三维守恒律有限元方法逼近光滑解的误差估计

我们对一个三维守恒律的显式有限元方法证明了H^1范数的二阶误差估计。     

High Resolution Schemes and Discrete Entropy Conditions for 2-D Linear Conservation Laws

In this paper, fully discrete entropy conditions of a class of high resolution schemes with the MmB property are discussed by using the theory of proper discrete entropy flux for the linear scalar conservation laws in two dimensions. The theoretical resluts show that the high resolution schemes satisfying fully discrete entropy conditions with proper discrete entropy flux cannot preserve second order accuracy in the case of two dimensions.     

半导体瞬态问题的修正迎风有限体积格式

半导体器件的瞬时状态由包含三个拟线性偏微分方程所组成的方程组的初边值问题来描述.其中电子位势方程足椭圆型的,电子和空穴浓度方程是对流扩散型的.对电子位势方程采用一次元有限体积法米逼近,对电子浓度和空穴浓度方程采用修正的迎风有限体积方法来逼近,并进行详细的理论分析,关于位势得到O(h Δt)阶的H1模误差估计结果,关于浓度得到O(h2 Δt)阶的L2模误差估计结果.最后,给出数值例子.     

一维定常型对流占优扩散方程的一类迎风有限体积格式

本文针对一维定常型对流占优扩散方程提出了一类迎风有限体积格式 .该格式对对流项具有二阶精度 ,对扩散项保持一阶精度 ,符合对流占优扩散问题强对流、弱扩散的特点 .     

A Note on Conservation Laws of Symplectic Difference Schemes for Hamiltonian Systems

In this paper we consider the necessary conditions of conservation laws of symplectic difference schemes for Hamiltonian systems and give an example which shows that there does not exist any centered symplectic difference scheme which preserves all Hamiltonian energy.     

Mortar Upwind Finite Volume Element Method with Crouzeix-Raviart Element for Parabolic Convection Diffusion Problems

In this paper,we study the semi-discrete mortar upwind finite volume element method with the Crouzeix-Raviart element for the parabolic convection diffusion problems. It is proved that the semi-discrete mortar upwind finite volume element approximations derived are convergent in the H~1-and L~2-norms.     

The Symmetrical Dissipative Difference Schemes for Quasi-Linear Hyperbolic Conservation Laws

In this paper we construct a new type of symmetrical dissipative difference scheme. Except discontinuity these schemes have uniformly second-order accuracy. For calculation using these, the simple-wave is very exact, the shock has high resolution, the programing is simple and the CPU time is economical. Since the paper [1] introduced that in some conditions Lax-Wendroff scheme would converge to nonphysical solution, many researchers have discussed this problem. According to preserving the monotonicity of the solution preserving monotonial schemes and TVD schemes have been introduced by Harten, et. According to property of hyperbolic wave propagation the schemes of split-coefficient matrix (SCM) and split-flux have been formed. We emphasize the dissipative difference scheme, and these schemes are dissipative on arbitrary conditions.     

非线性守恒方程的两种稳定化谱元方法

考虑非线性守恒方程的高阶数值解法,介绍了基于谱元法的两种稳定性方法,一种是谱粘性消去法(SVV),另一种是过滤法.在SVV方法中,我们推广并分析了传统的基于单区域的SVV算子的定义.在过滤法中,我们分析了SVV-H elm holtz过滤算子的性质.文中从分析和计算两方面对两种方法进行了比较,建立了两者之间的关系.最后通过一系列数值试验说明方法的有效性.     

二维双曲守恒律的大时间步长Godunov方法(英文)

考虑了关于二维守恒律的大时间步长Godunov方法.该方法是关于一维问题的自然推广.证明了文中给出的数值流函数下,该方法是守恒的.进一步还给出了近似Riemann解应满足的条件,并且证明了利用满足这些条件的近似Riemann解的大时间步长Godunov方法守恒.最后,补充证明了满足这些条件的近似Riemann解是满足熵条件的.     

The Modified Upwind Finite Difference Fractional Steps Method for Compressible Two-phase Displacement Problem

For compressible two-phase displacement problem,the modified upwind finite difference fractionalsteps schemes are put forward.Some techniques,such as calculus of variations,commutative law of multiplicationof difference operators,decomposition of high order difference operators,the theory of prior estimates and tech-niques are used.Optimal order estimates in L~2 norm are derived for the error in the approximate solution.Thismethod has already been applied to the numerical simulation of seawater intrusion and migration-accumulationof oil resources.     

