Chapter 1 Finite element methods for the LaplaceBeltrami operator
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2019 Elsevier B.V. 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 LaplaceBeltrami problem posed on an n-dimensional surface embedded in Rn+1: the parametric, trace, and narrow band methods. The parametric method entails constructing an approximating polyhedral surface whose faces comprise the finite element triangulation. The finite element method is then posed over the approximate surface in a manner very similar to standard FEM on Euclidean domains. In the trace method it is assumed that the given surface is embedded in an n + 1-dimensional domain which has itself been triangulated. An n-dimensional approximate surface is then constructed roughly speaking by interpolating over the triangulation of , and the finite element space over consists of the trace (restriction) of a standard finite element space on to . In the narrow band method the PDE posed on the surface is extended to a triangulated n + 1-dimensional band about 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 , which is never assumed better than of class C2, the manner in which is represented in theory and practice, and the properties of the resulting methods.
