A Posteriori Error Estimate of Two Grid Mixed Finite Element Methods for Semilinear Elliptic Equations

In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.

Hermite Finite Element Method for Vibration Problem of Euler-Bernoulli Beam on Viscoelastic Pasternak Foundation

Viscoelastic foundation plays a very important role in civil engineering. It can effectively disperse the structural load into the foundation soil and avoid the damage caused by the concentrated load. The model of Euler-Bernoulli beam on viscoelastic Pasternak foundation can be used to analyze the deformation and response of buildings under complex geological conditions. In this paper, we use Hermite finite element method to get the numerical approximation scheme for the vibration equation of viscoelastic Pasternak foundation beam. Convergence and error estimation are rigourously established. We prove that the fully discrete scheme has convergence order O(τ2+h4), where τis time step size and his space step size. Finally, we give four numerical examples to verify the validity of theoretical analysis.

A Priori Error Analysis for NCVEM Discretization of Elliptic Optimal Control Problem

In this paper, we propose the nonconforming virtual element method (NCVEM) discretization for the pointwise control constraint optimal control problem governed by elliptic equations. Based on the NCVEM approximation of state equation and the variational discretization of control variables, we construct a virtual element discrete scheme. For the state, adjoint state and control variable, we obtain the corresponding prior estimate in H1 and L2 norms. Finally, some numerical experiments are carried out to support the theoretical results.

What is the Most Effective Method of Error Correction in L2 Writing?

The efficacy of error correction and various kinds of correction approaches is one of the key issues in second language writing faced by both teachers and researchers. The current paper reviews the definition of error correction and examines the different views on whether error correction in L2 writing should be corrected. In particular, the paper discusses and analyses the three common correction methods: direct correction, peer feedback and indirect correction. Teachers are encouraged to weigh and analyze the advantages and disadvantages of these methods according to the current literature, employ the most beneficial error correction method in L2 writing, and adapt its suitability to their teaching context.

Analogue correction method of errors and its application to numerical weather prediction

In this paper, an analogue correction method of errors （ACE） based on a complicated atmospheric model is further developed and applied to numerical weather prediction （NWP）. The analysis shows that the ACE can effectively reduce model errors by combining the statistical analogue method with the dynamical model together in order that the information of plenty of historical data is utilized in the current complicated NWP model, Furthermore, in the ACE, the differences of the similarities between different historical analogues and the current initial state are considered as the weights for estimating model errors. The results of daily, decad and monthly prediction experiments on a complicated T63 atmospheric model show that the performance of the ACE by correcting model errors based on the estimation of the errors of 4 historical analogue predictions is not only better than that of the scheme of only introducing the correction of the errors of every single analogue prediction, but is also better than that of the T63 model.

On-Machine Measurement of the Straightness and Tilt Errors of a Linear Slideway Using a New Four-Sensor Method

Although there are some multi-sensor methods for measuring the straightness and tilt errors of a linear slideway, they need to be further improved in some aspects, such as suppressing measurement noise and reducing precondition.In this paper, a new four-sensor method with an improved measurement system is proposed to on-machine separate the straightness and tilt errors of a linear slideway from the sensor outputs, considering the influences of the reference surface profile and the zero-adjustment values. The improved system is achieved by adjusting a single sensor to di erent positions. Based on the system, a system of linear equations is built by fusing the sensor outputs to cancel out the e ects of the straightness and tilt errors. Three constraints are then derived and supplemented into the linear system to make the coe cient matrix full rank. To restrain the sensitivity of the solution of the linear system to the measurement noise in the sensor outputs, the Tikhonov regularization method is utilized. After the surface profile is obtained from the solution, the straightness and tilt errors are identified from the sensor outputs. To analyze the e ects of the measurement noise and the positioning errors of the sensor and the linear slideway, a series of computer simulations are carried out. An experiment is conducted for validation, showing good consistency. The new four-sensor method with the improved measurement system provides a new way to measure the straightness and tilt errors of a linear slideway, which can guarantee favorable propagations of the residuals induced by the noise and the positioning errors.

