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# The Finite Element Method, fem10

## Information for fem10

The goal of this course is to give basic knowledge of the theory and practice of the finite element method and its application to 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 as well as estimating the discretization error and how to use error estimates to adaptively refine the mesh (see the FEniCS gallery and the Body and Soul gallery). The practical part deals with computer implementation: element matrices, assembly, numerical integration, etc.

## News (2010):

Dec 31: The resilt of the course evaluation is here.

Nov 5: Reexamination: January 17, 09-13, 4523.

Nov 3:

Grading scheme for the examination: "p" is the total points of the examination and bonus points
F: p<16,
FX: 16<= p < 20,
E: 20<= p < 26,
D: 26<= p < 32,
C: 32<= p < 38,
B: 38<= p < 44,
A: 44<= p.

Exam paper Oct 19 pdf here

Oct 26:

 Please fill out the course evaluation form. It will take few minutes, but help us a lot to improve the course in future!

Oct 10: A misspell in the project description: Problem 2 of Part B should be written as: "... Solve the dual problem corresponding to Problem 2 of Part A: ... "

Oct 8: New deadline for the project: midnight Sunday 17 Oct.

Sep 15: Some misspells with the deadline dates are fixed now.

Sep 15: The Project is uploaded. The deadline is Tuesday October 12! Submitting the project after the deadline will give you the minimum grade.

Sep 14: The deadline for the Problem set A is moved to Sunday September 26. Note, that the deadline for the laboration is still Thuesday September 21!

Sep 1: Tasks for the laboration are uploaded to the homepage. Note: the deadline is Thuesday September 21.

Aug 31: Pdf books by Claes Johnson are available now. The books have some useful introduction to the finite elements.

## Teachers

Coordinator and lecturer is Murtazo Nazarov; email: `murtazo(at)csc.kth.se`
Teaching assistant is Niclas Jannson; email: `njansson(at)csc.kth.se`

## Office Hours

Murtazo Nazarov: Thursdays 9-10
Niclas Jannson: Wednesdays 8.30-9.30

## Examination

The total grade of this course will be the mean value of the grade of a written exam and a project (rounded up):
(1) Written exam: Tue 19 Oct, 14-19 (Q11, Q13, Q22, Q24, Q26)

(2) Laboration. Report should be handed in by Thuesday September 21.
Project. Report should be handed in by Tuesday October 12.
The project and laboration should be carried out individually or in groups of two.

2 sets of problems generate maximum 5 bonus points for the written exam if handed in by Thursday September 21 (Problem set A) and by Wednesday October 6 (Problem set B). Deadline is very sharp for both problem sets, which means that any solutions to the Problem A-B handed in after the deadline is ignored (= no bonus points for the written exam).
Problem set A: 8.13, 15.19, 15.20, 15.21, 15.22

Problem set B: 8.22, 15.45(a,b), 15.48, 15.49, 21.8
Important: Policy regarding deadlines for projects and problem sheets: Complementing material for Laboration and Problems A are allowed until Tuesday September 21. Deadline for Project: Tuesday October 12th at 15.00, is a sharp deadline, any reports handed in after the deadline can give maximum grade E.

## Literature

#### Course book (CDE)

"K. Eriksson, D. Estep, P. Hansbo, C. Johnson: Computational Differential Equations", Studentlitteratur, ISBN ISBN 91-44-49311-8. Price: 410 kr at kårens bokhandel.

Applied Mathematics Body & Soul II. Read chapter 52, 53.

Applied Mathematics Body & Soul III. Read chapter 76, 77.

Body and Soul Mathematical Simulation Technology. Claes Johnson.

## Laboration and Project

First page of reports should include: name, email and program for all group members.

Computer Sessions (F1-F5 tutorial for the project)
Puffin (used in the project)
Puffin Manual
DOLFIN (big brother of Puffin)
FEniCS (software project including both Puffin and DOLFIN)
Body and Soul (educational project including Puffin sessions, CDE book, other books,...)

Laplacian models (AMBS) (some PDE applications)
Robin boundary conditions
Robin boundary conditions in 2D

## Extra material

Old exams from KTH with solutions: aug06, jan06, dec05, dec02, feb03, maj03
Old exams from Chalmers: 00a, 00c, 01c, 02c, 03a, with solutions: 00a, 00c, 01c, 02c, 03a
Extra excercises:
Exercises (E1a) with solutions
Exercises (E1b) with solutions
Exercises (E2) with solutions
Exercises (E3) with solutions
Exercises (E4) (in swedish) with solutions
Problems (P1) (in swedish) with solutions

H.P. Langtangen, "Computational Partial Differential Equations Numerical Methods and Diffpack Programming" Springer Verlag, 2003. (Development of FEM programs; using numerical C++ library Diffpack)

S.C. Brenner and L.R. Scott, "The mathematical theory of Finite Element Methods", Springer Verlag, 1994. (Mathematical; elliptic problems, mixed methods,..)

