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Supercloseness of the Divergence-Free Finite Element Solutions on rectangular grids

Communications in Mathematics and Statistics 2013年第2期1卷 143-162页

作者： Yunqing Huang Shangyou Zhang Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan 411105China Department of Mathematical Sciences University of DelawareNewarkDE 19716USA

By the standard theory,the stable Qk+1,k−Qk,k+1/Qdck divergence-free element converges with the optimal order of approximation for the Stokes equations,but only order k for the velocity in H1-norm and the pressure in *** is due to one polynomial degree less in y direction for the first component of velocity,which is a Qk+1,k polynomial of x and *** this manuscript,we will show by supercloseness of the divergence free element that the order of convergence is truly k+1,for both velocity and *** special solutions(if the interpolation is also divergence-free),a two-order supercloseness is shown to *** tests are provided confirming the accuracy of the theory.

关键词： Mixed finite element Stokes equations Divergence-free element Quadrilateral element rectangular grids Supercloseness Superconvergence

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Bivariate Barycentric and Newton Rational Interpolation Over rectangular grids

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INTERNATIONAL JOURNAL OF PATTERN RECOGNITION AND ARTIFICIAL INTELLIGENCE 2022年第15期36卷 2259034-2259034页

作者： Cai, Zhidan Fang, Ming Li, Zhe Yang, Longfei Changchun Univ Sci & Technol Sch Artificial Intelligence Changchun 130022 Peoples R China Changchun Univ Sci Sch Math & StatTechnol Changchun 130022 Peoples R China

In this paper, we consider the problem of computing bivariate barycentric and Newton rational interpolation over rectangular grids. Given a set of rectangular grids, under the conditions that the degrees of the numerator and denominator of rational interpolant are prescribed, we provide a matrix method for calculating barycentric weight vector and denominator value vector, which, respectively, induce barycentric and Newton representations of rational interpolants. We aim at constructing the rational interpolants that globally approximate interpolated function. Numerical examples compare the maximum errors of barycentric rational interpolant, Newton rational interpolant, linear interpolant, cubic interpolant and spline interpolant. Numerical results show the higher accuracy of barycentric and Newton rational interpolation.

关键词： Bivariate rational interpolation barycentric weight vector denominator value vector rectangular grids

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Bandwidth edge counts for linear arrangements of rectangular grids

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Journal of Graph Theory 1998年第4期26卷

作者： Peter Fishburn Paul Wright AT&T Labs–Research Florham Park NJ 07932

Let G n , m denote a graph with nm vertices arranged in n rows and m ≥ max{ n , 2} columns with an edge { u , v } between vertices u and v if they are adjacent horizontally or vertically. The bandwidth of G n , m is known to equal n . We prove that the number of edges in a bandwidth-achieving linear arrangement f from the vertex set of G n , m onto {1, 2,…, nm } for which | f ( u ) - f ( v )| = n can range from 2( n - 1) + n ( m - n ) up to n ( m - 1). The lower bound is attained for a down-diagonal lexicographic linear arrangement; the upper bound is attained by a column-row lexicographic linear arrangement. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 195–202, 1997

关键词： linear arrangements graph bandwidth rectangular grids

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矩形网格上二元切触插值的表现公式

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高等学校计算数学学报 2003年第4期25卷 296-302页

作者：王家正焦建玲安徽教育学院数学系安徽教育学院数学系合肥 230061

1 引言文[3]利用切触Lagrange插值法,建立了矩形网格上切触Lagrange插值公式,不过作者给出的基本多项式是一个较为形式化的公式,如果利用该公式来求插值多项式是相当困难的,本文将给出利用广义Vandermonde逆矩阵表示的插值公式,为此采... 详细信息

1 引言文[3]利用切触Lagrange插值法,建立了矩形网格上切触Lagrange插值公式,不过作者给出的基本多项式是一个较为形式化的公式,如果利用该公式来求插值多项式是相当困难的,本文将给出利用广义Vandermonde逆矩阵表示的插值公式,为此采用如下记号:记R为实数集,Z为整数集,Z;为非负整数集,Z+为正整数集。

关键词： generalized Vandermonde matrix rectangular grids contact interpolation polynomial.

