Eligio Colmenares, Ricardo Oyarzúa, Francisco Piña:
A fully-DG method for the stationary Boussinesq system
In this work we present and analyze a finite element scheme yielding discontinuous Galerkin approximations to the solutions of the stationary Boussinesq system for the simulation of non-isothermal flow phenomena. The model consists of a Navier-Stokes type system, describing the velocity and the pressure of the fluid, coupled to an advection-diffusion equation for the temperature. The proposed numerical scheme is based on the standard interior penalty technique and an upwind approach for the nonlinear convective terms and employs the divergence-conforming Brezzi-Douglas-Marini (BDM) elements of order k for the velocity, discontinuous elements of order k - 1 for the pressure and discontinuous elements of order k for the temperature. Existence and uniqueness results are shown and stated rigorously for both the continuous problem and the discrete scheme, and optimal a priori error estimates are also derived. Numerical examples back up the theoretical expected convergence rates as well as the performance of the proposed technique.
This preprint gave rise to the following definitive publication(s):
Eligio COLMENARES, Ricardo OYARZúA, Francisco PIñA: A discontinuous Galerkin method for the stationary Boussinesq system. Computational Methods in Applied Mathematics, vol. 22, 4, pp. 797-820, (2022).