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  error estimate
Stability in l-norm and error estimate in I2-norm are obtained.
      
By using the elliptic Ritz-Volterra projection, the analysis of the error estimates for the finite element numerical solutions and the optimal H1-norm error estimate are demonstrated.
      
Calculation of maximum value of twice-differentiable function with a posteriori error estimate
      
An error estimate is derived, and an a priori choice of the regularization parameters is described.
      
An error estimate is obtained for the stable approximate solution obtained by solving a set of linear algebraic equations for the wavelet coefficients of the desired solution.
      
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  error estimates
In this paper, we prove the convergence of the nodal expansion method, a new numerical method for partial differential equations and provide the error estimates of approximation solution.
      
Furtheremore, we provide the error estimates of the approximate solutions of the Navier-Stokes Equations.
      
The ellipticH2-Volterra projection is induced and then used in the derivations of error estimates for semi-discrete and full-discreteH1-Galerkin methods.
      
The optimalL2,H1 andH2 norm error estimates are obtained.
      
Continuous time and discrete time Galerkin methods are introduced to approximate the solution and optimalH1 error estimates are obtained.
      
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  error estimation
The large time error estimation between the spectral approximate solution and the exact solution is obtained.
      
Phase-shifting measurement and its error estimation method were studied according to the holographic principle.
      
The difficulties of the error estimation in phase-shifting phase measurement were restricted by this error estimation method.
      
Meanwhile, the maximum error estimation method of phase-shifting phase measurement and its formula were proposed.
      
A formula is presented for the approximate calculation of the correlation coefficient with error estimation.
      
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  optimal error estimates
Optimal error estimates in L∞(J;H1(ω)) are proved, which implies an essential improvement to existed results.
      
Optimal error estimates in L2 and H1 norm are obtained for the approximation solution.
      
By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L2-norm are obtained.
      
Optimal error estimates of a locally one-dimensional method for the multidimensional heat equation
      
For the multidimensional heat equation in a parallelepiped, optimal error estimates inL2(Q) are derived.
      
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Let the axisymmetric loads formed by Mindlin's Point forces be distributed along the Z-axis in (0, L) of the elastic half-space. In addition to Boussinesq's solution, We can reduce the problem with the following boundary conditions1.Z=0,r≠0,σz=τrz=02.0≤z≤L,U(e,z)=a-e,(1≥e/a≈1)3.P=-2π[a∫Lτry(a,z)dz+∫0arσ(r,L)dr](1)to a Fredholm integral equation of the first kind.A theorem of error estimation for the approximate solution of such an equtaion is presented. Applied to piles analysis, this method probably has several...

Let the axisymmetric loads formed by Mindlin's Point forces be distributed along the Z-axis in (0, L) of the elastic half-space. In addition to Boussinesq's solution, We can reduce the problem with the following boundary conditions1.Z=0,r≠0,σz=τrz=02.0≤z≤L,U(e,z)=a-e,(1≥e/a≈1)3.P=-2π[a∫Lτry(a,z)dz+∫0arσ(r,L)dr](1)to a Fredholm integral equation of the first kind.A theorem of error estimation for the approximate solution of such an equtaion is presented. Applied to piles analysis, this method probably has several advantages over that by. R. Butterfield et al. i. e., the integral equation obtained is one-dimensional nonsiagular; the effect of initial stresses is taken into account; the unknown settlement funtion at the surface of the pile need not be prescribed; three-dimensional stress state in compressible piles is taken into account.

将由Mindlin集中力组成的轴对称载荷沿弹性半空间z轴[0,L]内分布,并迭加Boussinesq的解,就能使边界条件为1.Z=0,r≠0,σz=τrz=02.0≤z≤L,U(e,z)=a-e,(1≥e/a≈1)3.P=-2π[a∫Lτry(a,z)dz+∫0arσ(r,L)dr](1)的圆柱嵌入半空间的三维问题归结为一Fredholm第一种积分方程.本文给出了Fredholm第一种积分方程近似解误差估计的一个定理. 将本文所论述的方法,用于桩的分析,较R.Butterfield等人的方法为优越,即所得到积分方程是一维非奇异的、能考虑初应力的影响、不需要预先假定沉陷函数,并且考虑了可压缩桩中的三维应力状态.

