Isoparametric finite element approximation of Ricci curvature
Isoparametric finite element approximation of Ricci curvature
复制标题
里奇曲率的等参有限元近似
DOI：
10.1093/imanum/drs037
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发表时间：
2013
影响因子：
2.1
通讯作者：
中科院分区：
文献类型：
--
作者：
曲面有限元法可以用来逼近嵌入超曲面上的曲率，也可以用来离散几何偏微分方程。本文在等参有限元弱形式离散化的基础上，给出了任意维多面体超曲面上离散Ricci曲率的定义。我们证明了对于二维或三维超曲面的分段二次逼近，该定义以L2（Γ）范数的线性收敛阶逼近Γ的Ricci曲率。在Γ的分段线性逼近的情况下，利用光滑化方法，得到了在L2（Γ）模下的阶收敛性和在W1，2（Γ）模下的阶收敛性.
The surface finite element method can be used to approximate curvatures on embedded hypersurfaces and to discretize geometric partial differential equations. In this paper, we present a definition of discrete Ricci curvature on polyhedral hypersurfaces of arbitrary dimension based on the discretization of a weak formulation with isoparametric finite elements. We prove that for a piecewise quadratic approximation of a two- or three-dimensional hypersurface Γ ⊂ ℝn+1, this definition approximates the Ricci curvature of Γ with a linear order of convergence in the L2(Γ) norm. By using a smoothing scheme in the case of a piecewise linear approximation of Γ, we still get a convergence of order ⅔ in the L2(Γ) norm and of order ⅓ in the W1, 2(Γ) norm.