  Mineral precipitation and dissolution processes in a porous medium can alter
the structure of the medium at the scale of pores. Such changes make numerical
simulations a challenging task as the geometry of the pores changes in time in
an apriori unknown manner. To deal with such aspects, we here adopt a two-scale
phase-field model, and propose a robust scheme for the numerical approximation
of the solution. The scheme takes into account both the scale separation in the
model, as well as the non-linear character of the model. After proving the
convergence of the scheme, an adaptive two-scale strategy is incorporated,
which improves the efficiency of the simulations. Numerical tests are
presented, showing the efficiency and accuracy of the scheme in the presence of
anisotropies and heterogeneities.
