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<p>This paper presents the formulation and analysis of a mixed finite element method for a hemivariational inequality arising from the stationary convective Brinkman-Forchheimer extended Darcy (CBFeD) equations. This model extends the incompressible Navier-Stokes equations by incorporating both damping and pumping effects. The hemivariational inequality describes the flow of a viscous, incompressible fluid through a saturated porous medium, subject to a nonsmooth, nonconvex friction-type slip boundary condition. The incompressibility constraint is handled via a mixed variational formulation. We establish the existence and uniqueness of solutions by utilizing the pseudomonotonicity and coercivity properties of the underlying operators and provide a detailed error analysis of the proposed numerical scheme. Under suitable regularity assumptions, the method achieves optimal convergence rates with low-order mixed finite element pairs. The scheme is implemented using the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\text {P1b/P1}\)</EquationSource> <EquationSource Format="MATHML"><math> <mtext>P1b/P1</mtext> </math></EquationSource> </InlineEquation> element pair, and numerical experiments are presented to validate the theoretical results and confirm the expected convergence behavior.</p>

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Mixed Finite Element Method for a Hemivariational Inequality of Stationary Convective Brinkman-Forchheimer Extended Darcy Equations

  • 期刊论文

  • 发表日期：

    2026年5月21日

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  • Wasim Akram,
  • Manil T. Mohan

摘要

This paper presents the formulation and analysis of a mixed finite element method for a hemivariational inequality arising from the stationary convective Brinkman-Forchheimer extended Darcy (CBFeD) equations. This model extends the incompressible Navier-Stokes equations by incorporating both damping and pumping effects. The hemivariational inequality describes the flow of a viscous, incompressible fluid through a saturated porous medium, subject to a nonsmooth, nonconvex friction-type slip boundary condition. The incompressibility constraint is handled via a mixed variational formulation. We establish the existence and uniqueness of solutions by utilizing the pseudomonotonicity and coercivity properties of the underlying operators and provide a detailed error analysis of the proposed numerical scheme. Under suitable regularity assumptions, the method achieves optimal convergence rates with low-order mixed finite element pairs. The scheme is implemented using the \(\text {P1b/P1}\) P1b/P1 element pair, and numerical experiments are presented to validate the theoretical results and confirm the expected convergence behavior.