对流扩散方程三角形有限元解的一致估计

利用三角形线性元的积分恒等式,给出了二维非定常对流扩散方程的半离散有限元解和真解的一致最优误差估计,即误差与ε无关,而仅与右端f和初值u_0有关.     

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.     

求解奇异摄动问题混合有限元的最优一致收敛的统一方法

给出了在些Shiskin型网格[21,23,19,18]上,利用一个任意次的混合有限元方法在L2一模下得到奇异摄动问题解的最优一致收敛阶的一个统一方法,通过研究一个四阶问题,定常和不定常问题,我们显示了这个方法的一般性,结果显示非传统Shiskin型网格上的误差估计比传统Shiskin型网格上的误差估计更容易得到,但两种网格给出的误差估计是相容的,它们证明了Roos的猜想[21]是合理的。     

半线性椭圆问题的自适应有限元分析

本文既不利用非线性问题的单调性,也不利用有限元逼近的一致有界性进行半线性椭圆问题的自适应有限元分析.本文先推导有限元逼近解的先验和后验误差估计,然后,利用这些估计,分析一类自适应有限元算法,特别是,得到该算法的收敛率和复杂度.     

一类非线性Signorini问题的有限元逼近

本文讨论了一类与非线性单调算子相联系的变分不等式问题——一类非线性Signorini问题。证明了解的存在性和唯一性,给出解的一个表征性质。随后,构造了问题的一个有限元逼近格式;得到了有限元近似解的收敛性结果和误差估计,关键词:非线性单调算子,变分不等式,Signorini问题,有限元逼近,收敛性,误差估计。     

线性对流占优扩散方程的后验误差估计

对于线性对流占优扩散方程,采用特征线有限元方法离散时间导数项和对流项,用分片线性有限元离散空间扩散项,并给出了一致的后验误差估计,其中估计常数不依赖与扩散项系数。     

双线性非协调有限元逼近的梯度恢复后验误差估计

给出了二阶椭圆方程的双线性非协调有限元逼近的梯度恢复后验误差估计.该误差估计是在Q_1非协调元上得到的,并给出了误差的上下界.进一步证明该误差估计在拟一致网格上是渐进精确地.证明依赖于clement插值和Helmholtz分解,数值结果验证了理论的正确性.     

Stokes问题Q_2-P_1混合元外推方法

考虑Stokes问题的有限元解与精确解插值的Q2-P1混合元的渐近误差展开和分裂外推.首先利用积分恒等式技巧确定了微分方程精确解与有限元插值之间积分式的主项,其次再借助插值后处理和分裂外推技术,得到了比通常的误差估计提高两阶的收敛速度.     

三维Stokes问题各向异性混合元分析

本文提出了一个一般的立方体单元格式并将其应用到三维Stokes问题的混合有限元逼近,给出了各向异性插值误差估计,相容误差估计和LBB条件成立的验证,从而证明了其在不满足正则性和拟一致条件下的收敛性．另外我们还得到了其一个特殊收敛性质,即在解(u,p)∈(H3(Ω))3×H2(Ω)时,相容误差阶为O(h2max),比插值误差阶O(hmax)高一阶．     

无穷凹角区域各向异性问题的自然边界元与有限元耦合法

本文研究无穷凹角区域上一类各向异性问题的自然边界元与有限元耦合法.利用自然边界归化原理,获得圆弧或椭圆弧人工边界上的自然积分方程,给出了耦合的变分形式及其数值方法,以及逼近解的收敛性和误差估计,最后给出了数值例子,以示方法的可行性和有效性.     

Uniform error estimates for triangular finite element solutions of advection-diffusion equations

In this paper, the authors use the integral identities of triangular linear elements to prove a uniform optimal-order error estimate for the triangular element solution of two-dimensional time-dependent advection-diffusion equations. Also the authors introduce an interpolation postprocessing operator to get the superconvergence estimate under the ε weighted energy norm. The estimates above depend only on the initial and right data but not on the scaling parameter ε.     

齐型空间上的双线性Calderon-Zygmund奇异积分算子

文在齐型空间上引入双线性Calderon-Zygmund奇异积分算子的基本概念, 研究了其基本性质以及在L^1× L¹上的弱有界性.     

Bilinear space-time estimates for linearised KP-type equations on the three-dimensional torus with applications

A bilinear estimate in terms of Bourgain spaces associated with a linearised Kadomtsev-Petviashvili-type equation on the three-dimensional torus is shown. As a consequence, time localized linear and bilinear space-time estimates for this equation are obtained. Applications to the local and global well-posedness of dispersion generalised KP-II equations are discussed. Especially it is proved that the periodic boundary value problem for the original KP-II equation is locally well-posed for data in the anisotropic Sobolev spaces , if and ε>0.     

Convergence analysis of the LDG method applied to singularly perturbed problems

Considering a two‐dimensional singularly perturbed convection–diffusion problem with exponential boundary layers, we analyze the local discontinuous Galerkin (DG) method that uses piecewise bilinear polynomials on Shishkin mesh. A convergence rate O(N^‐1 lnN) in a DG‐norm is established under the regularity assumptions, while the total number of mesh points is O(N²). The rate of convergence is uniformly valid with respect to the singular perturbation parameter ε. Numerical experiments indicate that the theoretical error estimate is sharp. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

线性抛物方程的变网格各向异性双线性有限元方法

通过利用各向异性双线性元矩形剖分,结合变网格有限元思想,导出了线性抛物方程的全离散变网格各向异性双线性元有限元格式,并给出其L2模误差估计.     

The Computational Complexity of the Resultant Method for Solving Polynomial Equations

Under an assumption of distribution on zeros of the polynomials, we have given the estimate of computational cost for the resultant method. The result in that, in probability $1-\mu$, the computational cost of the resultant method for finding $ε$-approximations of all zeros is at most $$cd^2(log d+log\frac{1}{\mu}+loglog\frac{1}ε)$$, where the cost is measured by the number of f-evaluations. The estimate of cost can be decreased to $c(d^2logd+d^2log\frac{1}{\mu}+dloglog\frac{1}ε)$ by combining resultant method with parallel quasi-Newton method.     

ON STABILITY AND REGULARIZATION FOR BACKWARD HEAT EQUATION

Consider a backward heat equation in a bounded domain Ω (?) R2 with the noisy data in the initial time geometry. The aim is to find the temperature for 0 < ε < t < T. For this ill-posed problem, the authors give a continuous dependence estimate of the solution. Moreover, the convergence rate of the approximate solution is also given.     

The Gradient Superconvergence of Bilinear Finite Volume Element for Elliptic Problems

We study the gradient superconvergence of bilinear finite volume element (FVE) solving the elliptic problems. First, a superclose weak estimate is established for the bilinear form of the FVE method. Then, we prove that the gradient approximation of the FVE solution has the superconvergence property: where denotes the average gradient on elements containing point $P$ and $S$ is the set of optimal stress points composed of the mesh points, the midpoints of edges and the centers of elements.     

The strengthened C.B.S. inequality constant for second order elliptic partial differential operator and for hierarchical bilinear finite element functions

We estimate the constant in the strengthened Cauchy-Bunyakowski-Schwarz inequality for hierarchical bilinear finite element spaces and elliptic partial differential equations with coefficients corresponding to anisotropy (orthotropy). It is shown that there is a nontrivial universal estimate, which does not depend on anisotropy. Moreover, this estimate is sharp and the same as for hierarchical linear finite element spaces.This research was supported by the Grant Agency of Czech Republic under the contract No. 201/02/0595.