutta Methods 2

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Attachment

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Attachment

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4.2.1 standingWave.mp4 4.2.1 standingWave.mp4 5

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4.2.1 travellingWave.mp4 4.2.1 travellingWave.mp4 6

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4.2.1 travellingWave2.mp4 4.2.1 travellingWave2.mp4 7

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Assignment

HW 10 HW 10 2.5 8

assignment 3942062 1

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5 Conservation equations

5 Conservation equations 5 Conservation equations

Context Module Sub Header

Lecture 21: The transport term 1

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Skeleton-Notes-Lecture21.pdf Skeleton-Notes-Lecture21.pdf 2

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Notes-Lecture21.pdf Notes-Lecture21.pdf 3

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Lecture 22: DG for the diffusion term and operator-splitting methods 4

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Skeleton-Notes-Lecture22.pdf Skeleton-Notes-Lecture22.pdf 5

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Notes-Lecture22.pdf Notes-Lecture22.pdf 6

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OnOperatorSplittingMethods.pdf OnOperatorSplittingMethods.pdf 7

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Course Modules

1 Classification of PDEs

Lecture 1: Classification of PDEs 1

2 Second-Order Elliptic Equations

Lecture 2: Elliptic maximum principle 1

Lecture 3: From strong solutions to weak solutions 5

Lecture 4: Well posedness of weak formulations 9

Lecture 5: Finite Differences for Poisson's equation 14

Lecture 6: Conditiong of linear systems from Finite-Difference methods 19

Lecture 7: Finite Element for the Poisson's equation 24

Lecture 8: Linear Finite Element 28

Lecture 9: Quadrature formulas 31

Lecture 10: General A Priori Error Analysis of Galerkin Methods 38

Lecture 11: Interpolation with Finite Elements 41

Lecture 12: Error estimates: FE versus FD 45

Lecture 13: Boundary Approximations on Non-Polygonal Domains 48

Lecture 14: A Posteriori Error Analysis and Adaptive Finite Elements 52

Lecture 15: The Method of Manufactured Solutions 60

3 Second-Order Parabolic Equations

Lecture 16: Characteristic features 1

Lecture 17: Finite Element for the heat equation - Time-stepping methods Part I 4

Lecture 18: Finite Element for the heat equation - Time-stepping methods Part II 8

Lecture 19: Space-time discretization - additional comments 11

4 Second-Order Hyperbolic Equations

Lecture 20: The wave equation 1

5 Conservation equations

Lecture 21: The transport term 1

Lecture 22: DG for the diffusion term and operator-splitting methods 4