Welcome to TiddlyWiki created by Jeremy Ruston, Copyright © 2007 UnaMesa Association
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To get started with this blank TiddlyWiki, you'll need to modify the following tiddlers:
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Welcome to the Math 206D Wiki!
To navigate this page, simply follow the links on your left.
# [[Instructor]]: Who I am, and how to contact me.
# [[Teaching Assistant]]: There will no TA for this course.
# [[Location]]: Where the class will meet.
# [[Syllabus]]: Information about the course.
# [[Weekly Readings]]: Here I will post the readings for each week, and the lesson objectives. Please take a look at it before the class.
# [[Homework]]: I will post the homework here.
# [[Lecture Notes]]: I will post the notes from class here.
# [[Self-Assessment Questions]]: This will help you study for the midterm and the final exam.
# [[Exams]]: Information about the midterm and final exams.
Once you open a tiddler, you can close it by clicking on its link again. Alternatively you can use the close button over on the right.
This page is experimental, and hopefully more services will be added as the course progresses, so come back often.
Homework will be assigned on Thursday, and will be collected at the beginning of the class on the following Thursday. ''@@color(red): Late homework will not be accepted. If you have a predictable absence, you will need to hand it in earlier@@''.
''Assignments:''
#[[Homework 1|hw1.pdf]]: Due on October 8th, 2009.
#[[Homework 2|hw2.pdf]]: Due on October 29h, 2009.
!Carlos J. ~García-Cervera.
South Hall, Room 6707.
Office Phone: (805) 893 3681
Office Fax: (805) 893 2385
Office Hours: Tuesday and Thursday, from 11:25am-12:25pm.
Main Website: http://www.math.ucsb.edu/~cgarcia
In this course we will study mostly second order linear ordinary differential equations of the form
ay' '+by'+cy = f(x).
Since the equation is linear, it turns out that the set of solutions forms a finite dimensional vector space, which explains why we study Linear Algebra in this course.
These are the notes from class:
Tuesday and Thursday, 9:30-10:45am, NH 1111
[[Instructor]]
[[Teaching Assistant]]
[[Location]]
[[Syllabus]]
[[Weekly Readings]]
[[Homework]]
[[Lecture Notes]]
[[Self-Assessment Questions]]
[[Exams]]
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Try to answer the following questions without looking in the book. If you do not feel comfortable with them, please review the corresponding chapters in the book, and them attempt to solve them again.
The Finite Element Method
Tuesday & Thursday, 9:30-10:45pm, NH 1111
http://www.math.ucsb.edu/~cgarcia/Courses/Math118A_F09/
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''Textbook:'' //Numerical Solution of Partial Differential Equations using the Finite Element Method//, by Claes Johnson.
''Additional bibliography:''
# //Computational Differential Equations//, by by K. Eriksson, D. Estep, P. Hansbo, and C. Johnson.
# //Finite elements for electrical engineers//, by Peter P. Silvester and Ronald L. Ferrari.
# //The Mathematical Theory of Finite Element Methods//, by Susanne C. Brenner and L. Ridgway Scott.
# //A Multigrid Tutorial//, by William L. Briggs, Van Emden Henson, and Steve F. McCormick.
# //Multilevel Adaptive Methods for Partial Differential Equations//, by Steve F. McCormick.
# //Numerical Solution of Partial Differential Equations//, by K.W. Morton and D.F. Mayers.
# //Finite Element Method: Its basis and Fundamentals//, by O.C. Zienkiewicz, R.L. Taylor, and J.Z. Zhu.
For a review of Advanced Calculus, you may want to check out the following:
# //Vector calculus//, by Jerrold E. Marsden and Anthony J. Tromba.
''Course description:'' This is a graduate course in Numerical Analysis, with an emphasis in the solution of Partial Differential
Equations by the Finite Element Method (FEM). In this course we will study the FEM for elliptic, hyperbolic, and parabolic equations, and we
will also cover the boundary element method. Although previous knowledge in Numerical Analysis is recommended, it is not a
requirement for this course. Therefore I will cover some basic material, such as function approximation, and the solution of linear
systems of equations. Although the emphasis will be in applications, the course will have a strong theoretical component
''Prerequisites:'' Knowledge of a
computer language suitable for numerical computing: FORTRAN, C,
C++, or Matlab. Previous knowledge in Numerical Analysis is
recommended, although not necessary.
''Assignments and grading:'' Homework will be assigned every two
weeks. It will be handed out on
Thursdays, and will be collected at the beginning of the class on the
Thursday of the second following week. Late homeworks will not be
accepted. The homework will generally consist
of some theoretical questions, and some computational assignments. You
will be required to write a program to solve certain problems. The
program must be given to me as part of the assignment, together with
the output of the program, in the format indicated in the
assignment, and an interpretation of the results whenever
necessary. You can write the programs either in FORTRAN, C,
C++, or Matlab.
In addition to the homework, there will be a final project. Your final
grade for the course will be decided according to the following formula:
Final Grade = 30% Homework + 70% Final.
''Syllabus:'' During this course we will try to follow the following
schedule. However, much like everything said earlier, this is subject
to change:
# ''Week 1:'' Introduction to the FEM for elliptic problems.
## Variational formulation of a one-dimensional model
problem.
## FEM for the model problem with piecewise linear
functions.
## An error estimate for FEM for the model problem.
## FEM for the Poisson equation.
## The Hilbert spaces $L_2(\Omega)$, $H^1(\Omega)$, and
$H^1_0(\Omega)$.
## A geometric interpretation of FEM.
## A Neumann problem. Natural and essential boundary
conditions.
## Remarks on programming.
# ''Week 2:'' Abstract formulation of the finite element method
for elliptic problems.
## Introduction. The continuous problem.
## Discretization. An error estimate.
## The energy norm.
# ''Weeks 3 & 4:'' Some finite element spaces and approximation theory
for FEM.
## Introduction. Regularity requirements.
## Some examples of finite elements.
## Interpolation with piecewise linear functions in two
dimensions.
## Interpolation with polynomials of higher degree.
## Error estimates for FEM for elliptic problems.
## On the regularity of the exact solution.
# ''Week 5:'' Some applications to elliptic problems.
## The elasticity problem.
## Stokes problem.
## Maxwell's equations.
# ''Weeks 6 & 7:' Solving linear systems of equations.
## Gaussian elimination. Cholesky's method.
## Condition number.
## Iterative methods: Jacobi and Gauss-Seidel.
## The conjugate Gradient method.
## Preconditioning.
## Multigrid methods.
# ''Week 8:'' FEM for parabolic problems.
## A one dimensional model problem.
## Semi-discretization in space.
## Discretization in space and time.
## Numerical solution of ODEs.
## The discontinuous Galerkin Method.
# ''Week 9:'' Boundary element methods.
## Some integral equations.
### An integral equation for an exterior Dirichlet
problem using a single layer potential.
### An exterior Dirichlet problem with double layer
potential.
### An exterior Neumann problem with single layer
potential.
### Alternative integral equation formulations.
## Finite element methods.
### FEM for a Fredholm equation of the first kind.
### FEM for a Fredholm equation of the second kind.
# ''Week 11:'' FEM for hyperbolic problems.
## Transport in one dimension.
## The wave equation in one dimension.
## The wave equation in higher dimensions.
## A finite element method.
## Error estimates and adaptive error control.
There is no TA for this course.