  The Intrinsic Surface Finite Element Method (ISFEM) was recently proposed to
solve Partial Differential Equations (PDEs) on surfaces. ISFEM proceeds by
writing the PDE with respect to a local coordinate system anchored to the
surface and makes direct use of the resulting covariant basis. Starting from a
shape-regular triangulation of the surface, existence of a local
parametrization for each triangle is exploited to approximate relevant
quantities on the local chart. Standard two-dimensional FEM techniques in
combination with surface quadrature rules complete the ISFEM formulation thus
achieving a method that is fully intrinsic to the surface and makes limited use
of the surface embedding only for the definition of linear basis functions.
However, theoretical properties have not yet been proved. In this work, we
complement the original derivation of ISFEM with its complete convergence
theory and propose the analysis of the stability and error estimates by
carefully tracking the role of the geometric quantities in the constants of the
error inequalities. Numerical experiments are included to support the
theoretical results.
