Finite Element Methods

Information for fem06

The goal of this course is to give basic knowledge of the theory and practice of the finite element method and its application to the partial differential equations of physics and engineering sciences. The purpose is to give a balanced combination of theoretical and practical skills. The theoretical part is mainly concerned with the derivation of finite element formulations as well as estimating the discretization error and how to use error estimates to adaptively refine the mesh (see the FEniCS gallery and the Body and Soul gallery). The practical part deals with computer implementation: element matrices, assembly, numerical integration, etc.

Course PM

News

061205: Result of course evaluation is available (only 4 answers so far; you can still give feedback).

061106: Solutions to the written exam is updated (question 1c).

061025: Solutions to the written exam is now available.

061019: The "Hints and solutions..." have been updated for Problem 21.8(a)-(b) (which is one of the hand-in assignements).

061018: Re-exam is set to January 16 2007, 8-13 in E34. The next re-exam after that is in August 2007.

061018: Correction to Problem 3 for Project B; the zero order term should be in terms of the dual solution not u: Download updated project pm.

061017: Note that energy norm is defined on page 512 in the CDE book.

061016: The "Hints and solutions..." have been updated for Problem 21.8(a) (which is one of the hand-in assignements).

061010: In Problem 15.49 you may assume that the coefficient a>0 is constant.

061010: The error estimates in Problem 15.45b should be in the energy norm.

061009: Correction to Problem 3 for Project B; missing epsilon in the Laplacian term: Download updated project pm.

061003: Correction to Problem 2.1 for Porject B: Download updated project pm.

060929: Course evaluation form is avaliable here. Please fill out this form after the written exam, comments are appreciated.

060929: Important: Policy regarding deadlines for projects and problem sheets: Complementing material for Project A and Problems A are allowed until Friday October 20th. The second deadline: Friday October 20th at 15.00, is a sharp deadline for both parts A and B; which means that any solutions to the Problem A-B handed in after the deadline is ignored (= no bonus points for the written exam), and any project reports for Project A-B handed in after the deadline can give maximum grade 3.

060926: For your results to be reported you need to register electronically for this course as soon as possible!! (You may alternatively use the res-system directly)

060920: New deadline for Project A and Problems A: Monday Sep 25, 9.00.

060920: Note! Deadline for Project A and Problems A on Friday September 22!! Submit either electronically to Johan, or put in the box for this course in Studentexpeditionen on floor 2.

060913: The electronic registration (using the res-system) is now working. Please register as soon as possible!

060913: Links to hints and solutions to problems in the book is now available.

060913: Error in project pm: edge data is used in Problem 4 of Project A. Download updated project pm.

060831: Lecture notes 1-3 now available below in the weekly plan.

060809: Course starts Wednesday August 30, 8-10, in E1.

Registration

To register for this course you need to sign up electronically.

Teachers

Coordinator and lecturer is Johan Hoffman; email: jhoffman@csc.kth.se
Teaching assistant is Erik von Schwerin; email: schwerin@csc.kth.se

Office Hours

Johan Hoffman (room 4526): Mondays 9.00-10.00
Erik von Schwerin (room 4520): Tuesdays 9.00-10.00

In case there is teaching (lectures, excercises, or computer sessions) these hours, the office hours are cancelled.

Examination

The total grade of this course will be the mean value of the grade of a written exam and a project (rounded up):

(1) Written exam: Saturday October 21, 8-13, and
(2) Project: Report A should be handed in by Friday September 22, and Report B by Friday October 20.

The project should be carried out individually or in groups of two.

2 sets of problems generate maximum 5 bonus points for the written exam if handed in by Friday September 22 (Problem set A) and by Friday October 20 (Problem set B):

Problem set A: 8.13, 15.19, 15.20, 15.21, 15.22
Problem set B: 8.22, 15.45(a,b), 15.48, 15.49, 21.8

Important: Policy regarding deadlines for projects and problem sheets: Complementing material for Project A and Problems A are allowed until Friday October 20th. The second deadline: Friday October 20th at 15.00, is a sharp deadline for both parts A and B; which means that any solutions to the Problem A-B handed in after the deadline is ignored (= no bonus points for the written exam), and any project reports for Project A-B handed in after the deadline can give maximum grade 3.

