Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter November 2, 2018

Residual-based a posteriori error estimation for hp-adaptive finite element methods for the Stokes equations

Arezou Ghesmati, Wolfgang Bangerth and Bruno Turcksin

Abstract

We derive a residual-based a posteriori error estimator for the conforming hp-Adaptive Finite Element Method (hp-AFEM) for the steady state Stokes problem describing the slow motion of an incompressible fluid. This error estimator is obtained by extending the idea of a posteriori error estimation for the classical h-version of AFEM. We also establish the reliability and efficiency of the error estimator. The proofs are based on the well-known Clément-type interpolation operator introduced in [27] in the context of the hp-AFEM. Numerical experiments show the performance of an adaptive hp-FEM algorithm using the proposed a posteriori error estimator.

JEL Classification: 65N30; 65N15

Acknowledgment

This material is based upon work supported by the U.S. Department of Energy, Office of Science, under contract number DE-AC05-00OR22725. AG and WB’s work was supported by the National Science Foundation under award OCI−1148116 as part of the Software Infrastructure for Sustained Innovation (SI2) program. WB was also supported by the Computational Infrastructure in Geodynamics initiative (CIG), through the National Science Foundation under Awards No. EAR-0949446 and EAR−1550901, administered by The University of California, Davis.

This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

References

[1] M. Ainsworth and J. T. Oden, A posteriori error estimation in finite element analysis, Comput. Methods. Appl. Mech. Engrg. 142 (1997), 1–88.10.1002/9781118032824Search in Google Scholar

[2] M. Ainsworth and B. Senior, An adaptive refinement strategy for hp-finite-element computations, Appl. Numer. Math. 26 (1998), 165–178.10.1016/S0168-9274(97)00083-4Search in Google Scholar

[3] D. Arndt, W. Bangerth, D. Davydov, T. Heister, L. Heltai, M. Kronbichler, M.Maier, J.-P. Pelteret, B. Turcksin, and D. Wells, The deal.II Library, Version 8.5, J. Numer. Math. 25 (2017), No. 3, 137–146.10.1515/jnma-2017-0058Search in Google Scholar

[4] D. N. Arnold, F. Brezzi, and M. Fortin, A stable finite element for the Stokes equations, Calcolo21 (1984), 337–344.10.1007/BF02576171Search in Google Scholar

[5] I. Babuška, Error estimates for adaptive finite element computations, SIAM J. Math. Anal. 15 (1978), No. 4, 736–754.Search in Google Scholar

[6] I. Babuška and M. R. Dorr, Error estimates for the combined h- and p-versions of the finite element method, Numer. Math. 37 (1981), No. 2, 257–277.10.1007/BF01398256Search in Google Scholar

[7] I. Babuška and W. C. Rheinboldt, A posteriori error estimates for the finite element method, Int. J. Numer. Methods Engrg. 12 (1978), 1597–1615.10.1002/nme.1620121010Search in Google Scholar

[8] I. Babuška and W. C. Rheinboldt, Adaptive approaches and reliability estimations in finite element analysis, Comput. Meth. Appl. Mech. Engrg. 17 (1979), 519–540.10.1016/0045-7825(79)90042-2Search in Google Scholar

[9] I. Babuška, B. Szabó, and I.N. Katz, The p-version of the finite element method, SIAM J. Numer. Anal. 18 (1981), 515–545.10.1137/0718033Search in Google Scholar

[10] W. Bangerth, R. Hartmann, and G. Kanschat, deal.II a general purpose object oriented finite element library, ACM Trans. Math. Software33(4) (2007), 24/1–24/27.10.1145/1268776.1268779Search in Google Scholar

[11] W. Bangerth and O. Kayser-Herold, Data structures and requirements for hp finite element software, ACM Trans. Math. Softw. 36 (2009), No. 1, 4/1–4/31.10.1145/1486525.1486529Search in Google Scholar

[12] W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Birkhäuser Verlag, 2003.10.1007/978-3-0348-7605-6Search in Google Scholar

[13] C. Bernardi, R. G. Owens, and J. Valenciano, An error indicator for mortar element solution to the Stokes problem, SIAM J. Numer. Anal. 21 (2001), 857–886.10.1093/imanum/21.4.857Search in Google Scholar

[14] F. Brezzi, On the existence, uniqueness, and approximation of saddle-point problems arising from Lagrangian multipliers, RAIRO Anal. Numer. 8 (1974), 129–151.10.1051/m2an/197408R201291Search in Google Scholar

[15] M. Bürg, A residual-based a posteriori error estimator for the hp-finite element method for Maxwell’s equations, Appl. Numer. Math. 62 (2012), 922–940.10.1016/j.apnum.2012.02.007Search in Google Scholar

