Multisymplectic Structure－Preserving in Simple Finite Element Method in High Dimensional Case

In this paper, we study a finite element scheme of some semi-linear elliptic boundary value problems in high-dhnensjonal space. With uniform mesh, we find that, the numerical scheme derived from finite element method can keep a preserved multisymplectic structure.     

Analysis of an Implicit Fully Discrete Local Discontinuous Galerkin Method for the Time-Fractional Kdv Equation

In this paper, we consider a fully discrete local discontinuous Galerkin (LDG) finite element method for a time-fractional Korteweg-de Vries (KdV) equation. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. We show that our scheme is unconditionally stable and convergent through analysis. Numerical examples are shown to illustrate the efficiency and accuracy of our scheme.     

A Priori Error Estimates of Finite Element Methods for Linear Parabolic Integro-Differential Optimal Control Problems

In this paper, we study the mathematical formulation for an optimal control problem governed by a linear parabolic integro-differential equation and present the optimality conditions. We then set up its weak formulation and the finite element approximation scheme. Based on these we derive the a priori error estimates for its finite element approximation both in $H^1$ and $L^2$ norms. Furthermore, some numerical tests are presented to verify the theoretical results.     

Discrete Maximum Principle Based on Repair Technique for Finite Element Scheme of Anisotropic Diffusion Problems

In this paper, we construct a global repair technique for the finite element scheme of anisotropic diffusion equations to enforce the repaired solutions satisfying the discrete maximum principle. It is an extension of the existing local repair technique. Both of the repair techniques preserve the total energy and are easy to be implemented. The numerical experiments show that these repair techniques do not destroy the accuracy of the finite element scheme, and the computational cost of the global repair technique is lower than the local repair technique when the diffusion tensors are highly anisotropic.     

量子化学中有限元法的并行计算

在有限元方法的框架下,将量子化学中的基组展开方法和普通有限元方法结合起来,提出了一个新方案。在曙光1000并行机上,采用新方案,计算了Li₂、LiH、BH分子的基态总能量。新方案在相同计算量下,可获得比普通有限元方法高得多的精度。     

Superconvergence of mixed finite element approximations to 3-D Maxwell’s equations in metamaterials

Numerical simulation of metamaterials has attracted more and more attention since 2000, after the first metamaterial with negative refraction index was successfully constructed. In this paper we construct a fully-discrete leap-frog type finite element scheme to solve the three-dimensional time-dependent Maxwell’s equations when metamaterials are involved. First, we obtain some superclose results between the interpolations of the analytical solutions and finite element solutions obtained using arbitrary orders of Raviart–Thomas–Nédélec mixed spaces on regular cubic meshes. Then we prove the superconvergence result in the discrete l₂ norm achieved for the lowest-order Raviart–Thomas–Nédélec space. To our best knowledge, such superconvergence results have never been obtained elsewhere. Finally, we implement the leap-frog scheme and present numerical results justifying our theoretical analysis.     

A New Finite Volume Element Formulation for the Non-Stationary Navier-Stokes Equations

A semi-discrete scheme about time for the non-stationary Navier-Stokes equations is presented firstly, then a new fully discrete finite volume element (FVE) formulation based on macroelement is directly established from the semi-discrete scheme about time. And the error estimates for the fully discrete FVE solutions are derived by means of the technique of the standard finite element method. It is shown by numerical experiments that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the FVE method is feasible and efficient for finding the numerical solutions of the non-stationary Navier-Stokes equations and it is one of the most effective numerical methods among the FVE formulation, the finite element formulation, and the finite difference scheme.     

