ACM 256A, Topics in Finite Element Methods
Dr. Jeff Ovall

Office:  Firestone 209
Office Hours:  MW 3-4:30, or by appointment
Email: jovall@acm.caltech.edu
Course Location:  Firestone 306
Course Times:  MWF 9-9:50
Recommended Text: Finite Elements: Theory, fast solvers, and applications in solid mechanics (3rd Ed), by Dietrich Braess

Course Description

This course is intended to provide a solid introduction to the finite element method for solving partial differential equations -- with a heavier-than-usual treatment of a posteriori error estimation and adaptive refinement. In order to do this in one quarter, we will focus on second-order, linear, elliptic PDEs. As the simplest non-trivial case, we will consider piecewise linear finite elements in two dimensions in detail -- both theory and practice are most well-developed in this case, and it provides a good starting point for further investigation. Topics to be covered (not necessarily in order) include: Data structures and the assembly and solution of the corresponding linear systems will not be ignored entirely, but will not be given as much attention as they might deserve. If time permits, we may consider some of the following: non-coercive problems, mixed methods, more exotic finite elements, or topics of interest to you.

Homework/Grading

Your grade for this course will be based on eight, equally-weighted homework assignments. Their will be no midterm or final exams.
Homework Assignments: HW1   HW2   HW3   HW4   HW5   HW6   HW7   HW8
Homework Solutions: HW1s   HW2s   HW3s   HW4s   HW5s   HW6s   HW7s   HW8s

Course Calendar

Topics listed are tentative.
Week Monday Wednesday Friday
1
Jan 7-11
Organizational Meeting
Classification of 2nd order PDEs
Well-posedness
Example finite difference, finite volume and finite element discretizations for a simple Poisson problem
2
Jan 14-18
Variational formulation for general 2nd order elliptic BVPs
Basics from Banach and Hilbert Space Theory
Coercive condition and Lax-Milgram Theorem
Inf-sup condition and B-N-B Theorem
3
Jan 21-25
Martin Luther King Jr. Day
No meeting
Basics from Sobolev Space Theory
Homework 1 due
Galerkin methods for variational problems
Cea's Lemma and quasi-optimal approximation
The finite element approach
4
Jan 28-Feb 1
Linear and quadratic Lagrange finite elements
Standard assumptions on families of triangulations
Basic A Priori error estimates
Homework 2 due
More on A Priori error estimates
5
Feb 4-8
Some elliptic regularity results
The Aubin-Nitsche "trick"
Basic data structures
Homework 3 due
Barycentric cooridinates and quadrature
6
Feb 11-15
An overview of A Posteriori error estimation and adaptive refinement
A residual-based error estimator
Quasi-interpolation
A Weighted Poincare Inequality
Homework 4 due
A hierarchical basis error estimator
7
Feb 18-22
Presidents' Day
No meeting
Gradient recovery error estimators
Homework 5 due
Superconvergence and effectivity of gradient recovery estimators
Analysis of hierarchical basis error esitmators revisited
8
Feb 25-29
Functional error estimation and duality
Elliptic eigenvalue problems
Homework 6 due
An error estimator for eigenvalue problems
9
Mar 3-7
Saddle-point problems
Two mixed formulation for the Poisson problems
Discretization of saddle-point problems
Quasi-optimal approximation property
A stable and unstable primal mixed formulation for the Poisson problem
Homework 7 due
Two stable dual mixed formulations for the Poisson problem
Raviart-Thomas elements
10
Mar 10-14
The Stokes Problem
Some Appropriate Finite Elements
Grand, sweeping overview of what we have covered
Last day of classes
Homework 8 due
Jeff Ovall Applied and Computational Mathematics Triangle (2D Mesh Generator) PLTMG (2D Finite Element Package) FETK (General Finite Element Toolkit)