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The Hellan--Herrmann--Johnson Method with Curved Elements
SIAM Journal on Numerical Analysis ( IF 2.8 ) Pub Date : 2020-01-01 , DOI: 10.1137/19m1288723
Douglas N. Arnold , Shawn W. Walker

We study the finite element approximation of the Kirchhoff plate equation on domains with curved boundaries using the Hellan-Herrmann-Johnson (HHJ) method. We prove optimal convergence on domains with piecewise $C^{k+1}$ boundary for $k \geq 1$ when using a parametric (curved) HHJ space. Computational results are given that demonstrate optimal convergence and how convergence degrades when curved triangles of insufficient polynomial degree are used. Moreover, we show that the lowest order HHJ method on a polygonal approximation of the disk does not succumb to the classic Babuska paradox, highlighting the geometrically non-conforming aspect of the HHJ method.

中文翻译:

具有弯曲单元的 Hellan--Herrmann--Johnson 方法

我们使用 Hellan-Herrmann-Johnson (HHJ) 方法研究具有弯曲边界的域上 Kirchhoff 板方程的有限元近似。当使用参数化(弯曲)HHJ 空间时,我们证明了 $k \geq 1$ 的分段 $C^{k+1}$ 边界域的最优收敛。给出的计算结果证明了最佳收敛性以及当使用多项式次数不足的弯曲三角形时收敛性如何降低。此外,我们表明圆盘的多边形近似上的最低阶 HHJ 方法不会屈服于经典的 Babuska 悖论,突出了 HHJ 方法在几何上不符合的方面。
更新日期:2020-01-01
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