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Hessian recovery for finite element methods. (English) Zbl 1361.65087

Summary: In this article, we propose and analyze an effective Hessian recovery strategy for the Lagrangian finite element method of arbitrary order. We prove that the proposed Hessian recovery method preserves polynomials of degree \( k+1\) on general unstructured meshes and superconverges at a rate of \( O(h^k)\) on mildly structured meshes. In addition, the method is proved to be ultraconvergent (two orders higher) for the translation invariant finite element space of any order. Numerical examples are presented to support our theoretical results.

MSC:

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

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References:

[1] Agouzal, A.; Vassilevski, Yu., On a discrete Hessian recovery for \(P_1\) finite elements, J. Numer. Math., 10, 1, 1-12 (2002) · Zbl 1010.65009 · doi:10.1515/JNMA.2002.1
[2] Aguilera, N{\'e}stor E.; Morin, Pedro, On convex functions and the finite element method, SIAM J. Numer. Anal., 47, 4, 3139-3157 (2009) · Zbl 1204.65076 · doi:10.1137/080720917
[3] Bank, Randolph E.; Xu, Jinchao, Asymptotically exact a posteriori error estimators. I. Grids with superconvergence, SIAM J. Numer. Anal., 41, 6, 2294-2312 (electronic) (2003) · Zbl 1058.65116 · doi:10.1137/S003614290139874X
[4] Brenner, Susanne C.; Scott, L. Ridgway, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics 15, xviii+397 pp. (2008), Springer, New York · Zbl 1135.65042 · doi:10.1007/978-0-387-75934-0
[5] Cao, Weiming, Superconvergence analysis of the linear finite element method and a gradient recovery postprocessing on anisotropic meshes, Math. Comp., 84, 291, 89-117 (2015) · Zbl 1305.65226 · doi:10.1090/S0025-5718-2014-02846-9
[6] Ciarlet, Philippe G., The Finite Element Method for Elliptic Problems, xix+530 pp. (1978), North-Holland Publishing Co., Amsterdam-New York-Oxford · Zbl 0999.65129
[7] Evans, Lawrence C., Partial Differential Equations, Graduate Studies in Mathematics 19, xxii+749 pp. (2010), American Mathematical Society, Providence, RI · Zbl 1194.35001 · doi:10.1090/gsm/019
[8] gan X. Gan and J.E. Akin, Superconvergent second order derivative recovery technique and its application in a nonlocal damage mechanics model, Finite Elements in Analysis and Design 35 (2014), 118-127.
[9] Huang, Can; Zhang, Zhimin, Polynomial preserving recovery for quadratic elements on anisotropic meshes, Numer. Methods Partial Differential Equations, 28, 3, 966-983 (2012) · doi:10.1002/num.20669
[10] Huang, Yunqing; Xu, Jinchao, Superconvergence of quadratic finite elements on mildly structured grids, Math. Comp., 77, 263, 1253-1268 (2008) · Zbl 1195.65193 · doi:10.1090/S0025-5718-08-02051-6
[11] Kamenski, Lennard; Huang, Weizhang, How a nonconvergent recovered Hessian works in mesh adaptation, SIAM J. Numer. Anal., 52, 4, 1692-1708 (2014) · Zbl 1303.65101 · doi:10.1137/120898796
[12] Lakhany, A. M.; Whiteman, J. R., Superconvergent recovery operators: derivative recovery techniques. Finite element methods, Jyv\`“askyl\'”a, 1997, Lecture Notes in Pure and Appl. Math. 196, 195-215 (1998), Dekker, New York · Zbl 0902.65057
