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

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

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