Bibliography
Ad75
[1] R. A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich,
Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65.
AF03
[2] R. A. Adams and J. J. F. Fournier, Sobolev spaces, vol. 140 of Pure and Applied Mathematics (Amsterdam), Elsevier/Academic Press, Amsterdam, second ed., 2003.
AO00
[3] M. Ainsworth and J. T. Oden, A posteriori error estimation in finite element analysis,
Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New
York, 2000.
AFW10
[4] D. N. Arnold, R. S. Falk, and R. Winther, Finite element exterior calculus: from Hodge
theory to numerical stability, Bull. Amer. Math. Soc. (N.S.), 47 (2010), pp. 281–354.
BR78
[5] I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15 (1978), pp. 736–754.
BV84
[6] I. Babuška and M. Vogelius, Feedback and adaptive finite element solution of onedimensional boundary value problems, Numer. Math., 44 (1984), pp. 75–102.
BNZ05
[7] C. Bacuta, V. Nistor, and L. T. Zikatanov, Improving the rate of convergence of highorder finite elements on polyhedra. I. A priori estimates, Numer. Funct. Anal. Optim., 26 (2005),
pp. 613–639.
BR03
[8] W. Bangerth and R. Rannacher, Adaptive finite element methods for di↵erential equations,
Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2003.
BSW83
[9] R. E. Bank, A. H. Sherman, and A. Weiser, Refinement algorithms and data structures for
regular local mesh refinement, in Scientific computing (Montreal, Que., 1982), IMACS Trans.
Sci. Comput., I, IMACS, New Brunswick, NJ, 1983, pp. 3–17.
BX03
[10] R. E. Bank and J. Xu, Asymptotically exact a posteriori error estimators. I. Grids with
superconvergence, SIAM J. Numer. Anal., 41 (2003), pp. 2294–2312 (electronic).
BR01
[11] R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation
in finite element methods, Acta Numer., 10 (2001), pp. 1–102.
BDD04
[12] P. Binev, W. Dahmen, and R. DeVore, Adaptive finite element methods with convergence
rates, Numer. Math., 97 (2004), pp. 219–268.
89
90
BIBLIOGRAPHY
BDDP02
[13] P. Binev, W. Dahmen, R. DeVore, and P. Petrushev, Approximation classes for adaptive
methods, Serdica Math. J., 28 (2002), pp. 391–416. Dedicated to the memory of Vassil Popov
on the occasion of his 60th birthday.
BDN13
[14] A. Bonito, R. A. DeVore, and R. H. Nochetto, Adaptive finite element methods for
elliptic problems with discontinuous coefficients, SIAM J. Numer. Anal., 51 (2013), pp. 3106–
3134.
BS08
[15] S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, vol. 15
of Texts in Applied Mathematics, Springer, New York, third ed., 2008.
CKNS08
[16] J. Cascon, C. Kreuzer, R. H. Nochetto, and K. G. Siebert, Quasi-optimal convergence
rate for an adaptive finite element method, SIAM J. Numer. Anal., 46 (2008), pp. 2524–2550.
CD00
[17] M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains,
Arch. Ration. Mech. Anal., 151 (2000), pp. 221–276.
Da88
[18] M. Dauge, Elliptic boundary value problems on corner domains, vol. 1341 of Lecture Notes in
Mathematics, Springer-Verlag, Berlin, 1988.
Da92
[19]
Dauge_web
, Neumann and mixed problems on curvilinear polyhedra, Integral Equations Operator
Theory, 15 (1992), pp. 227–261.
[20] M. Dauge, Regularity and singularities in polyhedral domains.
rennes1.fr/monique.dauge/publis/Talk Karlsruhe08.pdf, April 2008.
https://perso.univ-
DG12
[21] A. Demlow and E. Georgoulis, Pointwise a posteriori error control for discontinuous
Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 50 (2012), pp. 2159–2181.
DGS11
[22] A. Demlow, J. Guzmán, and A. H. Schatz, Local energy estimates for the finite element
method on sharply varying grids, Math. Comp., 80 (2011), pp. 1–9.
DK15
[23] A. Demlow and N. Kopteva, Maximum-norm a posteriori error estimates for singularly
perturbed reaction-di↵usion problems, Numer. Math., (2015).
