Rigorous derivation of porous-media phase-field equations

The evolution of interfaces in Complex heterogeneous Multiphase Systems (CheMSs) plays a fundamental role in a wide range of scientific fields such as thermodynamic modelling of phase transitions, materials science, or as a computational tool for interfacial flow studies or material design. Here, we focus on phase-field equations in CheMSs such as porous media [1,2]. To the best of our knowledge, we present the first rigorous derivation of error estimates for fourth order, upscaled, and nonlinear evolution equations. For CheMs with heterogeneity $\epsilon$, we obtain the convergence rate $\epsilon^{1/4}$ , which governs the error between the solution of the new upscaled formulation and the solution of the microscopic phase-field problem [1,2]. This error behaviour has recently been validated computationally in [3]. Due to the wide range of application of phase-field equations, we expect this upscaled formulation to allow for new modelling, analytic, and computational perspectives for interfacial transport and phase transformations in CheMSs. [1] M. Schmuck $\&$ S. Kalliadasis, SIAM J. Appl. Math, accepted (2017). [2] M. Schmuck et al., Nonlinearity, 26(12):3259-3277 (2013). [3] A. Ververis $\&$ M. Schmuck, J. Comp. Phys., 344:485-498 (2017).