A Posteriori Error Estimates for the Laplace--Beltrami Operator on Parametric $C^2$ Surfaces

Abstract

We prove new a posteriori error estimates for surface finite element methods (SFEMs). SFEMs approximate solutions to PDEs posed on surfaces. Prototypical examples are elliptic PDEs involving the Laplace--Beltrami operator. Typically the surface is approximated by a polyhedral or higher-order polynomial approximation. The resulting finite element method exhibits both a geometric consistency error due to the surface approximation and a standard Galerkin error. A posteriori estimates for SFEMs require practical access to geometric information about the surface in order to computably bound the geometric error. It is thus advantageous to allow for maximum flexibility in representing surfaces in practical codes when proving a posteriori error estimates for SFEMs. However, previous a posteriori estimates using general parametric surface representations are suboptimal by one order on $C^2$ surfaces. Proofs of error estimates optimally reflecting the geometric error instead employ the closest point projection, which is defined using the signed distance function. Because the closest point projection is often unavailable or inconvenient to use computationally, a posteriori estimates using the signed distance function have notable practical limitations. We merge these two perspectives by assuming practical access only to a general parametric representation of the surface, but using the distance function as a theoretical tool. This allows us to derive sharper geometric estimators which exhibit improved experimentally observed decay rates when implemented in adaptive surface finite element algorithms.