  An energy stable finite element scheme within arbitrary Lagrangian Eulerian
(ALE) framework is derived for simulating the dynamics of millimetric droplets
in contact with solid surfaces. Supporting surfaces considered may exhibit
non--homogeneous properties which are incorporated into system through
generalized Navier boundary conditions (GNBC). Numerical scheme is constructed
such that the counterpart of (continuous) energy balance holds on the discrete
level. This ensures that no spurious energy is introduced into the discrete
system, i.e. the discrete formulation is stable in the energy norm. The newly
proposed scheme is numerically validated to confirm the theoretical
predictions. Of a particular interest is the case of droplet on a
non-homogeneous inclined surface. This case shows the capabilities of the
scheme to capture the complex droplet dynamics (sliding and rolling) while
maintaining stability during the long time simulation.
