Mixed Finite Element Method for a Hemivariational Inequality of Stationary convective Brinkman-Forchheimer Extended Darcy equations

TL;DR

This paper develops and analyzes a mixed finite element method for a hemivariational inequality modeling fluid flow in porous media with complex boundary conditions, extending existing models with new numerical and theoretical insights.

Contribution

It introduces a novel mixed finite element scheme for the hemivariational inequality of the CBFeD equations, including proof of existence, uniqueness, and optimal error estimates.

Findings

The method achieves optimal convergence rates under regularity assumptions.

Numerical experiments confirm the theoretical error estimates.

The scheme effectively handles nonsmooth, nonconvex boundary conditions.

Abstract

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 $\text{P1b/P1}$ element pair, and numerical experiments are presented to validate the theoretical results and confirm the expected convergence behavior.