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This paper presents a Crank–Nicolson leap-frog (CNLF) scheme for the unsteady incompressible magnetohydrodynamics (MHD) equations. The spatial discretization adopts the Galerkin finite element method (FEM), and the temporal discretization employs the CNLF method for linear terms and the semi-implicit method for nonlinear terms. The first step uses Stokes style's scheme, the second step employs the Crank–Nicolson extrapolation scheme, and others apply the CNLF scheme. We establish that the fully discrete scheme is stable and convergent when the time step is less than or equal to a positive constant. Firstly, we show the stability of the scheme by means of the mathematical induction method. Next, we focus on analyzing error estimates of the CNLF method, where the convergence order of the velocity and magnetic field reach second-order accuracy, and the pressure is the first-order convergence accuracy. Finally, the numerical examples demonstrate the optimal error estimates of the proposed algorithm. • Crank–Nicolson leap-frog (CNLF) scheme for the unsteady incompressible magnetohydrodynamics (MHD) equations. • The CNLF method for linear terms and the semi-implicit method for nonlinear terms. • Fully discrete scheme is stable and convergent. • The convergence order of the velocity and magnetic field reach second-order accuracy. [ABSTRACT FROM AUTHOR]

This paper aims to study an H 1 -Galerkin mixed finite element method (FEM) for Klein–Gordon–Zakharov (KGZ) equations with power law nonlinearity. The bilinear element and zero-order Raviart–Thomas element ( Q 11 / Q 10 × Q 01 ) are taken as approximation spaces for the original variables p , ϕ and the auxiliary variables u = ∇ p and ψ = ∇ ϕ , respectively. The supercloseness and superconvergence estimates of order O (h 2 + τ 2) are presented for p and ϕ in the H 1 norm and u and ψ in the L 2 norm, which improve the known results that the original variable ϕ can only reach optimal error estimate shown in numerical experiments without theoretical analysis. Here h is the subdivision parameter and τ is the time step. Finally, some numerical results are provided to support the theoretical analysis. • An H 1 -Galerkin mixed FEM for nonlinear KGZ equations is studied. • The superconvergence estimates are presented, which improve the known results. • The results are also valid to the nonconforming FE pair E Q 1 r o t / Q 10 × Q 01 . [ABSTRACT FROM AUTHOR]

In this paper, we investigate a fully discrete finite element scheme for the incompressible magnetohydrodynamic (MHD) equations with magnetic-current formulation that was introduced in Hu et al. (2016). We discretize the system by the semi-implicit Euler scheme in time and a mixed finite element approach together with finite element exterior calculus in space. The resulting scheme enjoys the structure-preserving feature that it can always produce an exactly divergence-free magnetic induction on the discrete level. The unique solvability and unconditional stability of the scheme are also proved rigorously. By utilizing the energy argument, error estimates for the velocity, magnetic induction, current density and induced electric field are further established under the low regularity hypothesis for the exact solutions. Numerical results are provided to verify the theoretical analysis and to show the effectiveness of the proposed scheme. • Error analysis of a fully discrete finite element method for the MHD equations is given. • The proposed scheme is linear, unconditional stability and structure-preserving. • Numerical examples are provided to verify the theoretical findings. [ABSTRACT FROM AUTHOR]