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The Kirchhoff plate equation on surfaces: the surface Hellan–Herrmann–Johnson method
IMA Journal of Numerical Analysis ( IF 2.1 ) Pub Date : 2021-07-07 , DOI: 10.1093/imanum/drab062
Shawn W Walker 1
Affiliation  

We present a mixed finite element method for approximating a fourth-order elliptic partial differential equation (PDE), the Kirchhoff plate equation, on a surface embedded in ${\mathbb {R}}^{3}$, with or without boundary. Error estimates are given in mesh-dependent norms that account for the surface approximation and the approximation of the surface PDE. The method is built on the classic Hellan–Herrmann–Johnson method (for flat domains), and convergence is established for $C^{k+1}$ surfaces, with degree $k$ (Lagrangian, parametrically curved) approximation of the surface, for any $k \geqslant 1$. Mixed boundary conditions are allowed, including clamped, simply-supported and free conditions; if free conditions are present then the surface must be at least $C^{2,1}$. The framework uses tools from differential geometry and is directly related to the seminal work of Dziuk, G. (1988) Finite elements for the Beltrami operator on arbitrary surfaces. Partial Differential Equations and Calculus of Variations, vol. 1357 (S. Hildebrandt & R. Leis eds). Berlin, Heidelberg: Springer, pp. 142–155. for approximating the Laplace–Beltrami equation. The analysis here is the first to handle the full surface Hessian operator directly. Numerical examples are given on nontrivial surfaces that demonstrate our convergence estimates. In addition, we show how the surface biharmonic equation can be solved with this method.

中文翻译:

曲面上的基尔霍夫板方程:曲面 Hellan-Herrmann-Johnson 方法

我们提出了一种混合有限元方法,用于在嵌入 ${\mathbb {R}}^{3}$ 的曲面上逼近四阶椭圆偏微分方程 (PDE),即 Kirchhoff 板方程,无论有无边界。误差估计以与网格相关的范数给出,该范数说明了表面近似和表面 PDE 的近似。该方法建立在经典的 Hellan-Herrmann-Johnson 方法(对于平面域)的基础上,并且建立了 $C^{k+1}$ 个曲面的收敛性,曲面的度数为 $k$(拉格朗日,参数弯曲)逼近, 对于任何 $k \geqslant 1$。允许混合边界条件,包括夹紧、简支和自由条件;如果存在自由条件,则曲面必须至少为 $C^{2,1}$。该框架使用来自微分几何的工具,并且与 Dziuk, G. (1988) 任意表面上的贝尔特拉米算子的有限元的开创性工作直接相关。偏微分方程和变分法,卷。1357(S. Hildebrandt & R. Leis 编)。柏林,海德堡:施普林格,第 142-155 页。用于逼近拉普拉斯-贝尔特拉米方程。这里的分析是第一次直接处理全表面Hessian算子。在非平凡表面上给出了数值示例,证明了我们的收敛估计。此外,我们展示了如何用这种方法求解表面双调和方程。海德堡:施普林格,第 142-155 页。用于逼近拉普拉斯-贝尔特拉米方程。这里的分析是第一次直接处理全表面Hessian算子。在非平凡表面上给出了数值示例,证明了我们的收敛估计。此外,我们展示了如何用这种方法求解表面双调和方程。海德堡:施普林格,第 142-155 页。用于逼近拉普拉斯-贝尔特拉米方程。这里的分析是第一次直接处理全表面Hessian算子。在非平凡表面上给出了数值示例,证明了我们的收敛估计。此外,我们展示了如何用这种方法求解表面双调和方程。
更新日期:2021-07-07
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