Zhichao Fang

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.

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.

Application of low-dimensional finite element method to fractional diffusion equation

Classical finite element method (FEM) has been applied to solve some fractional differential equations, but its scheme has too many degrees of freedom. In this paper, a low-dimensional FEM, whose number of basis functions is reduced by the theory of proper orthogonal decomposition (POD) technique, is proposed for the time fractional diffusion equation in two-dimensional space. The presented method has the properties of low dimensions and high accuracy so that the amount of computation is decreased and the calculation time is saved. Moreover, error estimation of the method is obtained. Numerical example is given to illustrate the feasibility and validity of the low-dimensional FEM in comparison with traditional FEM for the time fractional differential equations.

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.

A NEW CHARACTERISTIC EXPANDED MIXED METHOD FOR SOBOLEV EQUATION WITH CONVECTION TERM

In this paper, a new numerical method based on a new expanded mixed scheme and the characteristic method is developed and discussed for Sobolev equation with convection term. The hyperbolic part is handled by the characteristic method and the diffusion term ∇ ⋅ (a(x, t)∇u+b(x, t)∇ut) is approximated by the new expanded mixed method, whose gradient belongs to the simple square integrable (L2(Ω))2 space instead of the classical H(div; Ω) space. For a priori error estimates, some important lemmas based on the new expanded mixed projection are introduced. An optimal priori error estimates in L2-norm for the scalar unknown u and a priori error estimates in (L2)2-norm for its gradient λ, and its flux σ (the coefficients times the negative gradient) are derived. In particular, an optimal priori error estimate in H1-norm for the scalar unknown u is obtained.

A new expanded mixed element method for convection-dominated Sobolev equation.

We propose and analyze a new expanded mixed element method, whose gradient belongs to the simple square integrable space instead of the classical H(div; Ω) space of Chens expanded mixed element method. We study the new expanded mixed element method for convection-dominated Sobolev equation, prove the existence and uniqueness for finite element solution, and introduce a new expanded mixed projection. We derive the optimal a priori error estimates in L 2-norm for the scalar unknown u and a priori error estimates in (L 2)2-norm for its gradient λ and its flux σ. Moreover, we obtain the optimal a priori error estimates in H 1-norm for the scalar unknown u. Finally, we obtained some numerical results to illustrate efficiency of the new method.