Abstract | ||
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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 $\gamma$ embedded in $\mathbb{R}^{n+1}$: the parametric, trace, and narrow band methods. The parametric method entails constructing an approximating polyhedral surface $\Gamma$ whose faces comprise the finite element triangulation. The finite element method is then posed over the approximate surface $\Gamma$ in a manner very similar to standard FEM on Euclidean domains. In the trace method it is assumed that the given surface $\gamma$ is embedded in an $n+1$-dimensional domain $\Omega$ which has itself been triangulated. An $n$-dimensional approximate surface $\Gamma$ is then constructed roughly speaking by interpolating $\gamma$ over the triangulation of $\Omega$, and the finite element space over $\Gamma$ consists of the trace (restriction) of a standard finite element space on $\Omega$ to $\Gamma$. In the narrow band method the PDE posed on the surface is extended to a triangulated $n+1$-dimensional band about $\gamma$ 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 $\gamma$, which is never assumed better than of class $C^2$, the manner in which $\gamma$ is represented in theory and practice, and the properties of the resulting methods. |
Year | Venue | DocType |
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2019 | CoRR | Journal |
Volume | Citations | PageRank |
abs/1906.02786 | 0 | 0.34 |
References | Authors | |
0 | 3 |
Name | Order | Citations | PageRank |
---|---|---|---|
Andrea Bonito | 1 | 141 | 19.34 |
Alan Demlow | 2 | 162 | 21.97 |
Ricardo H. Nochetto | 3 | 907 | 110.08 |