Contraction property of adaptive hybridizable discontinuous Galerkin methods

Cockburn, Bernardo; Nochetto, Ricardo; Zhang, Wujun

doi:10.1090/mcom/3014

Contraction property of adaptive hybridizable discontinuous Galerkin methods
HTML articles powered by AMS MathViewer

by Bernardo Cockburn, Ricardo H. Nochetto and Wujun Zhang PDF

Math. Comp. 85 (2016), 1113-1141 Request permission

Abstract:

We establish the contraction property between consecutive loops of adaptive hybridizable discontinuous Galerkin methods for the Poisson problem with homogeneous Dirichlet condition. The contractive quantity is the sum of the square of the $L^2$-norm of the flux error, which is not even monotone, and a two-parameter scaled error estimator, which quantifies both the lack of $H(\mathrm {div},\Omega )$-conformity and the deviation from a gradient of the approximate flux. A distinctive and novel feature of this analysis, which enables comparison between two nested meshes, is the lifting of trace residuals from inter-element boundaries to element interiors.

References

A. Alonso, Error estimators for a mixed method, Numer. Math. 74 (1996), no. 4, 385–395. MR 1414415, DOI 10.1007/s002110050222
Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/02), no. 5, 1749–1779. MR 1885715, DOI 10.1137/S0036142901384162
Douglas N. Arnold, L. Ridgway Scott, and Michael Vogelius, Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), no. 2, 169–192 (1989). MR 1007396
Roland Becker and Shipeng Mao, An optimally convergent adaptive mixed finite element method, Numer. Math. 111 (2008), no. 1, 35–54. MR 2448202, DOI 10.1007/s00211-008-0180-8
Roland Becker and Shipeng Mao, Quasi-optimality of adaptive nonconforming finite element methods for the Stokes equations, SIAM J. Numer. Anal. 49 (2011), no. 3, 970–991. MR 2802555, DOI 10.1137/100802967
Roland Becker, Shipeng Mao, and Zhongci Shi, A convergent nonconforming adaptive finite element method with quasi-optimal complexity, SIAM J. Numer. Anal. 47 (2010), no. 6, 4639–4659. MR 2595052, DOI 10.1137/070701479
Peter Binev, Wolfgang Dahmen, and Ron DeVore, Adaptive finite element methods with convergence rates, Numer. Math. 97 (2004), no. 2, 219–268. MR 2050077, DOI 10.1007/s00211-003-0492-7
Andrea Bonito and Ricardo H. Nochetto, Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method, SIAM J. Numer. Anal. 48 (2010), no. 2, 734–771. MR 2670003, DOI 10.1137/08072838X
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR 1894376, DOI 10.1007/978-1-4757-3658-8
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
Carsten Carstensen and Ronald H. W. Hoppe, Convergence analysis of an adaptive nonconforming finite element method, Numer. Math. 103 (2006), no. 2, 251–266. MR 2222810, DOI 10.1007/s00211-005-0658-6
Carsten Carstensen, Daniel Peterseim, and Hella Rabus, Optimal adaptive nonconforming FEM for the Stokes problem, Numer. Math. 123 (2013), no. 2, 291–308. MR 3010181, DOI 10.1007/s00211-012-0490-8
J. Manuel Cascon, Christian Kreuzer, Ricardo H. Nochetto, and Kunibert G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 (2008), no. 5, 2524–2550. MR 2421046, DOI 10.1137/07069047X
J. M. Cascón and R. H. Nochetto, Quasioptimal cardinality of AFEM driven by nonresidual estimators, IMA J. Numer. Anal. 32 (2012), no. 1, 1–29.
Long Chen, Michael Holst, and Jinchao Xu, Convergence and optimality of adaptive mixed finite element methods, Math. Comp. 78 (2009), no. 265, 35–53. MR 2448696, DOI 10.1090/S0025-5718-08-02155-8
B. Cockburn, O. Dubois, J. Gopalakrishnan, and S. Tan, Multigrid for an HDG method, IMA J. Numer. Anal. 34 (2014), no. 4, 1386–1425. MR 3269430, DOI 10.1093/imanum/drt024
Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1319–1365. MR 2485455, DOI 10.1137/070706616
Bernardo Cockburn, Jayadeep Gopalakrishnan, and Francisco-Javier Sayas, A projection-based error analysis of HDG methods, Math. Comp. 79 (2010), no. 271, 1351–1367. MR 2629996, DOI 10.1090/S0025-5718-10-02334-3
