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, convection-diffusion-reaction 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 Lax-Milgram 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 | 8-10 | Pascal | Lecture |
Mohammad:10-12, L2035; questions |
| WED | 8-10 | Pascal | Exercise (Lecture W1) | Maximilian:10-12,
L2032-2; assignments |
| THU | 10-12 | Pascal | Lecture |
|
| FRI | 8-10 | 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 ${\mathbb R}^d,\,\, d>1$.
Reference literature:
Program
Lectures (preliminary plan)
| Week | Book chapters |
Contents |
|---|---|---|
|
3, Study w1 |
1, 3.1-3.3,
3.7-3.8 |
Classification of
PDEs, Math Tools, Power of abstraction, Reisz and
Lax-Milgram theorems |
| 4,
study W2 |
4, 5 |
Polynomial
approximation, Polynomial Interpolation, L_2
projection, Gauss quarature rule |
|
5, study W3 |
7 |
Two-points
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 |
Initial-boundary
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 convection-diffusion model problem. Selected problems from exams, 2012-2017. |
| 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 FEM-formulation with piecewise linear, continuous functions, and a uniform stepsize h=1/4. 3: The same as above, but with piecewise quadratic functions. Book: 2.1-2.5, 2.11, 2.12, 2.21, 2.22 |
| 5, study w3 |
Chapters 3-5:
Read through iterative methods of chapter 5(self
study not included in the exam). Book: 3.3,
4.1-4.4, 5.8-5.10
|
| 6,7
study w 4,5 |
Chapters 7-9:
Book: Problems in Chapters 7.1, 7.3, 7.9,
8.3-8.6, 9.3, 9.6, 9.9, |
| 8,
study w6 |
Chap 10-12: Lecture
Notes: Problems in Chapters 10-12. |
Demonstrated/recommended Exercises
| Week | Excersises:Problem
file (will be completed by exercises
from the Lecture notes) |
|---|---|
| 4,
study w2 |
Problem file: Problems 53-60 Book:
3.5-3.7, |
| 5, study w3 |
Problem file:
Problems 1-5, Book: 4.5-4.7, 5.15, 5.16 |
| 6,
study w4 |
Problem file:
Problems 6-12 Book: 7.3-7.8, 1.10,
7.16-7.19 |
| 7, study w5 |
Problem file:
Problems 13-20 Book: 8.10-8.14, 8.19,
9.6-9.9 |
| 8,
study w6 |
Problem file:
chosen problems from the list: 21-23, 26-27
Lecture Notes: 10.10, 10.11 |
| 8,
study w6 |
Problem file:
chosen problems from the list: 34-40 43-52
Lecture Notes: 11.9, 11.11 |
| 9,
study w7 |
Problem file:
chosen problems from the list: 43-52, Lecture
Notes:12.4, 12.9, 12.13, 12.14 |
| 10,
study w8 |