The Finite Element Method (SF2561), 7.5hp, Fall 2015
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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 discretization error and efficient adaptive algorithms that minimize 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 discretization 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.
Laboratory work
Registration
IMPORTANT: All KTH students need to register for the written exam using My Pages (Mina Sidor). All SU students and PhD students need to register for the written exam by an email to <ireneh@kth.se>.
DEADLINE: Sunday October 11
Direct any questions to: Irene Hanke <ireneh@kth.se>
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All students need to activate themselves in RAPP. Otherwise you are not registered on the course and you cannot register for the written exam.
Register for the course in RAPP
Direct any questions to: Irene Hanke <ireneh@kth.se>
Examination
The total grade of this course will be given by the written exam.
- Written exam (grade A-F) : Thursday October 27, 8-13.
- Laboratory work (Pass/Fail): The laboratory work should be carried out individually or in groups of two, but individual reports should be handed in. 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. Deadlines: Report A should be handed in by Friday October 2, and Report B by Friday October 16.
- 2 sets of problems generate maximum 5 bonus points for the written exam if handed in by the deadlines: Friday September 25 (Problem set A: 8.13, 15.19, 15.20, 15.21, 15.22), and Friday October 9 (Problem set B: 8.22, 15.45(a,b), 15.48, 15.49, 21.8).
- NOTE: In some electronic versions of the CDE book, the Chapter 13 "Calculus of Several Variables" may be missing. For the problem sets the correct chapters should be: Chapter 8 "Two-Point Boundary Value Problems", Chapter 15: "The Poisson Equation", Chapter 21: "The Power of Abstraction".
Old exams
Written exam 2014-10-30
Written exam 2012-10-18 [solutions]
Written exam 2010-10-19
Written exam 2007-01-16
Written exam 2006-10-21
Written exam 2006-08-25
Written exam 2006-01-21
Written exam 2005-12-16
Course leader
Office hours: Mondays 9:00-10:00 (Office 4429, Lindstedtsvägen 5)
Teaching assistant
Literature
Course book [CDE]: Eriksson, Estep, Hansbo, Johnson, "Computational Differential Equations", Studentlitteratur, (ISBN ISBN 91-44-49311-8), 1996. [Bokus] [Studentlitteratur] [Kårbokhandeln]
NOTE: In some electronic versions of the CDE book, the Chapter 13 "Calculus of Several Variables" may be missing. For the problem sets the correct chapters should be: Chapter 8 "Two-Point Boundary Value Problems", Chapter 15: "The Poisson Equation", Chapter 21: "The Power of Abstraction".
Course book reading instructions
Hint and solutions to some problems in the book
Software
Puffin (simple Matlab/Octave FEM software)
Extra material
Solutions to problems 8.9, 8.10, 8.11, 15.16
Description of FEM boundary conditions in 1D
Description of FEM boundary conditions in 2D
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.
Schedule
The weekly schedule is available here
Preliminary week plan
[NOTE: In some electronic versions of the CDE book, the Chapter 13 "Calculus of Several Variables" may be missing. For the problem sets the correct chapters should be: Chapter 8 "Two-Point Boundary Value Problems", Chapter 15: "The Poisson Equation", Chapter 21: "The Power of Abstraction".]
Week 1
Lecture 1: FEM for 1D boundary value problem
Reading: [Course book CDE 1-4,6,8.1]
[Lecture notes 1]
Lecture 2: 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: Boundary conditions, adaptive mesh refinement
Reading: [CDE 15.1,15.3,15.4]
[Lecture notes 3]
Lab 1: Implementation of 1D FEM [Matlab/Octave]
Week 3
Lecture 4: Error estimation
Reading: [CDE 5,8.2-8.6,14.2,15.2-15.3]
[Lecture notes 4]
Exercise 1: Galerkin method, boundary conditions
Suggested 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)]
[Hint and solutions to some problems in the book]
Week 4
Lecture 5: Duality, a posteriori error estimation
Reading: [CDE 15.5]
[Lecture notes 5]
Lecture 6: Abstract problem, well-posedness
Reading: [CDE 21,12]
[Lecture notes 6]
Deadline (Fri Sep 25) : Problem set A [CDE 8.13, 15.19, 15.20, 15.21, 15.22]
Week 5
Lecture 7: Initial value problem
Reading: [CDE 9.1-9.2,16,17]
[Lecture notes 7]
Exercise 2: Galerkin method, error estimation
Suggested problems: [CDE same as Exercise 1]
[Hint and solutions to some problems in the book]
Lab 2: Implementation of 2D FEM [Puffin]
Deadline (Fri Oct 2) : Lab Report A
Week 6
Exercise 3: Interpolation, error estimation
Suggested 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]
[Hint and solutions to some problems in the book]
Lecture 8: Convection-diffusion-reaction, space-time FEM, stabilized FEM
Reading: [CDE 18,19]
[Lecture notes 8]
Exercise 4: Stability
Suggested 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))]
[Hint and solutions to some problems in the book]
Deadline (Fri Oct 9) : Problem set B [CDE 8.22, 15.45(a,b), 15.48, 15.49, 21.8]
Week 7
Lecture 9: Course review and outlook
Exercise 5: Adaptivity, a posteriori error estimation, duality
Suggested problems: [CDE 18.1, 18.7, 18.9, 19.1, 19.2]
[Hint and solutions to some problems in the book]
Exercise 6: Abstract problem, Lax-Milgram theorem
Suggested problems: [CDE 21.1,21.2,21.8,21.9,21.11,21.12,21.13,21.14,21.17 (21.5,21.10)]
[Hint and solutions to some problems in the book]
Deadline (Fri Oct 16) : Lab Report B
Week 8
Preparation for written exam
Week 9
Written exam