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<p>A first-order in-time bounds-preserving finite element method is analyzed for solving the quasi-incompressible Cahn-Hilliard-Darcy system with the Flory-Huggins potential for two-phase flows of variable densities and viscosities in porous media. The proposed scheme is uniquely solvable, bounds-preserving, mass-conservative, and unconditionally energy stable. By exploiting the Darcy equations and bounds-preservation of the numerical solution, we obtain stability estimate of the pressure gradient. An inductive rough error estimate then gives the strict separation property of the numerical solution. Finally, a refined <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(l^{\infty }(H^1)\cap l^2(H^3)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>l</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>H</mi> <mn>1</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <msup> <mi>l</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>H</mi> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> error estimate establishes the optimal convergence rate for the order parameter in the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(H^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> norm.</p>

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Convergence Analysis of a Bounds-Preserving Numerical Scheme for the Quasi-Incompressible Cahn-Hilliard-Darcy System

  • 期刊论文

  • 发表日期：

    2026年7月21日

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  • Wenbin Chen,
  • Daozhi Han,
  • Qianqian Liu,
  • Xiaoming Wang

摘要

A first-order in-time bounds-preserving finite element method is analyzed for solving the quasi-incompressible Cahn-Hilliard-Darcy system with the Flory-Huggins potential for two-phase flows of variable densities and viscosities in porous media. The proposed scheme is uniquely solvable, bounds-preserving, mass-conservative, and unconditionally energy stable. By exploiting the Darcy equations and bounds-preservation of the numerical solution, we obtain stability estimate of the pressure gradient. An inductive rough error estimate then gives the strict separation property of the numerical solution. Finally, a refined \(l^{\infty }(H^1)\cap l^2(H^3)\) l ∞ ( H 1 ) ∩ l 2 ( H 3 ) error estimate establishes the optimal convergence rate for the order parameter in the \(H^1\) H 1 norm.