Final project

The presentations will take place on Monday December 9, starting at 12:00 in Burnside Hall 1120.


  • Assignment 1 [tex] due Friday September 20
  • Assignment 2 [tex] due Wednesday October 9
  • Assignment 3 [tex] due Wednesday November 6

    Class schedule

  • WF 11:35–12:55, Burnside Hall 1205

    Date Topics
    W 9/4 Bounded below operators. Inf-sup conditions. Bilinear forms. Lax-Milgram lemma [notes]
    F 9/6 Examples of variational problems. Petrov-Galerkin methods. Operator viewpoint [notes]
    W 9/11 Fourier-Galerkin methods and finite element methods in a simple setting [notes]
    F 9/13 Examples of finite element spaces. Finite elements [notes]
    W 9/18 Averaged Taylor polynomials. Sobolev's inequality [notes]
    F 9/20 Bramble-Hilbert lemma. Nodal interpolation for Lagrange finite elements [notes]
    W 9/25 Mesh refinement. Inverse estimates [notes]
    F 9/27 Lp-stability and conditioning [notes]
    W 10/2 Peetre's K-method of interpolation. K-functionals [notes]
    F 10/4 Interpolation between C and Lip, and between C and C1. Hölder-Lipschitz spaces [notes]
    W 10/9 Interpolation of operators. Moduli of continuity. Nikolsky-Besov spaces
    F 10/11 Simple observations on Besov spaces. Interpolation between Lp and W1,p
    W 10/16 Slobodecky norms. Approximation by trigonometric polynomials
    F 10/18 Zygmund spaces. Jackson and Bernstein theorems
    W 10/23 Moduli of smoothness. Marchaud inequality. Higher order Besov spaces
    F 10/25 Guest lecture by prof. Legrand
    W 10/30 Trigonometric approximation in Lp. Hardy inequalities
    F 11/1 Abstract approximation theory. Approximation spaces
    W 11/6 Representation theorems
    F 11/8 Quasi-interpolation operators
    W 11/13 Residual based a postieriori error estimators. Global upper bound
    F 11/15 Global lower bound. Oscillation
    W 11/20 Geometric error reduction
    F 11/22 Local discrete upper bound. Dörfler marking
    W 11/27 Convergence rates of adaptive finite element methods
    F 11/29 Complexity of local mesh refinements

    Course outline

    Instructor: Dr. Gantumur Tsogtgerel

    Topics: The primary goal of this course is to discuss some of the spectacular results that have been achieved during the last decade in the theory of adaptive finite element methods. However, more time will be spent on the necessary background material and various digressions from there. In particular, we will give solid introductions to finite element methods, the fundamentals of approximation theory, and adaptivity. An emphasis will be placed on a systematic application of harmonic analysis techniques. More precisely, the planned topics are

  • Linear elliptic problems, LBB conditions
  • Galerkin methods, finite elements, splines, wavelets
  • Polynomial approximation theory, direct and inverse estimates
  • Besov spaces and interpolation theory
  • Stability and conditioning of finite elements
  • Multilevel preconditioning
  • Wavelet characterization of function spaces
  • Nonlinear approximation
  • Optimally convergent adaptive wavelet methods
  • Residual based a posteriori error estimators
  • Mesh refinement procedures
  • Convergence rates of adaptive finite element methods
  • Complexity analysis
  • Eigenproblems (if time permits)

    Prerequisites: Basic numerical analysis, basic PDE theory, some familiarity with Sobolev spaces and functional analysis


  • Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods. Springer 2007
  • Rüdiger Verfürth, A posteriori error estimation techniques for finite element methods. Oxford 2013
  • 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

    Online resources

  • 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 on adaptive FEM by Rüdiger Verfürth (RUB)
  • Lecture notes on adaptive FEM coauthored by Ricardo Nochetto (Maryland)
  • Zürich summer school on a posteriori error control and adaptivity
  • Homepage of Albert Cohen (UPMC)
  • Homepage of Wolfgang Dahmen (RWTH Aachen)
  • Homepage of Ronald DeVore (TAMU)
  • FreeFEM project
  • FEniCS project
  • FETK - The Finite Element ToolKit
  • ALBERTA - An adaptive hierarchical finite element toolbox