A linearized Crank–Nicolson–Galerkin FEM for the time‐dependent Ginzburg–Landau equations under the temporal gauge

We propose a decoupled and linearized fully discrete finite element method (FEM) for the time‐dependent Ginzburg–Landau equations under the temporal gauge, where a Crank–Nicolson scheme is used for the time discretization. By carefully designing the time‐discretization scheme, we manage to prove the convergence rate , where τ is the time‐step size and r is the degree of the finite element space. Due to the degeneracy of the problem, the convergence rate in the spatial direction is one order lower than the optimal convergence rate of FEMs for parabolic equations. Numerical tests are provided to support our error analysis. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1279–1290, 2014     

Stability of the Crank–Nicolson–Adams–Bashforth scheme for the 2D Leray‐alpha model

We consider the stability of an efficient Crank–Nicolson–Adams–Bashforth method in time, finite element in space, discretization of the Leray‐α model. We prove finite‐time stability of the scheme in L², H¹, and H², as well as the long‐time L‐stability of the scheme under a Courant‐Freidrichs‐Lewy (CFL)‐type condition. Numerical experiments are given that are in agreement with the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1155–1183, 2016     

Convergence analysis of a linearized Crank–Nicolson scheme for the two‐dimensional complex Ginzburg–Landau equation

A linearized Crank–Nicolson‐type scheme is proposed for the two‐dimensional complex Ginzburg–Landau equation. The scheme is proved to be unconditionally convergent in the L² ‐norm by the discrete energy method. The convergence order is \begin{align*}\mathcal{O}(\tau^2+h_1^2+h^2_2)\end{align*}, where τ is the temporal grid size and h₁,h₂ are spatial grid sizes in the x ‐ and y ‐directions, respectively. A numerical example is presented to support the theoretical result. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A three‐level linearized finite difference scheme for the camassa–holm equation

The Camassa–Holm (CH) system is a strong nonlinear third‐order evolution equation. So far, the numerical methods for solving this problem are only a few. This article deals with the finite difference solution to the CH equation. A three‐level linearized finite difference scheme is derived. The scheme is proved to be conservative, uniquely solvable, and conditionally second‐order convergent in both time and space in the discrete L_∞ norm. Several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 451–471, 2014     

The finite volume method based on the Crouzeix–Raviart element for a fracture model

In this paper, we present and analyze a finite volume method based on the Crouzeix–Raviart element for the coupled fracture model, where the fluid flow is governed by Darcy's law in the one‐dimensional fracture and two‐dimensional surrounding matrix. In the numerical scheme, the pressure in the matrix and fracture is respectively approximated by the Crouzeix–Raviart elements and piecewise constant functions, and then the velocity is calculated by piecewise constant functions element by element. The existence and uniqueness of the numerical solution are discussed, and optimal order error estimates for both the pressure p and the velocity u are proved on general triangulations. We finally carry out numerical experiments, and results confirm our theoretical analysis.     

Analysis and computational method based on quadratic B‐spline FEM for the Rosenau–Burgers equation

L_∞‐error estimates for B‐spline Galerkin finite element solution of the Rosenau–Burgers equation are considered. The semidiscrete B‐spline Galerkin scheme is studied using appropriate projections. For fully discrete B‐spline Galerkin scheme, we consider the Crank–Nicolson method and analyze the corresponding error estimates in time. Numerical experiments are given to demonstrate validity and order of accuracy of the proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 877–895, 2016     

Fractional Crank–Nicolson–Galerkin finite element scheme for the time‐fractional nonlinear diffusion equation

This article presents a finite element scheme with Newton's method for solving the time‐fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank–Nicolson scheme based on backward Euler convolution quadrature. We discuss the existence‐uniqueness results for the fully discrete problem. A new discrete fractional Gronwall type inequality for the backward Euler convolution quadrature is established. A priori error estimate for the fully discrete problem in L²(Ω) norm is derived. Numerical results based on finite element scheme are provided to validate theoretical estimates on time‐fractional nonlinear Fisher equation and Huxley equation.     

A numerical method for the Landau–Lifshitz equation with magnetostriction

We present a numerical scheme for Landau–Lifshitz–Gilbert equation coupled with the equation of elastodynamics. The considered physical model describes the behaviour of ferromagnetic materials when magnetomechanical coupling is taken into account. The time‐discretization is based on the backward Euler method with projection. In the numerical approximation, the two equations are decoupled by a suitable linearization in order to solve the magnetic and mechanic part separately. The resulting semi‐implicit scheme is linear and allows larger time‐steps than explicit methods. We prove stability and error estimates for the presented time discretization in 2D. Finally, we test the accuracy of the scheme on an academic numerical example with known exact solution. Copyright © 2005 John Wiley & Sons, Ltd.     