A Modified Weak Galerkin Finite Element Method for Sobolev Equation

For Sobolev equation, we present a new numerical scheme based on a modified weak Galerkin finite element method, in which differential operators are approximated by weak forms through the usual integration by parts. In particular, the numerical method allows the use of discontinuous finite element functions and arbitrary shape of element. Optimal order error estimates in discrete $H^1$ and $L^2$ norms are established for the corresponding modified weak Galerkin finite element solutions. Finally, some numerical results are given to verify theoretical results.     

对流-扩散问题的6节点三角形单元流线迎风有限元法和自适应网格重分技术

提出了使用6节点三角形单元的流线迎风有限元法．该方法沿局部流线，直接用于输运控制方程的对流项．采用多个对流一扩散实例来评价该方法的有效性，结果显示该方法是单调的，并且不产生任何振荡．另外，自适应网格技术和该方法相结合后，进一步提高了解的精度，又减少了计算时间和对计算机内存的需求．     

Augmented Macro-Hybrid Mixed Finite Element Schemes for Elastic Contact Problems

On a setting of subdifferential models, variational augmented macro-hybrid mixed finite element schemes are formulated and analyzed for elastic unilateral contact problems with prescribed friction. Composition duality principles determine primal and dual mixed solvability, adopting coupling surjectivity for dualization. Macro-hybridization corresponds to nonoverlapping decompositions of elastic solid body systems, with displacement continuity and traction equilibrium transmission conditions dualized. In general, traction and displacement multipliers synchronize sub-bodies through nonmatching finite element interfaces. Three-field formulations give the basis for variational augmentation, in a sense of exact penalization, allowing speed-up of rates of convergence as well as proximation procedures of parallel numerical resolution algorithms.     

广义神经传播方程的一种修正混合有限元方法的误差分析

利用修正的H~1-Galerkin混合有限元方法研究了广义神经传播方程,论证了其半离散解的存在唯一性,得到了半离散解的最优阶误差估计,该方法的优点是不需验证LBB相容性条件.     

A New Class of Uniformly Second Order Accurate Difference Schemes for 2D Scalar Conservation Laws

In this paper, concerned with the Cauchy problem for 2D nonlinear hyperbolic conservation laws, we construct a class of uniformly second order accurate finite difference schemes, which are based on the E-schemes. By applying the convergence theorem of Coquel-Le Floch [1], the family of approximate solutions defined by the scheme is proven to converge to the unique entropy weak $L^{\infty}$-solution. Furthermore, some numerical experiments on the Cauchy problem for the advection equation and the Riemann problem for the 2D Burgers equation are given and the relatively satisfied result is obtained.     

Spectral Element Viscosity Methods for Nonlinear Conservation Laws on the Semi-Infinite Interval

In this paper we propose a spectral element: vanishing viscosity (SEW) method for the conservation laws on the semi-infinite interval. By using a suitable mapping, the problem is first transformed into a modified conservation law in a bounded interval, then the well-known spectral vanishing viscosity technique is generalized to the multi-domain case in order to approximate this trarsformed equation more efficiently. The construction details and convergence analysis are presented. Under a usual assumption of boundedness of the approximation solutions, it is proven that the solution of the SEW approximation converges to the uniciue entropy solution of the conservation laws. A number of numerical tests is carried out to confirm the theoretical results.     

A New Technique for Preserving Conservation Laws

Foundations of Computational Mathematics - This paper introduces a new symbolic-numeric strategy for finding semidiscretizations of a given PDE that preserve multiple local conservation laws. We...     

A New Finite Element Space for Expanded Mixed Finite Element Method

In this article, we propose a new finite element space Λ$_h$ for the expanded mixed finite element method (EMFEM) for second-order elliptic problems to guarantee its computing capability and reduce the computation cost. The new finite element space Λ$_h$ is designed in such a way that the strong requirement V$_h\subset$Λ$_h$ in [9] is weakened to {v$_h\in$V$_h$; divv$_h$=0}$\subset$Λ$_h$ so that it needs fewer degrees of freedom than its classical counterpart. Furthermore, the new Λ$_h$ coupled with the Raviart-Thomas space satisfies the inf-sup condition, which is crucial to the computation of mixed methods for its close relation to the behavior of the smallest nonzero eigenvalue of the stiff matrix, and thus the existence, uniqueness and optimal approximate capability of the EMFEM solution are proved for rectangular partitions in $\mathbb{R}^d, d=2,3$ and for triangular partitions in $\mathbb{R}^2$. Also, the solvability of the EMFEM for triangular partition in $\mathbb{R}^3$ can be directly proved without the inf-sup condition. Numerical experiments are conducted to confirm these theoretical findings.