Developing weak Galerkin finite element methods for the wave equation

In this article, we extend the recently developed weak Galerkin method to solve the second‐order hyperbolic wave equation. Many nice features of the weak Galerkin method have been demonstrated for elliptic, parabolic, and a few other model problems. This is the initial exploration of the weak Galerkin method for solving the wave equation. Here we successfully developed and established the stability and convergence analysis for the weak Galerkin method for solving the wave equation. Numerical experiments further support the theoretical analysis. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 868–884, 2017     

Implicit–explicit multistep characteristic finite element methods for nonlinear convection–diffusion equations

Implicit–explicit multistep characteristic methods are given for convection‐dominated diffusion equations. Multistep difference along characteristics of the one‐order hyperbolic part of the equation is used for discretization in time, and finite element method is used to discrete the space variables. The resulting schemes are consistent, stable and very efficient. Optimal‐rate of convergence is proved. Also, a note is given for a paper published earlier© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

A numerical technique for solution of the MRLW equation using quartic B-splines

Numerical scheme based on quartic B-spline collocation method is designed for the numerical solution of modified regularized long wave (MRLW) equation. Unconditional stability is proved using Von-Neumann approach. Performance of the method is checked through numerical examples. Using error norms L₂ and L_∞ and conservative properties of mass, momentum and energy, accuracy and efficiency of the new method is established through comparison with the existing techniques.     

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.     

Galerkin finite element method for one nonlinear integro-differential model

Galerkin finite element method for the approximation of a nonlinear integro-differential equation associated with the penetration of a magnetic field into a substance is studied. First type initial-boundary value problem is investigated. The convergence of the finite element scheme is proved. The rate of convergence is given too. The decay of the numerical solution is compared with the analytical results.     

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.     

Continuous Galerkin finite element methods for a forward‐backward heat equation

A space‐time finite element method is introduced to solve a model forward‐backward heat equation. The scheme uses the continuous Galerkin method for the time discretization. An error analysis for the method is presented. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 257–265, 1999     

High accuracy error estimates of a Galerkin finite element method for nonlinear time fractional diffusion equation

In this work, an effective and fast finite element numerical method with high-order accuracy is discussed for solving a nonlinear time fractional diffusion equation. A two-level linearized finite element scheme is constructed and a temporal–spatial error splitting argument is established to split the error into two parts, that is, the temporal error and the spatial error. Based on the regularity of the time discrete system, the temporal error estimate is derived. Using the property of the Ritz projection operator, the spatial error is deduced. Unconditional superclose result in H¹-norm is obtained, with no additional regularity assumption about the exact solution of the problem considered. Then the global superconvergence error estimate is obtained through the interpolated postprocessing technique. In order to reduce storage and computation time, a fast finite element method evaluation scheme for solving the nonlinear time fractional diffusion equation is developed. To confirm the theoretical error analysis, some numerical results are provided.     

Time discontinuous Galerkin space-time finite element method for nonlinear Sobolev equations

This article presents a complete discretization of a nonlinear Sobolev equation using space-time discontinuous Galerkin method that is discontinuous in time and continuous in space. The scheme is formulated by introducing the equivalent integral equation of the primal equation. The proposed scheme does not explicitly include the jump terms in time, which represent the discontinuity characteristics of approximate solution. And then the complexity of the theoretical analysis is reduced. The existence and uniqueness of the approximate solution and the stability of the scheme are proved. The optimalorder error estimates in L²(H¹) and L²(L²) norms are derived. These estimates are valid under weak restrictions on the space-time mesh, namely, without the condition k_n≥ch², which is necessary in traditional space-time discontinuous Galerkin methods. Numerical experiments are presented to verify the theoretical results.     

On discontinuous Galerkin finite element method for singularly perturbed delay differential equations

A nonsymmetric discontinuous Galerkin FEM with interior penalties has been applied to one-dimensional singularly perturbed problem with a constant negative shift. Using higher order polynomials on Shishkin-type layer-adapted meshes, a robust convergence has been proved in the corresponding energy norm. Numerical experiments support theoretical findings.     

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.     

Error analysis of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation

This article discusses a priori and a posteriori error estimates of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation. We use variational discretization concept to discretize the control variable and discontinuous piecewise linear finite elements to approximate the state and costate variable. Based on the error estimates of discontinuous Galerkin finite element method for the transport equation, we get a priori and a posteriori error estimates for the transport equation optimal control problem. Finally, two numerical experiments are carried out to confirm the theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1493–1512, 2017     

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.     

Numerical solution of fractional diffusion-wave equation based on fractional multistep method

This paper is devoted to application of fractional multistep method in the numerical solution of fractional diffusion-wave equation. By transforming the diffusion-wave equation into an equivalent integro-differential equation and applying Lubich’s fractional multistep method of second order we obtain a scheme of order O(τ^α+h²)$O({\tau }^{\alpha }+h^2)$ for 1?α?1.71832$1?\alpha ?1.71832$ where α$\alpha$ is the order of temporal derivative and τ$\tau$ and h denote temporal and spatial stepsizes. The solvability, convergence and stability properties of the algorithm are investigated and numerical experiment is carried out to verify the feasibility of the scheme.     

Weak Galerkin finite element method for valuation of American options

We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce an artificial boundary to make the computational domain bounded. For the optimal exercise boundary, which satisfies a nonlinear Volterra integral equation, it is resolved by a higher-order collocation method based on graded meshes. With the computed optimal exercise boundary, the front-fixing technique is employed to transform the free boundary problem to a one- dimensional parabolic problem in a half infinite area. For the other spatial domain boundary, a perfectly matched layer is used to truncate the unbounded domain and carry out the computation. Finally, the resulting initial-boundary value problems are solved by weak Galerkin finite element method, and numerical examples are provided to illustrate the efficiency of the method.     

Some details of the Galerkin finite element method

The incorporation of the Galerkin technique in the finite element method has removed the constraint of finding a variational formulation for many problems of mathematical physics. The method has been successfully applied to many areas and has received wide acceptance. However, in the process of transplanting the concept from the Galerkin method for the entire domain to the Galerkin finite element method, some formal details have been overlooked or glossed over in the literature. This paper considers some of these details, including a possible reason for integration by parts and the contribution of interelement discontinuity terms.     

Linear B-spline finite element method for the improved Boussinesq equation

In this paper, we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved Boussinesq equation. The finite element method with linear B-spline basis functions is used to discretize the nonlinear partial differential equation in space and derive a second order system involving only ordinary derivatives. It is shown that the coefficient matrix for the second order term in this system is invertible. Consequently, for the first time, the initial boundary value problem can be reduced to an explicit initial value problem to which many accurate numerical methods are readily applicable. Various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting, wave interaction and blow-up behavior.     

非结构网格下2D Riesz分数阶方程的Galerkin有限元方法

讨论了2D Riesz分数阶扩散方程的Galerkin有限元方法.基于非结构网格,采用Lagrange线性分片多项式作为基函数,详细描述了分数阶扩散方程的有限元实现.与现有方法相比,该方法有效地降低了计算成本,提高了刚度矩阵的精度.最后,数值算例验证了所提方法的有效性.     

Weak Galerkin finite element method for a class of quasilinear elliptic problems

The purpose of this paper is to study the weak Galerkin finite element method for a class of quasilinear elliptic problems. The weak Galerkin finite element scheme is proved to have a unique solution with the assumption that guarantees the corresponding operator to be strongly monotone and Lipschitz-continuous. An optimal error estimate in a mesh-dependent energy norm is established. Some numerical results are presented to confirm the theoretical analysis.