Instructor: Luke Olson
Time: T/Th 2:00-3:15pm
Class Location: 1103 Siebel Center
Office Hours: T/Th 3:00-5:00pm
Office: 4312 SC Phone: 244-8422 (x4-8422)
email: lukeo 
uiuc.edu
| Presenter | Date | Topic |
|---|
HW1 :: [.pdf]
HW2 :: [.pdf]
HW3 :: [.pdf]
HW4 :: [.pdf]
HW5 :: Read the following papers for 2005-11-15 and
2005-11-17:
Unified
Analysis of Discontinuous Galerkin Methods for Elliptic
Problems by Arnold, Brezzi, Cockburn, Marini
Discontinuous
Galerkin Methods for the Time-Domain Maxwell's Equations: An
Introduction by Hesthaven and Warburton.
HW 6 :: USEMe code
(at Rice)
Code from class: DG_Matlab_Helmholtz.tgz
Continuous Galerkin 2-D code for the Helmholtz problem is here: 2d_helmholtz.tgz
nodefiles.tgz
Comment on HW4: see
Warburton's paper for the warb&blend method
Hesthaven's paper for the electrostatic method
Comment on HW2: for jacobiderivative1d, the suggested theorems in the homework are fine, however a more condensed version is available if you consider equation (4.5) in the handout. Differentiating (4.5) yields dP^{a,b}_{n}/dx depending on only one term: P^{a+1,b+1}_{n-1}. Give it a try...
| Lecture | Date | Topic | Reference |
|---|---|---|---|
| 1 | August 25, 2005 | Motivation,History | [1-7] |
| 2 | August 30, 2005 | Why High-Order with Finite Differences | [6] |
| 3 | September 01, 2005 | Fourier Method and Finite Differences | [6] |
| 4 | September 01, 2005 | Variational Frameworks | [2] |
| 5 | September 08, 2005 | FEM, SEM, Weighted Residuals, Collocation | [2,1] |
| 6 | September 13, 2005 | The Sturm-Liouville Problem | none |
| 7 | September 20, 2005 | Jacobi Polynomials and Quadrature points | [3,1] |
| 8 | September 22, 2005 | Nodal versus Modal Representation | [3,1] |
| 9 | September 27, 2005 | Vandermonde Representation | [3,1] |
| 10 | September 29, 2005 | FEM/SEM details | [1],notes:[.pdf] |
| 11 | October 04, 2005 | Full 1-D SEM Code | [1,3,4],code,notes:[.pdf] |
| 12 | October 06, 2005 | 1-D Convergence Theory | [1 § 2.5],notes:[.pdf] |
| 13 | October 11, 2005 | 2-D tensor basis on quads, fast implementation | [1 § 3.1, 2 § 4], notes: [.pdf] [.odp] |
| 14 | October 13, 2005 | 2-D tensor basis on triangles, collapsing | [1 § 3.2] |
| 15 | October 18, 2005 | 2-D basis on triangles, Hesthaven and Fekete nodes | [1 § 3.3] |
| 16 | October 20, 2005 | 2-D basis on triangles, Hesthaven and Fekete nodes | [1 § 3.3] |
| 17 | October 25, 2005 | 2-D basis on triangles, Hesthaven and Fekete nodes | [1 § 3.3], notes: [.pdf] [.odp] |
| 18 | October 27, 2005 | Global Computational Operations | [1 § 4.2] |
| 19 | November 01, 2005 | Global Computational Operations | [1 § 4.2] |
| 20 | November 03, 2005 | Global Computational Operations | [1 § 4.2, notes] |
| 21 | November 08, 2005 | Discontinuous Galerkin | [1 § 7.5, notes] |
| 22 | November 10, 2005 | Discontinuous Galerkin | [1 § 7.5, notes] |
| 23 | November 15, 2005 | Discontinuous Galerkin Implementation | [1 § 7.5, notes] |
| 24 | November 17, 2005 | Discontinuous Galerkin Implementation | [notes] |
| 25 | November 29, 2005 | Discontinuous Galerkin Implementation | [notes] |
| 26 | December 01, 2005 | presentation | |
| - | December 06, 2005 | presentation | N/A |
| - | December 08, 2005 | presentation | N/A |
→ Opening Syllabus
[.odt]
[.pdf]
→ High-Order book review (SIAM)
[.html]
→ Selected lecture notes
Lecture 10:[.pdf]
Lecture 11:[.pdf]
Lecture 12:[.pdf]
Lecture 13:[.pdf]
[.odp]
→ Lecture 11: 1-D SEM. Simple Poisson. Dirichlet B.C.
Compressed files: [.tgz] [.zip]
Individual files:
build_operator.m
func_rhs.m
func_u.m
gll_data.m
global_connect.m
highorder_interp.m
jacobi1d_derivative.m
jacobi1d_Dhat.m
jacobi1d_D.m
jacobi1d_gamma.m
jacobi1d.m
jacobi1d_vdm.m
jacobi1d_zeros.m
main.m
mesh_data.m
ref_data.m
wrapper.m
→ Lecture 13: Plot the shape modes on quads in 2-d:
modal.m
September 27 UPDATE: Books are in at the bookstore...
Primary Text:
→ [1]
"Spectral/hp Element Methods for CFD" by Karniadakis and
Sherwin, **Second Edition**
→
[Publisher's Link]
→
(this text is comprehensive and
well-devloped both in theory and in computational detail)
Helpful Reference Texts:
→ [2]
"High-Order Methods for Incompressible Fluid Flow" by Deville,
Fischer, Mund
→
[Publisher's Link]
→
(this text gives a very good
overview of high-order methods with application to a large number of
problems. very nice read.)
→ [3]
"Finite and Spectral Element Methods using Matlab" by Pozrikidis,
→
[Publisher's Link]
→
(as the name suggests, this text
is a useful reference for certain computational aspects)
→ [4]
[Publisher's Link]
→
"High-Order Finite Element Methods" by Solin, Segeth, and Dolezel
→
(very solid reference for a FE framework to high-order methods)
→ [5]
[Publisher's Link]
→
"A Practical Guide to Pseudospectral Methods" by Fornberg
→
(excellent reference. very practical.)
other notes:
→ [6] "High-Order Nodal Methods" soon...: [.pdf]
→ [7]
[Author's Link]
→
"Chebyshev and Fourier Spectral Methods" by Fornberg
→
(Huge amount of information)
In this course, we will take a comprehesive look at high-order
approximation schemes. FOr many problems that arise in areas such
as aerodynamics, chemistry, oceanography, electormagnetics, and
more, basic finite difference and finite element methods cannot
effectively and efficiently handle the underlying the physics. As
such, high-order multidomain methods have emerged as prominent
contenders in resolving a wide variety of problems.
The goal of this coarse is to provide a framework for further
exploration in high-order methods. We will highlight the
advantageous properties of several approaches including collocation
shcemes, hp finite element methods, and constinuous and
discontinuous spectral element techniques. Fundamental building
blocks such as orthogonal polynomials will be studied along with
more computational topics that include modal versus nodal arguments,
quadrature considerations, triangles-quadrilaterals, and also
effective solution techniques (e.g. domain decomposition).
Parts of this class may be covered briefly in other courses, such as Numerical Analysis, Iterative Methods, or courses on Finite Elements and Numerical PDEs. A course in Numerical Analysis is essential with some exposure to finite difference or finite element approximations. There will be a large computational component in the homework and final project, so programming in Matlab, C, or C++ will be required. A basic knowledge of PDEs is also helpful, since this is a course on Numerical PDEs.