Bernardo Cockburn, Johnny Guzmán, and Haiying Wang, Superconvergent discontinuous Galerkin methods for second-order elliptic problems, Math. Comp. 78 (2009), no. 265, 1–24. MR 2448694, DOI 10.1090/S0025-5718-08-02146-7
B. Cockburn, R.H. Nochetto, and W. Zhang, Approximation class and quasi-optimality of adaptive hybridizable discontinuous Galerkin methods. Submitted.
Bernardo Cockburn and Francisco-Javier Sayas, Divergence-conforming HDG methods for Stokes flows, Math. Comp. 83 (2014), no. 288, 1571–1598. MR 3194122, DOI 10.1090/S0025-5718-2014-02802-0
Bernardo Cockburn, Weifeng Qiu, and Ke Shi, Conditions for superconvergence of HDG methods for second-order elliptic problems, Math. Comp. 81 (2012), no. 279, 1327–1353. MR 2904581, DOI 10.1090/S0025-5718-2011-02550-0
Bernardo Cockburn and Wujun Zhang, A posteriori error analysis for hybridizable discontinuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 51 (2013), no. 1, 676–693. MR 3033028, DOI 10.1137/120866269
Bernardo Cockburn and Wujun Zhang, A posteriori error estimates for HDG methods, J. Sci. Comput. 51 (2012), no. 3, 582–607. MR 2914423, DOI 10.1007/s10915-011-9522-2
Willy Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. MR 1393904, DOI 10.1137/0733054
Ricardo G. Durán and Maria Amelia Muschietti, An explicit right inverse of the divergence operator which is continuous in weighted norms, Studia Math. 148 (2001), no. 3, 207–219. MR 1880723, DOI 10.4064/sm148-3-2
R. H. W. Hoppe, G. Kanschat, and T. Warburton, Convergence analysis of an adaptive interior penalty discontinuous Galerkin method, SIAM J. Numer. Anal. 47 (2008/09), no. 1, 534–550. MR 2475951, DOI 10.1137/070704599
Jun Hu and Jinchao Xu, Convergence and optimality of the adaptive nonconforming linear element method for the Stokes problem, J. Sci. Comput. 55 (2013), no. 1, 125–148. MR 3030706, DOI 10.1007/s10915-012-9625-4
Ohannes A. Karakashian and Frederic Pascal, Convergence of adaptive discontinuous Galerkin approximations of second-order elliptic problems, SIAM J. Numer. Anal. 45 (2007), no. 2, 641–665. MR 2300291, DOI 10.1137/05063979X
Igor Kossaczký, A recursive approach to local mesh refinement in two and three dimensions, J. Comput. Appl. Math. 55 (1994), no. 3, 275–288. MR 1329875, DOI 10.1016/0377-0427(94)90034-5
Shipeng Mao, Xuying Zhao, and Zhongci Shi, Convergence of a standard adaptive nonconforming finite element method with optimal complexity, Appl. Numer. Math. 60 (2010), no. 7, 673–688. MR 2646469, DOI 10.1016/j.apnum.2010.03.010
Khamron Mekchay and Ricardo H. Nochetto, Convergence of adaptive finite element methods for general second order linear elliptic PDEs, SIAM J. Numer. Anal. 43 (2005), no. 5, 1803–1827. MR 2192319, DOI 10.1137/04060929X
Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal. 38 (2000), no. 2, 466–488. MR 1770058, DOI 10.1137/S0036142999360044
Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Convergence of adaptive finite element methods, SIAM Rev. 44 (2002), no. 4, 631–658 (2003). Revised reprint of “Data oscillation and convergence of adaptive FEM” [SIAM J. Numer. Anal. 38 (2000), no. 2, 466–488 (electronic); MR1770058 (2001g:65157)]. MR 1980447, DOI 10.1137/S0036144502409093
Ricardo H. Nochetto, Kunibert G. Siebert, and Andreas Veeser, Theory of adaptive finite element methods: an introduction, Multiscale, nonlinear and adaptive approximation, Springer, Berlin, 2009, pp. 409–542. MR 2648380, DOI 10.1007/978-3-642-03413-8_{1}2
H. Rabus, A natural adaptive nonconforming FEM of quasi-optimal complexity, Comput. Methods Appl. Math. 10 (2010), no. 3, 315–325. MR 2770297, DOI 10.2478/cmam-2010-0018
P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292–315. MR 0483555
Rob Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math. 7 (2007), no. 2, 245–269. MR 2324418, DOI 10.1007/s10208-005-0183-0
Rob Stevenson, The completion of locally refined simplicial partitions created by bisection, Math. Comp. 77 (2008), no. 261, 227–241. MR 2353951, DOI 10.1090/S0025-5718-07-01959-X