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 ${\mathbb R}^d,\,\, d>1$.
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 