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Comparison of Structure-Preserving Methods for the Cahn-Hilliard-Navier-Stokes Equations

Jimmy Kornelije Gunnarsson, Robert Klöfkorn

Abstract

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.

Comparison of Structure-Preserving Methods for the Cahn-Hilliard-Navier-Stokes Equations

Abstract

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 -adaptive meshes for both standalone Cahn-Hilliard and coupled systems shows significant computational savings without accuracy loss.
Paper Structure (6 sections, 5 theorems, 27 equations, 4 figures, 2 tables)

This paper contains 6 sections, 5 theorems, 27 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Discretization
  3. Discontinuous Galerkin
  4. Limiter and $hp$-adaptivity
  5. Numerics
  6. Summary and Outlook

Key Result

theorem 1

Suppose that $\Omega$ is convex and that the initial phase-field satisfies $\psi^0 \in H^1(\Omega)$ and $||\psi^0||_{L^\infty(\Omega)} \leq 1$. Then, if the mobility function $M(\psi)$ is defined as above, and $\int_{\Omega} M(\psi^0) + W(\psi^0) < C$ for $C > 0$, then the weak solution $\psi \in H^

Figures (4)

  • Figure 1: Example of a scaling limiter for local $\mathbb{P}^2$ phase-field $\psi_h$
  • Figure 2: Ex. \ref{['ex:stationary']} with $hp$-adaptivity for $p_{\min} = 1$ and $p_{\max} = 2$ at time $t = 0$.
  • Figure 3: Ex. \ref{['ex:droplets']}: Energy, mass, and boundedness for SWIPD-L.
  • Figure 4: Ex. \ref{['ex:stationary']}: Energy, mass, and boundedness for SWIPD-L.

Theorems & Definitions (11)

  • theorem 1: Boundedness Elliott:2000
  • lemma 1: Trace inequality, Riviere:2008
  • lemma 2: Trace inequality for mixed polynomial order
  • proof
  • theorem 2: Coercivity
  • proof
  • remark 1: mobility flux
  • corollary 1: Case $V_h^0$
  • proof
  • definition 1: Phase-field mass
  • ...and 1 more