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We approximate the solution of the equation $$ -\Delta_S u+u = f $$ on atwo-dimensional, embedded, orientable, closed surface $S$ where $-\Delta_S$denotes the Laplace Beltrami operator on $S$ by using continuous, piecewiselinear finite elements on a triangulation of $S$ with flat triangles. We showthat the $L^{\infty}$-error is of order $O(h^2|\log h|)$ as in thecorresponding situation in an Euclidean setting.

$L^{\infty}$-error estimate for the finite element method on two dimensional surfaces