Introduction to PDEs and Numerical Methods

Lecturers

Dr. Jaroslav Vondřejc

Assistant

(excercises)

Registration

Please register via StudIP

j.vondrejc@tu-bs.de

Schedule

Lecture: Wed 9:45-11:15 in room PK 3.4. (Pockelsstraße 3, Am Okerufer, Hochhaus, 205)

Exercises/Tutorials: Friday 13:15 - 14:45, room HS65.012 (except on the dates 04.11 and 18.11, the room for these dates is SN 19.4)

Start

Prerequisites

Target group

Certificates

Office hours

j.vondrejc@tu-bs.de

n.friedman@tu-bs.de

Exam

The exam is scheduled on 22/2/2017 at 10:30-12:30, in PK11.2. There are two examples of tests from previous year 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, differential operators, preliminaries
• Lecture 2: classification of PDEs, introductory examples of PDEs, analytical solution of ODEs and PDEs - from PDEs to ODEs, eigenfunctions, eigenvalues
• Lecture 3:Vector spaces, inner product and normed spaces, projection theory
• Lecture 4: Fourier series and its application to PDEs, see notes for exercise
• Lecture 5: About existence and uniqueness of solution, solution methods for PDEs, essential ODEs, introduction to the Finite Difference Method
• Lecture 6: Solving the heat equation with the Finite Difference method
• Lecture 7: Von Neumann stability analysis, stability, consistency, convergence
• Lecture 8 (section 2): Krylov subspace linear solvers (CG, GMRES) as Bubnov-Galerkin or Petrov-Galerkin method
• Lecture 9 (section 2 - continuation): Conjugate gradients, Gramm-Schmidt ortogonalisation
• Lecture 10 (section 3): Weak formulation of PDEs
• Lecture 11 (section 4): Finite element method in 1D (see also lecture given by Noemi last year)
• Lecture 12: Assembling FEM matrices (see also lecture given by Noemi last year)
• Lecture 13: Numerical integration (Gauss-Legendre quadrature)

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 assignment 1: Mark S. Gockenbach: Partial Differential Equations - Analytical and Numerical Methods, Chapter 1-2, Script 1.1, 1.2, deadline: 4.11.2015.
• Homework 1: Differential operators, the heat equation, seperation of variables, deadline: 04.11.2015., see solutions here.
• Reading assignment 2: Mark S. Gockenbach: Partial Differential Equations - Analytical and Numerical Methods, Chapter 3, deadline: 11.11.2015.
• Homework 2: Norms, inner products, projection theory, see solutions here
• (Recommended) reading assignment 3: Brad Osgood: The Fourier Transform and its Application, Chapters 1-2
• Homework 3: Fourier series and its application for solving PDEs, with solution here.
• Reading assignment 4: Mark S. Gockenbach: Partial Differential Equations - Analytical and Numerical Methods, Chapters 4.1, 4.2, 4.3, 5.1, 5.2, 5.3, Script:1.4, deadline: 23.11.2015.
• Homework 4: Finite Difference Method, operator properties, eigenvalues, deadline: 05.12.2016, solution here
• Reading assignment 5:Revise 1.4 in the script and read 1.5 , revise Chapter 5.1-5.3, deadline 30.11.2016
• Homework 5: FD on nonuniform mesh, numerical solution of the heat eq. with FD method, Von Neumann stability analysis, solution here
• Homework 6 (full version availbale): Hello world to FEniCS and Python, see hints; solution to linear systems with CG and MINRES/GMRES, solution is on bibucket repository (implementation) and here.
  
• Homework 7 (deadline Monday 9/1/2017): orthogonalisation, Conjugate gradients, function of a matrix, solution
• Reading assignment: read a caption about conjugate gradients in one of the books recommended in the lecture notes
• Homework 8 (deadline Monday 23/1/2017): weak formulations of PDEs. Solution.
• Homework 9 (deadline Monday 30/1/2017): projection and interpolation with piece-wise linear basis funtions. Solution can be found on bitbucket repository.
• Reading assignment: Gockenbach: Partial Differential Equations - Analytical and Numerical Methods, sections 5.4-5.6, 10.1 or Braess: Finite elements: Theory, fast solvers, and applications in elasticity theory, Chapter II, paragraph 8
• Homework 10 (last homework with deadline Friday 10/02/2017): Assembling local FEM matrices in 2D. The implementation has been uploaded to bitbucket repository. Part of the solution can be found here; the corresponding formulas can be found in the lecture drafts or e.g. in the book of D. Braess: Finite Elements, Chapter II, paragraph 8.