DMP and energy-law status for FEM-L, SIPG-L, and SWIP-L schemes

Determine whether the FEM-L, SIPG-L, and SWIP-L schemes introduced in earlier work satisfy a discrete maximum principle and a discrete energy law, beyond numerical evidence.

Background

The authors note that previous limiter-based schemes show boundedness numerically but lack formal proofs of a discrete maximum principle or energy dissipation.

A definitive theoretical result would clarify the scope of those methods and position them relative to the provable guarantees established in this paper.

References

For the FEM-L, SIPG-L, and SWIP-L schemes in , it is unclear whether the procedure satisfies a discrete maximum principle or an energy law; only numerical evidence is provided.

— A Discontinuous Galerkin Scheme for the Cahn-Hilliard Equations with Discrete Maximum Principle for Arbitrary Polynomial Order  (2604.00988 - Gunnarsson et al., 1 Apr 2026) in Subsection "Comparison to other schemes", Section 4 (item 3)