We study the numerical solution of a Cahn-Hilliard/Allen-Cahn system with strong coupling through state and gradient dependent non-diagonal mobility matrices. A fully discrete approximation scheme in space and time is proposed which preserves the underlying gradient flow structure and leads to dissipation of the free-energy on the discrete level. Existence and uniqueness of the discrete solution is established and relative energy estimates are used to prove optimal convergence rates in space and time under minimal smoothness assumptions. Numerical tests are presented for illustration of the theoretical results and to demonstrate the viability of the proposed methods.

翻译：本文研究通过状态和梯度相关的非对角迁移矩阵强耦合的Cahn-Hilliard/Allen-Cahn系统的数值求解。提出了一种在空间和时间上完全离散的近似格式，该格式保持了底层梯度流结构，并在离散层面上实现了自由能的耗散。建立了离散解的存在唯一性，并在最小光滑性假设下利用相对能量估计证明了空间和时间上的最优收敛速率。数值实验展示了理论结果，并证明了所提方法的有效性。