Math 7875-008: Selected Topics in Numerical Methods for PDE

Instructor: Yekaterina Epshteyn

Lectures: TH 10:45 - 12:05 pm, JWB 333

Office Hours (tentative, it may be some changes)

Thursday 12:10 - 1:50pm, or by appointment
Office: LCB 337
E-mail: epshteyn@math.utah.edu

Prerequisites: Math 6610 and Math 6620 (or equivalent graduate courses in Numerical Analysis).

Some References:

Dietrich Braess, Finite elements, Third Edition, Cambridge

Mats G. Larson and Fredrik Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Springer

Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM

Victor S. Ryaben'kii and Semyon V. Tsynkov, A Theoretical Introduction to Numerical Analysis, Chapman & Hall/CRC

Victor S. Ryaben'kii, Method of Difference Potentials and Its Applications, Springer Series in Computational Mathematics, 2002

Eitan Tadmor, A Review of Numerical Methods for Nonlinear Partial Differential Equations, pdf 

Kendall Atkinson, An Introduction to Numerical Analysis, Wiley

Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations, Second Edition, Cambridge University Press

E. Suli and D. Mayers, An Introduction to Numerical Analysis, Cambridge 2003

Stig Larsson and Vidar Thomee, Partial Differential Equations with Numerical Methods, Texts in Applied Mathematics, Springer

The Course

Preliminary/Tentative Examples of the Topics:   Finite Difference Methods (FD), Difference Potentials Methods (DPM) and Finite Element Methods (FEM) for the elliptic and parabolic problems will be the main emphasis of the course. Accuracy, stability, and efficiency of the algorithms will be studied from both theoretical and computational standpoints. Some applications to problems from Fluid Dynamics and Material Sciences will be considered as well.

7875-008 Tentative Plan (can be adjusted based on the course progress):

2) Prior to meetings:  Students are encouraged to read the suggested material, and to be prepared to discuss it during meetings.

3) Exercises:  Several exercises/questions (theoretical, computational or both) during series of the lectures will be given. Students will be encouraged to work on these exercises (this will help to understand the covered material better). Students will submit their solutions to only selected questions (students will usually have two or three weeks to work on the selected questions).

I will give my feedback (comments and suggestions) on the submitted solutions to these selected questions, but I will not assign/give a grade for these exercises.

4) Research papers:  Students will form teams and will be assigned to read research papers related to the topics of the course. Students will give lectures in class based on the papers that they read.

5) Computational work:  Computations/Numerical simulations during the course can be done using MATLAB, software produced by The MathWorks. The Matlab language provides extensive library of mathematical and scientific function calls entirely built-in. Matlab is available on Unix and Windows. The full set of manuals is on the web in html format. The "Getting Started" manual is a good place to begin and is available in Adobe PDF format.

Another good source is the book by Cleve Moler "Numerical Computing with Matlab" available here

ADA Statement

Evaluation and Course Work: Course participation (work on the selected exercises, participation and attendance of the lectures) 70%, and research paper presentations 30%

Letter Grade Information (More or Less Standard) 

Note: Updates based on the course progress will be posted on class website during semester. Students are responsible for checking it regularly.