Two‐Grid method for nonlinear parabolic equations by expanded mixed finite element methods

In this article, we develop a two‐grid algorithm for nonlinear reaction diffusion equation (with nonlinear compressibility coefficient) discretized by expanded mixed finite element method. The key point is to use two‐grid scheme to linearize the nonlinear term in the equations. The main procedure of the algorithm is solving a small‐scaled nonlinear equations on the coarse grid and dealing with a linearized system on the fine space using the Newton iteration with the coarse grid solution. Error estimation to the expanded mixed finite element solution is analyzed in detail. We also show that two‐grid solution achieves the same accuracy as long as the mesh sizes satisfy H = O(h^1/2). Two numerical experiments are given to verify the effectiveness of the algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Large time behavior for the non‐isentropic Navier–Stokes–Maxwell system

In this paper, we are concerned with the system of the non‐isentropic compressible Navier–Stokes equations coupled with the Maxwell equations through the Lorentz force in three space dimensions. The global existence of solutions near constant steady states is established, and the time‐decay rates of perturbed solutions are obtained. The proof for existence is due to the classical energy method, and the investigation of large‐time behavior is based on the linearized analysis of the non‐isentropic Navier–Stokes–Poisson equations and the electromagnetic part for the linearized isentropic Navier–Stokes–Maxwell equations. In the meantime, the time‐decay rates obtained by Zhang, Li, and Zhu [J. Differential Equations, 250(2011), 866‐891] for the linearized non‐isentropic Navier–Stokes–Poisson equations are improved. Copyright © 2016 John Wiley & Sons, Ltd.     

Decoupled characteristic stabilized finite element method for time‐dependent Navier–Stokes/Darcy model

In this article, we propose and analyze a new decoupled characteristic stabilized finite element method for the time‐dependent Navier–Stokes/Darcy model. The key idea lies in combining the characteristic method with the stabilized finite element method to solve the decoupled model by using the lowest‐order conforming finite element space. In this method, the original model is divided into two parts: one is the nonstationary Navier–Stokes equation, and the other one is the Darcy equation. To deal with the difficulty caused by the trilinear term with nonzero boundary condition, we use the characteristic method. Furthermore, as the lowest‐order finite element pair do not satisfy LBB (Ladyzhen‐Skaya‐Brezzi‐Babuska) condition, we adopt the stabilized technique to overcome this flaw. The stability of the numerical method is first proved, and the optimal error estimates are established. Finally, extensive numerical results are provided to justify the theoretical analysis.     

A combined finite volume–finite element scheme for the discretization of strongly nonlinear convection–diffusion–reaction problems on nonmatching grids

We propose and analyze in this paper a numerical scheme for nonlinear degenerate parabolic convection–diffusion–reaction equations in two or three space dimensions. We discretize the time evolution, convection, reaction, and source terms on a given grid, which can be nonmatching and can contain nonconvex elements, by means of the cell‐centered finite volume method. To discretize the diffusion term, we construct a conforming simplicial mesh with the vertices given by the original grid and use the conforming piecewise linear finite element method. In this way, the scheme is fully consistent and the discrete solution is naturally continuous across the interfaces between the subdomains with nonmatching grids, without introducing any supplementary equations and unknowns or using any interpolation at the interfaces. We allow for general inhomogeneous and anisotropic diffusion–dispersion tensors, propose two variants corresponding respectively to arithmetic and harmonic averaging, and use the local Péclet upstream weighting in order to only add the minimal numerical diffusion necessary to avoid spurious oscillations in the convection‐dominated case. The scheme is robust, efficient since it leads to positive definite matrices and one unknown per element, locally conservative, and satisfies the discrete maximum principle under the conditions on the simplicial mesh and the diffusion tensor usual in the finite element method. We prove its convergence using a priori estimates and the Kolmogorov relative compactness theorem and illustrate its behavior on a numerical experiment. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Two‐grid methods for semilinear interface problems

In this article, we consider two‐grid finite element methods for solving semilinear interface problems in d space dimensions, for d = 2 or d = 3. We consider semilinear problems with discontinuous diffusion coefficients, which includes problems containing subcritical, critical, and supercritical nonlinearities. We establish basic quasioptimal a priori error estimates for Galerkin approximations. We then design a two‐grid algorithm consisting of a coarse grid solver for the original nonlinear problem, and a fine grid solver for a linearized problem. We analyze the quality of approximations generated by the algorithm and show that the coarse grid may be taken to have much larger elements than the fine grid, and yet one can still obtain approximation quality that is asymptotically as good as solving the original nonlinear problem on the fine mesh. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A numerical method for solving the nonlinear fermi–pasta–ulam problem

An effective finite difference scheme for solving the nonlinear Fermi–Pasta–Ulam (FPU) problem is derived. The most important feature of the scheme inherits energy conservation property from the nonlinear FPU problem. The unique solvability and the convergence of the difference scheme are proved by the energy method. The convergence order is in the maximum norm, where τ is the temporal grid size and h is the spatial grid size, respectively. In addition, the stability of the difference scheme is obtained. Numerical results are presented to support the theoretical analysis and verify numerically the energy conservation property.© 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 187‐209, 2014     