Modification Design Method for an Enveloping Hourglass Worm Gear with Consideration of Machining and Misalignment Errors

The influences of machining and misalignment errors play a very critical role in the performance of the anti-backlash double-roller enveloping hourglass worm gear(ADEHWG).However,a corresponding efficient method for eliminating or reducing these errors on the tooth profile of the ADEHWG is seldom reported.The gear engagement equation and tooth profile equation for considering six different errors that could arise from the machining and gear misalignment are derived from the theories of differential geometry and gear meshing.Also,the tooth contact analysis(TCA) is used to systematically investigate the influence of the machining and misalignment errors on the contact curves and the tooth profile by means of numerical analysis and three-dimensional solid modeling.The research results show that vertical angular misalignment of the worm wheel(Δβ) has the strongest influences while the tooth angle error(Δα) has the weakest influences on the contact curves and the tooth profile.A novel efficient approach is proposed and used to minimize the effect of the errors in manufacturing by changing the radius of the grinding wheel and the approaching point of contact.The results from the TCA and the experiment demonstrate that this tooth profile design modification method can indeed reduce the machining and misalignment errors.This modification design method is helpful in understanding the manufacturing technology of the ADEHWG.

Monte Carlo Method for the Uncertainty Evaluation of Spatial Straightness Error Based on New Generation Geometrical Product Specification

Straightness error is an important parameter in measuring high-precision shafts. New generation geometrical product speeifieation（GPS） requires the measurement uncertainty characterizing the reliability of the results should be given together when the measurement result is given. Nowadays most researches on straightness focus on error calculation and only several research projects evaluate the measurement uncertainty based on ＂The Guide to the Expression of Uncertainty in Measurement（GUM）＂. In order to compute spatial straightness error（SSE） accurately and rapidly and overcome the limitations of GUM, a quasi particle swarm optimization（QPSO） is proposed to solve the minimum zone SSE and Monte Carlo Method（MCM） is developed to estimate the measurement uncertainty. The mathematical model of minimum zone SSE is formulated. In QPSO quasi-random sequences are applied to the generation of the initial position and velocity of particles and their velocities are modified by the constriction factor approach. The flow of measurement uncertainty evaluation based on MCM is proposed, where the heart is repeatedly sampling from the probability density function（PDF） for every input quantity and evaluating the model in each case. The minimum zone SSE of a shaft measured on a Coordinate Measuring Machine（CMM） is calculated by QPSO and the measurement uncertainty is evaluated by MCM on the basis of analyzing the uncertainty contributors. The results show that the uncertainty directly influences the product judgment result. Therefore it is scientific and reasonable to consider the influence of the uncertainty in judging whether the parts are accepted or rejected, especially for those located in the uncertainty zone. The proposed method is especially suitable when the PDF of the measurand cannot adequately be approximated by a Gaussian distribution or a scaled and shifted t-distribution and the measurement model is non-linear.

ERROR ESTIMATES OF GALERKIN METHOD FOR HIGH DIMENSIONAL STEADY STATE KURAMOTO-SIVASHINSKY EQUATION

Error estimates of Galerkin method for Kuramoto-Sivashingsky (K-S) equation in space dimension ≥3 are derived in the paper. These results furnish strong evidence for the computation of the solutions.

THE ERROR ESTIMATES OF HALLEY'S METHOD

In this paper we give an almost sharp error estimate of Halley’s iteration for the majorizing sequence. Compared with the corresponding results in [6,14], it is far better. Meanwhile,the convergence theorem is established .for Halley’s iteration in Banach spaces.

A New Regularized Minimum Error Thresholding Method

To overcome the shortcoming that the traditional minimum error threshold method can obtain satisfactory image segmentation results only when the object and background of the image strictly obey a certain type of probability distribution,one proposes the regularized minimum error threshold method and treats the traditional minimum error threshold method as its special case.Then one constructs the discrete probability distribution by using the separation between segmentation threshold and the average gray-scale values of the object and background of the image so as to compute the information energy of the probability distribution.The impact of the regularized parameter selection on the optimal segmentation threshold of the regularized minimum error threshold method is investigated.To verify the effectiveness of the proposed regularized minimum error threshold method,one selects typical grey-scale images and performs segmentation tests.The segmentation results obtained by the regularized minimum error threshold method are compared with those obtained with the traditional minimum error threshold method.The segmentation results and their analysis show that the regularized minimum error threshold method is feasible and produces more satisfactory segmentation results than the minimum error threshold method.It does not exert much impact on object acquisition in case of the addition of a certain noise to an image.Therefore,the method can meet the requirements for extracting a real object in the noisy environment.

Error Control Strategies for Numerical Integrations in Fast Collocation Methods

We propose two error control techniques for numerical integrations in fast multiscale collocation methods for solving Fredholm integral equations of the second kind with weakly singular kernels. Both techniques utilize quadratures for singular integrals using graded points. One has a polynomial order of accuracy if the integrand has a polynomial order of smoothness except at the singular point and the other has exponential order of accuracy if the integrand has an infinite order of smoothness except at the singular point. We estimate the order of convergence and computational complexity of the corresponding approximate solutions of the equation. We prove that the second technique preserves the order of convergence and computational complexity of the original collocation method. Numerical experiments are presented to illustrate the theoretical estimates.

THE LARGE TIME ERROR ESTIMATES OF FOURIER SPECTRAL METHOD FOR GENERALIZED BENJAMIN-BONA-MAHONY EQUATIONS

In this paper, a spectral method to analyze the generalized Benjamin Bona Mahony equations is used. The existence and uniqueness of global smooth solution of these equations are proved. The large time error estimation between the spectral approximate solution and the exact solution is obtained.

A NONLINEAR GALERKIN MIXED ELEMENT METHOD AND A POSTERIORI ERROR ESTIMATOR FOR THE STATIONARY NAVIER-STOKES EQUATIONS

A nonlinear Galerkin mixed element (NGME) method and a posteriori error exstimator based on the method are established for the stationary Navier-Stokes equations. The existence and error estimates of the NGME solution are first discussed, and then a posteriori error estimator based on the NGME method is derived.

The Error Estimates of Direct Discontinuous Galerkin Methods Based on Upwind-Baised Fluxes

In this paper, we study the error estimates for direct discontinuous Galerkin methods based on the upwind-biased fluxes. We use a newly global projection to obtain the optimal error estimates. The numerical experiments imply that L2 norms error estimates can reach to order k + 1 by using time discretization methods.

A POSTERIORI ERROR ESTIMATES OF FINITEELEMENT METHOD FOR PARABOLIC PROBLEMS

This paper deals with a-posteriori error estimates for piecewise linear finite element approximations of parabolic problems in two space dimensions. The analysis extends previous results for elliptic problems to the parabolic context.

Backward Time Difference Collocation Method and Error Analysis for the Semiconductor Problem

Collocation method is put forward to solve the semiconductor problem with heat-conduction, whose mathematical model is described by an initial and boundary problem for a nonlinear partial differential equation system. One elliptic equation is for the electric potential, and three parabolic equations are for the electron concentration, hole concentration and heat-conduction. Using the prior estimate and technique of differential equations, we obtained almost optimal error estimates in L2.

ERROR PROCESSING METHOD OF CYCLOIDAL GEAR MEASUREMENT USING 3D COORDINATES MEASURING MACHINE

An error processing method is presented based on optimization theory and microcomputer technique which can be successfully used in the cycloidal gear measurement on three dimensional coordinates measuring machine (CMM). In the procedure, the minimum quadratic sum of the normal deviation is used as the object function and the equidistant curve is dealed with instead of the teeth profile. CMM is a high accurate measuring machine which can provide a way to evaluate the accuracy of the cycloidal gear completely.

Source location error analysis and optimization methods

The efficiency of an optimization method for acoustic emission/microseismic（AE/MS） source location is determined by the compatibility of its error definition with the errors contained in the input data.This compatibility can be examined in terms of the distribution of station residuals.For an ideal distribution,the input error is held at the station where it takes place as the station residual and the error is not permitted to spread to other stations.A comparison study of two optimization methods,namely the least squares method and the absolute value method,shows that the distribution with this character constrains the input errors and minimizes their impact,which explains the much more robust performance by the absolute value method in dealing with large and isolated input errors.When the errors in the input data are systematic and/or extreme in that the basic data structure is altered by these errors,none of the optimization methods are able to function.The only means to resolve this problem is the early detection and correction of these errors through a data screening process.An efficient data screening process is of primary importance for AE/MS source location.In addition to its critical role in dealing with those systematic and extreme errors,data screening creates a favorable environment for applying optimization methods.

A New Algorithm of Roundness Error Separation Technique of Three-point Method

Through the analysis of roundness error separation technique of three-point method and based on the invariability and periodicity of the geometrical characteristic of measured round contour, a new matrix algorithm, which can be used to solve directly the roundness of the measured round contour without Fourier transform, is presented. On the basis of the research and analysis of the rotation error movement which is separated by using the three-point method, a mathematical equation is derived, which can be used to separate the eccentric motion of least square center of measured round contour and the pure rotation motion error of spindle in rotation motion. The correctness of this method is validated by means of simulation.