A. Logg, Automating the finite element method.

## Preliminary weekly plan

(lectures notes by Johan Hoffman are available in fem06)

All labs scheduled in the computer rooms Spellhallen and Sporthallen

### Week 1

Lecture 1: Fri 27 Aug, 13-15; V1
Poisson 1D, boundary conditions, weak formulation, Galerkin method, piecewise polynomials 1D (CDE 1-4,6,8.1).
l1-1 l1-2 l1-3 l1-4 l1-5 l1-6 l1-7 l1-8

### Week 2

Lecture 2: Man 30 Aug, 13-15; V3
Poisson 2D, assembly algorithm, FEM mesh, piecewise polynomials 2D, quadrature, affine mapping, implementation in Puffin (CDE 5.5,(7),13,14.1-14.2,14.4,15.1).
l2-1 l2-2 l2-3 l2-4 l2-5 l2-6 l2-7 l2-8 l2-9 l2-10

Excercise 1, Wed 1 Sep, 15-17; E33
Variational form of DE, Galerkin method, boundary conditions, corresponding system of equations: Problems: 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)

### Week 3

Lecture 3: Mon 6 Sep, 8-10; E3
Boundary conditions, adaptivity, residual, mesh refinement (CDE 15.1,15.3,15,4, Robin boundary conditions in 1D and 2D).
l3-1 l3-2 l3-3 l3-4 l3-5 l3-6 l3-7

Lab: Tue 7 Sep, 13-15;

Lecture 4: Wed 8 Sep, 8-10; E3
Interpolation, error estimation, higher order FEM (CDE 5,8.2-8.6,14.2,15.2-15.3).

l4-1 l4-2 l4-3 l4-4 l4-5 l4-6 l4-7 l4-8 l4-9 l4-10 l4-11 l4-12 l4-13 l4-14

Lab: Thu 9 Sep, 13-15;

Excercise 2, Fri 10 Sep, 13-15; E32
Variational form of DE in 2D, Galerkin method, boundary conditions, corresponding system of equations:

### Week 4

Lecture 5: Mon 13 Sep, 8-10; E3
Adaptivity, a priori, a posteriori, duality (CDE 15.5).
l5-1 l5-2 l5-3 l5-4 l5-5

Lecture 6: Thu 14 Sep, 15-17; E3
Abstract problem, Lax-Milgram (CDE 21,12).

l6-1 l6-2 l6-3 l6-4 l6-5 l6-6 l6-7

Lab: Wed 15 Sep, 15-17;

Lab: Thu 16 Sep, 13-15;

### Week 5

Lecture 7: Wed 20 Sep, 8-10; E3
Initial value problem, heat equation, wave equation, stability, theta-method (CDE 9.1-9.2,16,17).
l7-1 l7-2 l7-3 l7-4 l7-5 l7-6 l7-7 l7-8 l7-9 l7-10

Lab: Tue 21 Sep, 13-15;

Lab: Wed 22 Sep, 8-10;

Excercise 3, Wed 22 Sep, 15-17; D32
Interpolation, error estimation: Problems: 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

Lab: Thu 23 Sep, 08-10;

### Week 6

Lecture 8: Mon 27 Sep, 13-15; E3
Convection-diffusion-reaction equation, space-time FEM, stabilization (CDE 18,19).
l8-1 l8-2 l8-3 l8-4 l8-5 l8-6 l8-7 l8-8

Lab: Tue 28 Sep, 10-12;

Lab: Tue 28 Sep, 13-15;

Lab: Wed 29 Sep, 13-15;

Excercise 4, Thu 30 Sep, 8-10; E33
Error estimation, a priori, a posteriori, duality, stabilization: Problems: 18.1, 18.7, 18.9, 19.1, 19.2 Abstract problem, Lax-Milgram Theorem 21.1,21.2,21.8,21.9,21.11,21.12,21.13,21.14,21.17 (21.5,21.10)

### Week 7

Lecture 9: Mon 4 Oct, 13-15; E3
ALE, Navier-Stokes, overview/repetition (CDE ).
l9-1 l9-2 l9-3 l9-4 l9-5 l9-6 l9-7

Lab: Tue 5 Oct, 10-12;

Lab: Tue 5 Oct, 13-15;

Excercise 5, Wed 6 Oct, 15-17; E53
Repetion, preparation for written exam: Problems: extra problems, old exams

Lab: Thu 7 Oct, 13-15;

Written exam, Tue 19 Oct, 14-19 (Q11, Q13, Q22, Q24, Q26)

Sidansvarig: Murtazo Nazarov <murtazo@csc.kth.se>