Announcement

  • Information regarding the rest of the semester can be found on MyCourses and on Facebook.
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    Assignments

  • Assignment 1 [tex] due Thursday January 30
  • Assignment 2 [tex] due Tuesday February 18
  • Assignment 3 [tex] due Tuesday March 31
  • Lab assignment 1 due Friday April 3, 18:00 EDT
  • Assignment 4 [tex] due Tuesday April 14, 18:00 EDT
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    Class schedule

  • TR 13:05–14:25, Burnside Hall 920

    Date Topics
    T 1/7 Hilbert space method. Sobolev spaces. Strong derivative. Weak formulation.
    R 1/9 Weak derivative. Poisson problem. Friedrichs inequality.
    T 1/14 Boundary conditions. Variational and operator formulations. Ritz-Galerkin framework.
    R 1/16 Céa's lemma. Fourier-Galerkin methods.
    T 1/21 Finite elements in 1D.
    R 1/23 Finite elements in 2D. Averaged Taylor polynomials.
    T 1/28 Error of averaged Taylor polynomials. Truncated Riesz potential.
    R 1/30 Bramble-Hilbert lemma. Nodal interpolation for Lagrange finite elements.
    T 2/4 Mesh refinement. Discontinuous elements. Quasi-interpolation.
    R 2/6 Inverse estimates. Lp-stability. Conditioning.
    T 2/11 K-functionals.
    R 2/13 Real interpolation. Interpolation of Lebesgue spaces.
    T 2/18 Interpolation of Sobolev spaces.
    R 2/20 Approximation spaces.
    T 2/25 Abstract approximation theory.
    R 2/27 Application to finite elements. Zygmund spaces.
    3/2–3/6 Study break
    T 3/10 Besov spaces. Inf-sup conditions.
    R 3/12 Saddle point problems
    T 3/17 Class cancelled
    R 3/19 Class cancelled
    T 3/24 Class cancelled
    R 3/26 Class cancelled
    T 3/31 Petrov-Galerkin methods. Fortin operators.
    R 4/2 Finite elements for the Stokes problem. Raviart-Thomas elements.
    T 4/7 Variational crimes. Numerical quadrature.
    R 4/9 Nitsche's boundary penalty method.

    Online resources

  • Related courses: Math 579 Winter 2010, Math 765 Fall 2013
  • Lecture notes on FEM by Ronald Hoppe (Houston)
  • Lecture notes on FEM by Endre Süli (Oxford)
  • Course notes on basic FEM by Joseph E. Flaherty (RPI)
  • Lecture notes by Volker John (Berlin)
  • Lecture notes by Greg Fasshauer (IIT)
  • Lecture notes by Pascal Frey (Sorbonne)
  • FEM course at IC London
  • FEM sofware by Carsten Carstensen
  • FreeFEM project
  • FEniCS project
  • FETK - The Finite Element ToolKit
  • ALBERTA - An adaptive hierarchical finite element toolbox

    • Reference books

  • Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods. Springer 2007
  • Gilbert Strang and George Fix, An analysis of the finite element method. Wellesley-Cambridge 2008
  • Philippe G. Ciarlet, The finite element method for elliptic problems. SIAM 2002
  • Dietrich Braess, Finite elements: Theory, fast solvers and application in solid mechanics. Cambridge 2007

    • Course outline

    Instructor: Dr. Gantumur Tsogtgerel

    Prerequisite: MATH 475 and MATH 387 or permission of the instructor

    Note: If you plan to take this course without taking MATH 578, please consult with the instructor.

    Topics: The main focus of the course is going to be on mathematical analysis of finite element methods. If time permits, topics on adaptivity will be included.

    Calendar description: Numerical solution of initial and boundary value problems in science and engineering: ordinary differential equations; partial differential equations of elliptic, parabolic and hyperbolic type. Topics include Runge Kutta and linear multistep methods, adaptivity, finite elements, finite differences, finite volumes, spectral methods.

    Homework: Both analytical and computational. Assigned and graded roughly every two weeks.

    Course project: The course project consists of the student studying an advanced topic, implementing the relevant algorithms, experimenting, writing a report, and giving a presentation.

    Grading: Homework assignments 50% + Course project 50%.