Finite element methods

Period 2, 2013/2014

Department of Information Technology

Uppsala University

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Short Bio

• Patrick Henning, grew up in Germany

• Educated in Freiburg and Münster

• Working on Numerical Analysis

• PostDoc at Uppsala University since ’13

I started doing research on FEM in ’05. It is still the main focus in my

research.

Office: 2446

Phone: 018-471 2965

email: patrick.henning@it.uu.se

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About the course

• Finite element methods, 5hp

Homepage: http://www.it.uu.se/edu/course/homepage/fem/ht13/

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Registration

• If you you want to pass the course, you must be registered!

• Registration is possible at Studentportalen.

• Web-registration is possible until November 11, at the latest.

• Note that registration is a two step procedure consisting of

applying and, when you get accepted, a web-registration.

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Outline

• Goals

• Literature

• Schedule

• Assignments

• Examination

• Finite element method

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Learning goals

To pass the course the student shall be able to:

• formulate and solve (with a computer) second order elliptic boundary value problems in 1d

(one spatial dimension) using the finite element method.

• formulate and solve (with a computer) second order elliptic boundary value problems in 2d

(two spatial dimensions) with Dirichlet, Neumann, and Robin boundary conditions, using

the finite element method.

• derive a priori and a posteriori error bounds for elliptic equations in one and two spatial

dimensions, and be able to use these error bounds to construct adaptive algorithms for

local mesh refinement.

• solve parabolic and hyperbolic partial differential equations using the finite element

method in space and finite differences in time, and to compare different time stepping

algorithms and choose appropriate algorithms for the problem at hand.

• use finite element software to solve more complicated problems, such as coupled systems

of equations.

• evaluate different techniques for solving problems and be able to motivate when to use

existing software and when to write new code.

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Literature

Book: The Finite Element Method: Theory, Implementation, and

Practice by M.G. Larson and F. Bengzon, Texts in Computational

Science and Engineering, Springer, 2013.

• Compact, covers the course very nicely

• Was used several times before at Uppsala and at Umeå

• Several times revised (so few misprints/mistakes)

• Answers to exercises are published online

• Available in the student book stores

Other relevant books include:

Johnson, C., Numerical Solution of Partial Differential Equations by the

Finite Element Method, Studentlitteratur, 1987.

Eriksson, K., Estep, D., Hansbo, P., and Johnson, C. Computational

Differential Equations, Studentlitteratur, 1997.

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(Preliminiary) Schedule

Week 44:

30/10 lecture 1 Introduction + (LB) Chap.1, part 1

31/10 lecture 2 (LB) Chap.1, part 2

Week 45:

04/11 exercise 1 (LB) Chap.1 problems

05/11 lecture 3 (LB) Chap.2, part 1

07/11 lecture 4 (LB) Chap.2, part 2

Week 46:

11/11 exercise 2 (LB) Chap.2 problems

12/11 lecture 5 (LB) Chap.3, part 1

14/11 computer lab 1 Finite element method in 1D

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(Preliminiary) Schedule

Week 47:

19/11 lecture 6 (LB) Chap.3, part 2

20/11 exercise 3 (LB) Chap.3 problems

21/11 lecture 7 (LB) Chap.4, part 1

Week 48:

25/11 exercise 4 (LB) Chap.4 problems

26/11 lecture 8 (LB) Chap.4, part 2

27/11 computer lab 2 Finite element method in 2D

28/11 lecture 9 (LB) Chap.5, part 1

Week 49:

02/02 lecture 10 (LB) Chap.5, part 2

03/12 exercise 5 (LB) Chap.5 problems

05/12 laboration 3 Comsol multiphysics

06/12 lecture 11 discussion of problems from old exams

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(Preliminiary) Schedule

Week 50:

10/12 exercise 6 Questions

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Examination

Assignments

There will be two assignments. Each group can consist of one or two

students.

Deadline is 06/12, and final deadline (with corrections if needed) is

18/12.

Written exam

Exam date 18/12.

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The finite element method: History

• First approach ’40 (Courant)

• Developed in ’50 and ’60 (Engineers, structural mechanics)

• Mathematics foundation ’70 (Strang and Fix)

• Leading researcher Babuška ’60-present

• Swedish key contributors Thomée (Chalmers) and Johnson

(KTH)

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The finite element method: Applications

• Structural mechanics

• Electro magnetics

• Fluid mechanics

• Geophysics

• Biophysics

• General relativity

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The finite element method: Research

• Mathematics

• Mathematical modeling (equation, data, uncertainties)

• Functional analysis (existence, uniqueness, regularity)

• Discretization (elements, disc. func. spaces, time stepping)

• Error analysis (bounds, estimates)

• Adaptivity (refinement criteria)

• Implementation (cad, meshing, refining, linear algebra,

visualization)

• Parallel programming (efficiency, data structure)

• Computation

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Words of wisdom

Far better an approximate answer to the right question, which is often

vague, than the exact answer to the wrong question, which can always

be made precise. - John Tukey

Computers are incredibly fast, accurate and stupid. Human beings are

incredibly slow, inaccurate and brilliant. Together they are powerful

beyond imagination. - Albert Einstein

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