## Math 269C, Section 1, Spring 2016

## Advanced Numerical Analysis: The Finite Element Method

**Lecture Meeting Time:** Mon, Wed, Fri 3-3.50pm.

** Lecture Location:** MS 5138.

**Instructor:** Luminita A. Vese

**Office:** MS 7620D

**Office hours:** TBA (after the lecture, or by appointment).

**E-mail:** lvese@math.ucla.edu

**Textbook:**
(placed on reserve for 2 hours/overnight at SEL Library).

Claes Johnson, *Numerical solution of partial differential equations by
the finite element method*, Cambridge University Press, 1987 (or the 2nd edition, 2009).

**Other references:**

O. Axelsson and V. A. Barker, *Finite Element Solution of Boundary Value Problems: Theory and Computation*, Academic Press, London, 1984.
Braess, D. *Finite elements. Theory, fast solvers, and applications in solid mechanics.*
Translated from the 1992 German original by Larry L. Schumaker. Cambridge University Press, Cambridge, 1997.
S. C. Brenner and L. R. Scott, *The Mathematical Theory of Finite Element Methods*, Springer-Verlag, 1996.
F. Brezzi and M. Fortin, *Mixed and Hybrid Finite Element Methods*, Springer Series in Computational Mathematics, Vol 15, Springer-Verlag, New York, 1991.
Ciarlet, P.G. *The finite element method for elliptic problems.*
Studies in Mathematics and its Applications, Vol. 4. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.
P. G. Ciarlet and J. L. Lions, *Handbook of Numerical Analysis*, Vol. II, Finite Element Methods (Part I), North-Holland, 1991.
Girault, V., Raviart, P.-A. *Finite element methods for Navier-Stokes equations. Theory and algorithms.*
Springer Series in Computational Mathematics, 5. Springer-Verlag, Berlin-New York, 1986.
Girault, V., Raviart, P.-A. *Finite element approximation of the Navier-Stokes equations.* Springer-Verlag, Berlin-New York, 1981.
T. J. R. Hughes, *The Finite Element Method, Linear Static and Dynamic Finite Element Analysis*, Prentice-Hall, Englewood Cliffs, New Jersey, 1987, Dover, 2000.
A. R. Mitchell and R. Wait, *The Finite Element Method in Partial Differential Equations*, John Wiley & Sons, Ltd, 1977.
Pironneau, O. *Finite element methods for fluids.*
Wiley, New York; Masson, Paris, 1989.
H. R. Schwarz, *Finite Element Methods*, in Computational Mathematics and Applications, Academic Press, 1988.
W. G. Strang and G. J. Fix, *An Analysis of the Finite Element Method*, Wellesley Cambridge Press, 1973.
B. Szabo and I. Babuska, *Finite Element Analysis*, John Wiley & Sons, 1991.
R. Temam, *Navier-Stokes Equations, Theory and Numerical Analysis*, 3rd ed., North-Holland, 1984.
V. Thomee, *Galerkin Finite Element Methods for Parabolic Problems*, Springer Series in Computational Mathematics, Vol. 25, Springer Verlag, 1997.
O. C. Zienkiewicz, *The Finite Element Method*, 3rd ed, McGraw-Hill, New York, 1977.
O. C. Zienkiewicz and K. Morgan, *Finite Elements and Approximation, John Wiley & Sons, 1983.
**
O. C. Zienkiewicz and R. L. Taylor, **The Finite Element Method: Volume 1, The Basis*, Butterworth-Heinemann, 2001.
*
O. C. Zienkiewicz and R. L. Taylor, **The Finite Element Method: Volume 2, Solid Mechanics*, Butterworth-Heinemann, 2001.
*
O. C. Zienkiewicz and R. L. Taylor, **The Finite Element Method, Volume 3, Fluid Mechanics*, Butterworth-Heinemann, 2001.
* J. Tinsley Oden, Graham F. Carey, ** The Texas Finite Element Series,
Finite Elelemnts*, Vols: I-VI, Prentice Hall.
* H. Attouch, G. Buttazzo, and G. Michaille, **Variational Analysis in Sobolev and BV Spaces: applications to PDE's and optimization*, MPS-SIAM 2006.

**Enforced Requisites:** courses 115A, 151A, 151B.

**Useful Links:**
*PIC Lab: Boelter Hall 2817 and
Mathematical Sciences 3970
*

http://www.pic.ucla.edu/piclab/
*
MATLAB Documentation (thanks to Prof. C. Anderson, UCLA)
**
Class Web Page: http://www.math.ucla.edu/~lvese/269c.1.16s/
**
Numerical Analysis Qualifying Exam
** Getting started with MATLAB
*

**Assignments Policy:**
There will be several homework assignments on theoretical questions, and two
computer projects.

**Examinations:**
The last homework will be considered as a take-home final exam.

**Grading Policy:**
Homework and Projects 80%, Final 20%

**Homework Assignments, Projects & Practice Problems:**

**HW #1** (due on Wednesday, April 20)
__
HW1.pdf__ __
HW1.tex__

**HW #2** (due on Wednesday, May 4)
__
HW2.pdf__ __
HW2.tex__

**Computational Project 1** (due on Monday, May 9)
__
Project 1__
__
Project 1 latex file __

__
Useful Results__

**HW #3** (due on Friday, May 20)
__
HW 3__
__
HW3.tex__

**Computational Project 2** (due on Friday, June 3rd)
__
Project 2__
__
Project2.tex__

**HW #4** (due on Friday, June 3rd)
__
HW4.pdf__
__
HW4.tex__

**HW #5** (due on: TBA)
__
HW 5__
__
HW5.tex__

**ADDITIONAL NOTES AND PRACTICE PROBLEMS**

__
Error Estimates__
__
Theoretical notes__
__
Two problems with partial solutions__
__ Practice problems __
__ Evolution problems __