Project description for MATH 689, Spring 2016
Instructor: Dr. Alan Demlow
In your project you will investigate a current research topic and present an oral summary of
your findings to the class. Oral presentations will be 40 minutes long and will take place during
the last two class periods (April 28 and May 3, two presentations each), and our final exam
period (Tuesday, May 10 8-10 a.m.) or another suitable time agreed upon by the instructor
and students (three presentations). The basic expectation is that you will digest (at least)
one substantial research paper. Your presentation should consist of an outline of context and
previous results for the problem tackled in your paper, a statement of the problem, the main
results, and a synopsis of the proofs of the main results. If you do not have time to give full
proofs during your talk, be sure to at least explain the main technical challenges overcome in
the proofs. It would be appropriate to for example prove a key lemma and then explain more
briefly how it is used to obtain the overall results in the paper. Each student should choose a
research topic and make an appointment to discuss it with me outside of class by Tuesday, April
5. A list of suggested papers and topics is below; other papers are also fine if agreed upon with
me.
List of potential topics
1. Instance optimality of AFEM [7]
2. Plain convergence of AFEM [12, 15]
3. Convergence of AFEM for discontinuous Galerkin methods [5] Min
4. AFEM for eigenvalue problems [9, 8, 3] Justin
5. AFEM for Maxwell’s equations [16] Yanbo
6. Goal-oriented AFEM [11]
7. Muckenhoupt spaces, fractional diffusion [13, 14] Peng
8. AFEM for the Laplace-Beltrami problem [2]
9. AFEM for the Stokes equation [1, 10, 4] Yong
10. AFEM for controlling L2 errors: [6] Wenyu
References
[1] E. Bänsch, P. Morin, and R. H. Nochetto, An adaptive Uzawa FEM for the Stokes
problem: convergence without the inf-sup condition, SIAM J. Numer. Anal., 40 (2002),
pp. 1207–1229.
[2] A. Bonito, J. M. Cascon, P. Morin, and R. H. Nochetto, AFEM for Geometric
PDE: The Laplace-Beltrami Operator, in Analysis and Numerics of Partial Differential
Equations. In memory of Enrico Magenes, vol. 4 of Springer INdAM Series, Springer, 2013.
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[3] A. Bonito and A. Demlow, Convergence and optimality of higher-order adaptive finite
element methods for eigenvalue clusters, preprint, (2015).
[4] A. Bonito and D. Devaud, Adaptive finite element methods for the Stokes problem with
discontinuous viscosity, Math. Comp., 84 (2015), pp. 2137–2162.
[5] A. Bonito and R. H. Nochetto, Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method, SIAM J. Numer. Anal., 48 (2010), pp. 734–771.
[6] A. Demlow and R. Stevenson, Convergence and quasi-optimality of an adaptive finite
element method for controlling L2 errors, Numer. Math., 117 (2011), pp. 185–218.
[7] L. Diening, C. Kreuzer, and R. Stevenson, Instance Optimality of the Adaptive
Maximum Strategy, Found. Comput. Math., 16 (2016), pp. 33–68.
[8] D. Gallistl, An optimal adaptive FEM for eigenvalue clusters, Numer. Math., 130 (2015),
pp. 467–496.
[9] E. M. Garau, P. Morin, and C. Zuppa, Convergence of adaptive finite element methods
for eigenvalue problems, Math. Models Methods Appl. Sci., 19 (2009), pp. 721–747.
[10] Y. Kondratyuk and R. Stevenson, An optimal adaptive finite element method for the
Stokes problem, SIAM J. Numer. Anal., 46 (2008), pp. 747–775.
[11] M. S. Mommer and R. Stevenson, A goal-oriented adaptive finite element method with
convergence rates, SIAM J. Numer. Anal., 47 (2009), pp. 861–886.
[12] P. Morin, K. G. Siebert, and A. Veeser, A basic convergence result for conforming
adaptive finite elements, Math. Models Methods Appl. Sci., 18 (2008), pp. 707–737.
[13] R. H. Nochetto, E. Otárola, and A. J. Salgado, Piecewise polynomial interpolation
in Muckenhoupt weighted sobolev spaces and applications, Numer. Math., 132 (2016), pp. 85–
130.
[14]
, A PDE approach to space-time fractional parabolic problems, SIAM J. Numer. Anal,
(to appear).
[15] K. G. Siebert, A convergence proof for adaptive finite elements without lower bound, IMA
J. Numer. Anal., 31 (2011), pp. 947–970.
[16] L. Zhong, L. Chen, S. Shu, G. Wittum, and J. Xu, Convergence and Optimality of
adaptive edge finite element methods for time-harmonic Maxwell equations, Math. Comp.,
(To appear.).
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