This paper presents a method for calculating the buckling load and mode shape of aring-stiffened circular cylindrical shell subject to hydrostatic pressure.According to thecomposite structure method,a set of differential equations for the stability of rings andinterring shell segments are established separately.With the height of ring cross-section,ecceutric distance,sum of cross-sectional area ΣF_r and sum of bending rigidity ΣE_rI_(Gra)kept constant,the number of rings is assumed to tend to infinity,and an...

This paper presents a method for calculating the buckling load and mode shape of aring-stiffened circular cylindrical shell subject to hydrostatic pressure.According to thecomposite structure method,a set of differential equations for the stability of rings andinterring shell segments are established separately.With the height of ring cross-section,ecceutric distance,sum of cross-sectional area ΣF_r and sum of bending rigidity ΣE_rI_(Gra)kept constant,the number of rings is assumed to tend to infinity,and an equivalentorthotropic shell with appropriate elastic constants is obtained as a zero-order approxi-mation.Further we can seek for the solution of equations for the ring-stiffened shell inthe form of a series.The first term of the series corresponds to the zero-order approxi-mation,i.e.the solution of the above-mentioned equivalent orthotropic shell.Theremaining terms represent the correction of various orders,which may serve as anindication of the error arising from substituting the real ring-stiffened shell with theequivalent orthotropic one.Finally an example is given and a comparison is made betweenour results and those by other authors'methods.Our theoretical results indicate a goodagreement with the experimental data.

本文给出了以环肋加强的圆柱壳在液压作用下屈曲形态和临界载荷的计算方法.根据组合结构的方法,建立了一组肋和肋间壳段的稳定微分方程组.在肋的截面高度、偏心距、截面总面积ΣF_r、总抗弯刚度ΣE_rI_(G_ra)不变的前提下,而使环肋的数目趋于无穷大,从而得到了作为组合的环肋加强壳的初次近似的正交各向异性壳模型及其弹性关系.可以进一步寻求方程组的级数解,其首项代表零阶近似解,亦即上述等效正交各向异性壳的解,其余各项代表逐次渐近的修正解,或等效壳和真实的环肋加强组合壳解的误差.根据误差的估计可以给出简化为等效各向异性壳的判据.最后给出了算例并与其它作者的方法进行了比较.计算结果表明与实验符合得很好.

In this paper, using Muskhelishvili's singular integral equation theory, the authors turn gravity-influenced free surface flows into Riemann-Hilbert problem. Taking the length of the streamline of the boundary as a variable and the velocity potential of the boundary as a function to be determined, the authors avoided the difficulty that the angle of the curved fixed part is unknown. Following the application of the difference method and the finite element method, the authors developed a new numerical method...

In this paper, using Muskhelishvili's singular integral equation theory, the authors turn gravity-influenced free surface flows into Riemann-Hilbert problem. Taking the length of the streamline of the boundary as a variable and the velocity potential of the boundary as a function to be determined, the authors avoided the difficulty that the angle of the curved fixed part is unknown. Following the application of the difference method and the finite element method, the authors developed a new numerical method that is suited for complex solid boundaries and which overcomes the difficulties encountered when applying analytic function theory. Under known discharge, the convergence and stability of the method have been proved and an estimation of error has been obtained. The method has been successfully applied to the calculation of the flow past spillway buckets. The calculated values agree extremely well with the measured results.

本文利用的奇异积分方程理论,将自由面重力流化为解析函数的Riemann-Hilbert边值问题,推出积分方程。提出以固壁边界和自由面流线长度为自变量,边界势函数为未知函数的迭代求解思想,避开了曲壁边界流速方向为未知函数的困难。提出了一种适用于直线和曲线边界自由面重力流的新的数值方法。在流量已知的条件下,证明了该方法的收敛性、稳定性。并给出了一个误差估计式。

 
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