A pressure-robust lowest-order virtual element method for the natural convection problem on general mesh

Abstract

In this article, our focus lies on the natural convection model, which is formulated by coupling the incompressible Navier-Stokes equations with the heat equation. However, the traditional mixed finite element method applied to the incompressible Navier-Stokes equations has certain limitations: it tends to relax the divergence constraint, resulting in a lack of sufficient robustness when dealing with large irrotational forces in momentum balance, and the velocity errors inherent in those methods are often closely related to the continuous pressure, which in turn further compromise numerical accuracy. To address these issues, we propose a pressure-robust lowest-order mixed virtual element method for natural convection problem on general mesh. We employed a $H^1$-conforming virtual element space which can provide a point-wise discrete divergence-free velocity, and through the reconstruction of velocity test function and compatibility with the Helmholtz-Hodge decomposition, the influence of continuous pressure on velocity field is eliminated and the numerical oscillations caused by large irrotational body force term in momentum balance are overcome. We rigorously prove the $H^1$-error estimates for the velocity and temperature fields, as well as the $L^2$-error estimate for the pressure field. We then conduct numerical experiments to verify the accuracy of the theory and the effectiveness of our proposed method.