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矩形网格上二元有理插值的存在性问题

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高等学校计算数学学报 2004年第1期26卷 12-17页

作者：王家正安徽教育学院数学系合肥 230061

1 引言与引理有理函数是属于简单函数类,它比多项式复杂,用它近似表示函数时,却比多项式灵活,更能反映函数的一些特性。矩形网格上的二元多项式插值是适定的,但是二元有理函数插值却未必存在,对于二元有理函数插值的存在性问题,至今还... 详细信息

1 引言与引理有理函数是属于简单函数类,它比多项式复杂,用它近似表示函数时,却比多项式灵活,更能反映函数的一些特性。矩形网格上的二元多项式插值是适定的,但是二元有理函数插值却未必存在,对于二元有理函数插值的存在性问题,至今还未见到一般结果,文[2]也只是一些特殊类型的结果,本文是将文[2]的结果进行推广,从而得到较一般的结果,不过二元有理插值有许多算法,如Thielie型分叉连分式算法,二元对称型插值算法等等。本文旨在寻求尽可能接近显式的插值公式,进而揭示二元有理插值的内在结构,得到了矩形网格上二元有理插值函数存在的一个充要条件,同时给出了二元有理插值函数的一种表现形式。从而为对矩形网格上二元有理插值问题进行理论分析提供了一个有力工具。为此引入如下记号:

关键词： rectangular grids interpolation points rational interpolation ratio-nal function.

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DITHERING WITH BLUE NOISE

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PROCEEDINGS OF THE IEEE 1988年第1期76卷 56-79页

作者： ULICHNEY, RA MIT ELECTR RES LABCAMBRIDGEMA 02139 UNIV CALIF DAVIS DAVISCA 95616 UNIV ROCHESTER ROCHESTERNY 14627

Digital halftoning, also referred to as spatial dithering is a method of rendering the illusion of continuous-tone pictures on displays that are capable of only producing binary picture elements. The concept of blue noise-high-frequency white noise-is introduced and found to have desirable properties for halftoning. Efficient algorithms for dithering with blue noise, based on perturbed error diffusion, are developed. The nature of dither patterns produced is extensively examined in the frequency domain. Metrics for analyzing the frequency content of aperiodic patterns for both rectangular and hexagonal grids are developed; blue-noise dithering is found to be ideally suited for rectangular grids. Several carefully selected digitally produced examples are included

关键词： aperiodic patterns binary picture elements blue noise continuous-tone pictures frequency content frequency domain halftoning hexagonal grids high-frequency white noise perturbed error diffusion picture processing rectangular grids spatial dithering white noise

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A divergence-free finite element method for a type of 3D Maxwell equations

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APPLIED NUMERICAL MATHEMATICS 2012年第6期62卷 802-813页

作者： Huang, Jianguo Zhang, Shangyou Univ Delaware Dept Math Sci Newark DE 19716 USA Shanghai Jiao Tong Univ Dept Math Shanghai 200240 Peoples R China Shanghai Normal Univ Div Computat Sci E Inst Shanghai Univ Shanghai Peoples R China

We seek a divergence-free finite element solution for the magnetic field governed by the static Maxwell equations. As usual, the solution is represented as a curl of a vector potential. Typically, this vector potential is uniquely defined in a divergence-free space. The novelty of our method is that we use some simple but non-divergence-free finite element spaces. In this way, the finite element vector potential does not approximate the divergence-free vector, but its curl is divergence-free and is exactly the same solution obtained by the divergence-free finite element potential. Computationally, the finite element solution for the magnetic field is obtained directly as a certain weighted L-2-orthogonal projection within the divergence-free finite element subspace. Optimal order convergence is shown for the method. Numerical tests are provided. (C) 2011 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Maxwell equations Vector potential Divergence-free element rectangular grids

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Binary tomography on the isometric tessellation involving pixel shape orientation

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IET IMAGE PROCESSING 2020年第1期14卷 25-30页

