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All issues Volume 57 / No 1 (January-February 2023) ESAIM: M2AN, 57 1 (2023) 329-366 References
Open Access
Issue
ESAIM: M2AN
Volume 57, Number 1, January-February 2023
Page(s) 329 - 366
DOI https://doi.org/10.1051/m2an/2022082
Published online 21 February 2023
  1. M. Ainsworth and J.T. Oden, A posteriori error estimation in finite element analysis, in Pure and Applied Mathematics (New York), Wiley-Interscience, John Wiley & Sons, New York (2000). [Google Scholar]
  2. M. Arioli, E.H. Georgoulis and D. Loghin, Stopping criteria for adaptive finite element solvers. SIAM J. Sci. Comput. 35 (2013) A1537–A1559. [CrossRef] [Google Scholar]
  3. I. Babuška and B. Guo, The h–p version of finite element method, part 1: The basic approximation results. Comput. Mech. (1986) 21–41. [Google Scholar]
  4. I. Babuška and B. Guo, The h–p version of finite element method, part 2: General results and application. Comput. Mech. (1986) 21–41. [Google Scholar]
  5. I. Babuška and A. Miller, A feedback finite element method with a posteriori error estimation. I. The finite element method and some basic properties of the a posteriori error estimator. Comput. Methods Appl. Mech. Eng. 61 (1987) 1–40. [CrossRef] [Google Scholar]
  6. R.E. Bank, A. Parsania and S. Sauter, Saturation estimates for hp-finite element methods. Comput. Vis. Sci. 16 (2013) 195–217. [CrossRef] [MathSciNet] [Google Scholar]
  7. R. Becker and S. Mao, Convergence and quasi-optimal complexity of a simple adaptive finite element method. ESAIM:M2AN 43 (2009) 1203–1219. [CrossRef] [EDP Sciences] [Google Scholar]
  8. R. Becker, C. Johnson and R. Rannacher, Adaptive error control for multigrid finite element methods. Computing 55 (1995) 271–288. [CrossRef] [MathSciNet] [Google Scholar]
  9. P. Binev, Tree approximation for hp-adaptivity. SIAM J. Numer. Anal. 56 (2018) 3346–3357. [Google Scholar]
  10. P. Binev, W. Dahmen and R. DeVore, Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004) 219–268. [Google Scholar]
  11. J. Blechta, J. Málek and M. Vohralk, Localization of the W−1,q norm for local a posteriori efficiency. IMA J. Numer. Anal. 40 (2020) 914–950. [CrossRef] [MathSciNet] [Google Scholar]
  12. D. Braess and J. Schöberl, Equilibrated residual error estimator for edge elements. Math. Comput. 77 (2008) 651–672. [Google Scholar]
  13. D. Braess, V. Pillwein and J. Schöberl, Equilibrated residual error estimates are p-robust. Comput. Methods Appl. Mech. Eng. 198 (2009) 1189–1197. [Google Scholar]
  14. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, In vol. 15 of Springer Series in Computational Mathematics, Springer-Verlag, New York (1991). [CrossRef] [Google Scholar]
  15. M. Bürg and W. Dörfler, Convergence of an adaptive hp finite element strategy in higher space-dimensions. Appl. Numer. Math. 61 (2011) 1132–1146. [CrossRef] [MathSciNet] [Google Scholar]
  16. C. Canuto, R.H. Nochetto, R.P. Stevenson and M. Verani, Convergence and optimality of hp-AFEM. Numer. Math. 135 (2017) 1073–1119. [CrossRef] [MathSciNet] [Google Scholar]
  17. C. Canuto, R.H. Nochetto, R.P. Stevenson and M. Verani, On p-robust saturation for hp-AFEM. Comput. Math. Appl. 73 (2017) 2004–2022. [CrossRef] [MathSciNet] [Google Scholar]
  18. C. Canuto, R.H. Nochetto, R.P. Stevenson and M. Verani, A saturation property for the spectral-Galerkin approximation of a Dirichlet problem in a square. ESAIM:M2AN 53 (2019) 987–1003. [CrossRef] [EDP Sciences] [Google Scholar]
  19. C. Carstensen and S.A. Funken, Fully reliable localized error control in the FEM. SIAM J. Sci. Comput. 21 (1999/00) 1465–1484. [Google Scholar]
