Introduction to PDEs and Numerical Methods

# Introduction to PDEs and Numerical Methods

## General Information

Lecturer

Assistant

Dr. Hanna Veselovska

Registration

Code of Honour

Dr. Jaroslav Vondřejc

Schedule

Start

Prerequisites

Target group

Certificates

Office hours

Dr. Jaroslav Vondřejc

Exam

The exam date in summer semester is scheduled on Monday 26.8.2019 at 10:00-12:00 in the Institute of Scientific Computing.

Here is a help for preparation that will be updated during the semester with the topics that should be covered for the exam.

You can find here two examples for the exams from last semesters: test1 and test2.

## Literature and other material

• The draft of lectures that will be improved during the semester can be found here; it can be edited so it allows to fix bugs or do improvements - there is a version control system on top of it so do not worry that it fails.
• Code relating to lectures or exercises is available at Bitbucket repository.
• Gockenbach, M. S. (2010). Partial differential equations: analytical and numerical methods. Philadelphia, PA, USA: Society for Industrial and Applied Mathematics.
• Gockenbach, M. (2006). Understanding and implementing the finite element method. Philadelphia, USA: Society for Industrial and Applied Mathematics.
• Braess, D. (2007). Finite elements: Theory, fast solvers, and applications in solid mechanics. Cambridge University Press.
• Morton, K. & Mayers, D., (2005). Numerical solution of partial differential equations: an introduction, New York: Cambridge University Press.

## Lecture drafts

• Lecture 1 - Description of the course and motivation.
• Lecture 2 - Classification of PDEs, analytical solution.
• Lecture 3 (7/11) - Analytical solution to PDEs, Fourier series.
• Lecture 4 (14/11) - Fourier transforms.
• Lecture 5 (21/11) - Numerical solution to PDEs by finite differences.
• Lecture 6 (28/11) - Numerical analysis of finite difference schemes for heat equation.
• Lecture 7 (5/12) - Von Neumann or Fourier analysis of finite difference schemes.
• Lecture 8 (12/12) - Solution of linear systems by Krylov subspace methods - Conjugate gradients.
• Lecture 9 (19/12) - Solution of linear systems by Krylov subspace methods - GMRES.
• Lecture 10 (9/1) - Convergence of CG.
• Lecture 10 (16/1) - Finite element method (FEM).
• Lecture 10 (23/1) - FEM: Assembling system matrix.
• Lecture 10 (30/1) - FEM: Implementation of different boundary conditions.

## Homework assignments

During a course homeworks will be provided such that the students can learn from exercises. The assignments are elaborated as a team work, with groups consisting of 3-4 students. It is required to obtain 50 % of the points. To obtain full points explain your solutions self-consistently with all necessary intermediate conclusions and calculation. Make your code as simple as possible, use comments, figures, etc.

Written assignments should be given to Dr. Veselovska before Tutorial or put into Institute's post box, in the ground floor of Mühlenpfordtstr. 23. Electronic assignments (e.g. Matlab of Python code) should be submitted wire.pde(at)gmail.com.

• Assignment 1 (differential operators, classification of PDEs). Deadline: 2/11/2018
• Reading assingment: Gockenbach, M. S. (2010). Partial differential equations: analytical and numerical methods. Sections 3.3, 4.2, and 6.1.
• Assignment 2 (Fourier series) can be found on StudIP.
• Assignment 3 (Fourier transform) can be found on StudIP.
• Assignment 4 (Finite differences) can be found on StudIP.
• Assignment 5 (Finite differences 2) can be found on StudIP.