arXiv:2610.32270v1 Announce Type: new Abstract: We develop energy-stable parametric finite element methods for surface diffusion of solid films with moving contact lines on curved substrates. Discretizing both the film surface and the substrate patches allows the introduction of linear schemes that are unconditionally energy stable. A nonlinear variant of the schemes ensures that, in addition, the vertices of the substrate approximations remain attached to the true substrate throughout. Moreover, our framework can incorporate anisotropic surface energy densities, for both the film surface and the substrate, as well as different tangential motions. Numerical experiments in two and three space dimensions illustrate the practicality of the introduced methods.