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, applied sciences and engineering disciplines are also welcome to take the course.
After the first 2 weeks, the third year students of Physical Chemistry (Kf3) programe are joining the course.
Course cods and credit points:
For students registered at Chalmers:
TMA372, (7.5 points).
For students registered in GU: MMG800, (7.5 points).
For Kf3 students: MVE455, (4.5 poits).
For PhD students as in TMA372/MMG800 or the supervisor's suggestion.
Contents: The main topic of this course is: Application of finite element methods to PDEs. More specifically, we consider problems of, e.g., dynamical systems, Poisson's (Laplace; stationary heat) equation, heat conduction, wave propagation, convectiondiffusionreaction equations. The goal is twofold:
i) To provide the students with some theoretical backgroud: wellposedness (existence, uniqueness and stability), regularity, convergence analysis and conservation properties (when applicable) for classical PDEs. The highlights in this part are Poincare inequaly, Riesz representation and LaxMilgram theorems.
ii) To introduce the participants some modern approximation skills:
Approximation procedures
for PDEs (and ODEs)
using Finite Element Methods (FEMs) based on
Polynomial Approximation/Interpolation. This results to large,
sparse linear system of equations (that are solved using
techniques from numerical linear algebra).
Stability and error analysis
in both a priori and a posteriori regi,
in order to verify efficiency and reliability of the
considered FEMs.
There are two compulsary, bonus generating, home assignments that
are recommended as group work. They
contain both analytic approaches as well as
coding aspects ranging from iterative
algorithms to problems involving multiphysics programing.
Participating and following the course actively, you should gain some analytical intutions (e.g., "in which mathematical environment a solution to a particular problem can make sense", or "not all PDEs: as is, can be assigned a closed form solution"), and learn some approximation techniques 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 containing both theoretical aspects as stability and convergence of approximate solutions, as well as numerical analysis and implementations.
The course consists of 36 lecture hours (Lh) and 20 exercise hours (Eh):
3x2 hours/week: 5h lectures+1 h Excercises (Totally 35Lh+7Eh) and
1x2 hours exercise/week (Totally 14Eh).
Latest news
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:1315, 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
OBS! The 2018 version.
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),
Reference literature:
Program
Lectures (preliminary plan)
Week  Book chapters 
Contents 

4, Study week 1 
1, 3.13.3, 3.73.8, 4 
Classification of PDEs, Math
Tools, Power of abstraction, Reisz and LaxMilgram theorems:
Summary of Chapter 1.
Summary of Chapter 3.
(proofs: later!) Polynomial approximation. Galerkin spaces, Stiffness matrix. 
5, study Week 2 
5, 7 
Polynomial Interpolation, Gauss quarature rule, Boundary value problem (BVP), Variational formulation (VP), minimization problem (MP) "a priori" and "a posteriori" error estimates in energy norm. 
6, study Week 3 
7, 8 
Finite element approximations: Examples of
Convectiondiffusionadvection/absorption problems .
Scalar initial value problem (IVP): Exact solution, theoretical stability. Finite Differences for (IVP) . 
7, study Week 4 
8 
Continuous and discontinuous Galerkin and
a priori/a posteriori error estimates for
Galerkin methods for scalar initial value problems. .
Initialboundary value problems: one
space dimension.
Stability for the heat conduction, finite element approximation
for the heat equation,
error analysis..

8, study Week 5 
9, Lecture Notes:
Poisson in ${\mathbb R}^d, \, d>1$ 
Initialboundary value problems: one
space dimension.
The wave equation, conservation of energy,
finite elements for the wave equation.
Error estimates and adaptive
error control in the energy norm for the Poisson's equation.
A canonical example.

9, study Week 6 
Lecture Notes:
Heat in ${\mathbb R}^d, \, d>1$ 
Stability and piecewise
linear Galerkin approximation for the heat equation.
Error analysis of finite element methods for the heat equation. 
10, study Week 7 
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, 20152018. 
11, (study) Week 8 
Reserved material 
The week is reserved for
going through
uncovered material , answering questions, solving
some oldexams and other interesting problems. 
Recommended exercises
Week  Excersises 

5, study week w2 
1: Give a varitional formulation of u''+u' +u=f in (0,1), with
u'(0) =1 and 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 
6, study week 3 
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

7,8 study week 4,5 
Chapters 79:
Book: Problems in Chapters 7.1, 7.3, 7.9, 8.38.6, 9.3, 9.6, 9.9, 
9, study week 6 
Chap 1012:
Lecture Notes: Problems in Chapters 1012. 
Demonstrated/recommended Exercises
Week  Excersises From:Problem file, Book and Lecture note 

4, study w2 
Problem file: Problems 5360
Book: 3.133.15, 
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, 7.10, 7.167.19 
7, study w5 
Problem file:
Problems 1320
Book: 8.88.11, 8.16, 9.59.8 
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 
Reserve 