Optimal Hessian recovery using a biorthogonal system with an application to adaptive refinement

Abstract

We present a method for recovering the Hessian from a linear finite element approach to achieve a higher rate of convergence. This method uses an \(L^{2}\)-based projection as well as boundary modification to achieve and improve the convergence rate. The projection uses a biorthogonal system to make the computation more numerically efficient. We present numerical examples to illustrate the efficiency and optimality of our approach on different meshes. The performance of our approach on adaptively refined meshes is briefly explored. References R. E. Bank, A. H. Sherman, and A. Weiser. Some refinement algorithms and data structures for regular local mesh refinement. Scientific computing: Applications of mathematics and computing to the physical sciences. Ed. by R. S. Stepleman. North-Holland Publishing, 1983, pp. 3–17 R. E. Bank and J. Xu. Asymptotically exact a posteriori error estimators, Part I: Grids with superconvergence. SIAM J. Numer. Anal. 41 (2003), pp. 2294–2312. doi: 10.1137/S003614290139874X J. H. Bramble and A. H. Schatz. Higher order local accuracy by averaging in the finite element method. Math. Comput. 31.137 (1977), pp. 94–111. doi: 10.2307/2005782. S. A. Funken and A. Schmidt. Adaptive mesh refinement in 2D—An efficient implementation in Matlab. Comput. Meth. Appl. Math. 20.3 (2020), pp. 459–479. doi: doi:10.1515/cmam-2018-0220. H. Guo, Z. Zhang, and R. Zhao. Hessian recovery for finite element methods. Math. Comput. 86.306 (2017), pp. 1671–1692. url: https://www.jstor.org/stable/90004689 B.-O. Heimsund, X.-C. Tai, and J. Wang. Superconvergence for the gradient of finite element approximations by L2 projections. SIAM J. Numer. Anal. 40.4 (2002), pp. 1263–1280. doi: 10.1137/S003614290037410X. Y. Huang and N. Yi. The superconvergent cluster recovery method. J. Sci. Comput. 44 (2010), pp. 301–322. doi: 10.1007/s10915-010-9379-9 M. Ilyas, B. P. Lamichhane, and M. H. Meylan. A gradient recovery method based on an oblique projection and boundary modification. Proceedings of the 18th Biennial Computational Techniques and Applications Conference, CTAC-2016. Ed. by J. Droniou, M. Page, and S. Clarke. Vol. 58. ANZIAM J. Aug. 2017, pp. C34–C45. doi: 10.21914/anziamj.v58i0.11730 L. Kamenski and W. Huang. How A nonconvergent recovered Hessian works in mesh adaptation. SIAM J. Numer. Anal. 52.4 (2014), pp. 1692–1708. url: http://www.jstor.org/stable/24512164 M. Krízek and P. Neittaanmäki. Superconvergence phenomenon in the finite element method arising from averaging gradients. Numer. Math. 45 (1984), pp. 105–116. doi: 10.1007/BF01379664 B. P. Lamichhane and J. Shaw-Carmody. A gradient recovery approach for nonconforming finite element methods with boundary modification. Proceedings of the 19th Biennial Computational Techniques and Applications Conference, CTAC-2020. Ed. by W. McLean, S. Macnamara, and J. Bunder. Vol. 62. ANZIAM J. Feb. 2022, pp. C163–C175. doi: 10.21914/anziamj.v62.16032 A. Naga and Z. Zhang. A posteriori error estimates based on the polynomial preserving recovery. SIAM J. Numer. Anal. 42.4 (2004), pp. 1780–1800. doi: 10.1137/S0036142903413002 Z. Zhang and A. Naga. A new finite element gradient recovery method: Superconvergence property. SIAM J. Sci. Comput. 26.4 (2005), pp. 1192–1213. doi: 10.1137/S1064827503402837 O. C. Zienkiewicz and J. Z. Zhu. The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique. Int. J. Numer. Meth. Eng. 33 (1992), pp. 1331–1364. doi: 10.1002/nme.1620330702