A virtual element method for a convective Brinkman-Forchheimer problem coupled with a heat equation

TL;DR

This paper introduces a virtual element method for solving a coupled convective Brinkman-Forchheimer and heat equation model, providing stability, error estimates, and numerical validation.

Contribution

It presents a novel virtual element discretization for the coupled model, including proof of well-posedness and optimal error estimates.

Findings

Numerical tests confirm theoretical error estimates.

Method effectively handles thermal diffusion and viscosity variations.

Stable and accurate solutions demonstrated across different mesh types.

Abstract

We develop a virtual element method to solve a convective Brinkman-Forchheimer problem coupled with a heat equation. This coupled model may allow for thermal diffusion and viscosity as a function of temperature. Under standard discretization assumptions, we prove the well posedness of the proposed numerical scheme. We also derive optimal error estimates under appropriate regularity assumptions for the solution. We conclude with a series of numerical tests performed with different mesh families that complement our theoretical findings.