Abstract
It is an open question if the threshold condition
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: CRC 1283
Funding statement: The research of Lars Diening was partially supported by the DFG through the CRC 1283.
References
[1] L. Belenki, L. Diening and C. Kreuzer, Optimality of an adaptive finite element method for the 𝑝-Laplacian equation, IMA J. Numer. Anal. 32 (2012), no. 2, 484–510. 10.1093/imanum/drr016Search in Google Scholar
[2] P. Binev, W. Dahmen and R. DeVore, Adaptive finite element methods with convergence rates, Numer. Math. 97 (2004), no. 2, 219–268. 10.21236/ADA640658Search in Google Scholar
[3] P. Binev and R. DeVore, Fast computation in adaptive tree approximation, Numer. Math. 97 (2004), no. 2, 193–217. 10.21236/ADA640671Search in Google Scholar
[4] A. Bonito and R. H. Nochetto, Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method, SIAM J. Numer. Anal. 48 (2010), no. 2, 734–771. 10.1137/08072838XSearch in Google Scholar
[5] C. Carstensen, M. Feischl, M. Page and D. Praetorius, Axioms of adaptivity, Comput. Math. Appl. 67 (2014), no. 6, 1195–1253. 10.1016/j.camwa.2013.12.003Search in Google Scholar PubMed PubMed Central
[6] C. Carstensen, D. Peterseim and H. Rabus, Optimal adaptive nonconforming FEM for the Stokes problem, Numer. Math. 123 (2013), no. 2, 291–308. 10.1007/s00211-012-0490-8Search in Google Scholar
[7] C. Carstensen and H. Rabus, Axioms of adaptivity with separate marking for data resolution, SIAM J. Numer. Anal. 55 (2017), no. 6, 2644–2665. 10.1137/16M1068050Search in Google Scholar
[8] J. M. 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), no. 5, 2524–2550. 10.1137/07069047XSearch in Google Scholar
[9] J. M. Cascón and R. H. Nochetto, Quasioptimal cardinality of AFEM driven by nonresidual estimators, IMA J. Numer. Anal. 32 (2012), no. 1, 1–29. 10.1093/imanum/drr014Search in Google Scholar
[10] L. Diening and C. Kreuzer, Linear convergence of an adaptive finite element method for the 𝑝-Laplacian equation, SIAM J. Numer. Anal. 46 (2008), no. 2, 614–638. 10.1137/070681508Search in Google Scholar
[11] L. Diening, C. Kreuzer and R. Stevenson, Instance optimality of the adaptive maximum strategy, Found. Comput. Math. 16 (2016), no. 1, 33–68. 10.1007/s10208-014-9236-6Search in Google Scholar
[12] M. Feischl, T. Führer and D. Praetorius, Adaptive FEM with optimal convergence rates for a certain class of nonsymmetric and possibly nonlinear problems, SIAM J. Numer. Anal. 52 (2014), no. 2, 601–625. 10.1137/120897225Search in Google Scholar
[13] M. Feischl, M. Karkulik, J. M. Melenk and D. Praetorius, Quasi-optimal convergence rate for an adaptive boundary element method, SIAM J. Numer. Anal. 51 (2013), no. 2, 1327–1348. 10.1137/110842569Search in Google Scholar
[14] T. Gantumur, Adaptive boundary element methods with convergence rates, Numer. Math. 124 (2013), no. 3, 471–516. 10.1007/s00211-013-0524-xSearch in Google Scholar
[15] C. Kreuzer and M. Schedensack, Instance optimal Crouzeix–Raviart adaptive finite element methods for the Poisson and Stokes problems, IMA J. Numer. Anal. 36 (2016), no. 2, 593–617. 10.1093/imanum/drv019Search in Google Scholar
[16] C. Kreuzer and K. G. Siebert, Decay rates of adaptive finite elements with Dörfler marking, Numer. Math. 117 (2011), no. 4, 679–716. 10.1007/s00211-010-0324-5Search in Google Scholar
[17] R. H. Nochetto, K. G. Siebert and A. Veeser, Theory of adaptive finite element methods: An introduction, Multiscale, Nonlinear and Adaptive Approximation, Springer, Berlin (2009), 409–542. 10.1007/978-3-642-03413-8_12Search in Google Scholar
[18] R. H. Nochetto and A. Veeser, Primer of adaptive finite element methods, Multiscale and Adaptivity: Modeling, Numerics and Applications, Lecture Notes in Math. 2040, Springer, Heidelberg (2012), 125–225. 10.1007/978-3-642-24079-9_3Search in Google Scholar
[19] R. Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math. 7 (2007), no. 2, 245–269. 10.1007/s10208-005-0183-0Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston