Variational solution of fractional advection dispersion equations on bounded domains in ℝd

In this article, we discuss the steady state fractional advection dispersion equation (FADE) on bounded domains in ?^d. Fractional differential and integral operators are defined and analyzed. Appropriate fractional derivative spaces are defined and shown to be equivalent to the fractional dimensional Sobolev spaces. A theoretical framework for the variational solution of the steady state FADE is presented. Existence and uniqueness results are proven, and error estimates obtained for the finite element approximation. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 256–281, 2007     

High‐order local discontinuous Galerkin method for a fractal mobile/immobile transport equation with the Caputo–Fabrizio fractional derivative

In this article, a local discontinuous Galerkin (LDG) method is studied for numerically solving the fractal mobile/immobile transport equation with a new time Caputo–Fabrizio fractional derivative. The stability of the LDG scheme is proven, and a priori error estimates with the second‐order temporal convergence rate and the (k + 1) th order spatial convergence rate are derived in detail. Finally, numerical experiments based on P^k, k = 0, 1, 2, 3, elements are provided to verify our theoretical results.     

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 second order finite difference-spectral method for space fractional diffusion equations

A high order finite difference-spectral method is derived for solving space fractional diffusion equations,by combining the second order finite difference method in time and the spectral Galerkin method in space.The stability and error estimates of the temporal semidiscrete scheme are rigorously discussed,and the convergence order of the proposed method is proved to be O(τ2+Nα-m)in L2-norm,whereτ,N,αand m are the time step size,polynomial degree,fractional derivative index and regularity of the exact solution,respectively.Numerical experiments are carried out to demonstrate the theoretical analysis.     

Weak Galerkin finite element method for a class of time fractional generalized Burgers' equation

In this article, we use the weak Galerkin (WG) finite element method to study a class of time fractional generalized Burgers' equation. The existence of numerical solutions and the stability of fully discrete scheme are proved. Meanwhile, by applying the energy method, an optimal order error estimate in discrete L² norm is established. Numerical experiments are presented to validate the theoretical analysis.     

A mixed‐type Galerkin variational formulation and fast algorithms for variable‐coefficient fractional diffusion equations

We consider the variable‐coefficient fractional diffusion equations with two‐sided fractional derivative. By introducing an intermediate variable, we propose a mixed‐type Galerkin variational formulation and prove the existence and uniqueness of the variational solution over . On the basis of the formulation, we develop a mixed‐type finite element procedure on commonly used finite element spaces and derive the solvability of the finite element solution and the error bounds for the unknown and the intermediate variable. For the Toeplitz‐like linear system generated by discretization, we design a fast conjugate gradient normal residual method to reduce the storage from O(N²) to O(N) and the computing cost from O(N³) to O(NlogN). Numerical experiments are included to verify our theoretical findings. Copyright © 2017 John Wiley & Sons, Ltd.     

Superconvergence of the direct discontinuous Galerkin method for a time‐fractional initial‐boundary value problem

A time‐fractional reaction–diffusion initial‐boundary value problem with periodic boundary condition is considered on Q ? Ω × [0, T] , where Ω is the interval [0, l] . Typical solutions of such problem have a weak singularity at the initial time t = 0. The numerical method of the paper uses a direct discontinuous Galerkin (DDG) finite element method in space on a uniform mesh, with piecewise polynomials of degree k ≥ 2 . In the temporal direction we use the L1 approximation of the Caputo derivative on a suitably graded mesh. We prove that at each time level of the mesh, our L1‐DDG solution is superconvergent of order k + 2 in L²(Ω) to a particular projection of the exact solution. Moreover, the L1‐DDG solution achieves superconvergence of order (k + 2) in a discrete L²(Q) norm computed at the Lobatto points, and order (k + 1) superconvergence in a discrete H¹(Q) seminorm at the Gauss points; numerical results show that these estimates are sharp.     

A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers

In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well‐posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H¹ and L² norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.     

Numerical simulation of a class of fractional subdiffusion equations via the alternating direction implicit method

In this article, a new numerical technique is proposed for solving the two‐dimensional time fractional subdiffusion equation with nonhomogeneous terms. After a transformation of the original problem, standard central difference approximation is used for the spatial discretization. For the time step, a new fractional alternating direction implicit (FADI) scheme based on the L₁ approximation is considered. This FADI scheme is constructed by adding a small term, so it is different from standard FADI methods. The solvability, unconditional stability and H¹ norm convergence are proved. Finally, numerical examples show the effectiveness and accuracy of our proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 531–547, 2016     

Interior penalty discontinuous Galerkin methods with implicit time‐integration techniques for nonlinear parabolic equations

We prove existence and numerical stability of numerical solutions of three fully discrete interior penalty discontinuous Galerkin methods for solving nonlinear parabolic equations. Under some appropriate regularity conditions, we give the l²(H¹) and l^∞(L²) error estimates of the fully discrete symmetric interior penalty discontinuous Galerkin–scheme with the implicit θ ‐schemes in time, which include backward Euler and Crank–Nicolson finite difference approximations. Our estimates are optimal with respect to the mesh size h. The theoretical results are confirmed by some numerical experiments. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Orthogonal spline collocation scheme for multiterm fractional convection‐diffusion equation with variable coefficients

The orthogonal spline collocation (OSC) technique is an efficient way to solve a wide variety of problems that are modeled by ordinary and partial differential equations. In this article, by using OSC method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established for a class of two‐dimensional multiterm fractional convection‐diffusion reaction equation with variable coefficients. The optimal estimates in H^j (j = 0, 1, 2) norms at each time step are derived. Also, estimate in space is provided. At last, we provide some numerical results to verify the accuracy and efficiency of the proposed algorithm.     