DLSW11
[24] A. Demlow, D. Leykekhman, A. H. Schatz, and L. B. Wahlbin, Best approximation
1
property in the W1
norm for finite element methods on graded meshes, Math. Comp., (2011).
Dev98
[25] R. A. DeVore, Nonlinear approximation, in Acta numerica, 1998, vol. 7 of Acta Numer.,
Cambridge Univ. Press, Cambridge, 1998, pp. 51–150.
Dor96
[26] W. Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal.,
33 (1996), pp. 1106–1124.
DS80
[27] T. Dupont and R. Scott, Polynomial approximation of functions in Sobolev spaces, Math.
Comp., 34 (1980), pp. 441–463.
Er94
[28] K. Eriksson, An adaptive finite element method with efficient maximum norm error control
for elliptic problems, Math. Models Methods Appl. Sci., 4 (1994), pp. 313–329.
BIBLIOGRAPHY
91
EV15
[29] A. Ern and M. Vohralı́k, Polynomial-degree-robust a posteriori estimates in a unified setting
for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations, SIAM J.
Numer. Anal., 53 (2015), pp. 1058–1081.
Ev98
[30] L. C. Evans, Partial di↵erential equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.
FV06
[31] F. Fierro and A. Veeser, A posteriori error estimates, gradient recovery by averaging, and
superconvergence, Numer. Math., 103 (2006), pp. 267–298.
Gan16
[32] T. Gantumur, Convergence rates of adaptive methods, Besov spaces, and multilevel approximation, Found. Comput. Math., (2016).
GM14
[33] F. D. Gaspoz and P. Morin, Approximation classes for adaptive higher order finite element
approximation, Math. Comp., 83 (2014), pp. 2127–2160.
GT98
[34] D. Gilbarg and N. S. Trudinger, Elliptic Partial Di↵erential Equations of Second Order,
Springer-Verlag, Berlin, 2nd ed., 1998.
GNS05
[35] V. Girault, R. H. Nochetto, and R. Scott, Maximum-norm stability of the finite element
Stokes projection, J. Math. Pures Appl. (9), 84 (2005), pp. 279–330.
Gr76
[36] P. Grisvard, Behavior of the solutions of an elliptic boundary value problem in a polygonal
or polyhedral domain, in Numerical solution of partial di↵erential equations, III (Proc. Third
Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975), Academic Press, New York,
1976, pp. 207–274.
G85
[37] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Publishing Inc., Marshfield,
MA, 1985.
GLRS09
[38] J. Guzmán, D. Leykekhman, J. Rossmann, and A. H. Schatz, Hölder estimates for
Green’s functions on convex polyhedral domains and their applications to finite element methods,
Numer. Math., 112 (2009), pp. 221–243.
Ke74
[39] R. B. Kellogg, On the Poisson equation with intersecting interfaces, Applicable Anal., 4
(1974/75), pp. 101–129. Collection of articles dedicated to Nikolai Ivanovich Muskhelishvili.
MR10
[40] V. G. Maz’ya and J. Rossmann, Elliptic Equations in Polyhedral Domains, vol. 162 of
Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2010.
NS74
[41] J. A. Nitsche and A. H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp.,
28 (1974), pp. 937–958.
PW60
[42] L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains,
Arch. Rational Mech. Anal., 5 (1960), pp. 286–292 (1960).
RS82
[43] R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp., 38 (1982), pp. 437–445.
S98
[44] A. H. Schatz, Pointwise error estimates and asymptotic error expansion inequalities for the
finite element method on irregular grids. I. Global estimates., Math. Comp., 67 (1998), pp. 877–
899.
92
BIBLIOGRAPHY
SZ90
[45] L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying
boundary conditions, Math. Comp., 54 (1990), pp. 483–493.
Ste07
[46] R. Stevenson, Optimality of a standard adaptive finite element method, Found. Comput.
Math., 7 (2007), pp. 245–269.
Ste08
[47]
, The completion of locally refined simplicial partitions created by bisection, Math. Comp.,
77 (2008), pp. 227–241 (electronic).
Th97
[48] V. Thomée, Galerkin finite element methods for parabolic problems, vol. 25 of Springer Series
in Computational Mathematics, Springer-Verlag, Berlin, 1997.
Ver89
[49] R. Verfürth, A posteriori error estimators for the Stokes equations, Numer. Math., 55 (1989),
pp. 309–325.