Numerical analysis of a porous natural convection system with vorticity and viscous dissipation

TL;DR

This paper develops and analyzes a new divergence-free mixed finite element method for simulating natural convection with vorticity and viscous dissipation, providing stability, error estimates, and numerical validation.

Contribution

It introduces a novel formulation and finite element approach for Darcy--Brinkman equations coupled with thermal energy, addressing complex boundary conditions and vorticity.

Findings

Proved well-posedness of the numerical scheme.

Derived error estimates for the finite element approximation.

Validated the method with numerical examples in 2D and 3D.

Abstract

In this paper we propose and analyse a new formulation and pointwise divergence-free mixed finite element methods for the numerical approximation of Darcy--Brinkman equations in vorticity--velocity--pressure form, coupled with a transport equation for thermal energy with viscous dissipative effect and mixed Navier-type boundary conditions. The solvability analysis of the continuous and discrete problems is significantly more involved than usual as it hinges on Banach spaces needed to properly control the advective and dissipative terms in the non-isothermal energy balance equation. We proceed by decoupling the set of equations and use the Banach fixed-point theorem in combination with the abstract theory for perturbed saddle-point problems. Some of the necessary estimates are straightforward modifications of well-known results, while other technical tools require a more elaborated analysis. The velocity is approximated by Raviart--Thomas elements, the vorticity uses N\'ed\'elec spaces of the first kind, the pressure is approximated by piecewise polynomials, and the temperature by continuous and piecewise polynomials of one degree higher than pressure. Special care is needed to establish discrete inf-sup conditions since the curl of the discrete vorticity is not necessarily contained in the discrete velocity space, therefore suggesting to use two different Raviart--Thomas interpolants. A discrete fixed-point argument is used to show well-posedness of the Galerkin scheme. Error estimates in appropriate norms are derived, and a few representative numerical examples in 2D and 3D and with mixed boundary conditions are provided.