meshless local petrov galerkin formulation for problems in composite micromechanics in this paper we present the meshless local petrov galerkin formulation for the generalized plane strain problem with specific emphasis on micromechanics of composite materials containing material discontinuities the problem requires the introduction of an extra discrete degree of freedom the out of plane uniform normal strain the treatment of material discontinuity at the interface between the two phases of the composite is presented by means of direct imposition of interface boundary conditions the meshless local petrov galerkin method is used in the micromechanical model for predicting the elastic constants of the composite to our knowledge this is the first study in which the meshless local petrov galerkin method is formulated for the so called meshless local petrov galerkin method based micromechanical analysis examples are presented to illustrate the effectiveness of the current method and it is validated by comparing the results with available analytical and numerical solutions the current method has the potential for use in micromechanics especially for textile composites where the meshing of the unit cell has been quite difficult

the natural element method in solid mechanics the application of the natural element method nem 1 2 to boundary value problems in two dimensional small displacement elastostatics is presented the discrete model of the domain consists of a set of distinct nodes n and a polygonal description of the boundary in the natural element method the trial and test functions are constructed using natural neighbour interpolants these interpolants are based on the voronoi tessellation of the set of nodes n the interpolants are smooth c 1 everywhere except at the nodes where they are c 0 in one dimension nem is identical to linear nite elements the nem interpolant is strictly linear between adjacent nodes on the boundary of the convex hull which facilitates imposition of essential boundary conditions a methodology to model material discontinuities and non convex bodies cracks using nem is also described a standard displacement based galerkin procedure is used to obtain the discrete system of linear equations application of nem to various problems in solid mechanics which include the patch test gradient problems bimaterial interface and a static crack problem are presented excellent agreement with exact analytical solutions is obtained which exemplies the accuracy and robustness of nem and suggests its potential application in the context of other classes of problems crack growth plates and large deformations to name a few 1998 john wiley sons ltd

implementation of a maximin power clustering criterion to select near replicates for regression lack of fit tests abstract in earlier work we presented a maximin power clustering criterion to partition observations into groups of near replicates specifically the criterion selects near replicate clusters for use with christensen s tests for orthogonal between and within cluster lack of fit this article further explores implementation of this clustering criterion in particular a methodology is developed to determine a collection of candidate groupings to which the maximin power criterion can be applied

adjoint based optimal control using meshfree discretizations the paper at hand presents a combination of optimal control approaches for pdes with meshless discretizations applying a classical lagrangian type particle method to optimization problems with hyperbolic constraints several adjoint based strategies differing in the sequential order of optimization and discretization of the lagrangian or eulerian problem formulation are proposed and compared the numerical results confirm the theoretically predicted independence principle of the optimization approaches and show the expected convergence behavior moreover they exemplify the superiority of meshless methods over the conventional mesh based approaches for the numerical handling and optimization of problems with time dependent geometries and freely moving boundaries

alternative goodness of fit tests for linear models fan and huang 2001 presented a goodness of fit test for linear models based on fourier transformations of the residuals of the fitted model we present two more theoretically appealing tests in which the fourier transforms are incorporated into a fitted model we show that when suitably normalized the new test statistics have the same asymptotic distribution as fan and huang s test we propose modifications to the asymptotic normalization constants to improve the small sample sizes of our tests while retaining their asymptotic distributions small sample sizes and powers are examined via simulations an illustration is given

an exact test of fit for the gaussian linear model using optimal nonbipartite matching abstractfisher tested the fit of gaussian linear models using replicated observations we refine this method by 1 constructing near replicates using an optimal nonbipartite matching and 2 defining a distance that focuses on predictors important to the model s predictions near replicates may not exist unless the predictor set is low dimensional the test addresses dimensionality by betting that model failures involve a subset of predictors important in the old fit despite using the old fit to pair observations the test has exactly its stated level under the null hypothesis simulations show the test has reasonable power even when many spurious predictors are present

selecting check points for testing lack of fit in response surface models in modeling a response as a function of a set of explanatory variables one way to check lack of fit in a fitted model of the form e y x 1 when the true model is e y x 1 x22 2 is to use additional check points the question of choosing check points has not often been addressed so we propose a method of selecting check points to maximize the power of the lack of fit test two examples illustrate the procedure

testing linear regression function adequacy without replication the well known pure error lack of fit test which can be used to assess the adequacy of a linear regression model is generalized to accommodate the case of nonreplication the asymptotic null distribution of the proposed test statistic is derived also the proposed test statistic is shown to be asymptotically comparable under general alternatives to the test statistic obtained in the case of replication consistency properties associated with pseudo lack of fit and pure error mean squares are given which parallel those obtained in the case of replication in addition the test statistic is invariant with respect to location and scale changes made to the regression variables

