On $hp$-adaptive Structure-Preservation for the Cahn--Hilliard--Navier--Stokes Equations with Degenerate Mobility

We develop structure-preserving discontinuous Galerkin methods for the Cahn-Hilliard-Navier-Stokes equations with degenerate mobility. The proposed SWIPD-L and SIPGD-L methods incorporate parametrized mobility fluxes with edge-wise mobility treatments for enhanced coercivity-stability control. We prove coercivity for the generalized trilinear form and demonstrate optimal convergence rates while preserving mass conservation, energy dissipation, and the discrete maximum principle. Comparisons with existing SIPG-L and SWIP-L methods confirm similar stability. Validation on $hp$-adaptive meshes for both standalone Cahn-Hilliard and coupled systems shows significant computational savings without accuracy loss.