  Over the last few decades, phase-field equations have found increasing
applicability in a wide range of mathematical-scientific fields (e.g. geometric
PDEs and mean curvature flow, materials science for the study of phase
transitions) but also engineering ones (e.g. as a computational tool in
chemical engineering for interfacial flow studies). Here, we focus on
phase-field equations in strongly heterogeneous materials with perforations
such as porous media. To the best of our knowledge, we provide the first
derivation of error estimates for fourth order, homogenized, and nonlinear
evolution equations. Our fourth order problem induces a slightly lower
convergence rate, i.e., $\epsilon^{1/4}$, where $\epsilon$ denotes the
material's specific heterogeneity, than established for second-order elliptic
problems (e.g. \cite{Zhikov2006}) for the error between the effective
macroscopic solution of the (new) upscaled formulation and the solution of the
microscopic phase field problem. We hope that our study will motivate new
modelling, analytic, and computational perspectives for interfacial transport
and phase transformations in strongly heterogeneous environments.
