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.

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 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

* KTH students should register for the written exam using My Pages (Mina Sidor)
* PhD students and SU students should register at the Mathematics student expedition
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 October 3 (Problem set A: 8.13, 15.19, 15.20, 15.21, 15.22) and by Friday October 10 (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".

Course leader Johan Hoffman

Office hours: Mondays 9:00-10:00 (Office 4429, Lindstedtsvägen 5)

Teaching assistant Niyazi Cem Degirmenci

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]

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 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]

Lab 2: Implementation of 2D FEM [Puffin]

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]

Week 5 Lecture 7: Initial value problem Reading: [CDE 9.1-9.2,16,17]

Exercise 2: Galerkin method, error estimation Suggested problems: [CDE same as Exercise 1][Hint and solutions to some problems in the book]

Deadline (Fri Oct 3) : Problem set A [CDE 8.13, 15.19, 15.20, 15.21, 15.22]

Deadline (Fri Oct 3) : Lab Report A

Week 6 Lecture 8: Convection-diffusion-reaction, space-time FEM, stabilized FEM Reading: [CDE 18,19]

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]

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 10) : 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 17) : Lab Report B

Week 8 Preparation for written exam

Week 9 Written exam