We consider the quadrilateral $Q$₁ isoparametric element and establish an optimal error estimate in $H$¹ norm for the interpolation operator under a weaker mesh condition which admits anisotropic quadrilaterals and allows the quadrilateral to become a regular triangle in the sense of maximum angle condition [5,11].

Taking $h_m$ as the mesh width of a curved edge $\Gamma _m$ $(m=1,...,d$ ) of polygons and using quadrature rules for weakly singular integrals, this paper presents mechanical quadrature methods for solving BIES of the first kind of plane elasticity Dirichlet problems on curved polygons, which possess high accuracy $O(h_0^3)$ and low computing complexities. Since multivariate asymptotic expansions of approximate errors with power $h_i^3$ $(i=1,2,...,d)$ are shown, by means of the splitting extrapolations high precision approximations and a posteriori estimate are obtained.  

A numerical test case demonstrates that the Lobatto and the Gauss points are not natural superconvergent points of the cubic and the quartic finite elements under equilateral triangular mesh for the Poisson equation.

The Filled Function Method is a class of effective algorithms for continuous global optimization. In this paper, a new filled function method is introduced and used to solve integer programming. Firstly, some basic definitions of discrete optimization are given. Then an algorithm and the implementation of this algorithm on several test problems are showed. The computational results show the algorithm is effective.

To solve nonlinear complementarity problems (NCP), at each iteration, the classical proximal point algorithm solves a well-conditioned sub-NCP while the Logarithmic-Quadratic Proximal (LQP) method solves a system of nonlinear equations ($LQP$ $system$). This paper presents a practical LQP method-based prediction-correction method for NCP. The predictor is obtained via solving the $LQP$ $system$ approximately under significantly relaxed restriction, and the new iterate (the corrector) is computed directly by an explicit formula derived from the original LQP method. The implementations are very easy to be carried out. Global convergence of the method is proved under the same mild assumptions as the original LQP method. Finally, numerical results for traffic equilibrium problems are provided to verify that the method is effective for some practical problems.  

Using least parameters, we expand the step-transition operator of any linear multi-step method (LMSM) up to $O(\tau ^{s+5})$ with order $s=1$ and rewrite the expansion of the step-transition operator for $s=2$ (obtained by the second author in a former paper). We prove that in the conjugate relation $G_3^{\lambda\tau} \circ G_1^{\tau}=G_2^{\tau}\circ G_3^{\lambda\tau}$ with $G_1$ being an LMSM, (1) the order of $G_2$ can not be higher than that of $G_1$; (2) if $G_3$ is also an LMSM and $G_2$ is a symplectic $B$-series, then the orders of $G_1$, $G_2$ and $G_3$ must be $2$, $2$ and $1$ respectively.

In this paper, we discuss the finite volume element method of $P_1$-nonconforming quadrilateral element for elliptic problems and obtain optimal error estimates for general quadrilateral partition. An optimal cascadic multigrid algorithm is proposed to solve the nonsymmetric large-scale system resulting from such discretization. Numerical experiments are reported to support our theoretical results.

Subdivision algorithm (Stationary or Non-stationary) is one of the most active and exciting research topics in wavelet analysis and applied mathematical theory. In multidimensional non-stationary situation, its limit functions are both compactly supported and infinitely differentiable. Also, these limit functions can serve as the scaling functions to generate the multidimensional non-stationary orthogonal or biorthogonal semi-multiresolution analysis (Semi-MRAs). The spectral approximation property of multidimensional non-stationary biorthogonal Semi-MRAs is considered in this paper. Based on nonstationary subdivision scheme and its limit scaling functions, it is shown that the multidimensional nonstationary biorthogonal Semi-MRAs have spectral approximation order $r$ in Sobolev space $H^s({\mathbb R}^d)$, for all $r\geq s\geq 0$.

In this paper the effect of integral memory terms in the behavior of diffusion phenomena is studied. The energy functional associated with different models is analyzed and stability inequalities are established. Approximation methods for the computation of the solution of the integro-differential equations are constructed. Numerical results are included.

This paper provides a simplified derivation for error estimates of the TRUNC plate element. The error analysis for the problem with mixed boundary conditions is also discussed.