modified moving least squares with polynomial bases for scattered data approximation one common problem encountered in many fields is the generation of surfaces based on values at irregularly distributed nodes to tackle such problems we present a modified robust moving least squares mls method for scattered data smoothing and approximation the error functional used in the derivation of the classical mls approximation is augmented with additional terms based on the coefficients of the polynomial base functions this allows quadratic base functions to be used with the same size of the support domain as linear base functions resulting in better approximation capability the increased robustness of the modified mls method to irregular nodal distributions makes it suitable for use across many fields the analysis is supported by several univariate and bivariate examples

maximin clusters for near replicate regression lack of fit tests to assess the adequacy of a nonreplicated linear regression model christensen introduced the concepts of orthogonal between and within cluster lack of fit with corresponding optimal tests however the properties of these tests depend on the choice of near replicate clusters in this paper a graph theoretic framework is presented to represent candidate clusterings a clustering is then selected according to a proposed maximin power criterion from among the clusterings consistent with a specified graph on the predictor settings examples are given to illustrate the methodology

new boundary condition treatments in meshfree computation of contact problems due to the loss of kronecker delta properties in the meshfree shape functions the imposition of essential boundary conditions consumes significant cpu time in meshfree computation in this work two boundary condition treatments are proposed to enhance the computational efficiency of meshfree methods for contact problems the mixed transformation method is modified from the previous full transformation method by introducing a node partitioning and a mixed coordinate so that the matrix inversion and multiplication of coordinate transformation involves operations only on the sub degrees of freedom associated with the boundary group the boundary singular kernel method introduces singularities to the kernel functions of the essential and contact boundary nodes so that the corresponding coefficients of the singular kernel shape functions recover nodal values and consequently kinematic constraints can be imposed directly the effectiveness of the proposed methods is demonstrated in several numerical examples

on the optimality of bayesian change point detection by introducing suitable loss random variables of detection we obtain optimal tests in terms of the stopping time or alarm time for bayesian change point detection not only for a general prior distribution of change points but also for observations being a markov process moreover the optimal minimal average detection delay is proved to be equal to 1 for any possibly large average run length to false alarm if the number of possible change points is finite

a coupled element free galerkin boundary element method for stress analysis of two dimensional solids element free galerkin efg method is a newly developed meshless method for solving partial differential equations using moving least squares interpolants it is however computationally expensive for many problems a coupled efg boundary element be method is proposed in this paper to improve the solution efficiency a procedure is developed for the coupled efg be method so that the continuity and compatibility are preserved on the interface of the two domains where the efg and be methods are applied the present coupled efg be method has been coded in fortran the validity and efficiency of the efg be method are demonstrated through a number of examples it is found that the present method can take the full advantages of both efg and be methods it is very easy to implement and very flexible for computing displacements and stresses of desired accuracy in solids with or without infinite domains

hermite type moving least squares approximations the moving least squares approach first presented by mclain 1 is a method for approximating multivariate functions using scattered data information the method is using local polynomial approximations incorporating weight functions of different types some weights with certain singularities induce c interpolation approximation in n in this work we present a way of generalizing the method to enable hermite type interpolation namely interpolation to derivatives data as well the essence of the method is the use of an appropriate metric in the construction of the local polynomial approximations

frictional contact analysis of functionally graded materials with lagrange finite block method summary based on the one dimensional differential matrix derived from the lagrange series expansion the finite block method was recently developed to solve both the elasticity and transient heat conduction problems of anisotropic and functionally graded materials in this paper the formulation of the lagrange finite block method with boundary type in the strong form is presented and applied to non conforming contact problems for the functionally graded materials subjected to either static or dynamic loads the first order partial differential matrices are only needed both in the governing equations and in the neumann boundary condition by introducing the mapping technique a block of quadratic type is transformed from the cartesian coordinate of global system to the normalized coordinate with eight seeds time dependent partial differential equations are analyzed in the laplace transformed domain and the durbin s inversion method is applied to determine all the physical values in the time domain conforming and non conforming contacts are investigated by using the iterative algorithm with full load technique illustrative numerical examples are given and comparisons have been made with analytical solutions copyright 2015 john wiley sons ltd

