A Discontinuous Galerkin Scheme for the Cahn-Hilliard Equations with Discrete Maximum Principle for Arbitrary Polynomial Order
We propose a structure-preserving discontinuous Galerkin scheme for the Cahn--Hilliard equations with degenerate mobility based on the Symmetric Weighted Interior Penalty formulation. By evaluating the mobility at cell averages rather than as a piecewise polynomial, the proposed scheme preserves strict degeneracy and yields a coercivity constant that is independent of the mobility, removing the need for regularization. Moreover, we establish existence of discrete solutions even with degeneracy via a Leray--Schauder fixed-point argument, and show that the scheme satisfies a provable discrete maximum principle at arbitrary polynomial order $p$ when combined with the Zhang--Shu scaling limiter for $p > 0$ and from the scheme alone for $p = 0$. Mass conservation and energy dissipation are established for the unlimited scheme; for the limited variant, we discuss observed energy dissipation for $p \in \{1,2\}$ and potential theoretical solutions. Numerical experiments confirm optimal convergence rates of order $p$ in $H^1$ for $p \in \{1,2\}$, and of order $p+1$ in $L^2$ in a low amplitude regime with order $2$ for $p=2$ in high amplitude regime and time-dependent low amplitude regime, and validate structure-preserving properties with numerical results.