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arXiv · 2607.10724

A four-field auxiliary reformulation of a Cahn-Hilliard time step: Analysis and conforming finite element discretization

Marvin Fritz
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Abstract

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 $-\Delta c$ 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.

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Marvin Fritz. 2026-07-12. A four-field auxiliary reformulation of a Cahn-Hilliard time step: Analysis and conforming finite element discretization. https://arxiv.org/abs/2607.10724

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