球几何中子输运保正线性间断有限元格式

针对球几何中子输运方程线性间断有限元方法计算的负中子通量问题,构造了保正线性间断有限元格式,该格式保持中子角通量0阶矩和1阶矩。现有方法计算中子角通量非负时,采用传统的线性间断有限元方法,求解线性方程组;原方法计算出现负通量,则采用构造的保正格式,求解非线性方程组。编制了球几何中子输运问题保正格式程序模块,并集成到应用程序。数值算例表明构造的保正格式计算的中子通量非负,有效降低数值误差,提高数值计算的精度。     

Efficient parallel resolution of the simplified transport equations in mixed-dual formulation

A reactivity computation consists of computing the highest eigenvalue of a generalized eigenvalue problem, for which an inverse power algorithm is commonly used. Very fine modelizations are difficult to treat for our sequential solver, based on the simplified transport equations, in terms of memory consumption and computational time.A first implementation of a Lagrangian based domain decomposition method brings to a poor parallel efficiency because of an increase in the power iterations [1]. In order to obtain a high parallel efficiency, we improve the parallelization scheme by changing the location of the loop over the subdomains in the overall algorithm and by benefiting from the characteristics of the Raviart–Thomas finite element. The new parallel algorithm still allows us to locally adapt the numerical scheme (mesh, finite element order). However, it can be significantly optimized for the matching grid case. The good behavior of the new parallelization scheme is demonstrated for the matching grid case on several hundreds of nodes for computations based on a pin-by-pin discretization.     

有限差分-有限元混合方法的隐式格式研究

利用文[1]提出的有限差分-有限元混合方法,给出一种新的有限元隐式格式,它避免了在传统有限元方法中采用隐式格式时需要求解大型带状稀疏矩阵以及存储量大等问题,同时利用差分法中近似因式分解方法和PuliamTH等人为避免块对角矩阵的求解提出的对角化技术,以期提高计算效率。     

Convection-Diffusion Lattice Boltzmann Scheme for Irregular Lattices

In this paper, a lattice Boltzmann (LB) scheme for convection diffusion on irregular lattices is presented, which is free of any interpolation or coarse graining step. The scheme is derived using the axioma that the velocity moments of the equilibrium distribution equal those of the Maxwell–Boltzmann distribution. The axioma holds for both Bravais and irregular lattices, implying a single framework for LB schemes for all lattice types. By solving benchmark problems we have shown that the scheme is indeed consistent with convection diffusion. Furthermore, we have compared the performance of the LB schemes with that of finite difference and finite element schemes. The comparison shows that the LB scheme has a similar performance as the one-step second-order Lax–Wendroff scheme: it has little numerical diffusion, but has a slight dispersion error. By changing the relaxation parameter ω the dispersion error can be balanced by a small increase of the numerical diffusion.     

A finite element method for analysis of vibration induced by maglev trains

This paper developed a finite element method to perform the maglev train–bridge–soil interaction analysis with rail irregularities. An efficient proportional integral (PI) scheme with only a simple equation is used to control the force of the maglev wheel, which is modeled as a contact node moving along a number of target nodes. The moving maglev vehicles are modeled as a combination of spring-damper elements, lumped mass and rigid links. The Newmark method with the Newton–Raphson method is then used to solve the nonlinear dynamic equation. The major advantage is that all the proposed procedures are standard in the finite element method. The analytic solution of maglev vehicles passing a Timoshenko beam was used to validate the current finite element method with good agreements. Moreover, a very large-scale finite element analysis using the proposed scheme was also tested in this paper.     

Finite Element $θ$-Schemes for the Acoustic Wave Equation

In this paper, we investigate the stability and convergence of a family of implicit finite difference schemes in time and Galerkin finite element methods in space for the numerical solution of the acoustic wave equation. The schemes cover the classical explicit second-order leapfrog scheme and the fourth-order accurate scheme in time obtained by the modified equation method. We derive general stability conditions for the family of implicit schemes covering some well-known CFL conditions. Optimal error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the $L^2$-norm error over a finite time interval converges optimally as $\mathcal{O}(h^{p+1}+∆t^s)$, where $p$ denotes the polynomial degree, $s$=2 or 4, $h$ the mesh size, and $∆t$ the time step.     