  20. C. Carstensen, M. Feischl, M. Page and D. Praetorius, Axioms of adaptivity. Comput. Math. Appl. 67 (2014) 1195–1253. [PubMed] [Google Scholar]
  21. J.M. Cascón and R.H. Nochetto, Quasioptimal cardinality of AFEM driven by nonresidual estimators. IMA J. Numer. Anal. 32 (2012) 1–29. [CrossRef] [MathSciNet] [Google Scholar]
  22. J.M. Cascón, 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) 2524–2550. [CrossRef] [MathSciNet] [Google Scholar]
  23. P. Ciarlet Jr. and M. Vohralk, Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients. ESAIM:M2AN 52 (2018) 2037–2064. [CrossRef] [EDP Sciences] [Google Scholar]
  24. P. Daniel, Adaptive hp-finite elements with guaranteed error contraction and inexact multilevel solvers. Ph.D. thesis, Sorbonne University (2019). [Google Scholar]
  25. P. Daniel, A. Ern, I. Smears and M. Vohralk, An adaptive hp-refinement strategy with computable guaranteed bound on the error reduction factor. Comput. Math. Appl. 76 (2018) 967–983. [CrossRef] [MathSciNet] [Google Scholar]
  26. P. Daniel, A. Ern and M. Vohralk, An adaptive hp-refinement strategy with inexact solvers and computable guaranteed bound on the error reduction factor. Comput. Methods Appl. Mech. Eng. 359 (2020) 112607. [CrossRef] [Google Scholar]
  27. P. Destuynder and B. Métivet, Explicit error bounds in a conforming finite element method. Math. Comp. 68 (1999) 1379–1396. [CrossRef] [MathSciNet] [Google Scholar]
  28. V. Dolejší, A. Ern and M. Vohralík, hp-adaptation driven by polynomial-degree-robust a posteriori error estimates for elliptic problems. SIAM J. Sci. Comput. 38 (2016) A3220–A3246. [Google Scholar]
  29. W. Dörfler, A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33 (1996) 1106–1124. [Google Scholar]
  30. W. Dörfler and V. Heuveline, Convergence of an adaptive hp finite element strategy in one space dimension. Appl. Numer. Math. 57 (2007) 1108–1124. [CrossRef] [MathSciNet] [Google Scholar]
  31. C. Erath, G. Gantner and D. Praetorius, Optimal convergence behavior of adaptive FEM driven by simple (h − h/2)-type error estimators. Comput. Math. Appl. 79 (2020) 623–642. [CrossRef] [MathSciNet] [Google Scholar]
  32. A. Ern and M. Vohralk, Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs. SIAM J. Sci. Comput. 35 (2013) A1761–A1791. [Google Scholar]
  33. A. Ern and M. Vohralk, Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations. SIAM J. Numer. Anal. 53 (2015) 1058–1081. [Google Scholar]
  34. A. Ern and M. Vohralk, Stable broken H1 and H(div) polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions. Math. Comput. 89 (2020) 551–594. [Google Scholar]
  35. A. Ern, I. Smears and M. Vohralk, Discrete p-robust H(div)-liftings and a posteriori estimates for elliptic problems with H−1 source terms. Calcolo 54 (2017) 1009–1025. [CrossRef] [MathSciNet] [Google Scholar]
  36. G. Gantner, A. Haberl, D. Praetorius and B. Stiftner, Rate optimal adaptive FEM with inexact solver for nonlinear operators. IMA J. Numer. Anal. 38 (2018) 1797–1831. [CrossRef] [MathSciNet] [Google Scholar]
  37. M. Holst, R. Szypowski and Y. Zhu, Adaptive finite element methods with inexact solvers for the nonlinear Poisson-Boltzmann equation, in Domain Decomposition Methods in Science and Engineering XX, vol. 91 of Lect. Notes Comput. Sci. Eng., Springer, Heidelberg, (2013) 167–174. [CrossRef] [Google Scholar]
  38. P. Jiránek, Z. Strakoš and M. Vohralk, A posteriori error estimates including algebraic error and stopping criteria for iterative solvers. SIAM J. Sci. Comput. 32 (2010) 1567–1590. [CrossRef] [MathSciNet] [Google Scholar]
