使用双正交系统的最佳黑森恢复，并应用于自适应细化

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-03-17 DOI:10.21914/anziamj.v64.17971

Jordan A. Shaw-Carmody, B. Lamichhane

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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\nJ. 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.\nS. 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.\nH. 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\nB.-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.\nY. Huang and N. Yi. The superconvergent cluster recovery method. J. Sci. Comput. 44 (2010), pp. 301–322. doi: 10.1007/s10915-010-9379-9 \nM. 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,\nM. 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\nM. 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\nB. P. Lamichhane and J. Shaw-Carmody. A gradient recovery approach for nonconforming finite element methods with boundary modification. 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引用次数: 0

摘要

我们提出了一种从线性有限元方法中恢复 Hessian 的方法，以实现更高的收敛率。该方法使用基于 \(L^{2}\) 的投影以及边界修正来实现和提高收敛率。投影使用双正交系统，使计算在数值上更加高效。我们通过数值示例来说明我们的方法在不同网格上的效率和最优性。我们还简要探讨了我们的方法在自适应细化网格上的性能。E. Bank、A. H. Sherman 和 A. Weiser。常规局部网格细化的一些细化算法和数据结构。科学计算：数学和计算在物理科学中的应用》。R. S. Stepleman 编。North-Holland Publishing, 1983, pp.E. Bank and J. Xu.渐近精确后验误差估计器，第一部分：超收敛网格。SIAM J. Numer.Anal.41 (2003), pp.H. Bramble and A. H. Schatz.有限元法中的高阶局部精度平均法.Math.31.137 (1977).31.137 (1977)，pp. 94-111. Doi: 10.2307/2005782.S. A. Funken and A. Schmidt.二维自适应网格细化--Matlab 中的高效实现.Comput.Meth.应用数学。20.3 (2020), pp.有限元方法的Hessian恢复.Math.计算。86.306（2017），第 1671-1692 页。网址：https://www.jstor.org/stable/90004689B.-O.Heimsund, X.-C. Tai, and J. Wang.Heimsund, X.-C. Tai, and J. Wang.有限元近似 L2 投影梯度的超收敛性。SIAM J. Numer.Anal.40.4 (2002), pp.超融合聚类恢复方法.J. Sci.44 (2010), pp.基于斜投影和边界修正的梯度恢复方法.第 18 届计算技术与应用双年会 CTAC-2016 会议论文集》。Ed. by J. Droniou,M. Page, and S. Clarke.Page, and S. Clarke.第 58 卷。ANZIAM J. Aug. 2017, pp. C34-C45. doi: 10.21914/anziamj.v58i0.11730 L. Kamenski and W. Huang.非收敛恢复的 Hessian 如何在网格适应中起作用。SIAM J. Numer.Anal.52.4 (2014), pp. 1692-1708. url: http://www.jstor.org/stable/24512164M.Krízek and P. Neittaanmäki.平均梯度引起的有限元法超收敛现象。Numer.Numer.45 (1984), pp.P. Lamichhane and J. Shaw-Carmody.具有边界修正的不符合有限元方法的梯度恢复方法.第 19 届计算技术与应用双年会论文集 CTAC-2020.Ed. byW.McLean, S. Macnamara, and J. Bunder.第 62 卷。2022 年 2 月，第 C163-C175 页。DOI：10.21914/anziamj.v62.16032 A. Naga 和 Z. Zhang.基于多项式保全恢复的后验误差估计.SIAM J. Numer.Anal.42.4 (2004), pp.一种新的有限元梯度恢复方法：超收敛特性.SIAM J. Sci.26.4 (2005), pp.超融合补丁恢复与后验误差估计。第一部分：恢复技术。Int. J. Numer.J. Numer.Meth.Numer.33 (1992), pp.

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

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

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

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