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Original Articles

Convergence and quasi-optimality of an adaptive finite element method for semilinear elliptic problems on L2 errors

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Pages 2337-2354 | Received 15 Jul 2018, Accepted 24 Nov 2019, Published online: 10 Dec 2019

References

  • M. Arioli, E.H. Georgoulis, D. Loghin, Convergence of inexact adaptive finite element solvers for elliptic problems, Tech. Rep. RAL-TR-2009-021.
  • R. Becker, S. Mao, and Z. Shi, A convergent adaptive finite element method with optimal complexity, Electron. Trans. Numer. Anal. 30 (2008), pp. 291–304.
  • 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), pp. 484–510. doi: 10.1093/imanum/drr016
  • P. Binev, W. Dahmen, and R. Devore, Adaptive finite element methods with convergence rates, Numer. Math. 97 (2004), pp. 219–268. doi: 10.1007/s00211-003-0492-7
  • 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), pp. 2524–2550. doi: 10.1137/07069047X
  • Z. Chen and H. Wu, Selected Topics in Finite Element Methods, Science Press, Beijing, 2010.
  • P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
  • A. Demlow, Convergence of an adaptive finite element method for controlling local energy errors, SIAM J. Numer. Anal. 48 (2010), pp. 470–497. doi: 10.1137/080741458
  • 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. doi: 10.1007/s00211-010-0349-9
  • L. Diening and C. Kreuzer, Linear convergence of an adaptive finite element method for the p-Laplacian equation, SIAM J. Numer. Anal. 46 (2008), pp. 614–638. doi: 10.1137/070681508
  • W. Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal. 33 (1996), pp. 1106–1124. doi: 10.1137/0733054
  • 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. doi: 10.1142/S0218202594000194
  • 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), pp. 601–625. doi: 10.1137/120897225
  • E.M. Garau, P. Morin, and C. Zuppa, Convergence of an adaptive Kačanov FEM for quasi-linear problems, Appl. Numer. Math. 61 (2011), pp. 512-529. doi: 10.1016/j.apnum.2010.12.001
  • E.M. Garau, P. Morin, and C. Zuppa, Quasi-optimal convergence rate of an AFEM for quasi-linear problems of monotone type, Numer. Math. Theory Methods Appl. 5 (2012), pp. 131–156. doi: 10.4208/nmtma.2012.m1023
  • L. He and A. Zhou, Convergence and complexity of adaptive finite element methods for elliptic partial differential equations, Int. J. Numer. Anal. Model. 8 (2011), pp. 615–640.
  • M. Holst, S. Pollock, and Y. Zhu, Convergence of goal-oriented adaptive finite element methods for semilinear problems, Comput. Vis. Sci. 17 (2015), pp. 43–63. doi: 10.1007/s00791-015-0243-1
  • M. Holst, R. Szypowski, and Y.R. Zhu, Adaptive finite element methods with inexact solvers for the nonlinear Poisson–Boltzmann equation, Domain Decomp. Methods Sci. Eng. XX 91 (2011), pp. 167–174. doi: 10.1007/978-3-642-35275-1_18
  • M. Holst, G. Tsogtgerel, and Y. Zhu, Local convergence of adaptive methods for nonlinear partial differential equations, 2010. Available at https://arxiv.org/abs/1001.1382.
  • S. Kesavan, Topics in Functional Analysis and Applications, Wiley, New York, 1989.
  • H. Leng and Y. Chen, Convergence and quasi-optimality of an adaptive finite element method for optimal control problems on L2 errors, J. Sci. Comput. 73 (2017), pp. 438–458. doi: 10.1007/s10915-017-0425-8
  • X. Liao and R.H. Nochetto, Local a posteriori error estimates and adaptive control of pollution effects, Numer. Methods Partial Differ. Equ. 19 (2003), pp. 421–442. doi: 10.1002/num.10053
  • Q. Lin, H.H. Xie, and F. Xu, Multilevel correction adaptive finite element method for semilinear elliptic equation, Appl. Math. 60 (2015), pp. 527–550. doi: 10.1007/s10492-015-0110-x
  • K. Mekchay and R.H. Nochetto, Convergence of adaptive finite element methods for general second order linear elliptic PDEs, SIAM J. Numer. Anal. 43 (2005), pp. 1803–1827. doi: 10.1137/04060929X
  • P. Morin, R.H. Nochetto, and K.G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal. 38 (2000), pp. 466–488. doi: 10.1137/S0036142999360044
  • P. Morin, K.G. Siebert, and A. Veeser, Convergence of finite elements adapted for weaker norms, Int. J. Numer. Methods Eng. 18 (2007), pp. 323–341.
  • 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. doi: 10.1142/S0218202508002838
  • R.N. Nochetto, K.G. Siebert and A. Veeser, Theory of Adaptive Finite Element Methods: An Introduction. Multiscale, Nonlinear and Adaptive Approximation, Springer, Berlin, 2009, pp. 409–542.
  • L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comput. 54 (1990), pp. 483–493. doi: 10.1090/S0025-5718-1990-1011446-7
  • R. Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math. 7 (2007), pp. 245–269. doi: 10.1007/s10208-005-0183-0
  • M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4th ed., Springer, Berlin, 2000.
  • A. Veeser, Convergent adaptive finite elements for the nonlinear Laplacian, Numer. Math. 92 (2002), pp. 743–770. doi: 10.1007/s002110100377
  • R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, Chichester, 1996.
  • T.P. Wihler, Weighted L2-norm a posteriori error estimation of FEM in polygons, Int. J. Numer. Anal. Model., 4 (2007), pp. 100–115.

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