Finite Element Methods for the Laplace-Beltrami Operator

Abstract

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.

Figures (3)

• Figure 1: Two different configurations when $n=2$ illustrating that the number of elements sharing the same vertex could be arbitrarily large even when using triangles satisfying \ref{['e:shape_reg_init']}.
• Figure 2: Non-overlapping parametrizations ${\bf X}_T : \widehat{T} \rightarrow T$ of $\Gamma$ and $\boldsymbol{\mathcal{\chi}} _T : \widehat{T} \rightarrow \widetilde{T}$ of $\gamma$.
• Figure 3: Bulk mesh cutaway with associated trace mesh (left); blowup of a trace mesh showing small and narrow elements (right).

Theorems & Definitions (114)

• Lemma 1: weak form of the Laplace-Beltrami operator
• proof
• Lemma 2: Poincaré-Friedrichs inequality
• proof
• Lemma 3: regularity
• proof
• Lemma 4: regularity for $W^2_p$ surfaces
• proof
• Remark 5: parametric independence
• Lemma 6: curvatures of parallel surface