The Finite Element Method (SF2561), 7.5hp, Fall 2016

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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.

https://www.youtube.com/watch?v=TSxCFoBgxUw&list=PLsFk6Zk9M10FjIa_jyB5uqE_qydlsQYWc

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.

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>.

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) : Tuesday October 25
* 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: Friday September 30, Friday October 14.
* 2 sets of theoretical problems. Deadlines: Friday September 23, Friday October 7.
Laboratory work Lab PM

Theoretical problems Problem set A¶

Old exams Written exam 2014-10-30Written exam 2012-10-18 [solutions]Written exam 2010-10-19 Written exam 2007-01-16Written exam 2006-10-21Written exam 2006-08-25Written exam 2006-01-21Written exam 2005-12-16

Course leaders Sara Zahedi

Office hours: Mondays 9:00-10:00 (Office 3538, Lindstedtsvägen 25)

Johan Hoffman

[Office hours: TBA]

Thomas Frachon (Lab questions)

Office hours: Thursdays 9:00-10:00 (Office 3413, Lindstedtsvägen 25)

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

More books in the same series

Software Puffin (simple Matlab/Octave FEM software)

Puffin tutorial

Triangle

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 Project Courses (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