The Finite Element Method

Information for fem08

The goal of this course is to give basic knowledge of the theory 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, discretization error estimates and algorithms for adaptively refining the mesh (see the FEniCS gallery and the Body and Soul gallery). The practical part deals with computer implementation: assembly of element matrices, numerical integration, etc.

Project course continuation

A natural continuation of the finite element course is the Project Course in Scientific computing (DN2295), given in period 2 2008. There you can choose or come up with a project which goes more in depth into a topic or application. You will receive supervision from members of our research group (Johan Jansson, Johan Hoffman, ...).

News

081204:

The date and place for the re-exam have been set to 2009-01-12 13:15-18:15 in room 4523, floor 5, Osquarsbacken 2. Please email me to register for the exam. The re-exam will consist of a new chance for the written examination, you can also submit reports for the modules in December 2008-January 2009 to me.

081105:

Grades have been submitted to the student expedition. The grades are set by a sum of modules points (maximum 30p, where you need at least 14p to pass) and written exam points (maximum 30p, where you need at least 14p to pass) with at least a pass necessary on each part. This is the grading scale (maximum 60p):

FX: 24p E: 27p D: 34p C: 41p B: 47p A: 54p

Written exam paper 2008-10-21 and solutions 2008-10-21.

Please take some time to give some feedback in the course evaluation form (see below).

Thanks for making the course interesting and good luck in the future.

081027:

Home page for project course now up. Mail me if you want to take the course and haven't talked to me about it yet. First meeting is wednesday 2008-10-29 at 10:15 in 4523.

081020:

The deadline for modules 6-9 has been extended until October 24 due to the lab computer kd0 being unusable over the weekend. Note for the exam: there are no aids allowed.

081020:

The kd0 computer seems to have had some trouble over the weekend, I'm aware of this and will extend the module deadline for those who need it.

081010:

Updated examination question for module 9 (clarification).

081009:

Modules 8 and 9 have been updated (completed). See lecture notes for lecture 9 below for repetition/overview and information in preparation for the written part of the examination.

081009:

The first batch of modules have been graded and can be picked up in the lab on friday or from Murtazo's office.

081009:

All examination questions in the remaining modules are now up.

081007:

Typical examination questions in preparation for the written exam.

081003:

Note correction in module 5 for the velocity beta. It should be a counterclockwise rotating velocity field. The remaining modules will be up before monday. The question in module 6 will deal with implementation of adaptive mesh refinement with regard to an error indicator based on the residual.

080929:

Put up "office hours" at tuesdays 15-16 in room 1517. Take advantage of these extra sessions if you feel you're falling behind.

080925:

Last question is up for module 5 (apologies for delay). Remember deadline at october 3 for modules 4-5.

080924:

Updated error estimation module (number 5).

080917:

Uploaded notes for lecture 5 and 6. Updated module pages with FEniCS instructions.

080909:

If you can't login to our computers, go to the reception of the system group for CSC and get an account.

080905:

Examination questions in the first three course modules are now available, with deadline 19 September.

080902:

Uploaded notes for lecture 2.

080901:

Uploaded notes for lecture 1. Course modules are now accessible.

080901:

Course starts Tuesday September 2, 8-10, in E3.

Registration

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

Teachers

Coordinator and lecturer is Johan Jansson; email: jjan@csc.kth.se

Exercise assistant is Niclas Jansson; email: njansson@csc.kth.se

Lab assistant is Murtazo Nazarov; email: murtazo@csc.kth.se

Office Hours

Johan Jansson (1517): Tuesdays 15.00-16.00

Meet me in my office if you feel you're falling behind, then we can go through the parts which are not clear.

Examination

The examination consists of submissions of written reports and software for the course modules (FEM08 modules at www.icarusmath.com) and a written exam.

(1) Written exam: Tuesday October 21, 14-19. Typical examination questions.

(2) Reports for modules 1-3 are due September 19, modules 4-5 October 3, the rest of the modules October 20. Give your answers with source code and possible figures in a PDF. Mail the PDF to me (jjan@csc.kth.se) with the text "module 1-3" in the title.

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

Course evaluation form

Please fill out the course evaluation form after the written exam.

Press here to get course evaluation form:

The following message means that your submitted form has been received: Din utvärdering har tagits emot. Tack för hjälpen!

Literature

Course book (CDE)

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

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

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

Useful inequalities.

More books in the same series.

Project

Extra material

FEM course from 2006 (has older lecture notes, etc.)

Computer Sessions (older Octave/Matlab educational version of FEniCS)

Puffin

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,...)

FEM notes from a numerical analysis course in Uppsala (summarizes some parts of the book)

Laplacian models (AMBS) (some PDE applications)

Robin boundary conditions

Robin boundary conditions in 2D

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

Further reading

P. Hansbo, Beyond the elements of finite elements. (Solid mechanics, general algorithms for non-linear problems)

S.C. Brenner and L.R. Scott, "The mathematical theory of Finite Element Methods", Springer Verlag, 1994. (Mathematical; elliptic problems, mixed methods,..)

A. Logg, Automating the finite element method.

D. Estep, A short course on duality, adjoint operators, Green's functions, and a posteriori error analysis.

Navier-Stokes: Quick and Easy

Compressible Flow (p. 37- for compressible Euler in 2d/3d)

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.

Schedule

The lecture/exercise/lab schedule

Preliminary weekly plan

Week 1

Lecture 1:

Course overview, differential equations, Poisson 1D, boundary conditions, weak formulation, polynomial approximation, interpolation, L2 projection, Galerkin method, piecewise polynomials 1D (CDE 1-4,6,8.1). (modules 1-3).

Lecture 2:

Poisson 2D, assembly algorithm, FEM mesh, piecewise polynomials 2D, quadrature, affine mapping (CDE 5.5,(7),13,14.1-14.2,14.4,15.1). (modules 3-4)

Week 2

Lecture 3:

Boundary conditions, FEniCS (CDE 8.1.5, 15.1,15.3,15,4, Robin boundary conditions in 1D and 2D).

Lecture 4:

error estimation, adaptivity

Week 3

Lecture 5:

error estimation 2D, mesh operations (CDE 5,8.2-8.6,14.2,15.2-15.3).

Lecture 6:

Initial value problem, heat equation, space-time FEM, stability (CDE 9.1-9.2,16,17).

Week 4

Lecture 7 (see module 8 for "lecture notes"):

Convection-diffusion-reaction equation, stabilization, wave equation Problem 18.6 (CDE 18,19).

Week 5

Lecture 8 (see modules 9 for "lecture notes"):

Uniqueness and existence for elliptic PDE, Lax-Milgram (CDE 21,12).

Week 6

Lecture 9:

ALE, Navier-Stokes, overview/repetition (CDE ).

Week 7

Work on modules/projects.

Week 8

Exam.

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Sidansvarig: Johan Jansson <jjan@csc.kth.se>
Senast ändrad 10 oktober 2008
Tekniskt stöd: <webmaster@csc.kth.se>