Unconditionally optimal error estimates of an IMEX finite element method for time-dependent penetrative convection problem based on the exponential-SAV technique

Based on an exponential scalar auxiliary variable technique, in this paper, we study a new first-order Euler implicit-explicit (IMEX) finite element scheme for the time-dependent penetrative convection problem. In designing this numerical scheme, the nonlinear terms are explicitly treated such that only a series of algebraic equations with constant coefficients need to be solved at each time step. The unconditional stability of numerical scheme is proved and the unconditionally optimal error estimates of the velocity, temperature in $H^1$ and $L^2$ norms and the pressure in the discrete $L^2$ norm are derived without any CFL type condition by discussing two cases of $\tau \le h^2$ and $\tau \ge h^2$, respectively, where $\tau$ and $h$ are time step and mesh size. Finally, we give numerical results to confirm the theoretical analysis.