syl_dg

Announcements , Course Information , Outline , Homework and Project , Matlab Tutorial , Sample code , Internet Resources

Numerical Partial Differential Equations II:
Discontinuous Galerkin Methods

Announcements:

5/11: I put another code for wave equations HERE.

The new package can calculate the error and rate of convergence. The new one is modified from the old one and is transformed to fortran 90. However there are only a few comments in the code and it is a little difficult to understand it. In the unzipped folder, you can find a file named "inputsample" which tells you how to input the key information. Please answer "1" to the question "boudary treatment 0:No, 1:test1".

5/6: Homework 3 is HERE.

5/6: Due to the modification of the schedule of the "Biweekly Seminaron of Optimal Control", the classes on Monday will be 5/8, 5/22, 6/12.

3/30: You can use "polyfit" to fit the data and find the slope (order).

For example,

T=log10(n)
% n is the number of the intervals
[c,d]=polyfit(T,log10(err_lvl),1)
% err_lvl is the array of errors
set(axes,'fontsize',18)
hold on
plot(T,log10(err_lvl),'s');
err_fit=c(1)*T+c(2);
plot(T,err_fit);
xlabel('-log_{10} N');
ylabel('log_{10} err');
hold off

2/20: There is no class today (2/20). I put some sample code on my website. The code is written in Fortran 77. Please download the code and try to understand it.

top

Time and Place: M. 16:10-18:00(每單週的週一); T. 15:10-17:00, Math Building 3175

(為了與王辰樹老師的seminar錯開, 週一上課日期預定為3/13, 3/27, 4/17, 5/1, 5/8, 5/22, 6/12)

Lecturer: Min-Hung Chen
Office: Math Building Rm. 404; Ph. 2757575 Ext 65127
e-mail: mhchen@math.ncku.edu.tw
Office Hours: by appointment.
Web: http://www.math.ncku.edu.tw/~mhchen

Grades will be based upon: In-class participation 10%, Homework 30% and Final Project (Oral presentation and 10-page report) 60%

Text:

A short essay on discontinuous Galerkin methods:
Discontinuous Galerkin methods, ZAMM. Z. Angew. Math. Mech. 83 (2003), no. 11, 731-754.
Lecture Notes by Bernardo Cockburn:
"An Introduction to the Discontinuous Galerkin method for Convection-Dominated Problems"

References:

Finite Difference Schemes and Partial Differential Equations by John Strikwerda
The mathematical theory of finite element methods by Susanne Brenner and Ridgway Scott
The finite element method for elliptic problems by Philippe Ciarlet
Matrix computations by Charles Van Loan and Gene Golub
Numerical solution of partial differential equations by the finite element method by Claes Johnson
Introduction to numerical analysis by Josef Stoer and Roland Bulirsch
Numerical Methods for Conservation Laws by Randall LeVeque
Finite Volume Methods for Hyperbolic Problems by Randall LeVeque
Discontinuous Galerkin Methods, Lecture Notes in Computational Science and Engineering Vol. 11
Lecture Notes by Onno Bokhove and Jaap J. W. van der Vegt
"An Introduction to (dis)continuous Galerkin finite element methods"
The first paper about discontinuous galerkin methods-
Triangular mesh methods for the neutron transport equation by Reed and Hill

Extended Reading:

QUADRATURE-FREE IMPLEMENTATION OF THE DISCONTINUOUS GALERKIN METHOD FOR HYPERBOLIC EQUATIONS by Harold Atkins Chi-Wang Shu

Efficient Implementations of the Quadrature-Free Discontinuous Galerkin Method by David Lockard and Harold Atkins

A DG method for two-dimensional flow and transport in shallow water

top

Online Resources: Search Engine:

http://scholar.google.com.tw/

Lecture Notes and Slides:

Legendre Polynomials:

Matlab References:

top

Course Outline

Introduction
Ordinary differential equations
Linear Problems:
- The transport equation
- The wave equation
- Second-order elliptic problems
- The Oseen system
Convection-Dominated Problems
- The scalar conservation law in one space dimension
- The RKDG method for multimensional systems
- Convection-diffusion problems: The LDG method
The RKDG method for Maxwell's equations