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B. Riviere


CAAM 552 FINITE ELEMENT METHODS

Fall 2009

    Meeting time and place:T Th 2:30pm-3:50pm, KH 101
    Office Hours:T Th 1:30pm-2:30pm and by appointment (send email) DH 3022
    Content
    This course addresses the theory and implementation of finite element methods. Topics include weak solutions of partial differential equations, Sobolev spaces, approximation theory, convergence and reliability of the numerical methods. Continuous and discontinuous finite element methods are considered.
    Grades
    Homeworks (60%), exam due November 6 (20%) and paper review (20%)
    Textbooks
    The Mathematical Theory of Finite Element Methods, by Suzanne C. Brenner and L. Ridgeway Scott, Publisher Springer.
    Homeworks
    In general, you may discuss homework problems with classmates, but you have to write your solution individually. Some homeworks are pledged. The exam is pledged and individual.
    Homework 1: due September 3
    Homework 2: due September 15
    Homework 3: due October 6
    Homework 4: due October 27
    Homework 5: due December 3
    Lectures
    Lectures 1-2: Introduction to 1D weak solution, Galerkin method
    Lectures 3-4: Hilbert spaces, Lax-Milgram
    Lectures 5-7: Sobolev spaces
    Lectures 8-10: Variational formulations, regularity of elliptic problems
    Lectures 11-12: Meshes, finite elements, interpolants
    Lectures 13-16: Approximation error, reference elements
    Lectures 17-18: A posteriori error estimation
    Lecture 19: Effect of numerical integration
    Lecture 20: Discontinuous Galerkin method
    Lecture 21: Mass conservation (global v. local)
    Lecture 22: Implementation
    Lectures 23-24: Mixed methods
    Lectures 25-27: Paper reviews
    Additional reading
    Sobolev Spaces, by Robert A. Adams.
    The Finite Element Method for Elliptic Problems, by Philippe G. Ciarlet.
    Understanding and Implementing the Finite Element Method by Mark S. Gockenbach, Publisher SIAM, 2006.
    Handbook of Numerical Analysis: Volume II, Finite Element Methods by Philippe G. Ciarlet and Jacques-Louis Lions, Publisher North Holland 1991.
    Theory and Practice of Finite Elements by Alexandre Ern and Jean-Luc Guermond, Publisher Springer 2004.
    Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, by Beatrice Riviere, Publisher SIAM, 2008.
    Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, by Jan S. Hesthaven and Tim Warburton, Publisher Springer, 2007.
    Class web site Students are responsible for viewing the class web site regularly as material will be added to the site throughout the semester.
    Late policy There are no late homeworks. Late homeworks will incur penalties in increments of 10%. Homeworks are to be given during class on the due date. If the homework is turned in after the class is over, it is considered late.
    Disability Any student with a documented disability requiring accomodations in this course is encouraged to contact me after class or during office hours. Additionally, students also need to contact Disability Support Services in the Allen Center.