Course evaluation form

Please fill out the course evaluation form after the written exam. The following message means that your submitted form has been received: Din utvärdering har tagits emot. Tack för hjälpen!

Result of course evaluation.
(Only 4 answers so far; you can still give feedback)

Literature

Course book (CDE)

"K. Eriksson, D. Estep, P. Hansbo, C. Johnson: Computational Differential Equations",

Studentlitteratur, ISBN ISBN 91-44-49311-8. Price: 410 kr at kårens bokhandel.

Hints and solutions to some of the problems in the book.

Useful inequalities.

More books in the same series.

Project

Project PM

Part A deadline: Friday September 22

Part B deadline: Friday October 20

First page of reports should include: name, email and program for all group members.

Using your own computer: Matlab is avaliable at the library, and the PDE-toolbox is avaliable to download for free. Detailed information on the mesh representation availble here.

Computer Sessions (F1-F5 tutorial for the project)
Puffin (used in the project)
Puffin Manual
DOLFIN (big brother of Puffin)
FEniCS (software project including both Puffin and DOLFIN)
Body and Soul (educational project including Puffin sessions, CDE book, other books,...)

Laplacian models (AMBS) (some PDE applications)
Robin boundary conditions
Robin boundary conditions in 2D

Triangle

Extra material

Old exams from KTH with solutions: aug06, jan06, dec05, dec02, feb03, maj03
Old exams from Chalmers: 00a, 00c, 01c, 02c, 03a, with solutions: 00a, 00c, 01c, 02c, 03a
Extra excercises:
Exercises (E1a) with solutions
Exercises (E1b) with solutions
Exercises (E2) with solutions
Exercises (E3) with solutions
Exercises (E4) (in swedish) with solutions
Problems (P1) (in swedish) with solutions

MSc projects

You may be interested in a MSc project at the School of Computer Science and Communication (CSC). MSc projects at CSC are avaliable here.

Preliminary weekly plan

(lectures notes avaliable below)

Week 1

Lecture 1: Wed 30 Aug, 8-10; E3: Poisson 1D, boundary conditions, weak formulation, Galerkin method, piecewise polynomials 1D (CDE 1-4,6,8.1).
l1-1 l1-2 l1-3 l1-4 l1-5 l1-6 l1-7 l1-8
Lecture 2: Thu 31 Aug, 8-10; E3: Poisson 2D, assembly algorithm, FEM mesh, piecewise polynomials 2D, quadrature, affine mapping, implementation in Puffin (CDE 5.5,(7),13,14.1-14.2,14.4,15.1).
l2-1 l2-2 l2-3 l2-4 l2-5 l2-6 l2-7 l2-8 l2-9 l2-10
Lecture 3: Fri 1 Sep, 8-10; V2: Boundary conditions, adaptivity, residual, mesh refinement (CDE 15.1,15.3,15,4, Robin boundary conditions in 1D and 2D).
l3-1 l3-2 l3-3 l3-4 l3-5 l3-6 l3-7

