Abstract: Discrete Laplace-Beltrami operators on polyhedral surfaces play an important role for various applications in geometry processing and related areas like physical simulation or computer graphics. While discretizations of the weak Laplace-Beltrami operator are well-studied, less is known about the strong form. We present a principle for constructing strongly consistent discrete Laplace-Beltrami operators based on the cotan weights. The consistency order we obtain, improves previous results reported for the mesh Laplacian. Furthermore, we prove consistency of the discrete Willmore energies corresponding to the discrete Laplace-Beltrami operators.