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The Finite Element Method (SF2561), 7.5hp, Fall 2014
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The Finite Element Method
The Finite Element Method (FEM) is a numerical method for solving general differential equations. FEM was first developed for elasticity and structural analysis, but is today used as a universal computational method in all areas of science and engineering, including fluid mechanics, electromagnetics, biomechanics and financial mathematics. The mathematical framework of FEM is well developed, which allows for detailed estimation of the discretisation error and efficient adaptive algorithms that minimise the computational cost, and many FEM software implementations are available, both commercial and open source. This course will cover the theory of FEM, basic algorithms, and practical aspects including software implementation.
Examples of FEM simulations using the open source software FEniCS.
Course activities
- 9 lectures
- 2 laboratory sessions
- 6 exercise sessions
Course goals
The goal of this course is to give basic knowledge of the theory and practice of the finite element method (FEM) and its application to solve 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, estimating the discretisation error and to use error estimates to adaptively refine the mesh. The practical part deals with computer implementation of the method: matrix and vector assembly, numerical integration, etc.
Course PM
Lab PM
Registration
Register for the course in RAPP
Registration for written exam: opens September 15 and closes October 12
Examination
The total grade of this course will be given by the written exam.
- Written exam (grade A-F) : Thursday October 30, 8-13 (E32).
- Laboratory work (Pass/Fail): The laboratory work should be carried out individually or in groups of two, but individual reports should be handed in. Report A should be handed in by Friday October 3, and Report B by Friday October 17. Labs A and B consist of a set of compulsory problems, and a set of non-compulsory problems that give bonus points for the written exam if submitted in time for the deadline. 5 bonus points maximal can be obtained for the lab.
- 2 sets of problems generate maximum 5 bonus points for the written exam if handed in by Friday September 26 (Problem set A) and by Friday October 10 (Problem set B).
Course leader
Office hours: Mondays 9:00-10:00 (Office 4429, Lindstedtsvägen 5)
Teaching assistant
Office hours: Tuesdays 9:00-10:00 (Office 4431, Lindstedtsvägen 5)
Literature
Course book [CDE]: Eriksson, Estep, Hansbo, Johnson, "Computational Differential Equations", Studentlitteratur, (ISBN ISBN 91-44-49311-8), 1996. [Bokus] [Studentlitteratur] [Kårbokhandeln]
Course book reading instructions
Hint and solutions to some problems in the book
Software
Puffin (simple Matlab/Octave FEM software)
Extra material
FEniCS (advanced Python/C++ FEM software)
FEniCS-HPC (High Performance Computing C++ implementation of FEniCS)
The FEniCS book: Automated solution of differential equations by the finite element method
E. Süli, Lecture notes on finite element methods for partial differential equations
A. Ern and J.-L. Guermond. Theory and Practice of Finite Elements. Springer Series: Applied Mathematical Sciences, Vol. 159. 2004. [Chapter 1]
S. Brenner and R. Scott. The Mathematical Theory of Finite Element Methods. Springer Series: Texts in Applied Mathematics, Vol. 15. 2008. [E-book from KTH Library]
D. Estep Short course on duality, adjoint operators, Green's functions and a posteriori error analysis
A. Larcher and N. C. Degirmenci, Lecture Notes: The Finite Element Method
J. Hoffman and C. Johnson, Computational Turbulent Incompressible Flow, Springer Verlag, 2007
Add-on studies
SF2567 Project Course in Scientific Computing (talk to course leader)
DD2365 Advanced Computation in Fluid Mechanics
Projects
Contact the course leader for BSc/MSc projects related to the course.
Preliminary week plan
Week 1
Lecture 1 (Mon Sep 1, 13-15, V01): FEM for 1D boundary value problem
Reading: [Course book CDE 1-4,6,8.1]
[Lecture notes 1]
Lecture 2 (Tue Sep 2, 13-15, E32): FEM for 2D boundary value problem
Reading: [CDE 5.5,(7),13,14.1-14.2,14.4,15.1]
[Lecture notes 2]
Week 2
Lecture 3 (Mon Sep 8, 15-17, E32): Boundary conditions, adaptive mesh refinement
Reading: [CDE 15.1,15.3,15.4]
[Lecture notes 3]
Lab 1 (Fri Sep 12, 13-15, 5O1Spe): Implementation of 1D FEM [Matlab/Octave]
Week 3
Lecture 4 (Tue Sep 16, 15-17, E32): Error estimation
Reading: [CDE 5,8.2-8.6,14.2,15.2-15.3]
Lab 2 (Wed Sep 17, 10-12, 5O1Spe): Implementation of 2D FEM [Puffin]
Exercise 1 (Fri Sep 19, 13-15, D34): Galerkin method, boundary conditions
Problems: [CDE 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)]
Week 4
Lecture 5 (Thu Sep 25, 14-16, L51): Duality, a posteriori error estimation
Reading: [CDE 15.5]
Lecture 6 (Fri Sep 26, 14-16, E35): Abstract problem, well-posedness
Reading: [CDE 21,12]
Deadline (Fri Sep 26) : Problem set A
Week 5
Lecture 7 (Mon Sep 29, 10-12, D34): Initial value problem
Reading: [CDE 9.1-9.2,16,17]
Exercise 2 (Tue Sep 30, 10-12, E32): Galerkin method, error estimation
Problems: [CDE same as Exercise 1]
Deadline (Fri Oct 3) : Lab Report A
Week 6
Lecture 8 (Mon Oct 6, 10-12, V01): Convection-diffusion-reaction, space-time FEM, stabilized FEM Reading: [CDE 18,19]
Exercise 3 (Wed Oct 8, 8-10, V21): Interpolation, error estimation
Problems: [CDE 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]
Exercise 4 (Fri Oct 10, 13-15, E51): Stability
Problems: [CDE 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))]
Deadline (Fri Oct 10) : Problem set B
Week 7
Lecture 9 (Mon Oct 13, 10-12, E36): Course review and outlook
Exercise 5 (Wed Oct 15, 8-10, V01): Adaptivity, a posteriori error estimation, duality
Problems: [CDE 18.1, 18.7, 18.9, 19.1, 19.2]
Exercise 6 (Fri Oct 17, 13-15, D34): Abstract problem, Lax-Milgram theorem
Problems: [CDE 21.1,21.2,21.8,21.9,21.11,21.12,21.13,21.14,21.17 (21.5,21.10)]
Deadline (Fri Oct 17) : Lab Report B
Week 8
Preparation for written exam
Week 9
Written exam (Thu Oct 30, 8-13, E32)