作者： Nagy, Benedek Lukic, Tibor Eastern Mediterranean Univ Fac Arts & Sci Mersin 10 Famagusta North Cyprus Turkey Univ Novi Sad Fac Tech Sci Novi Sad 2100 Serbia

In this study, a tomography reconstruction problem of binary images is considered on the isometric grid. On this grid, the triangle pixels have two types of orientations, accordingly, the authors call them delta or nabla shape pixels. The proposed reconstruction method uses data of projections of three natural directions. They are the lane directions of the triangular tessellation (these directions are somewhat analogous to row/column directions on the rectangular grids). The projection ray, penetrating through a grid lane, now not passing through the middle of pixels (i.e. through the middle line of triangle shape pixels), as usually taken, but little bit shifted from the middle parallel to the lane. This method provides the exact information about the number of nabla and delta shape triangle pixels in each lane of the image. This additional information is included in the reconstruction process to improve the quality of reconstruction. They formulate the suggested model into an energy-minimisation problem and apply a gradient-based approach for its minimisation. They show and analyse various experimental results on test images. The presented approach shows both better quality reconstructions and shorter running time than the earlier approaches.

关键词： gradient methods image reconstruction minimisation computerised tomography reconstruction method natural directions lane directions triangular tessellation rectangular grids projection ray grid lane middle line triangle shape pixels exact information delta shape triangle pixels reconstruction process energy-minimisation problem test images quality reconstructions binary tomography isometric tessellation pixel shape orientation tomography reconstruction problem binary images isometric grid

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A symmetric mixed covolume method for the nonlinear parabolic problem

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2022年第3期68卷 1591-1611页

作者： Zhao, Xuan Liu, Zhengguang Shandong Univ Sch Math Jinan Peoples R China Hunan Key Lab Computat & Simulat Sci & Engn Xiangtan Peoples R China Sch Math & Computat Sci Xiangtan Peoples R China Shandong Normal Univ Sch Math & Stat Jinan Peoples R China

In this paper, we develop a symmetric mixed covolume scheme for the nonlinear parabolic equations. The existence and uniqueness of the scheme are proved by using the lowest order Raviart-Thomas mixed element space on rectangular grids. We carry out a rigorous error estimates for the constructed scheme, establishing the first-order convergence for both velocity and pressure. Numerical examples are provided to verify that the numerical results are all in consistent with the theoretical analysis.

关键词： Symmetric mixed covolume method rectangular grids Existence and uniqueness Nonlinear parabolic problem Error estimates

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Nonlinear Interaction of Friction and Interpolation Errors in Unsteady Flow Analyses

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JOURNAL OF HYDRAULIC ENGINEERING 2013年第4期139卷 397-409页

作者： Shimada, Masashi Vardy, Alan E. Univ Tsukuba Dept Life & Environm Sci Tsukuba Ibaraki Japan Univ Dundee Civil Engn Div Dundee DD1 4HN Scotland

Explicit analytical assessments of numerical errors in solutions of decaying standing waves in pipes are obtained using analyses based on the fixed-grid method of characteristics and are used as a quantitative basis for comparing alternative grid types, sizes, and shapes. It is shown that the influences of friction and interpolation combine in a nonlinear manner, but that the combination has the form of a product so that the two effects may be considered independently to a certain extent. The analysis is presented for finite pipes, thus creating both upper and lower limits for the range of possible standing wave frequencies. It is found that frequencies in the upper half of the range are strongly damped. The performances of diamond and rectangular grids are compared, but there is no clear winner because, for equivalent computer resources, each has both advantages and disadvantages. The overall purposes of the paper are to move toward methods of (1) relieving untrained software users from tasks related to grid selection and (2) assisting third parties in assessing the validity of numerical results undertaken by others. DOI: 10.1061/(ASCE)HY.1943-7900.0000685. (C) 2013 American Society of Civil Engineers.

关键词： Pipe flows Water hammer Numerical errors Method of characteristics Time-line interpolation Friction Diamond grids rectangular grids

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