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The Finite Element Method (SF2561), 7.5hp, Fall 2014
[Work in progress]
News
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
Problem set
Course leader
Office hours
Mondays 9:00-10:00 (Office 4429, Lindstedtsvägen 5)
Literature
Course book (CDE): Eriksson, Estep, Hansbo, Johnson, "Computational Differential Equations", Studentlitteratur, (ISBN ISBN 91-44-49311-8), 1996. [Bokus] [Studentlitteratur]
Course book reading instructions
Software
Puffin (simple Matlab/Octave FEM software)
Extra material
FEniCS (advanced Python/C++ FEM software)
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
Add-on studies
DN2295 Project Course in Scientific Computing (talk to course leader)
DN2275 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 [CDE 1-4,6,8.1]
Lecture 2 (Tue Sep 2, 13-15, E32): FEM for 2D boundary value problem [CDE 5.5,(7),13,14.1-14.2,14.4,15.1]
Week 2
Lecture 3 (Mon Sep 8, 15-17, E32): Boundary conditions, adaptive mesh refinement [CDE 15.1,15.3,15.4]
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 [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)
Deadline (Fri Sep 19) : Lab Report A
Week 4
Lecture 5 (Thu Sep 25, 14-16, L51): Duality, a posteriori error estimation [CDE 15.5]
Lecture 6 (Fri Sep 26, 14-16, E35): Abstract problem, well-posedness [CDE 21,12]
Deadline (Fri Sep 26) : Problem set A
Week 5
Lecture 7 (Mon Sep 29, 10-12, D34): Initial value problem [CDE 9.1-9.2,16,17]
Exercise 2 (Tue Sep 30, 10-12, E32)
Deadline (Fri Oct 3) : Lab Report B
Week 6
Lecture 8 (Mon Oct 6, 10-12, V01): Convection-diffusion-reaction, space-time FEM, stabilization [CDE 18,19]
Exercise 3 (Wed Oct 8, 8-10, V21)
Exercise 4 (Fri Oct 10, 13-15, E51)
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)
Exercise 6 (Fri Oct 17, 13-15, D34)
Deadline (Fri Oct 17) : Lab Report C
Week 8
Week 9
Written exam (Thu Oct 30, 8-13, E32)