On h-adaptive Structure-Preserving for the Cahn–Hilliard–Navier–Stokes Equations with Degenerate Mobility

Jul 16, 2026

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We develop structure-preserving discontinuous Galerkin discretizations for the Cahn–Hilliard–Navier–Stokes system with degenerate mobility, targeting robust boundedness and stability under challenging interface dynamics. The proposed SWIPD-L and SIPGD-L schemes incorporate parametrized mobility fluxes and edge-wise mobility treatments, yielding coercivity for the generalized trilinear form. The methods preserve mass conservation, enforce energy dissipation, and satisfy a discrete maximum principle. Comparisons with existing SIPG-L and SWIP-L confirm similar stability, while h-adaptive tests show significant computational savings without accuracy loss.

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Introduction

Discretization

h-adaptive Structure-Preserving Methods for the CHNS Equations

arXiv :2602 .22861v3 [math .NA] 13 Jul 2026 On 􀁨 􀁰-adaptive Structure-Preservion for the Cahn–Hilliard–Navier–Stokes Equations with Degenerate Mobility Jimmy Kornelije Gunnarsson [0009−0000−7610−6360] and Robert Klfkorn [0000−0001−9664−0333] Abstract We develop structure-preserving discontinuous Galerkin methods for the Cahn-Hilliard-Navier-Stokes equations with degenerate mobility. The proposed SWIPDL 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 ℎ􀀿-adaptive meshes for both standalone Cahn-Hilliard and coupled systems shows significant computational savings without accuracy loss. 1 Introduction Consider the spatial domain Ω ⊂ R􀀳 , 􀀳 = 2, 3, with Lipschitz continuous boundary 􀁭Ω and the time interval (0, 􀀩 ] for 􀀩 ∈ R+ . The Cahn-Hilliard (CH) equations model phase separation in binary mixtures for a phase-field 􀁫 : Ω × (0, 􀀩 ] → [−1, 1] following: 􀁭 􀁃􀁫 + ∇ · (􀁫u) − ∇ · 􀀥􀀴−1 (􀀢(􀁫)∇􀁨) = 0, (1) 􀁨 + 􀀘􀀽2Δ􀁫 − 􀀬′(􀁫) = 0, (2) where 􀀥􀀴 > 0 is the Peclet number, 􀁵 is an advection field, 􀁨 is the chemical potential,􀀬 (􀁫) := 14 (1 − 􀁫2 )2 is the double-well potential, and 􀀢(􀁫) := max {1 − 􀁫2 , 0} is the Jimmy Kornelije Gunnarsson Centre for Mathematical Sciences, Lund University, Box 117, 22100 Lund, Sweden e-mail: jimmy_[kornelije.gunnarsson@math.lu.se](kornelije.gunnarsson@math.lu.se) Robert Klfkorn Centre for Mathematical Sciences, Lund University, Box 117, 22100 Lund, Sweden e-mail: robertk@ [math.lu.se](math.lu.se) 2 Jimmy Kornelije Gunnarsson and Robert Klfkorn mobility function. We equip Eq. (1) with homogeneous Neumann boundary conditions 􀁮 · ∇􀁫 = 0 and 􀁮 · ∇􀁨 = 0 on 􀁭Ω . We present below a special case of a fundamental theorem for boundedness. Theorem 1 (Boundedness [5]) Suppose that Ω is convex and that the initial phasefield satisfies 􀁫0 ∈ 􀀝 1 (Ω) and ||􀁫0 || 􀀡 ∞ (Ω) ≤ 1. Then, if the mobility function 􀀢 (􀁫) is defined as above, and ∫Ω 􀀢 (􀁫0 ) + 􀀬(􀁫0 ) < 􀀘 for 􀀘 > 0, then the weak solution 􀁫 ∈ 􀀝1 (Ω) of the CH equation satisfies a weak maximum principle, i.e., ||􀁫|| 􀀡 ∞ (Ω) ≤ 1 for 􀁃 ∈ (0, 􀀩] . While Theorem 1, proven in [5, Thm. 1], guarantees a weak maximum principle atthe continuum level for a degenerate mobility, achieving discrete boundedness remains a significant challenge. The standard Finite Element Method (FEM), symmetric weighted interior penalty (SWIP), and symmetric interior penalty Galerkin (SIPG) discretizations fail to preserve boundedness without additional stabilization [9], even when using the degenerate mobility 􀀢 . Recent work, however, has demonstrated that boundedness can be numerically achieved through carefully designed limited Galerkin schemes, including FEM-L for FEM, and for Discontinuous Galerkin (DG) schemes with SIPGL, and SWIP-L [9] with the degenerate CH equations. In this work, we extend these structure-preserving methods by introducing new mobility fluxes using the DG formulations with intersection-wise mobility treatments. These extend the SWIP-L and SIPG-L methods introduced in [9], by allowing for better control of the coercivity-stability balance. Boundedness of the CH equations is in particular important when coupled to the incompressible Navier-Stokes (NS) equations: 􀁤 (􀁫) 􀀀􀁭􀁃􀁵 + 􀁵 · ∇􀁵 􀀁 + 􀁐 · ∇􀁵 + ∇ · 􀀀􀀥I − 2􀁠 (􀁫)􀀙 (􀁵)􀀁 = − 􀀬􀀴1􀀘􀀽 􀁫∇􀁨 (3) ∇ · u = 0 (4) where 􀁤 (􀁫) := 1+2􀁫 􀁤 1 + 1−2􀁫 􀁤 2 is the density, 􀁐 := 􀁤~~ ~~2−2􀁤~~ ~~1 􀀥􀀴 −1􀀢 (􀁫)∇􀁨 is the mass flux,􀀥 is the pressure, 􀁠 (􀁫) := 1+2􀁫 􀁠 1 + 1−2􀁫 􀁠 2 is the viscosity, and 􀀙 (􀁵) = 12 (∇􀁵 + ∇􀁵􀀩 ) is the rate of deformation tensor, where I denotes the identity tensor. Moreover, 􀁤 􀀹 and 􀁠 􀀹 are the characteristic densities and vi

FAQs

What problem do the SWIPD-L and SIPGD-L methods address?

They address discrete boundedness and stability for the Cahn–Hilliard(-Navier–Stokes) equations when the mobility is degenerate, where standard schemes may fail without additional stabilization.

Which structural properties does the proposed discretization preserve?

The methods preserve mass conservation, enforce energy dissipation, and satisfy a discrete maximum principle, while also providing coercivity for the generalized trilinear form.

How is the mobility handled to improve stability?

The schemes introduce parametrized mobility fluxes and edge-wise (intersection-wise) mobility treatments, allowing better control of the coercivity–stability balance compared with earlier SIPG-L and SWIP-L approaches.

Jul 16, 2026

Upload by Finn