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In this paper, the unsteady incompressible thermomicropolar fluid (UITF) equations are considered. Theoretically, some a priori regularity conclusions are presented firstly, which seem to be not available in the literatures. Numerically, a penalty finite element method (PFEM) for the UITF equations is studied, the Euler semi-implicit temporal semi-discrete method of the penalty UITF equations is proposed, the stability and the L 2 - H 1 error estimates of the temporal discrete solutions are proved. Finally, the stability and the L 2 - H 1 error estimates for the finite element fully-discrete approximation of the penalty UITF equations are rigorously proved. The accuracy and efficiency of the fully-discrete PFEM are demonstrated by some numerical examples. • This paper focus on penalty method for unsteady incompressible thermomicropolar fluid equations. • Some a priori regularity conclusions are presented. • Stability and L 2 − H 1 error estimates of Euler semi-implicit method are proposed. • Stability and L 2 − H 1 error estimates of fully-discrete method are proved. [ABSTRACT FROM AUTHOR]

In this paper, we study fully discrete stabilized mixed finite element methods for solving chemotaxis equations on surfaces. By defining two global fluxes as new vector-valued variables, the original equations are firstly transformed into an equivalent first order mixed form. Then, employing the stabilized mixed finite element method for the spatial discretization, and the backward Euler and backward differential formulation of second order (BDF2) schemes for the temporal discretization, respectively, we propose the first and second order fully discrete stabilized finite element methods for the models. Furthermore, by utilizing a modified scalar auxiliary variable (SAV) method to balance the errors in the stabilizing and decoupling processes, we prove the stabilities and the inf–sup conditions of the stabilized methods. Finally, several numerical examples are presented to verify the efficiencies of the proposed methods. • By defining two new global fluxes, new mixed finite element methods are established. • By utilizing SAV method, we prove two schemes' stabilities and inf–sup conditions. • The method proposed here is better for solving the convective dominated problem. [ABSTRACT FROM AUTHOR]

To avoid solving a saddle-point system, in this paper, we study two-level Arrow–Hurwicz finite element methods for the steady bio-convection flows problem which is coupled by the steady Navier–Stokes equations and the steady advection–diffusion equation. Using the mini element to approximate the velocity, pressure, and the piecewise linear element to approximate the concentration, we use the linearized Arrow–Hurwicz iteration scheme to obtain the coarse mesh solution and use three different one-step Stokes/Oseen/Newton linearized scheme to obtain the fine mesh solution. The optimal error estimate O (h + H 2 + χ m / 2) of the velocity and concentration in the H 1 -norm and the pressure in the L 2 -norm are derived, where h and H are fine and coarse mesh sizes, respectively, and χ m / 2 denotes the iteration error with 0 < χ < 1. Numerical results are given to support the theoretical analysis and confirm the efficiency of the proposed two-level methods. • Two-level Arrow–Hurwicz (A–H) finite element methods are studied for the steady bio-convection flows problem. • We do not need to solve a saddle-point system to get the numerical solutions. • Optimal error estimates are derived for the proposed two-level A-H Stokes/Oseen/Newton methods. [ABSTRACT FROM AUTHOR]

In this paper, a linearized fully discrete scheme combining the variable-time-step two-step backward differentiation formula (VSBDF2) in time and the nonconforming finite element methods (FEMs) in space is constructed and analyzed for the nonlinear complex Ginzburg–Landau equation. A novel convergence analysis approach is proposed, which shows that the H 1 -norm error of this scheme can reach optimal order in space and sharp second-order convergence in time. The key ingredients of our analysis are temporal–spatial error splitting approach, a new positive definiteness property of discrete orthogonal convolution (DOC) kernels with respect to complex sequences, an energy projection operator and a discrete Laplace operator Δ h related to the nonconforming element space, and a relationship between Δ h and classical L 2 projection operator in complex space. Furthermore, through a specific treatment of the error equation and the combination technique of interpolating operator and projection operator, we prove that the proposed scheme has the global superconvergence result of order O (τ 2 + h 2) in H 1 -norm for the first time, which further improves the efficiency of numerical computation. Here τ and h are the maximum time and spatial step sizes, respectively. It should be pointed out that all these error results are rigorously established under mild assumptions of the time step, and get rid of the usual grid ratio constraints between temporal and spatial step sizes in linearized scheme. At last, numerical experiments are provided to verify the theoretical analysis. • A linearized variable-time-step BDF2 nonconforming FEM is proposed for GL equation. • Unconditional optimal error estimates and superconvergence analysis are presented. • The order reduction in time is avoided and mild time-step assumptions are demanded. • Rigorous proofs and detailed numerical results are provided. [ABSTRACT FROM AUTHOR]

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]

