The Kirchhoff plate equation on surfaces: the surface Hellan–Herrmann–Johnson method
The Kirchhoff plate equation on surfaces: the surface Hellan–Herrmann–Johnson method
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DOI：
10.1093/imanum/drab062
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发表时间：
2021-08
影响因子：
2.1
通讯作者：
Shawn W. Walker
中科院分区：
文献类型：
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作者：
Shawn W. Walker
本文提出了一种混合有限元方法来逼近四阶椭圆型偏微分方程（PDE），Kirchhoff板方程，在嵌入${\mathbb {R}}^{3}$的表面上，有边界或无边界.误差估计在网格相关的规范，占表面近似和表面偏微分方程的近似。该方法是建立在经典的Hellan-Herrmann-约翰逊方法（平坦域），并建立收敛$C^{k+1}$表面，与度$k$（拉格朗日，参数弯曲）近似的表面，任何$k \geqslant 1$。混合边界条件是允许的，包括固定、简支和自由条件;如果存在自由条件，则表面必须至少为$C^{2，1}$。该框架使用微分几何的工具，并与Dziuk，G。1988年：《Beltrami算子在任意曲面上的有限元》。偏微分方程和变分法，第1357卷（S。希尔德布兰特河Leis eds）.柏林，海德堡：Springer，pp. 142-155.近似拉普拉斯-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。数值例子给出了非平凡的表面，证明我们的收敛估计。此外，我们展示了如何表面双调和方程可以用这种方法来解决。
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.