  A pure and incompressible material is confined between two plates such that
it is heated from below and cooled from above. When its melting temperature is
comprised between these two imposed temperatures, an interface separating
liquid and solid phases appears. Depending on the initial conditions, freezing
or melting occurs until the interface eventually converges towards a stationary
state. This evolution is studied numerically in a two-dimensional configuration
using a phase-field method coupled with the Navier-Stokes equations. Varying
the control parameters of the model, we exhibit two types of equilibria:
diffusive and convective. In the latter case, Rayleigh-B\'enard convection in
the liquid phase shapes the solid-liquid front, and a macroscopic topography is
observed. A simple way of predicting these equilibrium positions is discussed
and then compared with the numerical simulations. In some parameter regimes, we
show that multiple equilibria can coexist depending on the initial conditions.
We also demonstrate that, in this bi-stable regime, transitioning from the
diffusive to the convective equilibrium is inherently a nonlinear mechanism
involving finite-amplitude perturbations.