This paper introduces a consistent projection finite method for the non-stationary incompressible thermally coupled MHD equations, which buoyancy affects because temperature differences in the flow cannot be neglected. It is a fully discrete projection method. The unconditional stability and error estimates of the velocity, pressure, temperature, and magnetic field under some assumptions are given. Then some numerical experiments are prescribed to support the effectiveness of the consistent projection finite element method. • A consistent projection finite method for thermally coupled MHD equations is given. • The buoyancy affects because temperature differences in the flow are considered. • It is a fully discrete projection method. • The unconditional stability and error estimates under some assumptions are given. [ABSTRACT FROM AUTHOR]

This paper is devoted to the unconditional optimal error analysis of a linearized, decoupled and conservative Galerkin finite element method (FEM) for the Klein–Gordon–Schödinger equations. The present work has two main contributions: One is that the mass and energy conservation properties of the scheme are proved, and a priori boundedness of the numerical solutions is derived. The other one is that a novel approach is developed and analyzed, which can avoid the use of popular error splitting technique and the restrictions between the time step size and mesh size required in the previous works, and may lead to the unconditional optimal error estimates for the related variables. Some numerical results are also provided to confirm the theoretical analysis and show the efficiency of the proposed scheme. • A linearized, decoupled and conservative Galerkin scheme is proposed for KGS equations. • A novel approach of error estimate is proposed and studied for KGS equations. • The unconditionally optimal error estimates in L 2 -norm are obtained skillfully. [ABSTRACT FROM AUTHOR]

• This paper introduces numerical schemes for approximation of solutions to some nonhomogeneous boundary value problems associated to the nonlinear Korteweg-de Vries equation (KdV) and a system of two coupled KdV-type equations derived by Gear and Grimshaw posed on a bounded interval. • The numerical results show the persistence of the behavior not uniform in time of the control functions computed. • The controllability problems for nonlinear KdV-type systems we consider have been intensively studied for the model equations addressed in this paper, but in the past, emphasis have been placed on theoretical results. In this paper we fill this gap by exploring this problem from the numerical point of view. • Numerical methods are valuable tools for studying dynamics of complicated evolution systems and for exploring open questions on controllability of dispersive-type equations. This paper addresses the approximation of solutions to some non-homogeneous boundary value problems associated with the nonlinear Korteweg-de Vries equation (KdV) and a system of two coupled KdV-type equations derived by Gear and Grimshaw posed on a bounded interval. An efficient Galerkin scheme that combines a finite element strategy for space discretization with a second-order implicit scheme for time-stepping is employed to approximate time dynamics of model equations studied. Several numerical experiments, including boundary controllability problems for nonlinear KdV and GG equations, are presented for different final states to show the performance of the numerical strategies proposed. The numerical results with nonlinear models agree with previous analytic theory and show the persistence of the behavior not uniform in time of the control functions computed already observed by Rosier [22] in the case of the linear KdV equation. [ABSTRACT FROM AUTHOR]

The paper is concerned with the strong approximation of a stochastic time-fractional Allen-Cahn equation driven by an additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time; namely, a Caputo fractional derivative of order α ∈ (0 , 1) , and a Riemann–Liouville fractional integral operator of order γ ∈ [ 0 , 1 ] applied to a Gaussian noise. We approximate the model by a standard piecewise linear finite element method (FEM) in space and the classical Grünwald–Letnikov method in time (for both time-fractional operators), and the noise by the L 2 -projection. Spatially semidiscrete and fully discrete schemes are analyzed and strong convergence rates are obtained by exploiting the temporal Hölder continuity property of the solution. Numerical experiments are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents an efficient and accurate differential quadrature (DQ) large amplitude free vibration analysis of laminated composite thin beams on nonlinear elastic foundation. Beams under consideration have elastically restrained against rotation and in-plane immovable edges. Elastic foundation has cubic nonlinearity with shearing layer. We impose the boundary conditions directly into the governing equations in spite of the conventional DQ method and without any extra efforts. A direct iterative method is used to solve the nonlinear eigenvalue system of equations after transforming the governing equations into the frequency domain. The fast rate of convergence of the method is shown and their accuracy is demonstrated by comparing the results with those for limit cases, i.e. beams with classical boundary conditions, available in the literature. Besides, we develop a finite element program to verify the results of the presented DQ approach and to show its high computational efficiency. The effects of different parameters on the ratio of nonlinear to linear natural frequency of beams are studied. [Copyright &y& Elsevier]

This paper presents a finite element method (FEM) for simulating open channel network flows. Using a recursive formula derived from element equations, a relationship for channel junctions is established and a system equation is obtained. The system equation is much smaller in size in comparison with the simultaneous solution approach and its solution provides boundary conditions for single channels. The FEM is applied to solve individual channels, and a double sweep method is employed to save computation time. To demonstrate the potential of the approach, two channel networks were computed. It is shown that the results obtained with the proposed FEM are very close to those given by the widely used Preissmann scheme, indicating that it is an effective method to model flows in open channels. [Copyright &y& Elsevier]