Finite element methods

Period 3, 2008/2009

Department of Information Technology

Uppsala University

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Short Bio

• Axel Målqvist, grew up in Aneby, Småland• Educated at Chalmers• PostDoc in US• Ass. Prof. Uppsala University since ’07

I have been interested in FEM since ’97 and I started doingresearch on FEM in ’01. It is still the main focus in my research.

Office: 2440Phone: 018-4716287email: axel.malqvist@it.uu.se

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About the course

• Finite element methods, 6hp, F• Finite element methods NV1, 7.5hp, FRI, NVP

Homepage: http://www.it.uu.se/edu/course/homepage/fem/vt09/

Outline: Literature, Schedule, Assignments, Examination, Finiteelement method

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Literature

New book: A first course in finite elements: lecture notes byM.G. Larson and F. Bengzon, Umeå University.

• Compact, covers the course very nicely• Has been used successfully at Umeå University• Not a final version, misprints• No answers to exercises• It is free!

Other relevant books include:

Johnson, C., Numerical Solution of Partial Differential Equationsby the Finite Element Method, Studentlitteratur, 1987.Eriksson, K., Estep, D., Hansbo, P., and Johnson, C.Computational Differential Equations, Studentlitteratur, 1997.

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Schedule

Week 4:20/01 lecture 1 Introduction + (MGL) Chap.1 pp 1 - 821/01 lecture 2 (MGL) Chap.1 pp 9 - 1822/01 exercise 1 (MGL) Chap.1 problems

Week 5:27/01 lecture 3 (MGL) Chap.2 pp 23-3028/01 lecture 4 (MGL) Chap.2 pp 31-3829/01 exercise 2 (MGL) Chap.2 problems30/01 computer lab 1 Finite element method in 1D

Week 6:03/02 lecture 5 (MGL) Chap.3 pp 43-5104/02 lecture 6 (MGL) Chap.3 pp 52-6205/02 exercise 3 (MGL) Chap.3 problems06/02 lecture 7 (MGL) Chap.4 pp 65-71

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Schedule

Week 7:09/02 lecture 8 (MGL) Chap.4 pp 72-8110/02 lecture 9 Guest lecturer Ed Fontes Comsol AB12/02 exercise 4 (MGL) Chap.4 problems13/02 computer lab 2 Finite element method in 2D

Week 8:16/02 lecture 10 (MGL) Chap.5 pp 87-9217/02 lecture 11 (MGL) Chap.5 pp 92-95

Week 9:24/02 exercise 5 (MGL) Chap.5 problems25/02 laboration 3 Femlab25/02 lecture 12 Exam problems27/02 exercise 6 Questions

13/03 Examination

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Assignments

There will be two (6 hp) or three (7.5 hp) assignments. One ortwo in each group. You need (G) to on these to pass the course.On A1 and A2 you can also get (VG) if your solution is very goodand handed in before deadline.

• Assignment 1 deadline for bonus 27/02• Assignment 2 deadline for bonus 10/03• Assignment 3, only 7.5hp version, no bonus

A (VG) gives 1 bonus mark on the exam. Maximum number ofbonus marks is therefore 2.

Final deadline for assignments 13/03.

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Examination

Written exam 13/03. Same exam for all students.• 5 questions, (5*8 marks=40 marks)• Maximum of 2 bonus marks• Grade 3=18 marks, grade 4=25 marks, and grade 5=32

marks• The course has slightly different focus this year i.e. all

questions on old exams are not relevant.

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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 Babuska ’60-present• Swedish key contributors Thomee and Johnson (Chalmers)

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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)• Discretization (elements, disc. func. spaces, time stepping)• Error analysis (bound, 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 isoften vague, than the exact answer to the wrong question, whichcan always be made precise. - John Tukey

Computers are incredibly fast, accurate and stupid. Humanbeings are incredibly slow, inaccurate and brilliant. Togetherthey are powerful beyond imagination. - Albert Einstein

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