[16] M. Bürg and W. Dörfler, Convergence of an adaptive hp finite element strategy in higher space-dimensions, Appl. Numer. Math. 61 (2011), 1132–1146.10.1016/j.apnum.2011.07.008Search in Google Scholar

[17] Ph. Clément, Approximation by finite element functions using local regularization, RAIRO Anal. Numer. 9 (1975), 77–84.10.1051/m2an/197509R200771Search in Google Scholar

[18] M. Costabel, M. Dauge, and C. Schwab, Exponential convergence of hp-FEM for Maxwell’s equations with weighted regularization in polygonal domains, M3AS15 (2005), No. 4, 575–622.Search in Google Scholar

[19] M. Dauge, Stationary Stokes and Navier–Stokes systems on two- or three-dimensional domains with corners, Part I: Linearized equations, SIAM J. Math. Anal. 20 (1989), 74–97.10.1137/0520006Search in Google Scholar

[20] L. Demkowicz, W. Rachowicz, and Ph. Devloo, A fully automatic hp-adaptivity, J. Sci. Comput. 17 (2002), 127–155.10.1023/A:1015192312705Search in Google Scholar

[21] W. Dörfler and V. Heuveline, Convergence of an adaptive hp finite element strategy in one space dimension, Appl. Numer. Math. 57 (2007), 1108–1124.10.1016/j.apnum.2006.10.003Search in Google Scholar

[22] T. Eibner and J.M. Melenk, An adaptive strategy for hp-FEM based on testing for analyticity, Comput. Mech. 39 (2007), 575–595.10.1007/s00466-006-0107-0Search in Google Scholar

[23] A. Ern and J. L. Guermond, Theory and Practice of Finite Elements, Springer, 2013.Search in Google Scholar

[24] V. Girault and P. A. Raviart, Finite Element Approximation of the NavierStokes Equations. Series in Computational Mathematics, Springer, 1986.10.1007/978-3-642-61623-5_4Search in Google Scholar

[25] V. Heuveline and R. Rannacher, Duality-based adaptivity in the hp-finite element method, J. Numer. Math. 11 (2003), No. 2, 95–113.10.1515/156939503766614126Search in Google Scholar

[26] P. Houston, D. K. Schötzau, and T. P.Wihler, hp-adaptive discontinuous Galerkin finite element methods for the Stokes problem, In: European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS), 2004.10.1007/978-3-642-18775-9_46Search in Google Scholar

[27] J. M. Melenk, hp-interpolation of nonsmooth functions and an application to hp-a posteriori error estimation, SIAM J. Numer. Anal. 43 (2005), 127–155.10.1137/S0036142903432930Search in Google Scholar

[28] J. M. Melenk and B. I. Wohlmuth, On residual-based a posteriori error estimation in hp-FEM, Adv. Comput. Math. 15 (2001), 311–331.10.1023/A:1014268310921Search in Google Scholar

[29] J.M. Melenk and C. Schwab, hp-FEM for reaction–diffusion equations, robust exponential convergence, SIAM J. Numer. Anal. 35 (1998.), 1520–1557.10.1137/S0036142997317602Search in Google Scholar

[30] W. Rachowicz, J.T. Oden, and L. Demkowicz, Toward a universal h-p adaptive finite element strategy, Part 3. Design of h-p meshes, Comput. Meth. Appl. Mech. Engrg. 77 (1989), No. 1-2, 181–212.10.1016/0045-7825(89)90131-XSearch in Google Scholar

[31] D. Schötzau and C. Schwab, Exponential convergence in a Galerkin least squares hp-FEM for Stokes flow, IMA J. Numer. Anal. 21 (2001), 53–80.10.1093/imanum/21.1.53Search in Google Scholar

[32] C. Schwab, p- andhp-Finite Element Methods, Clarendon Press, Oxford, 1998.Search in Google Scholar

[33] L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comput. 54 (1990), 483–493.10.1090/S0025-5718-1990-1011446-7Search in Google Scholar

[34] B. Szabó and I. Babuška, Finite Element Analysis, Wiley, 1991.Search in Google Scholar

[35] R. Verfürth, A posteriori error estimator for the Stokes equations, Numer. Math. 55 (1989), 309–325.10.1007/BF01390056Search in Google Scholar

[36] R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley, Chichester, 1996.Search in Google Scholar

[37] T. P.Wihler, An hp-adaptive strategy based on continuous Sobolev embeddings, J. Comput. Appl. Math. 235 (2011), 2731–2739.10.1016/j.cam.2010.11.023Search in Google Scholar

Received: 2018-04-20
Revised: 2018-10-14
Accepted: 2018-10-27
Published Online: 2018-11-02
Published in Print: 2019-12-18

© 2019 Walter de Gruyter GmbH, Berlin/Boston