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

Course leader

Johan Hoffman 

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)

Puffin tutorial

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

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, stabilized FEM [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

Preparation for written exam 

Week 9

Written exam (Thu Oct 30, 8-13, E32)