AI 中文总结
提出Cahn-Hilliard时间步长的四场辅助重构，受二维分解启发，通过辅助场表示特定标量，推导等效连续公式及协调有限元离散化，证明相关估计，数值实验验证了相场收敛等特性。
AI 中文摘要
我们提出了时间离散Cahn-Hilliard步长的四场辅助重构。其构造受二维Rafetseder-Zulehner分解基础的标量迹结构启发，通过辅助标量场p和辅助向量场u将标量-Δc表示为2p + div u的形式。所得辅助问题是一个混合二阶Stokes/弹性型系统，相场的演化方程保持其标准质量守恒梯度流结构。我们推导了与经典凸分裂混合时间步长等效的连续四场公式。还给出了协调有限元离散化，并证明了相场变量的一步空间估计以及包含显式弱拉普拉斯恢复缺陷的稳定辅助块估计。数值实验验证了在人工设置中的预期相场收敛、质量守恒、能量衰减以及辅助块的投影一致性。
英文摘要
We propose a four-field auxiliary reformulation of a time-discrete Cahn-Hilliard step. The construction is motivated by the scalar trace structure underlying two-dimensional Rafetseder-Zulehner decompositions and represents the scalar quantity $-Δc$ in the form $2p+div\,u$ through an auxiliary scalar field $p$ and an auxiliary vector field $u$. The resulting auxiliary problem is a mixed second-order Stokes/elasticity-type system, while the evolution equation for the phase field retains its standard mass-conserving gradient-flow structure. We derive a continuous four-field formulation that is equivalent to the classical convex-splitting mixed time step. We also state a conforming finite element discretization and prove one-step spatial estimates for the phase-field variables together with a stable auxiliary block estimate containing an explicit weak-Laplacian recovery defect. The numerical experiments verify the expected phase-field convergence in a manufactured setting, mass conservation, energy decay, and projected consistency of the auxiliary block.
