An analysis of weak Galerkin finite element method for a steady state Boussinesq problem

Abstract

In this article, we present and analyze a weak Galerkin finite element method (WG-FEM) for the coupled Navier–Stokes/temperature (or Boussinesq) problems. In this WG-FEM, discontinuous functions are applied to approximate the velocity, temperature, and the normal derivative of temperature on the boundary while piecewise constants are used to approximate the pressure. The stability, existence and uniqueness of solution of the associated WG-FEM are proved in detail. An optimal a priori error estimate is then derived for velocity in the discrete H1 and L2 norms, pressure in the L2 norm, temperature in the discrete H1 and L2 norms, and the normal derivative of temperature in H−1/2 norm. Finally, to complete this study some numerical tests are presented which illustrate that the numerical errors are consistent with theoretical results.