Introduction to PDEs and Numerical Methods

Lecturers

Dr. Jaroslav Vondřejc

Assistant

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Dr. Noemi Friedman

Schedule

Start

Prerequisites

Target group

Certificates

Office hours

Dr. Noemi Friedman

Dr. Jaroslav Vondřejc

Exam

Next written exam: 10:30-12:00, 20th of August, 2018, Mühlenpfordtstrasse 23, 8th floor, Seminarraum WiRe,

Exam results can be checked at 10:00, 28th of August, , 2018, Mühlenpfordtstrasse 23, 8th floor, Seminarraum WiRe

Written exam 22.02.2018, 10:00-11:30, Room PK 15.1 (Universitätsplatz 3, Physik-Hörsaal im Auditorium maximum, 007)

Here is a help for preparation with the topics that should be covered for the exam.

You can find two examples for the exam here: test1 and test2.

• A script for the lecture is available HERE.
• bitbucket repository with codes, where you can find some simple examples, turorials, etc. For installing or using Fenics, you can use following hints.

Lecture 1: Introduction, motivation, brush-up of differential operators

Lecture 2: Classification of PDEs, elementary PDEs, the heat equation, analytical solution of the heat equation by separation of variables

Lecture 3: Essential functional analysis, projection theory, Fourier series

Lecture 4: More on projection theory and Fourier-series in the compex domain, essential ODEs, existence and uniqueness of the solution

Lecture 5: Introduction to the finite difference method, solving the heat equation with the finite difference method 1.

Lecture 6: Solving the heat equation with the finite difference method 2.

Lecture 7-8: Stability, consistency, convergence, the heat eq. in higher dimension

Lecture 9-10 (section 2): Linear systems - Krylov subspace linear solvers (CG, GMRES) as variational methods for linear systems

Lecture 11: Introduction to the weighted residual methods

Lecture 12: The Galerkin method and FEM in 1D

Lecture 13: FEM in 1 and 2d, Isoparametric map

Lecture 14: Numerical integration

To obtain full points explain your solutions thoroughly and self-consistently with all necessary intermediate conclusions and calculation steps as to leave no doubt about the correctness and your understanding. Structure programmes nicely and with comments and argue why you think that it works. Support your reasons with necessary plots, examples, etc. so that it becomes obvious.

Reading assignments are strongly recommended for easy flow of understanding!

• 1. Reading assignment: Chapter 1 and 2 in M.S. Gockenbach - Partial Differential Equations, Analytical and Numerical Methods, Deadline: 27.10.2017., see solutions here.
• 1. Assignment: Differential operators, classification of PDEs, deadline: 08.11.2017
• 2. Reading assignment: First two chapters in the script, Chapter 3 in M.S. Gockenbach - Partial Differential Equations, Analytical and Numerical Methods, Deadline: 08.11.2017
• 2. Assignment: Fourier series, seperation of variables, Deadline: noon, 15.11.2017, see solutions here.
• 3. Reading assignment: Chapter 5.1, 5.2 and 5.3 in M.S. Gockenbach - Partial Differential Equations, Analytical and Numerical Methods, Deadline: 15.11.2017., also recommended but not obligatory: Chapter 4 in the same book
• 3. Assignment: Fourier-series in the complex domain, Solving ODEs with Fourier, uniqueness and existence of solution, Deadline: beginning of the lecture, 22. 11.2017, see solutions here.
• 4. Reading assignment: Script 1.4, Deadline: 22.11.2017.
• 4. Assignment: The Finite Difference method and its application to Boundary Value Problems (BVPs), deadline: beginning of the lecture, 06.12.2017, see solutions here.
• 5. Reading assignment: Script 1.5 and reread previous reading assignments, Deadline: 29.11.2017.
• 5. Assignment: Von Neumann stability analysis and solving the instationary heat equation with the FD method, deadline: beginning of the lecture, 13.12.2017, see solutions here.
• 6. Assignment: Conjugate gradients as Galerkin approximation, scalar product. Deadline: deadline: beginning of the lecture, 20.12.2017; solution is here and on bitbucket repository.
• 7. Assignment: Conjugate gradients as Galerkin approximation with orthogonal and A-orthogonal basis. Deadline: deadline: beginning of the lecture, 10.1.2018; solution is here and on bitbucket repository.
• 8. Reading assignment: Chapter 3.1 in the script,Chapter 5.4, 5.5 in M.S. Gockenbach - Partial Differential Equations
• 9. Reading assignment: Repeat to read 8. Reading assignment + read Chapter 5.6 and 6.4 and 6.5 of M.S. Gockenbach - Partial Differential Equations and Chapter 3.2, 3.3 and 3.4 of the script. Deadline: 24.01.2018.
• 8. Assignment: Basics of functional analysis, the weak formulation formulation of BVPs, the Finite Element Method. Deadline: 31.01.2018., see solutions here.
• 10. Reading assignment: Chapter 8.4 and 10.1-10.2 in M.S. Gockenbach - Partial Differential Equations, deadline: 31.01.2018.
• 9. Assignment: FEM and the isoparametric map, numerical integration, Deadline: 14.02.2018, see solutions here.
• 11. Reading assignment: Numerical integration in 1D: Gauß quadrature, Michael T. Heath: Scientific Computing, Chapter 7. Chapter 8.1-8.3, Deadline: 02.02.2018.