In this paper, we consider the initial-boundary value problem for the three-dimensional incompressible magnetohydrodynamic equations. We first establish certain regularity results for the solution (u, p, H) under the assumption that u, H ∈ L⁴(0,T;H¹(Ω)³). Next, we consider the fully discrete first-order Euler semi-implicit scheme based on the mixed finite element method. Then we obtain the L²-unconditional convergence of this scheme by using the negative norm technique.

This paper considers the stability and convergence results for the Euler implicit/explicit scheme applied to the spatially discretized two-dimensional (2D) time-dependent Navier-Stokes equations. A Galerkin finite element spatial discretization is assumed, and the temporal treatment is implicit/explict scheme, which is implicit for the linear terms and explicit for the nonlinear term. Here the stability condition depends on the smoothness of the initial data $u_0\in H^\alpha$ , i.e., the time step condition is $\tau \leq C_0$ in the case of $\alpha =2$ , $\tau |\log h|\leq C_0$ in the case of $\alpha =1$ and $\tau h^{-2}\leq C_0$ in the case of $\alpha =0$ for mesh size $h$ and some positive constant $C_0$ . We provide the $H^2$ -stability of the scheme under the stability condition with $\alpha =0,1,2$ and obtain the optimal $H^1-L^2$ error estimate of the numerical velocity and the optimal $L^2$ error estimate of the numerical pressure under the stability condition with $\alpha =1,2$ .

This paper provides an error analysis for the Crank–Nicolson extrapolation scheme of time discretization applied to the spatially discrete stabilized finite element approximation of the two-dimensional time-dependent Navier–Stokes problem, where the finite element space pair $(X_h,M_h)$ for the approximation $(u_h^n,p_h^n)$ of the velocity $u$ and the pressure $p$ is constructed by the low-order finite element: the $Q_1-P_0$ quadrilateral element or the $P_1-P_0$ triangle element with mesh size $h$ . Error estimates of the numerical solution $(u_h^n,p_h^n)$ to the exact solution $(u(t_n),p(t_n))$ with $t_n\in (0,T]$ are derived.

A fully discrete penalty finite element method is presented for the two-dimensional time-dependent Navier-Stokes equations. The time discretization of the penalty Navier-Stokes equations is based on the backward Euler scheme; the spatial discretization of the time discretized penalty Navier-Stokes equations is based on a finite element space pair $(X_h,M_h)$ which satisfies some approximate assumption. An optimal error estimate of the numerical velocity and pressure is provided for the fully discrete penalty finite element method when the parameters $\epsilon ,~\Delta t$ and $h$ are sufficiently small.