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大家知道,用有限元求解微分方程时,如果微分方程的解足够光滑,边界比较接近拆线,采用高阶元比采用低阶元网格可以稀得多,因此这时采用高阶元比低阶元有利得多,但在实际计算中,边界常常比较复杂,解在边界附近变化较大,这时只采用高阶元在边界 相似文献
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苏煜城 《高等学校计算数学学报》1979,(1)
§1.引言 在数学物理问题中存在着这样一类摄动问题,它们的小参数ε是含于微分方程的高阶导数项中,当ε=0时摄动方程将退化为低阶的微分方程或者同阶的微分方程,此时定解条件发生变化,要失去部分的或者全部的定解条件,退化方程的解一般不满足所失去的定解条件,因此摄动问题的解在失去定解条件的那一部分(或者全部)边界附近,当 相似文献
3.
一般的微积分教材 ,是这样定义高阶与低阶无穷小的 :设在同一变化过程中 ,α,β是无穷小量 ,若1° lim βα=0 ,就说 β是比 α高阶的无穷小 ;2° lim βα=∞ ,就说 β是比 α低阶的无穷小 .对此 ,不少人认为 2°是多余的 ,以为β是比α高阶的无穷小 ,就意味着α是比β低阶的无穷小 ,将 1°、 2°合并为一条 .果真 ,近年来有些高等数学教材 ,就是用 1°一条来定义 β是比 α高阶的无穷小 (或说 α是比 β低阶的无穷小 ) .笔者认为这是值得商榷的 ,因为无穷小的比较 ,首先是指无穷小的比 ,这样 ,β是比α高阶的无穷小未必有α是比β的低阶… 相似文献
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本文以复合材料的Reddy高阶理论为基础,引进一个位移函数Φ,将原来求解的微分方程组转化为一个高阶微分方程,得到了四边简支情况下的Navier型解,和一对边简支另一对边任意情况下的Levy型解.文中列举了算例进行比较,其数值结果和文献上已有结果相吻合,表明本文采用的解法是可靠的.Reddy高阶理论未知数不多,但精度比一阶剪切变形理论要好,计算时无需用剪切修正系数,计算较为简单. 相似文献
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本以复合材料的Reddy高阶理论为基础,引进一个位移函数Φ,将原来求解的微分方程组转化为一个高阶微分方程,得到了四边简支情况下的Navier型解,和一对边简支另一对边任意情况下的Levy型解,中列举了算例进行比较,其数值结果和献上已有结果相吻合,表明本采用的解法是可靠的,Reddy高阶理论未知数不多,但精度比一阶剪切变形理论要好,计算时地需剪切修正系数,计算较为简单。 相似文献
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在高等数学微分方程一章中,介绍了解常系数线性微分方程组的消无法,它是解常系数线性微分方程组的最初等的方法.消元法的基本思想是用微分法消去方程中某些未知函数及其各阶导数,最后得到只含一个未知函数的高阶常系数微分方程.解出这个高阶方程的解后,再根据消元过程,一般不用积分就可求出其余的未知函数.对于未知函数较少的小型微分方程组,采用消元法较为简便.对于未知函数较多时就得寻求更为有效的方法.本文对常系数线性齐次微分方程组的消无法和矩阵法作对比介绍.在掌握线性代数的知识后,用矩阵法解常系数线性齐次微分方程组较为方便. 相似文献
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伴有边界摄动的向量高阶非线性边值问题的奇摄动 总被引:1,自引:0,他引:1
倪守平 《数学物理学报(A辑)》1984,(3)
本文获得某类伴有边界摄动的向量高阶非线性微分方程边值问题解的存在性及其渐近估计式。 相似文献
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对多变量线性系统,本文给出了求其逆向系统的一种新方法,这种方法将逆向系统计算中高阶矩阵的求逆转化为通过初等变换求低阶矩阵的规范型,比以往的方法更加简单有效且易于编程计算。本文结合系统的可观测空间与不可观测空间的情况,给出了一种特定的等价变换,得到了比通常更低阶的逆向系统。 相似文献
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<正> 作者最近这几年已做出許多工作,涉及边界問題解法,即或者是对于某些类常微分积分微分方程初值問題解法或者是对于“全导数”問题解法. 最近,完全依賴C.魏尔斯特拉斯关于用多項式逼近連續函数的古典定理,作者已做出对于高阶积分微分方程的积分一种多項式法,在其中从所考虑的問題到等价的积分方程輔助系的变換起着重要作用. 相似文献
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A singularly perturbed one-dimensional convection-diffusion problem is solved numerically by the finite element method based on higher order polynomials. Numerical solutions are obtained using S-type meshes with special emphasis on meshes which are graded (based on a mesh generating function) in the fine mesh region. Error estimates in the ε-weighted energy norm are proved. We derive an 'optimal' mesh generating function in order to minimize the constant in the error estimate. Two layer-adapted meshes defined by a recursive formulae in the fine mesh region are also considered and a new technique for proving error estimates for these meshes is presented. The aim of the paper is to emphasize the importance of using optimal meshes for higher order finite element methods. Numerical experiments support all theoretical results. 相似文献
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Nelida Črnjarić-Žic Senka Maćešić Bojan Crnković 《Annali dell'Universita di Ferrara》2007,53(2):199-215
Most of the standard papers about the WENO schemes consider their implementation to uniform meshes only. In that case the
WENO reconstruction is performed efficiently by using the algebraic expressions for evaluating the reconstruction values and
the smoothness indicators from cell averages. The coefficients appearing in these expressions are constant, dependent just
on the scheme order, not on the mesh size or the reconstruction function values, and can be found, for example, in Jiang and
Shu (J Comp Phys 126:202–228, 1996). In problems where the geometrical properties must be taken into account or the solution
has localized fine scale structure that must be resolved, it is computationally efficient to do local grid refinement. Therefore,
it is also desirable to have numerical schemes, which can be applied to nonuniform meshes. Finite volume WENO schemes extend
naturally to nonuniform meshes although the reconstruction becomes quite complicated, depending on the complexity of the grid
structure. In this paper we propose an efficient implementation of finite volume WENO schemes to nonuniform meshes. In order
to save the computational cost in the nonuniform case, we suggest the way for precomputing the coefficients and linear weights
for different orders of WENO schemes. Furthermore, for the smoothness indicators that are defined in an integral form we present
the corresponding algebraic expressions in which the coefficients obtained as a linear combination of divided differences
arise. In order to validate the new implementation, resulting schemes are applied in different test examples.
