Siriguleng He

Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction–diffusion problem☆

Abstract In this article, a finite difference/finite element algorithm, which is based on a finite difference approximation in time direction and finite element method in spatial direction, is presented and discussed to cast about for the numerical solutions of a time-fractional fourth-order reaction–diffusion problem with a nonlinear reaction term. To avoid the use of higher-order elements, the original problem with spatial fourth-order derivative need to be changed into a second-order coupled system by introducing an intermediate variable σ = Δ u . Then the fully discrete finite element scheme is formulated by using a finite difference approximation for time fractional and integer derivatives and finite element method in spatial direction. The unconditionally stable result in the norm, which just depends on initial value and source item, is derived. Some a priori estimates of L 2 -norm with optimal order of convergence O ( Δ t 2 − α + h m + 1 ) , where Δ t and h are time step length and space mesh parameter, respectively, are obtained. To confirm the theoretical analysis, some numerical results are provided by our method.

A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative

In this article, we develop a two-grid algorithm based on the mixed finite element (MFE) method for a nonlinear fourth-order reaction-diffusion equation with the time-fractional derivative of Caputo-type. We formulate the problem as a nonlinear fully discrete MFE system, where the time integer and fractional derivatives are approximated by finite difference methods and the spatial derivatives are approximated by the MFE method. To solve the nonlinear MFE system more efficiently, we propose a two-grid algorithm, which is composed of two steps: we first solve a nonlinear MFE system on a coarse grid by nonlinear iterations, then solve the linearized MFE system on the fine grid by Newton iteration. Numerical stability and optimal error estimate O ( k Δ 2 - α + h r + 1 + H 2 r + 2 ) in L 2 -norm are proved for our two-grid scheme, where k Δ , h and H are the time step size, coarse grid mesh size, and fine grid mesh size, respectively. We implement the two-grid algorithm, and present the numerical results justifying our theoretical error estimate. The numerical tests also show that the two-grid method is much more efficient than solving the nonlinear MFE system directly.

A New Mixed Element Method for a Class of Time-Fractional Partial Differential Equations

A kind of new mixed element method for time-fractional partial differential equations is studied. The Caputo-fractional derivative of time direction is approximated by two-step difference method and the spatial direction is discretized by a new mixed element method, whose gradient belongs to the simple (L 2(Ω)2) space replacing the complex H(div; Ω) space. Some a priori error estimates in L 2-norm for the scalar unknown u and in (L 2)2-norm for its gradient σ. Moreover, we also discuss a priori error estimates in H 1-norm for the scalar unknown u.

Analysis of mixed finite element methods for fourth-order wave equations

Mixed finite element methods, explicit and implicit in time, for a fourth-order wave equation are considered in this paper. The optimal error estimates in the L^2 norm for velocity and moment and in the H^1 norm and L^2 norm for displacement are derived. These error estimates are proved by using a special interpolation operator on quasi-uniform rectangular meshes. The stabilities of the two schemes are also analyzed. In addition, three other kinds of mixed scheme are constructed. Numerical examples are provided to verify the theoretical results.

Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation☆

Abstract In this article, a fully discrete local discontinuous Galerkin (LDG) method with high-order temporal convergence rate is presented and developed to look for the numerical solution of nonlinear time-fractional fourth-order partial differential equation (PDE). In the temporal direction, for approximating the fractional derivative with order α ∈ ( 0 , 1 ) , the weighted and shifted Grunwald difference (WSGD) scheme with second-order convergence rate is introduced and for approximating the integer time derivative, two step backward Euler method with second-order convergence rate is used. For the spatial direction, the LDG method is used. For the numerical theories, the stability is derived and a priori error results are proved. Further, some error results and convergence rates are calculated by numerical procedure to illustrate the effectiveness of proposed method.

