Optimal   \(\boldsymbol{{L^2}}\)   Error Estimates of Unconditionally Stable Finite Element Schemes for the Cahn–Hilliard–Navier–Stokes System

Cai, Wentao; Sun, Weiwei; Wang, Jilu; Yang, Zongze

doi:10.1137/22m1486844

Cited by 9 publications

(2 citation statements)

References 38 publications

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“…Meanwhile, let us note that the technique developed in this work can be extended to another Hodge-decomposition based finite element method [25]. We shall also mention [3,34], which investigate the optimal error analysis of Cahn-Hilliard-Navier-Stokes system and magnetohydrodynamic equations.…”

Section: Discussionmentioning

confidence: 99%

Optimal Error Analysis of a Hodge-Decomposition Based Finite Element Method for the Ginzburg-Landau Equations in Superconductivity

Xie

2024

JCM

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Section: Discussionmentioning

confidence: 99%

Optimal Error Analysis of a Hodge-Decomposition Based Finite Element Method for the Ginzburg-Landau Equations in Superconductivity

Xie

2024

JCM

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“…Remark In the error analysis of many constant density problems [5,8,17,18], only the

{L}^2-

norms and

{H}^1-

norms of the numerical solutions need to be analyzed. However, as mentioned in [4], it is difficult to establish error analysis on variable density problems, not only due to the strong nonlinear coupling terms, but also due to the hyperbolic conservation of the density equation.…”

Section: Error Analysismentioning

confidence: 99%

Stability and temporal error estimate of scalar auxiliary variable schemes for the magnetohydrodynamics equations with variable density

Chen,

He,

Chen

2023

Numerical Methods Partial

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In this article, we construct first‐ and second‐order semidiscrete schemes for the magnetohydrodynamics (MHD) equations with variable density based on scalar auxiliary variable (SAV) approach. These schemes are decoupled, unconditionally energy stable and only solve a sequence of linear differential equations at each time step. We carry out a rigorous error analysis for the first‐order SAV scheme in two‐dimensional case. Some numerical experiments are presented to verify the accuracy and stability.

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Fully decoupled unconditionally stable Crank–Nicolson leapfrog numerical methods for the Cahn–Hilliard–Darcy system

Gao,

Han

2024

Numerical Methods Partial

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We develop two totally decoupled, linear and second‐order accurate numerical methods that are unconditionally energy stable for solving the Cahn–Hilliard–Darcy equations for two phase flows in porous media or in a Hele‐Shaw cell. The implicit‐explicit Crank–Nicolson leapfrog method is employed for the discretization of the Cahn–Hiliard equation to obtain linear schemes. Furthermore the artificial compression technique and pressure correction methods are utilized, respectively, so that the Cahn–Hiliard equation and the update of the Darcy pressure can be solved independently. We establish unconditionally long time stability of the schemes. Ample numerical experiments are performed to demonstrate the accuracy and robustness of the numerical methods, including simulations of the Rayleigh–Taylor instability, the Saffman–Taylor instability (fingering phenomenon).

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Optimal \(\boldsymbol{{L^2}}\) Error Estimates of Unconditionally Stable Finite Element Schemes for the Cahn–Hilliard–Navier–Stokes System

Cited by 9 publications

References 38 publications

Optimal Error Analysis of a Hodge-Decomposition Based Finite Element Method for the Ginzburg-Landau Equations in Superconductivity

Optimal Error Analysis of a Hodge-Decomposition Based Finite Element Method for the Ginzburg-Landau Equations in Superconductivity

Stability and temporal error estimate of scalar auxiliary variable schemes for the magnetohydrodynamics equations with variable density

Fully decoupled unconditionally stable Crank–Nicolson leapfrog numerical methods for the Cahn–Hilliard–Darcy system

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