Approximative Green's Functions on Surfaces and Pointwise Error Estimates for the Finite Element Method

In this paperwe give a new proof of the L-infinity-error estimate for the finite element approximation of the Laplace-Beltrami equation with an additional lower order term on a surface. While the proof available in the literature uses the method of perturbed bilinear forms from Schatz and Wahlbin, we adapt Scott's proof from an Euclidean setting to the surface case. Furthermore, in contrast to the literature we use an approximative Green's function on the surface instead of an exact Green's function which is obtained by lifting an Euclidean Green's function locally from the tangent plane to the surface.