Aspects of Finite Element Discretizations for Solving the Boussinesq Approximation of the Navier-Stokes Equations

Summary

We consider stable discretizations for solving the Boussinesq approximation of the stationary, incompressible Navier-Stokes equations in the twodimensional case. For the continuous problem the right hand side f ∈ L ²(Ω)² of the momentum equation can be splitted in the form f = ∇Φ + curl Ψ, where a variation of Φ does not change the velocity u. For the discrete problem this property is only true in the limit case, h tends to zero, unless exact divergencefree trial functions for approximating the velocity field are used. The main objective of the paper is to discuss the influence of this phenomenon on the accuracy of the approximated velocity field u _h when using only discrete divergencefree trial functions. For some benchmark problems the results of numerical calculations are also presented.