MATH 578: Numerical Partial Differential Equations

* Notes on all methods

• Introduction to Computational PDEs by de Sterck, Ullrich. link to pdf

* Finite Difference Methods

• Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, by LaVeque SIAM
• Numerical Solution of Partial Differential Equations, by Morton and Mayers Cambridge

* Finite Volume Methods

• Finite Volume Methods for Hyperbolic Problems, by LeVeque SIAM

* Finite Element Methods

• Theory, Fast Solvers, and Applications in Solid Mechanics, by BraessCambridge
• The Mathematical Theory of Finite Element Methods, Third edition, by Brenner and Scott Springer
• Numerical Solution of Partial Differential Equations by the Finite Element Method, by JohnsonDover
• The FEniCS Book, Automated Solution of Differential Equations by the Finite Element Method, eds. Logg, Mardal, and Wells link to pdf

Course Description:   This course covers the basics of finite difference schemes, finite volume schemes, and finite element methods. Additional topics (e.g. a posteriori error estimation, discontinuous Galerkin methods, etc) may be covered based on student interest and time constraints.You do not need to be an expert in PDEs or in coding. But you should have a course in numerical analysis as your background, be comfortable with differential equations, and have some coding experience. We'll be covering numerical methods for parabolic, hyperbolic and elliptic equations. We'll discuss the mathematical background of the numerical methods as well as its implementation.

Computation

*Python

I'll be using Python for the examples in class. The course homeworks and examples in class will be in Python with the libraries numpy, scipy and matplotlib for assignments. Python has a very gentle learning curve, so you should feel at home even if you've never done any work in Python. While it is possible to do the finite difference HW's in MATLAB, I highly recommend you do these in Python, since Python will be needed once we move to the Finite Element package Dolfin from the FEniCS Project.

*Python and Numpy Help

• Python tutorial (https://docs.python.org/3/tutorial/index.html)
• Facts and myths about Python names and values (http://nedbatchelder.com/text/names.html)
• Dive into Python 3 (http://www.diveinto.org/python3/)
• Project Euler (https://projecteuler.net/problems) (Lots of practice problems)
• Introduction to Python for Science (https://github.com/djpine/pyman)
• The SciPy lectures (http://scipy­lectures.github.io/)
• The Numpy MedKit (http://mentat.za.net/numpy/numpy_advanced_slides/)
• The Numpy User Guide (http://web.mit.edu/dvp/Public/numpybook.pdf) by Travis Oliphant
• Numpy/Scipy documentation (http://docs.scipy.org/doc/)
• More in this reddit thread (http://www.reddit.com/r/Python/comments/1lgxbf/best_tutorial_to_learn_numpy/)
• Spyder (https://github.com/spyder­ide/spyder) (a Python IDE, like Matlab)
• An introduction to Numpy and SciPy (http://www.engr.ucsb.edu/~shell/che210d/numpy.pdf)
• 100 Numpy exercises (http://www.loria.fr/~rougier/teaching/numpy.100/index.html)

*Virtual Machine

Virtual Box Image Username: mathgeeks Password: abc123

Grading:  
45% Homework
45% Projects
10% Participation

Homeworks:
Homework 0
Homework 1 Due: September 13
Update 1: Sept 2nd. Added domain for Problem 1. Added a bonus problem. Solution Code
Homework 2 Due: September 27 September 29
Update 1: Sept 20th. Fixed missing negative sign on RHS in Problem 5.
Update 2: Sept 26th. Due Date changed to Sept 29. parabolic_non_linear.py  cn_2d_neumann.py
Homework 3 Due: October 11
Update 1: Oct 6th. Fixed sign issue in problem 3, equation numbered 2. hw3_fd hw3_fv
Homework 4 Due: November 3
Homework 5 Due: November 24   elements.dat  vertices.dat  dom.xml  dom_physical_region.xml

Projects:
Midtem Project Due: November 8. You can work in groups of 2.  Addendum
Final Project Due: December 13. You can work in groups of 2. 

Syllabus  

Sample files/ Class Examples

numpy_matplotlib_example   explicit scheme for heat equation  explicit scheme for heat equation with mat-vec operations  Method of Lines Example 
1D FEM  1D FEM in Dolfin  1D FEM mesh (needed for Dolfin example) 

Handwritted and probably illegible notes: 1-3  4-6  FEM-1  FEM-2 

Final Project Presentations: December 13

Time	Title	Presenter(s)
12:30	Stokes Flow through a Backward Facing Step	Robert Malakhov
12:45	Adjoint Based Error Estimation and Mesh Refinement	Juan Diego Colmenares Fernandez
1:00	Using proper orthogonal decomposition to obtain solution from a parametrized partial differential equation	Joel Upston
1:15	Comparison of Least Squares and Mixed Method for Stationary 2D Convection-Diffusion Equation	Shu Wang and Anastassiya Semenova
1:30	Clement's Interpolation	Adam Frederickson
1:45	Hermite interpolation method on the overset grids	Oleksii Beznosov
2:00	Petro-Galerkin 3-Dimensional Burgers Equation	Brad Philipbar