This page contains the program of the course: lectures, exercise sessions and computer labs. Other information, such as learning outcomes, teachers, literature and examination, are in a separate course PM.
A good knowledge of calculus (single and several variables), linear algebra, ordinary differential equations and Fourier analysis.
Lectures: Mondays, Wednesdays and Thursdays.
I. M. Asadzasdeh, An Introduction to the Finite Element Method (FEM) for Differential Equations; Part I. Chalmers and GU, 2018. (Compendium ; available in Chalmers Book Store : Cremona).
II. M. Asadzadeh, An Introduction to the Finite Element Method (FEM) for Differential Equations; PartII_draft_FEM_version7A.pdf (Chapters 10-12).
Lecture notes and Course material :
- K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Computational Differential Equations, Studentlitteratur 1996.
- M. Asadzadeh, Lecture Notes in Fourier Analysis, (pdf).
- S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Second edition, Springer 2002.
- C. Johnson, Numerical solutions of partial differential equations by the finite element method, reprinted by Dover, 2008
- M. Taylor, Partial Differential equations (basic theory), Springer 1996.
- W. Strauss, Partial Differential equations, An inroduction, 2008.
- Tobin A. Driscoll, Learning MATLAB, ISBN: 978-0-898716-83-2
- (The book is published by SIAM; available online below in the list of computational literature)
- English-Swedish mathematical dictionary
The schedule of the course is in TimeEdit.
|Jan 20||Chapters 1, (2.2),||Classification of PDEs. Derivation of heat and wave equations|
|Jan 22||3.1, 3.3, 3.5||Vector spaces, n differentiable-and integrable functions, Sobolev spaces|
|Jan 23||3.6-3.8||Basic inequalities, power of abstraction, Riesz and Lax-Milgram theorems.|
|Jan 27||4.1, (4.2), 4.3||Polynomial approximation, Forward Euler for IVP, Galerkin for BVP, FDM.|
|Jan 29||5.1, 5.2||Preliminaries, Lagrange Interpolation|
|Jan 30||5.3||Numerical Integration, quadrature rule|
|Feb 03||7.2-7.3||Finite element method (FEM) , Error estimates in energy norm.|
|Feb 05||7.4||FEM for convection-diffusion-absorption BVPs.|
|Feb 06||8.1-8.3||IVP: solution formula, stability, FDM, Galerkin methods (continuous/discontinuous)|
|Feb 10||8.4||A posteriori error estimates error estimates for cG(1) and dG(0), adaptivity for dG(0).|
|Feb 12||8.5-(8.6)||A priori error estimates for dG(0). (Parabolic case).|
|Feb 13||9.1||Initial Boundary Value Problems (IBVP): Heat equation|
|Feb 17||9.2||Initial Boundary Value Problems (IBVP): Wave equation|
|Feb 19||9.3||Initial Boundary Value Problems (IBVP): Convection-Diffusion problems|
|Feb 20||10.1-10.3||Approximation in several variables. (Construction of finite element spaces)|
|Feb 24||10.4, 11.1-11.3||Interpolation, Poisson equation, Fundamental solution, stability, cG(1) Error estimates|
|Feb 26||12.1-12.2||IBVP in higher dimensions , Herat equation, stability|
|Feb 27||12.3, 12.5||Finite element for heat equation . Wave equation FEM for Heat and wave equations in higher dimensions,|
|March 02||Reserve||Reserved time (for jump overs).|
|March 05||Advancing||Some advanced estimates|
|March 06||Previous Exams||Solving problems from some previous exams|
Recommended exercises (to demonstrate)
|Jan 24||Problem File: New_Problems.pdf Problems 53-60. Book: 3.13, 3.15|
|Jan 31||Problem File: New_Problems.pdf Problems 1-5. Book: 4.5-4.7, 5.15, 5.16|
|Feb 07||Problem File: New_Problems.pdf Problems 6-12. Book:7.3-7.8, 7.10, 7.16-7.19|
|Feb 14||Problem File: New_Problems.pdf Problems 13-20. Book: 8.8, 8.11, 8.16, 9.5-9.8|
|Feb 21||Problem File: New_Problems.pdf Problems 21-23, 26-27. Book: 10.10, 10.11|
|Feb 28||Problem File: New_Problems.pdf Problems 34-40. Book: 11.9, 11.11|
|March 06||Problem File: New_Problems.pdf Problems 43-52. Book: 12.4, 12.9, 12-13, 12.14|
Recommended Exercises (self-study):
|Study Week (SW)||Exercises|
|SW2||1: Give a varitional formulation of -u''+u' +u=f in (0,1), with u'(0) =1 and 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
|SW3||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|
|SW4,5||Chapters 7-9: Book: Problems in Chapters 7.1, 7.3, 7.9, 8.3-8.6, 9.3, 9.6, 9.9,|
|SW6||Chap 10-12: Lecture Notes: Problems in Chapters 10-12.|
You may work in a group of 2 persons but hand in only one report for the group.