相似文献
14.
Error analysis of staggered finite difference finite volume schemes on unstructured meshes
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Qingshan Chen 《Numerical Methods for Partial Differential Equations》2017,33(4):1159-1182
This work combines the consistency in lower‐order differential operators with external approximations of functional spaces to obtain error estimates for finite difference finite volume schemes on unstructured nonuniform meshes. This combined approach is first applied to a one‐dimensional elliptic boundary value problem on nonuniform meshes, and a first‐order convergence rate is obtained, which agrees with the results previously reported. The approach is also applied to the staggered Marker‐and‐Cell scheme for the two‐dimensional incompressible Stokes problem on unstructured meshes. A first‐order convergence rate is obtained, which improves over a previously reported result in that it also holds on unstructured meshes. For both problems considered in this work, the convergence rate is one order higher on meshes satisfying special requirements. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1159–1182, 2017 相似文献
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在各向异性网格下首先研究了二阶椭圆特征值问题算子谱逼近的若干抽象结果.然后将这些结果具体应用于线性和双线性Lagrange型协调有限元,得到了与传统有限元网格剖分下相同的最优误差估计,从而拓宽了已有的成果. 相似文献
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R. Webster 《Numerical Linear Algebra with Applications》2010,17(4):655-676
This paper investigates the effectiveness of two different Algebraic Multigrid (AMG) approaches to the solution of 4th‐order discrete‐difference equations for incompressible fluid flow (in this case for a discrete, scalar, stream‐function field). One is based on a classical, algebraic multigrid, method (C‐AMG) the other is based on a smoothed‐aggregation method for 4th‐order problems (SA‐AMG). In the C‐AMG case, the inter‐grid transfer operators are enhanced using Jacobi relaxation. In the SA‐AMG case, they are improved using a constrained energy optimization of the coarse‐grid basis functions. Both approaches are shown to be effective for discretizations based on uniform, structured and unstructured, meshes. They both give good convergence factors that are largely independent of the mesh size/bandwidth. The SA‐AMG approach, however, is more costly both in storage and operations. The Jacobi‐relaxed C‐AMG approach is faster, by a factor of between 2 and 4 for two‐dimensional problems, even though its reduction factors are inferior to those of SA‐AMG. For non‐uniform meshes, the accuracy of this particular discretization degrades from 2nd to 1st order and the convergence factors for both methods then become mesh dependent. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
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JayanthaPasdunkoraleA. IanW.Turner 《计算数学(英文版)》2005,23(1):1-16
An unstructured mesh finite volume discretisation method for simulating diffusion in anisotropic media in two-dimensional space is discussed. This technique is considered as an extension of the fully implicit hybrid control-volume finite-element method and it retains the local continuity of the flux at the control volume faces. A least squares function recon-struction technique together with a new flux decomposition strategy is used to obtain an accurate flux approximation at the control volume face, ensuring that the overall accuracy of the spatial discretisation maintains second order. This paper highlights that the new technique coincides with the traditional shape function technique when the correction term is neglected and that it significantly increases the accuracy of the previous linear scheme on coarse meshes when applied to media that exhibit very strong to extreme anisotropy ratios. It is concluded that the method can be used on both regular and irregular meshes,and appears independent of the mesh quality. 相似文献
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A. I. Zadorin 《Computational Mathematics and Mathematical Physics》2008,48(9):1634-1645
Linear and quadratic spline interpolation methods for a one-variable function with a boundary-layer component are examined. It is shown that the interpolation method for such a function leads to considerable errors when applied on a uniform mesh. The error of linear and quadratic spline interpolations on meshes that are refined in the boundary layer is estimated. Numerical results are presented. 相似文献
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Finite element modelling of hydrostatic compaction where the applied pressure acts normal to the deformed surface requires a geometric nonlinear formulation and follower load terms [1, 5, 7]. These concepts are applied to high order [6] (p-FEM) elements with hierarchic shape functions. Applying the blending function method allows to precisely describe curved boundaries on coarse meshes. High order elements exhibit good performance even for high aspect ratios and strong distortion and therefore allow an efficient discretization of thin-walled structures. Since high order finite elements are less prone to locking effects a pure displacement-based formulation can be chosen. After introducing the basic concept of the p-version the application of follower loads to geometrically nonlinear high order elements is presented. For the numerical solution the displacement based formulation is linearized yielding the basis for a Newton-Raphson iteration. The accuracy and performance of the high order finite element scheme is demonstrated by a numerical example. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献