Welcome to the class website for Finite Element Methods for Partial Differential Equations. Finite elements provide a class of numerical methods for approximating the solutions of partial differential equations. In this course we will cover both fundamental mathematical concepts and in practice how to develop and apply finite element methods to specific problems. We will develop methods for Elliptic, Parabolic, and Hyperbolic PDEs as well as for non-linear problems.
Please be sure to read the prerequisites and grading policies for the class. Also see the syllabus for more details.
Prerequisites:
A working knowledge of advanced calculus, linear algebra, and partial differential equations will be assumed.
| Slides | ||
|---|---|---|
| (please submit any typos here) | ||
| - | Introduction to FEM and Ritz-Galerkin Approximation | [PDF] [GoogleSlides] |
| - | Finite Element Spaces | [PDF] [GoogleSlides] |
| - | Sobolev Spaces | [PDF] [GoogleSlides] |
| - | Variational Formulations and Elliptic PDEs | [PDF] [GoogleSlides] |
| - | Finite Element Approximation Properties and Convergence | [PDF] [GoogleSlides] |
| - | Elasticity Theory | [PDF] [GoogleSlides] |
| - | Mixed Methods | [PDF] [GoogleSlides] |
| - | Elasticity Theory: Numerical Example | [PDF] [GoogleSlides] |
Supplemental Materials:
- Python General Tutorial | Python tutorial at Codecademy | Python 3.7 documentation
- Anaconda Python Environment | Virtual Environment
- Numpy Python Package Tutorial | Jupyter Notebooks: Python Interface
- Example Python Code:
- Neville's Method: [PDF] [Python Code] [Jupyter Notebook]
Class Annoucements:
- The Mathematical Theory of Finite Element Methods (third edition) by S. Brenner & R. Scott, available on-line from UCSB at [link].
- Additional background: A First Course in Finite Elements, J. Fish and T. Belytschko, [link].
- A book that you might find helpful for background on Real Analysis is Analysis, by Lieb and Loss [link].
Homework Assignments:
Turn all homeworks in by Canvas by 5pm on the due date.
Numbered exercises are labelled as follows:
(BRS): The Mathematical Theory of Finite Element Methods (third edition) by S. Brenner & R. Scott [link].
(JFB): A First Course in Finite Elements by J. Fish & T. Belytschko [link].
HW1: (Due Fri, Oct 4) (JFB) Ch3: 1,9,14,15; (BRS) Ch0: 1,3,9.
HW2: (Due Fri, Oct 11) (JFB) Ch4: 3,6,8; (BRS) Ch0: 11,12,14.
HW3: (Due Fri, Oct 18) Numerical exercises [PDF].
HW4: (Due Fri, Nov 1) (JFB) Ch5: 1,2,15; Ch6: 2,4; (BRS) Ch3: 3,10,13; Ch 5: 7, 14.
HW5: (Due Fri, Nov 15) (JFB) Ch7: 1,2,4; Ch8: 1,2,4; (BRS) Ch 5: 1,2,8,10.