关于J－对称微分算子的J－自伴扩张的若干注记

本文给出了一条解析描述Ｊ－对称微分算子Ｊ－自伴扩张的新途径．我们借助方程Ｔ（ｙ）＝λｏｙ的解，而不是如文［３］利用方程＋（ｙ）＇＝－ｙ的解来描述Ｊ－对称微分算式的所有Ｊ－自伴域在奇异端点的边条件，不过我们假设生成的最小算子具非空正则域．我们对主要定理给出了若干有趣的注，得到了二阶极限圆情形的有趣结果．     

关于J－对称微分算子的J－自伴扩张的若干注记

本文给出了一条解析描述Ｊ－对称微分算子Ｊ－自伴扩张的新途径．我们借助方程Ｔ（ｙ）＝λｏｙ的解，而不是如文［３］利用方程＋（ｙ）＇＝－ｙ的解来描述Ｊ－对称微分算式的所有Ｊ－自伴域在奇异端点的边条件，不过我们假设生成的最小算子具非空正则域．我们对主要定理给出了若干有趣的注，得到了二阶极限圆情形的有趣结果．     

具反射边界条件的非自伴非紧的抽象边值问题

研究具反射边界条件，非自伴非紧算子的抽象边界问题，证明了它等价一个Wiener-Hopf方程，并证明了方程的适定性。     

二阶微分算子积的自伴性

本文讨论了由正则和奇异的二阶对称微分算式生成的微分算子的积算子的自伴性,得到了积算子为自伴算子时边条件应满足的充分而必要的条件及若干其他结果．     

度量图上两个微分算子乘积的自伴性

本文研究了度量图上二阶及四阶局部微分算子积的自伴顶点条件.在研究闭区间[a,b]上积算子自伴性的基础上,运用度量图上高阶局部微分算子的自伴顶点条件得到了积算子自伴的充分必要条件.此外,给出了积算子自伴与原算子自伴之间的关系.     

伪自伴量子系统的酉演化与绝热定理

经典量子系统的哈密尔顿是自伴算子.哈密尔顿算符的自伴性不仅确保了系统遵循酉演化,而且也保证了它自身具有实的能量本征值.但是,确实有一些物理系统,其哈密尔顿是非自伴的,但也具有实的能量本征值,这种具有非自伴哈密尔顿的系统就是非自伴量子系统.具有伪自伴哈密尔顿的系统是一类特殊的非自伴量子系统,其哈密尔顿相似于一个自伴算子.本文研究伪自伴量子系统的酉演化与绝热定理.首先,给出了伪自伴算子定义及其等价刻画;其次,对于伪自伴哈密尔顿系统,通过构造新内积,证明了伪自伴哈密尔顿在新内积下是自伴的,并给出了系统在新内积下为酉演化的充分必要条件.最后,建立了伪自伴量子系统的绝热演化定理及与绝热逼近定理.     

二项自伴向量微分算子的本质谱

首先利用算子比较的方法,研究了二项自伴向量微分算子的本质谱,得到了这类微分算子的本质谱分布范围;然后利用算子分解定理,得到了这类算子谱的离散性的一个充分条件;最后得到了Sturm-Liouville算子和Schr?dinger算子的本质谱范围,以及这两类算子谱的离散性的一个充分条件.     

具纯离散谱的u-标算子

本文研究一类具纯离散谱的非自伴算子,证明了该类算子在弱拓扑意义下可以特征展开的充分必要条件是该类算子是u-标的(u-scalar),又等价于该类算子拟仿射相似于自伴算子.并给出例子,说明其在弱拓扑意义下可以特征展开,但不属于经典的标型谱算子(Spectral operator of scalar type).     

m个微分算式乘积的自伴边界条件

本文假设n阶正则对称微分算式l(y)的幂算式lm(y)在L2[a,∞)中是部分分离的,首先刻画了由幂算式lm(y)生成的微分算子T(lm)的自伴边界条件.然后,在L2[a,∞)中,借助T(lm)的自伴域的这种刻画形式,研究了m个由n阶微分算式l(y)生成的微分算子Tk(l)(k=1,2,…,m;m∈z,m≥2)乘积的自伴性问题,获得了乘积算子Tm(l)…T2(l)T1(l)是自伴算子的充要条件.     

算子张量积及C_2上的初等算子

本文给出了形如的张量积算子成为自伴算子，Ｃ＿ｐ类算子，有限秩算子及一秩算子的充分必要条件，特别，作为应用，得到Ｈｉｌｂｅｒｔ－Ｓｃｈｍｉｄｔ类Ｃ＿２（Ｈ）上初等算子成为自伴算子，Ｃ＿ｐ类算子的充分必要条件．     

线性椭圆问题最低次混合有限元方法的最优最大模误差估计

本文对线性椭圆问题的最低次混合元方法提出了构造混合元空间的充分条件,并建立了新的插值算子.据此得到了混合元解,伴随向量函数及其散度的最优最大模误差估计.     

Generalized Rayleigh quotient and finite element two-grid discretization schemes

This study discusses generalized Rayleigh quotient and high efficiency finite element discretization schemes. Some results are as follows: 1) Rayleigh quotient accelerate technique is extended to nonselfadjoint problems. Generalized Rayleigh quotients of operator form and weak form are defined and the basic relationship between approximate eigenfunction and its generalized Rayleigh quotient is established. 2) New error estimates are obtained by replacing the ascent of exact eigenvalue with the ascent of fin...     

对流扩散方程特征线三角元法的一致估计

利用三角形线性元的积分恒等式,给出了二维非定常对流占优扩散方程的特征线有限元解和真解的一致最优估计,并利用插值后处理算子,得到了有限元解梯度的一致超收敛估计,即只与初值和右端项有关,而与ε无关.     

SUPERAPPROXIMATION PROPERTIES OF THE INTERPOLATION OPERATOR OF PROJECTION TYPE AND APPLICATIONS

AbstractSome superapproximation and ultra-approximation properties in function, gradient and two-order derivative approximations are shown for the interpolation operator of projection type on two-dimensional domain. Then, we consider the Ritz projection and Ritz-Volterra projection on finite element spaces, and by means of the superapproximation elementary estimates and Green function methods, derive the superconvergence and ultraconvergence error estimates for both projections, which are also the finite element approximation solutions of the elliptic problems and the Sobolev equations, respectively.     

The Euler-Galerkin finite element method for nonlocal diffusion problems with a p-Laplace-type operator

This paper is devoted to the study of the finite element method for a class of non-linear nonlocal diffusion problems associated with p-Laplace-type operator. Using the Euler–Galerkin finite element method, the convergence and a priori error estimates for the semi-discrete as well as fully-discrete formulations are established.     

Well-Posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises

The paper studies the well-posedness and optimal error estimates of spectral finite element approximations for the boundary value problems of semi-linear elliptic SPDEs driven by white or colored Gaussian noises. The noise term is approximated through the spectral projection of the covariance operator, which is not required to be commutative with the Laplacian operator.Through the convergence analysis of SPDEs with the noise terms replaced by the projected noises, the well-posedness of the SPDE is established under certain covariance operator-dependent conditions. These SPDEs with projected noises are then numerically approximated with the finite element method. A general error estimate framework is established for the finite element approximations. Based on this framework, optimal error estimates of finite element approximations for elliptic SPDEs driven by power-law noises are obtained. It is shown that with the proposed approach, convergence order of white noise driven SPDEs is improved by half for one-dimensional problems, and by an infinitesimal factor for higher-dimensional problems.     

Error estimates of the DtN finite element method for the exterior Helmholtz problem

A priori error estimates are established for the DtN (Dirichlet-to-Neumann) finite element method applied to the exterior Helmholtz problem. The error estimates include the effect of truncation of the DtN boundary condition as well as that of the finite element discretization. A property of the Hankel functions which plays an important role in the proof of the error estimates is introduced.     

Guaranteed error bounds for finite element approximations of noncoercive elliptic problems and their applications

In this paper, we discuss with guaranteed a priori and a posteriori error estimates of finite element approximations for not necessarily coercive linear second order Dirichlet problems. Here, ‘guaranteed’ means we can get the error bounds in which all constants included are explicitly given or represented as a numerically computable form. Using the invertibility condition of concerning elliptic operator, guaranteed a priori and a posteriori error estimates are formulated. This kind of estimates plays essential and important roles in the numerical verification of solutions for nonlinear elliptic problems. Several numerical examples that confirm the actual effectiveness of the method are presented.     

含弥散平面两相渗流驱动问题有限元方法的误差分析

研究了平面两相渗流可压缩问题含弥散情形的矩形有限元格式．引进一类插值算子,通过插值函数证明了有限元解的最优误差估计．     

Recovery Type a Posteriori Error Estimates of Fully Discrete Finite Element Methods for General Convex Parabolic Optimal Control Problems

This paper is concerned with recovery type a posteriori error estimates of fully discrete finite element approximation for general convex parabolic optimal control problems with pointwise control constraints. The time discretization is based on the backward Euler method. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. We derive the superconvergence properties of finite element solutions. By using the superconvergence results, we obtain recovery type a posteriori error estimates. Some numerical examples are presented to verify the theoretical results.