Sobolev方程的一类各向异性非协调有限元逼近

在各向异性网格下,分别讨论了Sobolev方程在半离散和全离散格式下的一类非协调有限元逼近,得到了与传统有限元方法相同的误差估计和一些超逼近性质.同时在半离散格式下,通过构造具有各向异性特征的插值后处理算子得到了整体超收敛结果.     

双曲积分微分方程的各向异性非协调有限元逼近

讨论了双曲积分微分方程在半离散格式下的一类各向异性非协调有限元逼近,得到了与传统有限元方法相同的最优误差估计和超逼近性质.同时利用插值后处理技术得到了整体超收敛结果.     

一类非线性双曲型方程的非协调有限元方法

研究了一类非线性双曲型方程的非协调有限元方法,在不需要传统的Ritz投影的情况下,得到了半离散格式下的误差估计及超收敛结果.     

双曲型方程的一类各向异性非协调有限元逼近

在各向异性条件下,讨论了双曲型方程的一类非协调有限元逼近,给出了半离散格式下的最优误差估计.同时通过新的技巧和精细估计得到了一些超逼近性质和超收敛结果.     

细菌模型的非协调有限元分析

将非协调元应用于描述细菌传播的反应扩散方程组的初边值问题.借助单元的一些特性和非协调误差估计技巧,分别在半离散和全离散有限元格式下,研究了其数值解与精确解的误差估计,得到了最优的误差估计以及超逼近结果.     

带弱奇异核的抛物型积分微分方程的非协调有限元方法

研究了带弱奇异核的抛物型积分微分方程的非协调有限元方法,在不需要Ritz-Volterra投影的情况下,在半离散和全离散的格式下分别得到了与协调有限元方法相同的误差估计.     

非定常线性化Navier-Stokes方程的非协调流线扩散有限元法分析

对非定常线性化Navier-Stokes方程提出了非协调流线扩散有限元方法．用向后Euler格式离散时间,用流线扩散法处理扩散项带来的非稳定性．速度采用不连续的分片线性逼近,压力采用分片常数逼近．得到了离散解的存在唯一性以及在一定范数意义下离散解的稳定性和误差估计．     

非线性抛物积分微分方程的类Wilson非协调元分析

在半离散和全离散格式下讨论非线性抛物积分微分方程的类Wilson非协调有限元逼近.当问题的精确解u∈H3(Ω)/H4(Ω)时,利用该元的相容误差在能量模意义下可以达到O(h2)/O(h3)比其插值误差高一阶和二阶的特殊性质,再结合协调部分的高精度分析及插值后处理技术,并借助于双线性插值代替传统有限元分析中不可缺少的Ritz-Volterra投影导出了半离散格式下的O(h2)阶超逼近和超收敛结果.同时分别得到了向后Euler全离散格式下的超逼近性和Crank-Nicolson全离散格式下的最优误差估计.     

强阻尼波动方程的H1-Galerkin混合有限元超收敛分析

研究了强阻尼波动方程的H¹-Galerkin混合有限元方法的超收敛性. 借助于协调线性三角形元已有的分析估计式, 直接利用插值算子代替原始变量 u 的 Ritz 投影和应力变量 p 的 Ritz-Volterra 投影,对半离散和全离散格式, 得到了u在 H¹(Ω) 模和 p 在 H(div;Ω) 模意义下比以往文献高一阶的超逼近和超收敛结果.     

非线性发展方程的无网格比高精度有限元方法

对于几类非线性的发展型方程——非线性抛物方程、非线性Schr?dinger方程、非线性Sobolev方程、非线性双曲方程,本文从协调有限元方法、非协调有限元方法、混合有限元方法等不同角度,利用不同技巧深入系统地研究了其线性化的全离散格式的构造、无网格比约束下的超逼近和超收敛分析.     

On residually finite groups of finite general rank

Following A. I.Mal’tsev, we say that a group G has finite general rank if there is a positive integer r such that every finite set of elements of G is contained in some r-generated subgroup. Several known theorems concerning finitely generated residually finite groups are generalized here to the case of residually finite groups of finite general rank. For example, it is proved that the families of all finite homomorphic images of a residually finite group of finite general rank and of the quotient of the group by a nonidentity normal subgroup are different. Special cases of this result are a similar result of Moldavanskii on finitely generated residually finite groups and the following assertion: every residually finite group of finite general rank is Hopfian. This assertion generalizes a similarMal’tsev result on the Hopf property of every finitely generated residually finite group.     

