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Moon, K.-S.; von Schwerin, E.; Szepessy, A.; Tempone, R.

Convergence rates for an adaptive dual weighted residual finite element algorithm. (English) Zbl 1101.65103

BIT 46, No. 2, 367-407 (2006).
Error estimates for functionals are considered, i.e. \(\langle F, u-u_n\rangle\) with \(F\) being a functional and \(u_n\) a finite element approximation of \(u\) in \(d\)-space. Error representations of the form \[ \text{error}=\sum_{\text{elements}} \text{error density}\times h^{2+d}_{\text{element}} \] are derived. The representation show in which part of the domain refinements bring the best profit, and how the error indicators become approximately equidistributed. Moreover, convergence and a geometrical decay of the error is a consequence.
Reviewer: Dietrich Braess (Bochum)

Cited in 1 Review
Cited in 6 Documents

MSC:

65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs

Keywords:

adaptive finite elements; mesh refinement; error estimates; convergence
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\textit{K. S. Moon} et al., BIT 46, No. 2, 367--407 (2006; Zbl 1101.65103)
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References:

[1] M. Ainsworth and J. T. Oden, A posteriori error estimation in finite element analysis, Comput. Methods Appl. Mech. Eng., 142 (1997), pp. 1–88. · Zbl 0895.76040
[2] I. Babuška, J. Hugger, T. Strouboulis, K. Copps, and S. K. Gangaraj, The asymptotically optimal meshsize function for bi-p degree interpolation over rectangular elements, J. Comput. Appl. Math., 90 (1998), pp. 185–221. · Zbl 0939.65133
[3] I. Babuška, A. Miller, and M. Vogelius, Adaptive methods and error estimation for elliptic problems of structural mechanics, in Adaptive Computational Methods for Partial Differential Equations, pp. 57–73, SIAM, Philadelphia, PA, 1983.
[4] I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15 (1978), pp. 736–754. · Zbl 0398.65069
[5] I. Babuška, T. Strouboulis, and S. K. Gangaraj, Guaranteed computable bounds for the exact error in the finite element solution. I. One dimensional model problem, Comput. Methods Appl. Mech. Eng., 176 (1999), pp. 51–79. · Zbl 0936.65094
[6] I. Babuška and M. Vogelius, Feedback and adaptive finite element solution of one-dimensional boundary value problems, Numer. Math., 44 (1984), pp. 75–102. · Zbl 0574.65098
[7] R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comput., 44 (1985), pp. 283–301. · Zbl 0569.65079
[8] N. S. Bakhvalov, On the optimality of linear methods for operator approximation in convex classes of functions, USSR Comput. Math. Math. Phys., 11 (1971), pp. 244–249. · Zbl 0254.41013
[9] R. Becker and R. Rannacher, A feed-back approach to error control in finite element methods: Basic analysis and examples, East-West J. Numer. Math., 4 (1996), pp. 237–264. · Zbl 0868.65076
[10] 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. · Zbl 1105.65349
[11] P. Binev, W. Dahmen, and R. DeVore, Adaptive finite element methods with convergence rates, Numer. Math., 97 (2004), pp. 219–268. · Zbl 1063.65120
[12] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, 15, Springer, New York, 1994. · Zbl 0804.65101
[13] A. Cohen, W. Dahmen, and R. DeVore, Adaptive wavelet methods for elliptic operator equations: Convergence rates, Math. Comput., 70 (2001), pp. 25–75. · Zbl 0980.65130
[14] F. Christian and G. Santos, A posteriori estimators for nonlinear elliptic partial differential equations, J. Comput. Appl. Math., 103 (1999), pp. 99–114. · Zbl 0948.65116
[15] R. A. DeVore, Nonlinear approximation, Acta Numer., 7 (1998), pp. 51–150. · Zbl 0931.65007
[16] W. Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal., 33 (1996), pp. 1106–1124. · Zbl 0854.65090
[17] H. Edelsbrunner and D. R. Grayson, Edgewise subdivision of a simplex, ACM Symposium on Computational Geometry, (Miami, FL, 1999), Discrete Comput. Geom., 24 (2000), pp. 707–719. · Zbl 0968.51016
[18] K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Introduction to adaptive methods for differential equations, Acta Numer., 4 (1995), pp. 105–158. · Zbl 0829.65122
[19] C. Johnson and A. Szepessy, Adaptive finite element methods for conservation laws based on a posteriori error estimates, Commun. Pure Appl. Math., 48 (1995), pp. 199–234. · Zbl 0826.65088
[20] L. Machiels, A. T. Patera, J. Peraire, and Y. Maday, A general framework for finite element a posteriori error control: application to linear and nonlinear convection-dominated problems, in Proceedings of ICFD Conference on Numerical Methods for Fluid Dynamics, Numerical Methods for Fluid Dynamics VI, Oxford, England, 1998, ISBN 0952492911.
[21] Y. Maday, A. T. Patera, and J. Peraire, A general formulation for a posteriori bounds for output functionals of partial differential equations; applications to the eigenvalue problem, C. R. Acad. Sci., Paris, Sér. I, Math., 328 (1999), pp. 823–828. · Zbl 0933.65129
[22] K.-S. Moon, Adaptive algorithms for deterministic and stochastic differential equations, Doctoral thesis, ISBN 91-7283-553-2, Royal Institute of Technology, Stockholm, 2003, http://www.nada.kth.se/oon
[23] K.-S. Moon, A. Szepessy, R. Tempone, and G. E. Zouraris, Convergence rates for adaptive approximation of ordinary differential equations, Numer. Math., 96 (2003), pp. 99–129. · Zbl 1054.65082
[24] K.-S. Moon, A. Szepessy, R. Tempone, and G. E. Zouraris, Convergence rates for adaptive weak approximation of stochastic differential equations, Stoch. Anal. Appl., 23 (2005), pp. 511–558. · Zbl 1079.65012
[25] K.-S. Moon, A. Szepessy, R. Tempone, and G. E. Zouraris, Hyperbolic differential equations and adaptive numerics, in Theory and Numerics of Differential Equations, Durham 2000, J. F. Blowey, J. P. Coleman, and A. W. Craig, eds., Springer, Berlin, 2001. · Zbl 0991.65089
[26] P. Morin, R. H. Nochetto, and K. G. Siebert, Convergence of adaptive finite element methods, SIAM Rev., 44 (2002), pp. 631–658. Revised reprint of ”Data oscillation and convergence of adaptive FEM”, [SIAM J. Numer. Anal., 38 (2000), pp. 466–488] · Zbl 1016.65074
[27] R. Stevenson, An optimal adaptive finite element method, SIAM J. Numer. Anal., 42 (2005), pp. 2188–2217. · Zbl 1081.65112
[28] A. Szepessy, R. Tempone, and G. E. Zouraris, Adaptive weak approximation of stochastic differential equations, Commun. Pure Appl. Math., 54 (2001), pp. 1169–1214. · Zbl 1024.60028
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