Improved eigenvalues for combined dynamical systems using a modified finite element discretization scheme

New approaches are presented to discretize an arbitrarily supported linear structure carrying various lumped attachments. Specifically, the exact eigendata, i.e., the exact natural frequencies and mode shapes, of the linear structure without the lumped attachments are first used to modify its finite element mass and stiffness matrix so that the eigensolutions of the discretized system coincide with the exact modes of vibration. This is achieved by identifying a set of minimum changes in the finite element system matrices and enforcing certain constraint conditions. Once the updated matrices for the linear structure are found, the finite element assembling technique is then used to include the lumped attachments by adding their parameters to the appropriate elements in the modified mass and stiffness matrices. Numerical experiments show that for the same number of elements, the proposed scheme returns higher natural frequencies that are substantially more accurate than those given by the finite element model. Alternatively, the proposed discretization scheme allows one to efficiently and accurately determine the higher natural frequencies of a combined system without increasing the number of elements in the finite element model.     

A Linear Energy-Preserving Finite Volume Element Method for the Improved Korteweg−de Vries Equation

In this paper, we introduce a linearized energy-preserving scheme which preserves the discrete global energy of solutions to the improved Korteweg?deVries equation. The method presented is based on the finite volume element method, by resorting to the variational derivative to transform the improved Korteweg?deVries equation into a new form, and then designing energy-preserving schemes for the transformed equation. The proposed scheme is much more efficient than the standard nonlinear scheme and has good stability. To illustrate its efficiency and conservative properties, we also compare it with other nonlinear schemes. Finally, we verify the efficiency and conservative properties through numerical simulations.     

On (essentially) non-oscillatory discretizations of evolutionary convection–diffusion equations

Finite element and finite difference discretizations for evolutionary convection–diffusion–reaction equations in two and three dimensions are studied which give solutions without or with small under- and overshoots. The studied methods include a linear and a nonlinear FEM-FCT scheme, simple upwinding, an ENO scheme of order 3, and a fifth order WENO scheme. Both finite element methods are combined with the Crank–Nicolson scheme and the finite difference discretizations are coupled with explicit total variation diminishing Runge–Kutta methods. An assessment of the methods with respect to accuracy, size of under- and overshoots, and efficiency is presented, in the situation of a domain which is a tensor product of intervals and of uniform grids in time and space. Some comments to the aspects of adaptivity and more complicated domains are given. The obtained results lead to recommendations concerning the use of the methods.     

局部时间步长间断有限元方法求解三维欧拉方程

使用间断有限元方法求解三维流体力学方程.空间剖分采用非结构四面体网格,为了克服显格式在单元网格尺寸差别较大时计算效率低下的问题,在格式中采用局部时间步长技术(LTS),即控制方程在空间、时间上积分得到一种单步格式,既可以局部计算每个单元又避免了Runge-Kutta高精度格式处理三维问题时存储量过大的问题.为了提高流体力学方程计算精度,在计算单元边界的数值流通量时使用任意高阶精度方法(ADER).数值算例表明格式稳定有效.     

A finite element method for the numerical solution of the coupled Cahn–Hilliard and Navier–Stokes system for moving contact line problems

In this paper, a semi-implicit finite element method is presented for the coupled Cahn–Hilliard and Navier–Stokes equations with the generalized Navier boundary condition for the moving contact line problems. In our method, the system is solved in a decoupled way. For the Cahn–Hilliard equations, a convex splitting scheme is used along with a P1-P1 finite element discretization. The scheme is unconditionally stable. A linearized semi-implicit P2-P0 mixed finite element method is employed to solve the Navier–Stokes equations. With our method, the generalized Navier boundary condition is extended to handle the moving contact line problems with complex boundary in a very natural way. The efficiency and capacity of the present method are well demonstrated with several numerical examples.     

非结构网格上的迎风有限元格式

在非结构网格上提出一种基于修正积分区域的迎风有限元格式,它与一阶迎风差分格式相当,可应用于构造各种不同的数值格式。     

一维双曲守恒律的龙格-库塔控制体积间断有限元方法

给出数值求解一维双曲守恒律方程的新方法——龙格-库塔控制体积间断有限元方法(RKCVDFEM),其中空间离散基于控制体积有限元方法,时间离散基于二阶TVB Runge-Kutta技术,有限元空间选取为分段线性函数空间.理论分析表明,格式具有总变差有界(TVB)的性质,而且空间和时间离散形式上具有二阶精度.数值算例表明,数值解收敛到熵解并且对光滑解的收敛阶是最优的,优于龙格-库塔间断Galerkin方法(RKDGM)的计算结果.