Week 2

Excercise 1, Mon 4 Sep, 15-17; E51, E52: Variational form of DE, Galerkin method, boundary conditions, corresponding system of equations: Problems: 4.21, 6.2, 6.8, 6.9, 6.10, 6.11, 6.14, 8.6, 8.7, 8.9, 8.10, 13.30, 15.14, 15.16, 15.19, 15.20, 15.21, 15.44, 15.45, (4.22, 4.24, 4.25, 6.12)
Excercise 2, Tue 5 Sep, 15-17; D31, D32: Error estimation, a priori, a posteriori, duality: Problems: same as above + extra problems.
Excercise 3, Wed 6 Sep, 8-10; E31, E32: Interpolation, error estimation: Problems: 5.8, 5.9, 5.13, 5.14, 5.17, 5.23, 5.24, 8.11, 8.12, 8.13, 8.16, 8.17, 8.18, 8.20, 8.21, 8.22, 8.23, 8.37, 8.40, 8.41, 14.9, 15.48, 15.49
Excercise 4, Thu 7 Sep, 10-12; D52, E53: Stability: Problems: 9.4, 9.5, 9.7, 9.10, 9.11, 9.14, 9.15, 9.16, 9.43, 10.18, 10.21, 10.24(a), 10.27, 10.28, 10.30, 16.14, 16.15, 16.18, 17.19, 17.20, 17.27, 17.21, (9.23, 9.24, 9.25, 9.40, 9.41, 9.42, 16.19, 16.20, 17.37(a))
Computer session 1, Fri 8 Sep, 8-10; 4V4Gul, 4V6Bru: Project

Week 3

Lecture 4: Tue 12 Sep, 15-17; E3: Interpolation, error estimation, higher order FEM (CDE 5,8.2-8.6,14.2,15.2-15.3).
l4-1 l4-2 l4-3 l4-4 l4-5 l4-6 l4-7 l4-8 l4-9 l4-10 l4-11 l4-12 l4-13 l4-14
Lecture 5: Wed 13 Nov, 8-10; E2: Adaptivity, a priori, a posteriori, duality (CDE 15.5).
l5-1 l5-2 l5-3 l5-4 l5-5
Lecture 6: Thu 14 Nov, 10-12; E3: Abstract problem, Lax-Milgram (CDE 21,12).
l6-1 l6-2 l6-3 l6-4 l6-5 l6-6 l6-7
Computer session 2, Fri 15 Sep, 13-15; 4V4Gul, 4V6Bru: Project

Week 4

Excercise 5, Thu 19 Sep, 15-17; E33, E34: Error estimation, a priori, a posteriori, duality, stabilization: Problems: 18.1, 18.7, 18.9, 19.1, 19.2
Lecture 7: Wed 20 Sep, 8-10; E2: Initial value problem, heat equation, wave equation, stability, theta-method (CDE 9.1-9.2,16,17).
l7-1 l7-2 l7-3 l7-4 l7-5 l7-6 l7-7 l7-8 l7-9 l7-10
Computer session 3, Thu 21 Sep, 15-17; 4V4Gul, 4V6Bru: Project
Submit Project A and Problems A, Fri 22 Sep.

Week 5

Lecture 8: Tue 26 Sep, 15-17; E3: Convection-diffusion-reaction equation, space-time FEM, stabilization Problem 18.6 (CDE 18,19).
l8-1 l8-2 l8-3 l8-4 l8-5 l8-6 l8-7 l8-8
Excercise 6, Wed 27 Sep, 8-10; E32, E33: Abstract problem, Lax-Milgram Theorem 21.1,21.2,21.8,21.9,21.11,21.12,21.13,21.14,21.17 (21.5,21.10)
Computer session 4, Thu 28 Sep, 15-17; 4V4Gul, 4V6Bru: Project

Week 6

Computer session 5, Mon 2 Oct, 8-10; 5O1Spe, 5O2Spo: Project
Excercise 7, Tue 3 Oct, 15-17; D31, D32: Repetion, preparation for written exam: Problems: extra problems, old exams
Lecture 9: Wed 4 Oct, 10-12; M3: ALE, Navier-Stokes, overview/repetition (CDE ).
l9-1 l9-2 l9-3 l9-4 l9-5 l9-6 l9-7
Computer session 6, Thu 5 Oct, 15-17; 4V4Gul, 4V6Bru: Project

Week 7

Time for work on Project B and Problem B.

Week 8

Submit Project B and Problems B, Fri 20 Oct.
Written exam, Sat 21 Oct, 8-13 (D31, D32, D33, D34, D35, 4V4Gul, 4V6Bru).

Sidansvarig: Johan Hoffman <jhoffman@csc.kth.se>
Senast ändrad 9 augusti 2006
Tekniskt stöd: <webmaster@csc.kth.se>