Abstract
We prove convergence and optimal complexity of an adaptive mixed finite element algorithm, based on the lowest-order Raviart–Thomas finite element space. In each step of the algorithm, the local refinement is either performed using simple edge residuals or a data oscillation term, depending on an adaptive marking strategy. The inexact solution of the discrete system is controlled by an adaptive stopping criterion related to the estimator.
Article PDF
Similar content being viewed by others
References
Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis., Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, and Tracts, vol. xx , (English). Wiley, Chichester (2000)
Alonso A.: Error estimators for a mixed method. Numer. Math. 74(4), 385–395 (1996)
Babuška, I., Strouboulis, Th.: The finite element methods and its reliability, vol xii, 2001 (English). Clarendon Press, Oxford (2001)
Babuška I., Vogelius M.: Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math. 44, 75–102 (1984)
Becker R., Johnson C., Rannacher R.: Adaptive error control for multigrid finite element methods. Computing 55, 271–288 (1995)
Becker, R., Mao, S., Shi, Z.-C.: A convergent adaptive finite element method with optimal complexity, ETNA (accepted for publication).
Binev P., Dahmen W., DeVore R.: Adaptive finite element methods with convergence rates. Numer. Math. 97(2), 219–268 (2004)
Braess D., Verfürth R.: A posteriori error estimators for the Raviart–Thomas element. SIAM J. Numer. Anal 33(6), 2431–2444 (1996)
Bramble J.H., Pasciak J.E.: New estimates for multilevel algorithms including the v-cycle. Math. Comput. 60(202), 447–471 (1995)
Brenner S.C.: A multigrid algorithm for the lowest-order Raviart–Thomas mixed triangular finite element method. SIAM J. Numer. Anal. 29(3), 647–678 (1992)
Carstensen C.: A posteriori error estimate for the mixed finite element method. Math. Comput. 66(218), 465–476 (1997)
Carstensen C., Hoppe R.H.W.: Error reduction and convergence for an adaptive mixed finite element method. Math. Comput. 75(255), 1033–1042 (2006)
Cascon J.M., Nochetto R.H., Siebert K.G.: Design and convergence of AFEM in H(div). Math. Models Methods Appl. Sci. 17(11), 1849–1881 (2007) MR MR2372340
Cascon J.M., Kreuzer Ch., Nochetto R.N., Siebert K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46(5), 2524–2550 (2008)
Chen, L., Holst, M., Xu, J.: Convergence and optimality of adaptive mixed finite element methods. Tech report, Pennstate (2006)
Chen, L., Nochetto, R.H., Xu, J.: Multilevel methods on graded bisection grids. Tech report, University of Maryland (2007)
Ciarlet P.G.: The finite element method for elliptic problems. Studies in Mathematics and its Applications, vol. 4. North-Holland Publishing Company, Amsterdam (1978)
Cohen A., Dahmen W., DeVore R.: Adaptive wavelet methods for elliptic operator equations: Convergence rates. Math. Comput. 70(233), 27–75 (2001)
DeVore R.: Nonlinear approximation. In: Iserles, A. (eds) Acta Numerica 1998, vol. 7, pp. 51–150. Cambridge University Press, London (1998)
Dörfler W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)
Eriksson K., Estep D., Hansbo P., Johnson C.: Introduction to adaptive methods for differential equations. In: Iserles, A. (eds) Acta Numerica 1995, pp. 105–158. Cambridge University Press, London (1995)
Mitchell W.F.: A comparison of adaptive refinement techniques for elliptic problems. ACM Trans. Math. Softw. (TOMS) 15, 326–347 (1989)
Morin P., Nochetto R.H., Siebert K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38(2), 466–488 (2000)
Peisker P., Braess, D.: Uniform convergence of mixed interpolated elements for Reissner-Mindlin plates. RAIRO Modél Math. Anal. Numér. 26(5):557–574 MR MR1177387 (93j:73070)
Raviart, P., Thomas, J.: A mixed finite element method for 2nd order elliptic problems. Lecture Notes on Mathematics, vol. 606, pp. 292–315. Springer, Berlin (1977)
Stevenson R.: An optimal adaptive finite element method. SIAM J. Numer. Anal. 42(5), 2188–2217 (2005)
Stevenson R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 7(2), 245–269 (2007)
Verfürth R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley/Teubner, New York/Stuttgart (1996)
Wohlmuth B.I., Hoppe R.H.W.: A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart–Thomas elements. Math. Comput. 68(228), 1347–1378 (1999)
Wu H., Chen Z.: Uniform convergence of multigrid v-cycle on adaptively refined finite element meshes for second order elliptic problems. Sci. China Ser. A Math. 49(10), 1405–1429 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Becker, R., Mao, S. An optimally convergent adaptive mixed finite element method. Numer. Math. 111, 35–54 (2008). https://doi.org/10.1007/s00211-008-0180-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-008-0180-8