L2 and pointwise a posteriori error estimates for FEM for elliptic PDEs on surfaces

Surface finite element methods (SFEMs) are widely used to solve surface partial differential equations arising in applications including crystal growth, fluid mechanics and computer graphics. A posteriori error estimators are computable measures of the error and are used to implement adaptive mesh refinement. Previous studies of a posteriori error estimation in SFEM have mainly focused on bounding energy norm errors. In this work, we derive a posteriori L-2 and pointwise error estimates for piecewise linear SFEM for the Laplace-Beltrami equation on implicitly defined surfaces. There are two main error sources in SFEM, a `Galerkin error' arising in the usual way for finite element methods, and a 'geometric error' arising from replacing the continuous surface by a discrete approximation when writing the finite element equations. Our work includes numerical estimation of the dependence of the error bounds on the geometric properties of the surface. We provide also numerical experiments where the estimators have been used to implement an adaptive FEM over surfaces with different curvatures.