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

Abstract In this paper, the H 2 -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 H 2 -stability results of the above numerical schemes are established. Firstly, we prove that the implicit and semi-implicit schemes are the H 2 -unconditional stable. Then we show that the implicit/explicit scheme is H 2 -almost unconditional stable with the initial data that belong to H 1 and H 2 , and the similar results are obtained for the semi-implicit/explicit scheme in the case of the initial data that belong to L 2 . Furthermore, we show that the explicit scheme is the H 2 -conditional stable. Finally, some numerical examples are provided to verify the established theoretical findings and confirm the corresponding H 2 stability analysis of the different numerical schemes.