arXiv 1906.027862019-06-06math.NAcs.NA

Laplace-Beltrami算子的有限元方法

Finite Element Methods for the Laplace-Beltrami Operator

Andrea Bonito, Alan Demlow, Ricardo H. Nochetto

AI总结：

本文综述了求解嵌入曲面上的Laplace-Beltrami问题的三种有限元方法（参数、迹、窄带），给出最优先验误差估计及后验估计，并强调曲面正则性与方法性质的关系。

摘要：

在曲面上提出的偏微分方程出现在许多应用中。在本综述中，我们描述了三种流行的有限元方法，用于逼近嵌入在$\mathbb{R}^{n+1}$中的$n$维曲面$\gamma$上的Laplace-Beltrami问题的解：参数方法、迹方法和窄带方法。参数方法需要构造一个逼近多面体曲面$\Gamma$，其面构成有限元三角剖分。然后，有限元方法在近似曲面$\Gamma$上以与欧几里得域上的标准有限元方法非常相似的方式提出。在迹方法中，假设给定曲面$\gamma$嵌入在一个$n+1$维域$\Omega$中，该域本身已被三角剖分。然后，粗略地说，通过在$\Omega$的三角剖分上插值$\gamma$来构造一个$n$维近似曲面$\Gamma$，并且$\Gamma$上的有限元空间由$\Omega$上的标准有限元空间在$\Gamma$上的迹（限制）组成。在窄带方法中，曲面上的偏微分方程被扩展到围绕$\gamma$的三角剖分的$n+1$维带中，该带的宽度与三角剖分中元素的直径成正比。在所有情况下，我们为最低阶有限元方法提供了最优的先验误差估计，并且我们还为参数和迹方法提供了后验误差估计。我们的介绍特别关注曲面$\gamma$的正则性（从未假设优于$C^2$类）、$\gamma$在理论和实践中表示的方式以及所得方法的性质之间的关系。

英文摘要：

Partial differential equations posed on surfaces arise in a number of applications. In this survey we describe three popular finite element methods for approximating solutions to the Laplace-Beltrami problem posed on an $n$-dimensional surface $γ$ embedded in $\mathbb{R}^{n+1}$: the parametric, trace, and narrow band methods. The parametric method entails constructing an approximating polyhedral surface $Γ$ whose faces comprise the finite element triangulation. The finite element method is then posed over the approximate surface $Γ$ in a manner very similar to standard FEM on Euclidean domains. In the trace method it is assumed that the given surface $γ$ is embedded in an $n+1$-dimensional domain $Ω$ which has itself been triangulated. An $n$-dimensional approximate surface $Γ$ is then constructed roughly speaking by interpolating $γ$ over the triangulation of $Ω$, and the finite element space over $Γ$ consists of the trace (restriction) of a standard finite element space on $Ω$ to $Γ$. In the narrow band method the PDE posed on the surface is extended to a triangulated $n+1$-dimensional band about $γ$ whose width is proportional to the diameter of elements in the triangulation. In all cases we provide optimal a priori error estimates for the lowest-order finite element methods, and we also present a posteriori error estimates for the parametric and trace methods. Our presentation focuses especially on the relationship between the regularity of the surface $γ$, which is never assumed better than of class $C^2$, the manner in which $γ$ is represented in theory and practice, and the properties of the resulting methods.

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