Mixed-Primal Methods for Natural Convection Driven Phase Change with Navier–Stokes–Brinkman Equations

Abstract

In this paper we consider a steady phase change problem for non-isothermal incompressible viscous flow in porous media with an enthalpy-porosity-viscosity coupling mechanism, and introduce and analyze a Banach spaces-based variational formulation yielding a new mixed-primal finite element method for its numerical solution. The momentum and mass conservation equations are formulated in terms of velocity and the tensors of strain rate, vorticity, and stress; and the incompressibility constraint is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the stress and the velocity. The resulting continuous formulation for the flow becomes a nonlinear perturbation of a perturbed saddle point linear system. The energy conservation equation is written as a nonlinear primal formulation that incorporates the additional unknown of boundary heat flux. The whole mixed-primal formulation is regarded as a fixed-point operator equation, so that its well-posedness hinges on Banach’s theorem, along with smallness assumptions on the data. In turn, the solvability analysis of the uncoupled problem in the fluid employs the Babuška–Brezzi theory, a recently obtained result for perturbed saddle-point problems, and the Banach–Nečas–Babuška Theorem, all them in Banach spaces, whereas the one for the uncoupled energy equation applies a nonlinear version of the Babuška–Brezzi theory in Hilbert spaces. An analogue fixed-point strategy is employed for the analysis of the associated Galerkin scheme, using in this case Brouwer’s theorem and assuming suitable conditions on the respective discrete subspaces. The error analysis is conducted under appropriate assumptions, and selecting specific finite element families that fit the theory. We finally report on the verification of theoretical convergence rates with the help of numerical examples.