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 | L^{p}-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 C^{1}. 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 L^{p} and W^{1,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 L^{p}. 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 |
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
Prerequisites: Basic numerical analysis, basic PDE theory, some familiarity with Sobolev spaces and functional analysis