[13] Lakkis, Omar; Pryer, Tristan, A finite element method for second order nonvariational elliptic problems, SIAM J. Sci. Comput., 33, 2, 786-801 (2011) · Zbl 1227.65114 · doi:10.1137/100787672
[14] Lakkis, Omar; Pryer, Tristan, A finite element method for nonlinear elliptic problems, SIAM J. Sci. Comput., 35, 4, A2025-A2045 (2013) · Zbl 1362.65126 · doi:10.1137/120887655
[15] Naga, Ahmed; Zhang, Zhimin, A posteriori error estimates based on the polynomial preserving recovery, SIAM J. Numer. Anal., 42, 4, 1780-1800 (electronic) (2004) · Zbl 1078.65098 · doi:10.1137/S0036142903413002
[16] Naga, A.; Zhang, Z., The polynomial-preserving recovery for higher order finite element methods in 2D and 3D, Discrete Contin. Dyn. Syst. Ser. B, 5, 3, 769-798 (2005) · Zbl 1078.65108 · doi:10.3934/dcdsb.2005.5.769
[17] easymesh B. Niceno, EasyMesh Version 1.4: A Two-Dimensional Quality Mesh Generator, http://www-dinma.univ.trieste.it/nirftc/research/easymesh.
[18] Neilan, Michael, Finite element methods for fully nonlinear second order PDEs based on a discrete Hessian with applications to the Monge-Amp\`ere equation, J. Comput. Appl. Math., 263, 351-369 (2014) · Zbl 1301.65124 · doi:10.1016/j.cam.2013.12.027
[19] Nitsche, Joachim A.; Schatz, Alfred H., Interior estimates for Ritz-Galerkin methods, Math. Comp., 28, 937-958 (1974) · Zbl 0298.65071
[20] Ovall, Jeffrey S., Function, gradient, and Hessian recovery using quadratic edge-bump functions, SIAM J. Numer. Anal., 45, 3, 1064-1080 (2007) · Zbl 1149.65088 · doi:10.1137/060648908
[21] Pouliot, B.; Fortin, M.; Fortin, A.; Chamberland, {\'E}., On a new edge-based gradient recovery technique, Internat. J. Numer. Methods Engrg., 93, 1, 52-65 (2013) · Zbl 1352.65536 · doi:10.1002/nme.4374
[22] Picasso, Marco; Alauzet, Fr{\'e}d{\'e}ric; Borouchaki, Houman; George, Paul-Louis, A numerical study of some Hessian recovery techniques on isotropic and anisotropic meshes, SIAM J. Sci. Comput., 33, 3, 1058-1076 (2011) · Zbl 1232.65147 · doi:10.1137/100798715
[23] Vallet, M.-G.; Manole, C.-M.; Dompierre, J.; Dufour, S.; Guibault, F., Numerical comparison of some Hessian recovery techniques, Internat. J. Numer. Methods Engrg., 72, 8, 987-1007 (2007) · Zbl 1194.76145 · doi:10.1002/nme.2036
[24] Wahlbin, Lars B., Superconvergence in Galerkin finite element methods, Lecture Notes in Mathematics 1605, xii+166 pp. (1995), Springer-Verlag, Berlin · Zbl 0826.65092
[25] Wu, Haijun; Zhang, Zhimin, Can we have superconvergent gradient recovery under adaptive meshes?, SIAM J. Numer. Anal., 45, 4, 1701-1722 (2007) · Zbl 1152.65104 · doi:10.1137/060661430
[26] Xu, Jinchao; Zhang, Zhimin, Analysis of recovery type a posteriori error estimators for mildly structured grids, Math. Comp., 73, 247, 1139-1152 (electronic) (2004) · Zbl 1050.65103 · doi:10.1090/S0025-5718-03-01600-4
[27] Zhang, Zhimin; Naga, Ahmed, A new finite element gradient recovery method: superconvergence property, SIAM J. Sci. Comput., 26, 4, 1192-1213 (electronic) (2005) · Zbl 1078.65110 · doi:10.1137/S1064827503402837
[28] Zienkiewicz, O. C.; Zhu, J. Z., The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique, Internat. J. Numer. Methods Engrg., 33, 7, 1331-1364 (1992) · Zbl 0769.73084 · doi:10.1002/nme.1620330702
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