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och datavetenskap

The Finite Element Method, fem11


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 CTL development and the Body and Soul gallery). The practical part deals with computer implementation: element matrices, assembly, numerical integration, etc.

News (2011):

Noc 7: The resilt of the course evaluation is here.

Oct 20:

Please fill out the course evaluation form. It will take few minutes, but help us a lot to improve the course in future!

Oct 11: New deadline for the project: Wed Oct 19, midnight!

Sep 14: Uploaded the Project.

Sep 9: There are changes in the Course schedule : some labs are changed into seminars.

Sep 8: Uploaded the lecture notes.


Coordinator and lecturer: Murtazo Nazarov; email:

Assistant: Niyazi Cem Degirmenci; email:

Office Hours

Murtazo Nazarov (4519): Tuesdays 09:00-10:00.
Niyazi Cem Degirmenci: Mondays 08:00-09:00.


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: Mon 17 Oct, 14-19 (V01, V11, V23)

(2) Laboration. Report should be handed in by Thuesday September 21.
Project. Report should be handed in by Tuesday October 12.
The project should be carried out individually or in groups of two.

The laboration and problem sets should be carried out and handed individually.

2 sets of problems generate maximum 5 bonus points for the written exam if handed in by Thursday September 21 (Problem set A) and by Wednesday October 6 (Problem set B). Deadline is very sharp for both problem sets, which means that any solutions to the Problem A-B handed in after the deadline is ignored (= no bonus points for the written exam).

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 Laboration and Problems A are allowed until Tuesday September 21. Deadline for Project: Tuesday October 12th at 15.00, is a sharp deadline, any reports handed in after the deadline can give maximum grade E.

Laboration and Project

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)
FEniCS project
Body and Soul (educational project including Puffin sessions, CDE book, other books,...)


K. Eriksson, D. Estep, P. Hansbo, C. Johnson: Computational Differential Equations. Studentlitteratur, ISBN ISBN 91-44-49311-8.

Alexandre Ern, Jean-Luc Guermond. Theory and Practice of Finite Elements. ISBN 978-0-387-20574-8

Claes Johnson. Numerical Solution of Partial Differential Equations by the Finite Element Method.

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

Useful inequalities.

Lecture notes 1-5

Lecture notes 6-8

The notes are almost the same as Hoffman's notes from fem06.

Preliminary weekly plan

Course schedule

Week 1

Course overview, differential equations, Poisson 1D, boundary conditions, weak formulation, Galerkin method, piecewise polynomials 1D (CDE 1-4,6,8.1).

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).

Week 2

Boundary conditions, adaptivity, residual, mesh refinement (CDE 15.1,15.3,15,4, Robin boundary conditions in 1D and 2D).

Week 3

Interpolation, error estimation, higher order FEM (CDE 5,8.2-8.6,14.2,15.2-15.3).

Adaptivity, a priori, a posteriori, duality (CDE 15.5).

Abstract problem, Lax-Milgram (CDE 21,12).

Week 4

Initial value problem, heat equation, wave equation, stability, theta-method (CDE 9.1-9.2,16,17).

Week 5

Convection-diffusion-reaction equation, space-time FEM, stabilization (CDE 18,19).

Week 6

Compressible Navier-Stokes, stabilization techniques, overview/repetition (CDE ). Standard stability estimates.
Copyright © Sidansvarig: Murtazo Nazarov <>
Uppdaterad 2011-11-07