Beatrice Riviere

Spring 2023

    Meeting time and place:T Th 1:00pm-2:15pm, DCH 1075
    Office Hours:by appointment (send email to: riviere at rice dot edu) DH 3019
    Course Objectives
    This course covers various numerical methods for solving partial differential equations: aspects of finite difference methods, finite element methods, finite volume methods, mixed finite element methods, discontinuous Galerkin methods, and virtual methods. Both theoretical convergence and practical implementation of the methods are studied for elliptic and parabolic problems.
    Course Outcomes
    Upon completion of the course, students have a good understanding of various numerical methods including finite difference, finite element methods and finite volume methods. They will have developed their own codes for solving elliptic and parabolic equations in 1D and 2D using those methods.
    Homeworks (100%).
    Homework 1: posted on Jan 21, due Feb 2
    Homework 2: posted on Feb 7, due Feb 23
    Homework 3: posted on March 1, due March 21
    Homework 4: posted on April 5, due April 20
    Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems by Randall J. LeVeque, SIAM, 2007.
    Understanding and Implementing the Finite Element Method by Mark S. Gockenbach, SIAM, 2006.
    In general, you may discuss homework problems with classmates, but you have to write your solution individually. Some homeworks are pledged. Homeworks will contain both theoretical and computational problems. Students are strongly encouraged to start their homeworks early. Matlab, Python or C will be the default programming environment.
    Additional reading
    Numerical Analysis of Partial Differential Equations by Charles Hall and Thomas Porsching, Prentice Hall (1990).
    Sobolev Spaces, by Robert A. Adams.
    The Mathematical Theory of Finite Element Methods, by Suzanne C. Brenner and L. Ridgeway Scott, Publisher Springer.
    The Finite Element Method for Elliptic Problems, by Philippe G. Ciarlet.
    Handbook of Numerical Analysis: Volume II, Finite Element Methods by Philippe G. Ciarlet and Jacques-Louis Lions, North Holland, NY (1991).
    Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, by Beatrice Riviere, Publisher SIAM.
    Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, by Jan S. Hesthaven and Tim Warburton, Publisher Springer.
    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
    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. Late homeworks will incur penalties in increments of 10%. Homeworks that are more than one week late will not be graded.
    If you have a documented disability that may affect academic performance, you should make sure this documentation is on file with Disability Resource Center (Allen Center, Room 111 / / x5841) to determine the accommodations you need, and meet with me to discuss your accommodation needs.
    CAAM 452 and CAAM 536
    Undergraduates should sign up for CAAM 452 and graduate students should sign up for CAAM 536. The only difference between the courses is the workload and assessment of the students. Graduate students will have a higher workload (more homework problems) than the undergraduate students.
    Title IX
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