Math 269C, Section 1, Spring 2017

Advanced Numerical Analysis: The Finite Element Method

Lecture Meeting Time: Mon, Wed, Fri 1-1.50pm.
Lecture Location: MS 5118.

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.1.17s/
  • Numerical Analysis Qualifying Exam
  • Getting started with MATLAB

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

    Grading Policy: Homework 75%; Projects 25%.

    Homework Assignments, Projects & Practice Problems:

    HW #1 (due on Friday, April 14) HW1.pdf HW1.tex

    HW #2 (due on Monday, April 24) HW2.pdf HW2.tex

    Computational Project 1 (due on Monday, May 1st) Project1.pdf Project1.tex

    Useful Results

    HW #3 (due on ) HW3.pdf

    Computational Project 2 (due on ) Project 2

    HW #4 (due on ) HW4.pdf

    HW #5 (due on ) HW5.pdf

    ADDITIONAL NOTES AND PRACTICE PROBLEMS
  • Error Estimates
  • Theoretical notes
  • Two problems with partial solutions
  • Practice problems
  • Evolution problems