A Divergence-Conforming DG-Mixed Finite Element Method for the Stationary Boussinesq Problem

Abstract

In this work we propose and analyze a new fully divergence-conforming finite element method for the numerical simulation of the Boussinesq problem, describing the motion of a non-isothermal incompressible fluid subject to a heat source. We consider the standard velocity-pressure formulation for the fluid flow equation and the dual-mixed one for the heat equation. In this way, the unknowns of the resulting formulation are given by the velocity, the pressure, the temperature and the gradient of the latter. The corresponding Galerkin scheme makes use of a nonconforming exactly divergence-free approach to approximate the velocity and pressure, and employs standard $H(\mathrm{div})$-conforming elements for the gradient of the temperature and discontinuous elements for the temperature. Since here we utilize a dual-mixed formulation for the heat equation, the temperature Dirichlet boundary condition becomes natural, thus there is no need of introducing a sufficiently small discrete lifting to prove well-posedness of the discrete problem. Moreover, the resulting numerical scheme yields exactly divergence-free velocity approximations; thus, it is probably energy-stable without the need of modifying the underlying differential equations, and provide an optimal convergent approximation of the temperature gradient. The analysis of the continuous and discrete problems is carried out by means of a fixed-point strategy under a sufficiently small data assumption. We derive optimal error estimates in the mesh size for smooth solutions and provide several numerical results illustrating the performance of the method and confirming the theoretical rates of convergence.