Assignment 1: See the file. For this assignment write a short yet detailed report, not exceeding ten pages, explaning your work and sumbit it by the end of study week 4 (Deadline: Friday February 14) . Use MATLAB to do the coding parts. Hints: For problem 1 you need to read chapter 7. problem 2 consider only the case a=4. A good starting point for problem 3 might be the Matlab code, which solves -u''=f, u(0)=u(1)=0 using cG(1).
If you don't have access to FEM-LAB, then you may skip FEM-LAB comparisons.
Assignment 2: Can be found here. Hand in report of your work beginning of study week 7
(Deadline: Monday March 2).
- Learning MATLAB, Tobin A. Driscoll. Provides a brief introduction to Matlab to the one who already knows computer programming. Available as e-book from Chalmers library.
Physical Modeling in MATLAB 3/E, Allen B. Downey
The book is free to download from the web. The book gives an introduction for those who have not programmed before. It covers basic MATLAB programming with a focus on modeling and simulation of physical systems.
- To pass this course you should pass the written exam and the assignments 1 and 2.
- To pass each assignment you need to get at least 1 point in that assigment.
- Assignments 1 and 2 have total of 3, respectively, 5 poits. Hence maximum bonus poits for them are 2 and 4, respectively.
- The two compulsory home assignments should be handed in before the due times above. They are generating max 6 (2+4) bonus points.
- For full points in assigment 2 you need to use a posteriori estimates and perform adaptive mesh-generation.
- Final exam is compulsory, written, and consists of 6 questions (5 problems + 1 theorm) with a maximum score of 30 (=6x5) points. This means that the proportion between the points in home assignments and the exam is 6/30=1/5=20%.
- The theory question is chosen from a list that will appear later in the web-site of the course.
- As for the proof of Lax-Milgram theorem, you may use the proof in lecture notes I.
- No aids are allowed.
- You should be able to state and explain all definitions and theorems given in the course and also apply them in problem solving (but you don't need to give the proofs for theorems that you use).
- Grades are set according to the table below.
Grades Chalmers Points Grades GU Points - <15 - 3 15-21 G 15-26 4 22-28 VG 27- 5 29-
Bring ID and receipt for your student union fee.
- The following link will tell you all about the examination room rules at Chalmers: Examination room instructions
- Solutions for exam problems will be posted in the course web-site.
- You will be notified the result of your exam by email from LADOK (This is done automatically as soon as the exams have been marked and the results are registered.)
- The exams will then be kept at the students' office in the Mathematical Sciences building.
- Check that the number of points and your grade given on the exam and registered in LADOK coincide.
- Complaints of the marking should be written and handed in at the office. There is a form you can use, ask the in the office student admin).
The proof of one of the following theorems/lemmas will be asked in exam:
1. Theorem 3.11: In a Dirichlet BVP the Variational formulation (V F) is equivalent to the minimisation problem (MP).
2. Theorem 3.15: Reisz (Lax-Milgram) theorem [MP has a unique solution].
3. Theorem 5.1: Interpolation error in maximum-norm.
4. Theorem 5.4: Lagrange interpolation error.
5. Theorem 7.1: A priori error estimate for Dirichlet BVP in energy norm.
6. Theorem 7.3: A posteriori error estimate for Dirichlet BVP in energy norm.
7. Theorem 8.3: A posteriori error estimate for cG(1) for IVP.
8. Theorem 8.6: A priori error estimate for cG(1) for IVP (state auxiliary results without proof).
9. Theorem 9.3: Energy estimate for IBVP
10. Theorem 9.5: Conservation of energy for 1-space dimensional wave equation.
11. Theorem 10.3: L2-projection error.
12. Theorem 10.4: L2-error for Laplace equation.
13. Theorem 11.3: cG(1) a priori errors estimate for Poisson equation (gradient estimate).
14. Theorem 11.4: cG(1) a priori errors estimate for Poisson equation (solution estimate).
15. Theorem 12.2 : A priori error estimate for solution of heat equation in higher dim.
16. Theorem 12.3 : A priori error estimate for gradient of solution of heat equation in
17. Theorem 12.5: Conservation of energy for the wave equation in higher dim.
18. Theorem 12.7: A priori error estimate for semi-discrete problem for wave equation higher dim.
The syllabus page shows a table-oriented view of course schedule and basics of course grading. You can add any other comments, notes or thoughts you have about the course structure, course policies or anything else.
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