  We derive error estimates for the piecewise linear finite element
approximation of the Laplace--Beltrami operator on a bounded, orientable,
$C^3$, surface without boundary on general shape regular meshes. As an
application, we consider a problem where the domain is split into two regions:
one which has relatively high curvature and one that has low curvature. Using a
graded mesh we prove error estimates that do not depend on the curvature on the
high curvature region. Numerical experiments are provided.
