Optimal error estimates of a second-order projection finite element method for magnetohydrodynamic equations

In this paper, we propose and analyze a temporally second-order accurate, fully discrete finite element method for the magnetohydrodynamic (MHD) equations. A modified Crank--Nicolson method is used to discretize the model and appropriate semi-implicit treatments are applied to the fluid convection term and two coupling terms. These semi-implicit approximations result in a linear system with variable coefficients for which the unique solvability can be proved theoretically. In addition, we use a decoupling projection method of the Van Kan type \cite{vankan1986} in the Stokes solver, which computes the intermediate velocity field based on the gradient of the pressure from the previous time level, and enforces the incompressibility constraint via the Helmholtz decomposition of the intermediate velocity field. The energy stability of the scheme is theoretically proved, in which the decoupled Stokes solver needs to be analyzed in details. Optimal-order convergence of $\mathcal{O} (\tau^2+h^{r+1})$ in the discrete $L^\infty(0,T;L^2)$ norm is proved for the proposed decoupled projection finite element scheme, where $\tau$ and $h$ are the time stepsize and spatial mesh size, respectively, and $r$ is the degree of the finite elements. Existing error estimates of second-order projection methods of the Van Kan type \cite{vankan1986} were only established in the discrete $L^2(0,T;L^2)$ norm for the Navier--Stokes equations. Numerical examples are provided to illustrate the theoretical results.