Mixed two‐grid finite difference methods for solving one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations

The aim of this paper is to propose mixed two‐grid finite difference methods to obtain the numerical solution of the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. The finite difference equations at all interior grid points form a large‐sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a family of finite difference methods for discretizing the spatial and time derivatives. The obtained system has been solved by two‐grid method, where the two‐grid method is used for solving the large‐sparse linear systems. Also, in the proposed method, the spectral radius with local Fourier analysis is calculated for different values of h and Δt. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical approximation of the generalized regularized long wave equation using Petrov–Galerkin finite element method

The generalized regularized long wave (GRLW) equation has been developed to model a variety of physical phenomena such as ion‐acoustic and magnetohydrodynamic waves in plasma, nonlinear transverse waves in shallow water and phonon packets in nonlinear crystals. This paper aims to develop and analyze a powerful numerical scheme for the nonlinear GRLW equation by Petrov–Galerkin method in which the element shape functions are cubic and weight functions are quadratic B‐splines. The proposed method is implemented to three reference problems involving propagation of the single solitary wave, interaction of two solitary waves and evolution of solitons with the Maxwellian initial condition. The variational formulation and semi‐discrete Galerkin scheme of the equation are firstly constituted. We estimate rate of convergence of such an approximation. Using Fourier stability analysis of the linearized scheme we show that the scheme is unconditionally stable. To verify practicality and robustness of the new scheme error norms L₂, L_∞ and three invariants I₁, I₂, and I₃ are calculated. The computed numerical results are compared with other published results and confirmed to be precise and effective.     

Unconditional superconvergent analysis of a new mixed finite element method for Ginzburg–Landau equation

In this article, unconditional superconvergent analysis of a linearized fully discrete mixed finite element method is presented for a class of Ginzburg–Landau equation based on the bilinear element and zero‐order Nédélec's element pair (Q₁₁/Q₀₁ × Q₁₀). First, a time‐discrete system is introduced to split the error into temporal error and spatial error, and the corresponding error estimates are deduced rigorously. Second, the unconditional superclose and optimal estimate of order O(h² + τ) for u in H¹‐norm and p = ?u in L²‐norm are derived respectively without the restrictions on the ratio between h and τ, where h is the subdivision parameter and τ, the time step. Third, the global superconvergent results are obtained by interpolated postprocessing technique. Finally, some numerical results are carried out to confirm the theoretical analysis.     

An energy‐preserving Crank–Nicolson Galerkin method for Hamiltonian partial differential equations

A semidiscretization based method for solving Hamiltonian partial differential equations is proposed in this article. Our key idea consists of two approaches. First, the underlying equation is discretized in space via a selected finite element method and the Hamiltonian PDE can thus be casted to Hamiltonian ODEs based on the weak formulation of the system. Second, the resulting ordinary differential system is solved by an energy‐preserving integrator. The relay leads to a fully discretized and energy‐preserved scheme. This strategy is fully realized for solving a nonlinear Schrödinger equation through a combination of the Galerkin discretization in space and a Crank–Nicolson scheme in time. The order of convergence of our new method is if the discrete L²‐norm is employed. An error estimate is acquired and analyzed without grid ratio restrictions. Numerical examples are given to further illustrate the conservation and convergence of the energy‐preserving scheme constructed.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1485–1504, 2016     

Decoupled energy‐law preserving numerical schemes for the Cahn–Hilliard–Darcy system

We study two novel decoupled energy‐law preserving and mass‐conservative numerical schemes for solving the Cahn‐Hilliard‐Darcy system which models two‐phase flow in porous medium or in a Hele–Shaw cell. In the first scheme, the velocity in the Cahn–Hilliard equation is treated explicitly so that the Darcy equation is completely decoupled from the Cahn–Hilliard equation. In the second scheme, an intermediate velocity is used in the Cahn–Hilliard equation which allows for the decoupling. We show that the first scheme preserves a discrete energy law with a time‐step constraint, while the second scheme satisfies an energy law without any constraint and is unconditionally stable. Ample numerical experiments are performed to gauge the efficiency and robustness of our scheme. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 936–954, 2016     

A linearized,decoupled, and energy‐preserving compact finite difference scheme for the coupled nonlinear Schrödinger equations

In this article, a decoupled and linearized compact finite difference scheme is proposed for solving the coupled nonlinear Schrödinger equations. The new scheme is proved to preserve the total mass and energy which are defined by using a recursion relationship. Besides the standard energy method, an induction argument together with an H¹ technique are introduced to establish the optimal point‐wise error estimate of the proposed scheme. Without imposing any constraints on the grid ratios, the convergence order of the numerical solution is proved to be of with mesh size h and time step τ. Numerical results are reported to verify the theoretical analysis, and collision of two solitary waves are also simulated. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 840–867, 2017