  39. C. Kreuzer and K.G. Siebert, Decay rates of adaptive finite elements with Dörfler marking. Numer. Math. 117 (2011) 679–716. [CrossRef] [MathSciNet] [Google Scholar]
  40. C. Kreuzer and A. Veeser, Oscillation in a posteriori error estimation. Numer. Math. 148 (2021) 43–78. [CrossRef] [MathSciNet] [Google Scholar]
  41. W.F. Mitchell, Adaptive refinement for arbitrary finite-element spaces with hierarchical bases. J. Comput. Appl. Math. 36 (1991) 65–78. [CrossRef] [MathSciNet] [Google Scholar]
  42. W.F. Mitchell and M.A. McClain, A comparison of hp-adaptive strategies for elliptic partial differential equations (long version). NISTIR 7824, National Institute of Standards and Technology (2011). [Google Scholar]
  43. P. Morin, R.H. Nochetto and K.G. Siebert, Convergence of adaptive finite element methods. SIAM Rev. 44 (2002) 631–658. Revised reprint of “Data oscillation and convergence of adaptive FEM”. SIAM J. Numer. Anal. 38 (2000) 466–488. [CrossRef] [MathSciNet] [Google Scholar]
  44. P. Morin, R.H. Nochetto and K.G. Siebert, Local problems on stars: a posteriori error estimators, convergence, and performance. Math. Comp. 72 (2003) 1067–1097. [Google Scholar]
  45. P. Morin, K.G. Siebert and A. Veeser, A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci. 18 (2008) 707–737. [Google Scholar]
  46. J. Papež, Z. Strakoš and M. Vohralk, Estimating and localizing the algebraic and total numerical errors using flux reconstructions. Numer. Math. 138 (2018) 681–721. [CrossRef] [MathSciNet] [Google Scholar]
  47. J. Papež, U. Rüde, M. Vohralk and B. Wohlmuth, Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach. Recovering mass balance in any situation. Comput. Methods Appl. Mech. Eng. 371 (2020) 113243. [CrossRef] [Google Scholar]
  48. C.-M. Pfeiler and D. Praetorius, Dörfler marking with minimal cardinality is a linear complexity problem. Math. Comp. 89 (2020) 2735–2752. [CrossRef] [MathSciNet] [Google Scholar]
  49. A. Quarteroni and A. Valli, Numerical approximation of partial differential equations, In Vol. 23 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin (1994). [CrossRef] [Google Scholar]
  50. V. Rey, C. Rey and P. Gosselet, A strict error bound with separated contributions of the discretization and of the iterative solver in non-overlapping domain decomposition methods. Comput. Methods Appl. Mech. Eng. 270 (2014) 293–303. [CrossRef] [Google Scholar]
  51. J.E. Roberts and J.-M. Thomas, Mixed and hybrid methods, in Handbook of Numerical Analysis, Vol II, North-Holland, Amsterdam (1991) 523–639. [CrossRef] [Google Scholar]
  52. D. Schötzau and C. Schwab, Exponential convergence of hp-FEM for elliptic problems in polyhedra: mixed boundary conditions and anisotropic polynomial degrees. Found. Comput. Math. 18 (2018) 595–660. [CrossRef] [MathSciNet] [Google Scholar]
  53. E.G. Sewell, Automatic generation of triangulations for piecewise polynomial approximation. ProQuest LLC, Ann Arbor, MI (1972). Ph.D. thesis–Purdue University. [Google Scholar]
  54. R. Stevenson, An optimal adaptive finite element method. SIAM J. Numer. Anal. 42 (2005) 2188–2217. [CrossRef] [MathSciNet] [Google Scholar]
  55. R. Stevenson, Optimality of a standard adaptive finite element method. Found. Comput. Math. 7 (2007) 245–269. [Google Scholar]
  56. R. Verfürth, A posteriori error estimation techniques for finite element methods, in Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford (2013). [Google Scholar]
  57. M. Vohralk, Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods. Math. Comp. 79 (2010) 2001–2032. [CrossRef] [MathSciNet] [Google Scholar]

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