关于几乎可解群

本文推广了关于局部有限群的Asar定理及p.Hall—Kulatilaka,Kargapolov定理.     

Preconditioning spectral element schemes for definite and indefinite problems

Spectral element schemes for the solution of elliptic boundary value problems are considered. Preconditioning methods based on finite difference and finite element schemes are implemented. Numerical experiments show that inverting the preconditioner by a single multigrid iteration is most efficient and that the finite difference preconditioner is superior to the finite element one for both definite and indefinite problems. A multigrid preconditioner is also derived from the finite difference preconditioner and is found suitable for the CGS acceleration method. It is pointed out that, for the finite difference and finite element preconditioners, CGS does not always converge to the accurate algebraic solution. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 535–543, 1999     

The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions

Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space. For the approximation to this equation with the initial and boundary value conditions using the finite difference method, the difficulty is how to construct matched finite difference schemes at all the inner grid points. In this paper, two finite difference schemes are constructed for the problem. The accuracy is second-order in time and first-order in space. The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a three-level linearized finite difference scheme. The Browder fixed point theorem is used to prove the existence of the nonlinear implicit finite difference scheme. The conservation, boundedness, stability, convergence of these schemes are discussed and analyzed by the energy method together with other techniques. The two-level nonlinear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent. Some numerical examples illustrate the efficiency of the proposed finite difference schemes.     

Conforming and nonconforming harmonic finite elements

ABSTRACT

Instead of using the full polynomial space, a conforming and a nonconforming finite element methods are designed where only harmonic polynomials (a much smaller space) are employed in the computation. The conforming quadratic harmonic polynomial finite element is defined only on a special triangular grid. The nonconforming quadratic harmonic finite element is defined on general triangular grids. The optimal order of convergence is proved for both finite element methods, and confirmed by numerical computations. In addition, numerical comparisons with the standard conforming and nonconforming finite elements are presented.     

三维热传导型半导体问题的交替方向特征有限元方法

提出交替方向特征有限元方法,对电场位势方程采用混合元格式,对电子,空穴浓度方程采用交替方向特征有限元格式,对温度方程提出交替方向格式.应用向量积计算及先验估计理论和技巧,得到最佳的L2误差估计.     

Finite integration method for partial differential equations

A finite integration method is proposed in this paper to deal with partial differential equations in which the finite integration matrices of the first order are constructed by using both standard integral algorithm and radial basis functions interpolation respectively. These matrices of first order can directly be used to obtain finite integration matrices of higher order. Combining with the Laplace transform technique, the finite integration method is extended to solve time dependent partial differential equations. The accuracy of both the finite integration method and finite difference method are demonstrated with several examples. It has been observed that the finite integration method using either radial basis function or simple linear approximation gives a much higher degree of accuracy than the traditional finite difference method.     

基于二重网格的定常Navier-Stokes方程的局部和并行有限元算法

对二维定常的不可压缩的Navier-Stokes方程的局部和并行算法进行了研究．给出的算法是多重网格和区域分解相结合的算法,它是基于两个有限元空间:粗网格上的函数空间和子区域的细网格上的函数空间．局部算法是在粗网格上求一个非线性问题,然后在细网格上求一个线性问题,并舍掉内部边界附近的误差相对较大的解．最后,基于局部算法,通过有重叠的区域分解而构造了并行算法,并且做了算法的误差分析,得到了比标准有限元方法更好的误差估计,也对算法做了数值试验,数值结果通过比较验证了本算法的高效性和合理性．     

真商群为s-自对偶群的有限p-群

如果有限群G的每个子群与G的某个商群同构,则称群G为s-自对偶群.如果s-自对偶群G的每个商群与G的某个子群同构,则称群G为自对偶群.本文分类了每个真商群均为s-自对偶群的有限p-群.作为推论,本文还分类了每个真截段均为s-自对偶群的有限p-群,每个真商群均为自对偶群的有限p-群,以及每个真截段均为自对偶群的有限p-群...     

二维非饱和土壤水分运动问题的半离散有限体积元模拟

本文利用基于重心对偶剖分的有限体积元法建立了二维非饱和土壤水分运动问题的数值逼近格式,讨论了离散有限体积元解的存在唯一性,并给出了最优误差估计的证明.最后给出数值算例,模拟结果表明,利用有限体积元格式来求解二维非饱和土壤水分运动问题是可靠的,且该格式具有稳定性和可实用性.