A Coupling Method of New EMFE and FE for Fourth-Order Partial Differential Equation of Parabolic Type

We propose and analyze a new numerical method, called a coupling method based on a new expanded mixed finite element (EMFE) and finite element (FE), for fourth-order partial differential equation of parabolic type. We first reduce the fourth-order parabolic equation to a coupled system of second-order equations and then solve a second-order equation by FE method and approximate the other one by a new EMFE method. We find that the new EMFE method’s gradient belongs to the simple square integrable space, which avoids the use of the classical H(div; Ω) space and reduces the regularity requirement on the gradient solution . For a priori error estimates based on both semidiscrete and fully discrete schemes, we introduce a new expanded mixed projection and some important lemmas. We derive the optimal a priori error estimates in and -norm for both the scalar unknown and the diffusion term γ and a priori error estimates in -norm for its gradient and its flux (the coefficients times the negative gradient). Finally, we provide some numerical results to illustrate the efficiency of our method.

A new expanded mixed method for parabolic integro-differential equations

A new expanded mixed scheme is studied and analyzed for linear parabolic integro-differential equations. The proposed methods gradient belongs to the simple square integrable space replacing the classical H ( div; ? ) space. The new expanded mixed projection is introduced, the existence and uniqueness of solution for semi-discrete scheme are proved and the fully discrete error estimates based on both backward Euler scheme are obtained. Moreover, the optimal a priori error estimates in L 2 and H 1 -norm for the scalar unknown u and the error results in L 2 ( ? ) -norm for its gradient λ, and its flux ? (the coefficients times the negative gradient) are derived. Finally, some numerical results are calculated to verify our theoretical analysis.

A New Linearized Crank-Nicolson Mixed Element Scheme for the Extended Fisher-Kolmogorov Equation

We present a new mixed finite element method for solving the extended Fisher-Kolmogorov (EFK) equation. We first decompose the EFK equation as the two second-order equations, then deal with a second-order equation employing finite element method, and handle the other second-order equation using a new mixed finite element method. In the new mixed finite element method, the gradient ∇u belongs to the weaker (L 2(Ω))2 space taking the place of the classical H(div; Ω) space. We prove some a priori bounds for the solution for semidiscrete scheme and derive a fully discrete mixed scheme based on a linearized Crank-Nicolson method. At the same time, we get the optimal a priori error estimates in L 2 and H 1-norm for both the scalar unknown u and the diffusion term w = −Δu and a priori error estimates in (L 2)2-norm for its gradient χ = ∇u for both semi-discrete and fully discrete schemes.

Second-order approximation scheme combined with H1-Galerkin MFE method for nonlinear time fractional convection–diffusion equation

Abstract In this article, a second-order approximation scheme combined with an H 1 -Galerkin mixed finite element (MFE) method for solving nonlinear convection–diffusion equation with time fractional derivative is proposed and analyzed. By introducing an auxiliary variable, a coupled system is formulated, then the spatial direction is approximated by H 1 -Galerkin MFE method and the temporal fractional derivative and integer derivative are discretized by second-order weighted and shifted Grunwald difference (WSGD) formula and linearized second-order difference scheme, respectively. The optimal priori error estimates in L 2 and H 1 -norm for the unknown function and the auxiliary variable with second-order convergent rate in time are obtained. Compared to the commonly used L1-approximation with ( 2 − α ) th-order convergence rate, our method can arrive at the order 2 in time. What is more, compared with the standard finite element method, our method can well approximate the auxiliary variable. Finally, the detailed computational process of the studied numerical algorithm is shown and a nonlinear numerical example with calculated data and some figures is provided to verify our theoretical analysis.

A Novel Characteristic Expanded Mixed Method for Reaction-Convection-Diffusion Problems

A novel characteristic expanded mixed finite element method is proposed and analyzed for reaction-convection-diffusion problems. The diffusion term is discretized by the novel expanded mixed method, whose gradient belongs to the square integrable space instead of the classical space and the hyperbolic part is handled by the characteristic method. For a priori error estimates, some important lemmas based on the novel expanded mixed projection are introduced. The fully discrete error estimates based on backward Euler scheme are obtained. Moreover, the optimal a priori error estimates in - and -norms for the scalar unknown and a priori error estimates in -norm for its gradient and its flux (the coefficients times the negative gradient) are derived. Finally, a numerical example is provided to verify our theoretical results.