  Biological cells utilize membranes and liquid-like droplets, known as
biomolecular condensates, to structure their interior. The interaction of
droplets and membranes, despite being involved in several key biological
processes, is so far little understood. Here, we present a first numerical
method to simulate the continuum dynamics of droplets interacting with
deformable membranes via wetting. The method combines the advantages of the
phase-field method for multi-phase flow simulation and the arbitrary
Lagrangian-Eulerian (ALE) method for an explicit description of the elastic
surface. The model is thermodynamically consistent, coupling bulk hydrodynamics
with capillary forces, as well as bending, tension, and stretching of a thin
membrane. The method is validated by comparing simulations for single droplets
to theoretical results of shape equations, and its capabilities are illustrated
in 2D and 3D axisymmetric scenarios.
