Abstract
This paper presents an optimal nonconforming adaptive finite element algorithm and proves its quasi-optimal complexity for the Stokes equations with respect to natural approximation classes. The proof does not explicitly involve the pressure variable and follows from a novel discrete Helmholtz decomposition of deviatoric functions.
Similar content being viewed by others
References
Brenner, S.C., Carstensen, C.: Encyclopedia of computational mechanics, chapter 4. In: Finite Element Methods. Wiley, New York (2004)
Binev, P., Dahmen, W., De Vore, R.: Adaptive finite element methods with convergence rates. Numer. Math. 97, 219–268 (2004)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New-York (1991)
Becker, R., Mao, S.: An optimally convergent adaptive mixed finite element method. Numer. Math. 111, 35–54 (2008)
Becker, R., Mao, S.: Adaptive nonconforming finite elements for the Stokes equations. SIAM J. Numer. Anal. 49(3), 970–991 (2011)
Bänsch, E., Morin, P., Nochetto, R.H.: An adaptive Uzawa FEM for the Stokes problem: convergence without the inf-sup condition. SIAM J. Numer. Anal. 40(4), 1207–1229 (2002)
Becker, R., Mao, S., Shi, Z.: A convergent nonconforming adaptive finite element method with quasi-optimal complexity. SIAM J. Numer. Anal. 47(6), 4639–4659 (2010)
Braess, D.: Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, Cambridge (2001)
Brenner, S.C., Scott, L.R.: The Mathematical theory of finite element methods, vol. 15, 3rd edn. In: Texts in Applied Mathematics. Springer, New York (2008)
Carstensen, C., Hoppe, R.H.W.: Convergence analysis of an adaptive nonconforming finite element method. Numer. Math. 103(2), 251–266 (2006)
Carstensen, C., Hoppe, R.H.W.: Error reduction and convergence for an adaptive mixed finite element method. Math. Comp. 75(255):1033–1042 (electronic) (2006)
Cascon, J.M., Kreuzer, Ch., Nochetto, R.H., Siebert, K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46(5), 2524–2550 (2008)
Carstensen, C., Peterseim, D., Schedensack, M: Comparison results of three first-order finite element methods for the Poisson model problem. Preprint 831, DFG Research Center Matheon Berlin (2011)
Crouzeix, M., Raviart, P.A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7(R-3), 33–75 (1973)
Carstensen, C., Rabus, H.: An optimal adaptive mixed finite element method. Math. Comp. 80(274), 649–667 (2011)
Carstensen, C., Rabus, H.: The adaptive nonconforming fem for the pure displacement problem in linear elasticity is optimal and robust. SIAM J. Numer. Anal., accepted (2012)
Dari, E., Durán, R., Padra, C.: Error estimators for nonconforming finite element approximations of the Stokes problem. Math. Comp. 64(211), 1017–1033 (1995)
Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)
Gudi, T.: A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comp. 79(272), 2169–2189 (2010)
Hu, J., Shi, Z., Xu, J.: Convergence and optimality of the adaptive morley element method. Numerische Mathematik 1–22 (2012). doi:10.1007/s00211-012-0445-0
Hu, J., Xu, J.: Convergence of adaptive conforming and nonconforming finite element methods for the perturbed Stokes equation. Research Report. School of Mathematical Sciences and Institute of Mathematics, Peking University (2007). Available at www.math.pku.edu.cn:8000/var/preprint/7297.pdf
Hu, J., Xu, J.: Convergence and optimality of the adaptive nonconforming linear element method for the Stokes problem. J. Sci. Comput. 1–24 (2012). doi:10.1007/s10915-012-9625-4
Morin, P., Nochetto, R. H., Siebert, K. G.: Convergence of adaptive finite element methods. SIAM Rev. 44(4):631–658 (electronic) (2003) 2002. Revised reprint of “Data oscillation and convergence of adaptive FEM”. SIAM J. Numer. Anal. 38(2):466–488 (2000) (electronic); MR1770058 (2001g:65157)
Mao, S., Zhao, X., Shi, Z.: Convergence of a standard adaptive nonconforming finite element method with optimal complexity. Appl. Numer. Math. 60, 673–688 (2010)
Rabus, H.: A natural adaptive nonconforming FEM of quasi-optimal complexity. Comput. Methods Appl. Math. 10(3), 315–325 (2010)
Stevenson, R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 7(2), 245–269 (2007)
Stevenson, R.: The completion of locally refined simplicial partitions created by bisection. Math. Comput. 77(261), 227–241 (2008)
Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)
Author information
Authors and Affiliations
Corresponding author
Additional information
Carsten Carstensen was supported by the World Class University (WCU) program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology R31-2008-000-10049-0. Daniel Peterseim was supported by the DFG research center Matheon ‘Mathematics in the key technologies’. Hella Rabus was supported by the DFG research group 797 ‘Analysis and Computation of Microstructure in Finite Plasticity’.
Rights and permissions
About this article
Cite this article
Carstensen, C., Peterseim, D. & Rabus, H. Optimal adaptive nonconforming FEM for the Stokes problem. Numer. Math. 123, 291–308 (2013). https://doi.org/10.1007/s00211-012-0490-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-012-0490-8