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Finite element methods
Period 2, 2013/2014
Department of Information Technology
Uppsala University
Finite element methods, Uppsala University, Sweden, 30th October 2013 – p. 1
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
Finite element methods, Uppsala University, Sweden, 30th October 2013 – p. 2
About the course
• Finite element methods, 5hp
Homepage: http://www.it.uu.se/edu/course/homepage/fem/ht13/
Finite element methods, Uppsala University, Sweden, 30th October 2013 – p. 3
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.
Finite element methods, Uppsala University, Sweden, 30th October 2013 – p. 4
Outline
• Goals
• Literature
• Schedule
• Assignments
• Examination
• Finite element method
Finite element methods, Uppsala University, Sweden, 30th October 2013 – p. 5
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.
Finite element methods, Uppsala University, Sweden, 30th October 2013 – p. 6
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.
Finite element methods, Uppsala University, Sweden, 30th October 2013 – p. 7
(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
Finite element methods, Uppsala University, Sweden, 30th October 2013 – p. 8
(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
Finite element methods, Uppsala University, Sweden, 30th October 2013 – p. 9
(Preliminiary) Schedule
Week 50:
10/12 exercise 6 Questions
Finite element methods, Uppsala University, Sweden, 30th October 2013 – p. 10
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.
Finite element methods, Uppsala University, Sweden, 30th October 2013 – p. 11
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)
Finite element methods, Uppsala University, Sweden, 30th October 2013 – p. 12
The finite element method: Applications
• Structural mechanics
• Electro magnetics
• Fluid mechanics
• Geophysics
• Biophysics
• General relativity
Finite element methods, Uppsala University, Sweden, 30th October 2013 – p. 13
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
Finite element methods, Uppsala University, Sweden, 30th October 2013 – p. 14
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
Finite element methods, Uppsala University, Sweden, 30th October 2013 – p. 15
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