Existence and uniqueness of minimizers for L2-constrained problems related to fractional Kirchhoff equation

In this paper, we discuss the existence and uniqueness of solutions of the constrained variational problem with respect to the fractional Kirchhoff equation. For the exponent p<p_*(s,N), a complete classification with respect to p for the existence of solutions of the fractional Kirchhoff functional on the L²-normalized manifold was given. Furthermore, all these solutions are unique up to translations, and our methods depend only on some simple energy estimates.     

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L_∞ ‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.     

Parallel Galerkin domain decomposition procedures for parabolic equation on general domain

Parallel Galerkin domain decomposition procedures for parabolic equation on general domain are given. These procedures use implicit Galerkin method in the subdomains and simple explicit flux calculation on the interdomain boundaries by integral mean method or extrapolation method to predict the inner‐boundary conditions. Thus, the parallelism can be achieved by these procedures. These procedures are conservative both in the subdomains and across interboundaries. The explicit nature of the flux prediction induces a time‐step limitation that is necessary to preserve stability, but this constraint is less severe than that for a fully explicit method. L²‐norm error estimates are derived for these procedures. Compared with the work of Dawson and Dupont [Math Comp 58 (1992), 21–35], these L²‐norm error estimates avoid the loss of H^?1/2 factor. Experimental results are presented to confirm the theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Weak Galerkin finite element methods for a fourth order parabolic equation

This paper is devoted to a newly developed weak Galerkin finite element method with the stabilization term for a linear fourth order parabolic equation, where weakly defined Laplacian operator over discontinuous functions is introduced. Priori estimates are developed and analyzed in L² and an H² type norm for both semi‐discrete and fully discrete schemes. And finally, numerical examples are provided to confirm the theoretical results.     

Well-posedness of equations with fractional derivative

We study the well-posedness of the equations with fractional derivative D^αu（t） = Au（t） ＋ f（t）,0≤ t ≤ 2π, where A is a closed operator in a Banach space X, α 〉 0 and D^α is the fractional derivative in the sense of Weyl. Using known results on LP-multipliers, we give necessary and/or sufficient conditions for the LP-well-posedness of this problem. The conditions we give involve the resolvent of A and the Rademacher boundedness. Corresponding results on the well-posedness of this problem in periodic Besov spaces, periodic Triebel-Lizorkin spaces and periodic Hardy spaces are also obtained.     

A mixed and discontinuous Galerkin finite volume element method for incompressible miscible displacement problems in porous media

The incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations, the pressure‐velocity equation and the concentration equation. In this article, we present a mixed finite volume element method for the approximation of pressure‐velocity equation and a discontinuous Galerkin finite volume element method for the concentration equation. A priori error estimates in L^∞(L²) are derived for velocity, pressure, and concentration. Numerical results are presented to substantiate the validity of the theoretical results. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Numerical solution of two-dimensional time fractional cable equation with Mittag-Leffler kernel

The main motive of this article is to study the recently developed Atangana-Baleanu Caputo (ABC) fractional operator that is obtained by replacing the classical singular kernel by Mittag-Leffler kernel in the definition of the fractional differential operator. We investigate a novel numerical method for the nonlinear two-dimensional cable equation in which time-fractional derivative is of Mittag-Leffler kernel type. First, we derive an approximation formula of the fractional-order ABC derivative of a function t^k using a numerical integration scheme. Using this approximation formula and some properties of shifted Legendre polynomials, we derived the operational matrix of ABC derivative. In the author of knowledge, this operational matrix of ABC derivative is derived the first time. We have shown the efficiency of this newly derived operational matrix by taking one example. Then we solved a new class of fractional partial differential equations (FPDEs) by the implementation of this ABC operational matrix. The two-dimensional model of the time-fractional model of the cable equation is solved and investigated by this method. We have shown the effectiveness and validity of our proposed method by giving the solution of some numerical examples of the two-dimensional fractional cable equation. We compare our obtained numerical results with the analytical results, and we conclude that our proposed numerical method is feasible and the accuracy can be seen by error tables. We see that the accuracy is so good. This method will be very useful to investigate a different type of model that have Mittag-Leffler fractional derivative.     

A Discontinuous Galerkin Method for Fractional Differential Equations

We investigate the numerical treatment of fractional derivatives using a discontinuous Galerkin approach. We consider fractional differential equations in one dimension. The most common definitions, the Riemann-Liouville and Caputo derivatives, as well as a third definition are considered. Different types of initial condition statements are investigated for the extended discontinuous Galerkin method. We obtain numerical results on the order of convergence in the L₂ and H¹ norms. We compare the classical discontinuous Galerkin method with our adapted approach and show results for the different types of fractional derivatives. Further work was carried out on parabolic differential equations with fractional time derivatives, which is to be published cf. [2]. A keystone in the further research is the development of analytical tools for theoretical convergence results. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new alternating‐direction finite element method for hyperbolic equation

A modified backward difference time discretization is presented for Galerkin approximations for nonlinear hyperbolic equation in two space variables. This procedure uses a local approximation of the coefficients based on patches of finite elements with these procedures, a multidimensional problem can be solved as a series of one‐dimensional problems. Optimal order H₀¹ and L² error estimates are derived. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007