H2-stability of the first order Galerkin method for the Boussinesq equations with smooth and non-smooth initial data

In this paper, the H2-stability of the first order fully discrete Galerkin finite element methods for the Boussinesq equations with smooth and non-smooth initial data is presented. The finite element spatial discretization for the Boussinesq equations is based on the mixed finite element method, and the temporal treatments of the spatial discrete Boussinesq equations include the implicit scheme, the semi-implicit scheme, the implicit/explicit scheme and the explicit scheme. The H2-stability results of the above numerical schemes are established. Firstly, we prove that the implicit and semi-implicit schemes are the H2-unconditional stable. Then we show that the implicit/explicit scheme is H2-almost unconditional stable with the initial data that belong to H1 and H2, and the similar results are obtained for the semi-implicit/explicit scheme in the case of the initial data that belong to L2. Furthermore, we show that the explicit scheme is the H2-conditional stable. Finally, some numerical examples are provided to verify the established theoretical findings and confirm the corresponding H2 stability analysis of the different numerical schemes.