This is a first course on partial differential equations (PDEs) intended for upper undergraduate and master students following engineering programs at Chalmers and math and physics programs at the University of Gothenburg. Due to recommendation from their advisers, PhD students in computational/applied math and applied sciences and engineering are also welcome to take this course.
After the first 2 weeks, students of Physical Chemistry (Kf3) programe are joining the course.
The course code is:
for students registered at Chalmers:
TMA372,
for students registered in GU: MMG800.
For Kf3 students: MVE455.
Contents:
The main topic of this course is:
Application of finite element methods to problems of, e.g.,
dynamical systems, Poisson's (Laplace; stationary heat) equation,
heat conduction, wave
propagation, convectiondiffusionreaction equations.
The goal is to provide the participants with some
i) Theoretical backgroud: wellposedness (existence, uniqueness
and stability), regularity,
convergence analysis and conservation properties (when applicable).
The highlights in this part are Reisz representation and LaxMilgram theorems.
ii) Modern approximation skills: Approximation procedures
to various PDEs (ODEs)
using various Finite Element Methods, Polynomial Interpolation,
Quadrature rules,
and the solution techniques of the resulting large,
sparse linear system of equations. Error estimates
in both a priori and a posteriori settings.
Compulsary home assignments,(in group)
contain both analytic approaches as well as
coding aspects, ranging from iterative
algorithms to problems involving complex multiphysics programing.
This can provide, and also challenge, the students
implementation skills.
Participating and following the course, actively, you should gain some analytical intutions (e.g., not all PDEs: as is, can be assigned a closed form solution), and learn some approximation skill to solve the PDEs (of e.g., your own field of interest), consisting of knowledge on correct weak/variational formulation, and a great deal about finite element analysis consisting of both theoretical aspects as stability and convergence of approximate solutions, as well as numerical analysis and implementations.
The course consists of 36 lecture hours, 20 exercise hours and gives 7.5 points.
Latest news
The schedule for the course can be found via the link to webTimeEdit top of the page.
Below is the concise schedule:
Day  Time  Place  Remarks  Office Hours 

MON  810  Pascal  Lecture 
Mohammad:1012, L2035; questions 
WED  810  Pascal  Exercise (Lecture W1)  Maximilian:1012, L20322; assignments 
THU  1012  Pascal  Lecture  
FRI  810  Pascal  Lecture/Exercise 
Teachers
Course coordinator:
Teaching assistants:
Lab supervisor:
Course literature
M. Asadzadeh, An Introduction to the Finite Element Method (FEM) for Differential Equations. Available in Cremona
M. Asadzadeh, Lecture Notes: An Introduction to the Finite Element Method (FEM) for Differential Equations: Part II Problems in ${\mathbb R}^d,\,\, d>1$. ( Chapters 1012; will appear by Feb 15),
Reference literature:
Program
Lectures (preliminary plan)
Week  Book chapters 
Contents 

3, Study w1 
1, 3.13.3, 3.73.8 
Classification of PDEs, Math
Tools, Power of abstraction, Reisz and LaxMilgram theorems 
4, study W2 
4, 5 
Polynomial approximation,
Polynomial Interpolation, L_2 projection, Gauss quarature rule 
5, study W3 
7 
Twopoints boundary value problems,
Finite element approximations,
"a priori" and "a posteriori" error estimates in energy norm.

6, study W4 
8 
Continuous and discontinuous Galerkin and
a priori/a posteriori error estimates for
Galerkin methods for scalar initial value problems.

7, study W5 
9 
Initialboundary value problems: one
space dimension.
Stability for the heat conduction, finite element approximation
for the heat equation,
error analysis. The wave equation, conservation of energy,
finite elements for the wave equation.

8, study W6 
Lecture Notes:
Poisson/Heat in ${\mathbb R}^d, \, d>1$ 
Error estimates and adaptive
error control in the energy norm for the Poisson's equation. Stability and piecewise linear Galerkin approximation for the heat equation. Error analysis of finite element methods for the heat equation. 
9, study W7 
Lecture Notes: Heat/Wave
in ${\mathbb R}^d, \, d>1$ 
Conservation of energy. Wave equation as a system of IVP. Finite elements for the wave equation. Finite element method for a convectiondiffusion model problem. Selected problems from exams, 20122017. 
Week  Excersises 

4, study w2 
1: Give a varitional formulation of u''+u=f in (0,1), with
u(0)=u(1)=0. 2: Write a FEMformulation with piecewise linear, continuous functions, and a uniform stepsize h=1/4. 3: The same as above, but with piecewise quadratic functions. Book: 2.12.5, 2.11, 2.12, 2.21, 2.22 
5, study w3 
Chapters 35:
Read through iterative methods of chapter 5(self study not included in the exam).
Book: 3.3, 4.14.4, 5.85.10

6,7 study w 4,5 
Chapters 79:
Book: Problems in Chapters 7.1, 7.3, 7.9, 8.38.6, 9.3, 9.6, 9.9, 
8, study w6 
Chap 1012:
Lecture Notes: Problems in Chapters 1012. 
Demonstrated/recommended Exercises
Week  Excersises:Problem file (will be completed by exercises from the Lecture notes) 

4, study w2 
Problem file: Problems 5360 Book: 3.53.7,

5, study w3 
Problem file: Problems 15,
Book: 4.54.7, 5.15, 5.16 
6, study w4 
Problem file: Problems
612 Book: 7.37.8, 1.10, 7.167.19 
7, study w5 
Problem file:
Problems 1320
Book: 8.108.14, 8.19, 9.69.9 
8, study w6 
Problem file:
chosen problems from the list: 2123, 2627 Lecture Notes: 10.10, 10.11 
8, study w6 
Problem file:
chosen problems from the list: 3440 4352 Lecture Notes: 11.9, 11.11 
9, study w7 
Problem file:
chosen problems from the list: 4352, Lecture Notes:12.4, 12.9, 12.13, 12.14 
10, study w8 