<p>The paper presents a decoupled, conservative virtual element method for inductionless magnetohydrodynamic equations. A semi-implicit Euler formulation is employed for time derivative terms, while the mixed virtual element method is used for spatial discretization. The approach utilizes the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10092_2025_635_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textbf{H}^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-conforming Stokes-like virtual element and discontinuous piecewise polynomials to approximate velocity and pressure, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10092_2025_635_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textbf{H}(\text{ div})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">H</mi> <mo stretchy="false">(</mo> <mspace width="0.333333em" /> <mtext>div</mtext> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-conforming virtual element and discontinuous piecewise polynomials for the current density and electric potential, respectively. In the fully discretized scheme, the velocity and pressure are decoupled from the current density and electric potential. Additionally, the discrete velocity and current density are pointwise divergence-free. Optimal error estimates are established for the velocity, current density, and electric potential. Finally, numerical experiments are conducted to validate the effectiveness of the theoretical analysis.</p>

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A decoupled, conservative virtual element method for inductionless magnetohydrodynamic equations

  • Yimin Luo,
  • Qili Tang,
  • Tianwen Wang

摘要

The paper presents a decoupled, conservative virtual element method for inductionless magnetohydrodynamic equations. A semi-implicit Euler formulation is employed for time derivative terms, while the mixed virtual element method is used for spatial discretization. The approach utilizes the \(\textbf{H}^1\) H 1 -conforming Stokes-like virtual element and discontinuous piecewise polynomials to approximate velocity and pressure, \(\textbf{H}(\text{ div})\) H ( div ) -conforming virtual element and discontinuous piecewise polynomials for the current density and electric potential, respectively. In the fully discretized scheme, the velocity and pressure are decoupled from the current density and electric potential. Additionally, the discrete velocity and current density are pointwise divergence-free. Optimal error estimates are established for the velocity, current density, and electric potential. Finally, numerical experiments are conducted to validate the effectiveness of the theoretical analysis.