A Fully-Mixed Finite Element Method for the n-Dimensional Boussinesq Problem with Temperature-Dependent Parameters

Abstract

AbstractIn this paper, we introduce and analyze a high-order, fully-mixed finite element method for the free convection of n-dimensional fluids, {n\in\{2,3\}}, with temperature-dependent viscosity and thermal conductivity. The mathematical model is given by the coupling of the equations of continuity, momentum (Navier–Stokes) and energy by means of the Boussinesq approximation, as well as mixed thermal boundary conditions and a Dirichlet condition on the velocity. Because of the dependence on the temperature of the fluid properties, several additional variables are defined, thus resulting in an augmented formulation that seeks the rate of strain, pseudostress and vorticity tensors, velocity, temperature gradient and pseudoheat vectors, and temperature of the fluid. Using a fixed-point approach, smallness-of-data assumptions and a slight higher-regularity assumption for the exact solution provide the necessary well-posedness results at both continuous and discrete levels. In addition, and as a result of the augmentation, no discrete inf-sup conditions are needed for the well-posedness of the Galerkin scheme, which provides freedom of choice with respect to the finite element spaces. In particular, we suggest a combination based on Raviart–Thomas, Lagrange and discontinuous elements for which we derive optimal a priori error estimates. Finally, several numerical examples illustrating the performance of the method and confirming the theoretical rates of convergence are reported.