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

a perturbation method for stochastic meshless analysis in elastostatics a stochastic meshless method is presented for solving boundary value problems in linear elasticity that involves random material properties the material property was modelled as a homogeneous random field a meshless formulation was developed to predict stochastic structural response unlike the finite element method the meshless method requires no structured mesh since only a scattered set of nodal points is required in the domain of interest there is no need for fixed connectivities between nodes in conjunction with the meshless equations classical perturbation expansions were derived to predict second moment characteristics of response numerical examples based on one and two dimensional problems are presented to examine the accuracy and convergence of the stochastic meshless method a good agreement is obtained between the results of the proposed method and monte carlo simulation since mesh generation of complex structures can be a far more time consuming and costly effort than the solution of a discrete set of equations the meshless method provides an attractive alternative to finite element method for solving stochastic mechanics problems copyright 2001 john wiley sons ltd

sequential change detection revisited in sequential change detection existing performance measures differ significantly in the way they treat the time of change by modeling this quantity as a random time we introduce a general framework capable of capturing and better understanding most well known criteria and also propose new ones for a specific new criterion that constitutes an extension to lorden s performance measure we offer the optimum structure for detecting a change in the constant drift of a brownian motion and a formula for the corresponding optimum performance

a meshfree enriched finite element method for compressible and near incompressible elasticity summary in this paper a two dimensional displacement based meshfree enriched fem me fem is presented for the linear analysis of compressible and near incompressible planar elasticity the me fem element is established by injecting a first order convex meshfree approximation into a low order finite element with an additional node the convex meshfree approximation is constructed using the generalized meshfree approximation method and it possesses the kronecker delta property on the element boundaries the gradient matrix of me fem element satisfies the integration constraint for nodal integration and the resultant me fem formulation is shown to pass the constant stress test for the compressible media the me fem interpolation is an element wise meshfree interpolation and is proven to be discrete divergence free in the incompressible limit to prevent possible pressure oscillation in the near incompressible problems an area weighted strain smoothing scheme incorporated with the divergence free me fem interpolation is introduced to provide the smoothing on strains and pressure with this smoothed strain field the discrete equations are derived based on a modified hu washizu variational principle several numerical examples are presented to demonstrate the effectiveness of the proposed method for the compressible and near incompressible problems copyright 2012 john wiley sons ltd

treatment of material discontinuity in the element free galerkin method abstract a method is presented for application of the element free galerkin method efg to solid mechanics problems containing material discontinuities by using the efg method the trial and test functions for the weak form are constructed with moving least square interpolants in each material domain additional constraints are imposed at the variational level to ensure satisfaction of interface conditions as a result only nodal data no elemental connectivity is needed given c 1 continuity of the interpolant weighting function the dependent variable and its gradient are continuous in each material domain two linear elastic bi material examples are presented to illustrate the effectiveness of the method

a lack of fit test for the mean function in a generalized linear model abstract a supremum type statistic based on partial sums of residuals is proposed to test the validity of the mean function of the response variable in a generalized linear model the proposed test does not need a partition of the space of covariates to handle the case of nonreplication the new test is consistent against the alternative under which the deterministic part of the model is not the one specified in the null hypothesis the asymptotic null distribution of the test statistic can be approximated through simulations this approximation is valid even if the variance of the response variable is misspecified in the model for practical sample sizes the adequacy of this large sample approximation to the null distribution of the test statistic is carefully examined power comparisons with other lack of fit tests are also performed to show the advantage of the test in particular we find that the new procedure is rather sensitive to detect a misspecified mean function if the assumed mean functio

checking for lack of fit in linear models with parametric variance functions i adapt near replicate lack of fit tests for checking the adequacy of a linear model for means to check for lack of fit in a standard log linear model for variances a graphical method for checking lack of fit in the variance model is also proposed in addition an overall test of lack of fit in means or variances is presented three examples are used to illustrate the techniques

higher order piezoelectric plate theory derived from a three dimensional variational principle a three dimensional mixed variational principle is used toderive a kth order two dimensional lineartheory for an anisotropichomogeneouspiezoelectric pzt plate themechanicaldisplacements theelectricpotential theinplane components of the stress tensor and the in plane components of the electric displacement are expressed as a e niteseries of orderk inthethickness coordinate bytakinglegendrepolynomialsas thebasisfunctions however the transverse shear stress the transverse normal stress and the transverse electric displacement are expressed as a e nite series of order k 2 in the thickness coordinate the formulation accounts for the double forces without moments that may change the thickness of the plate results obtained by using the plate theory are given for the bending of a cantilever thick plate loaded on the top and the bottom surfaces by uniformly distributed 1 normal tractions and 2 tangential tractions results are also computed for the bending of a cantilever thick pzt beam loaded by 1 a uniformly distributed charge density on the top and the bottom surfaces and 2 equal and opposite normal tractions distributed uniformly only on a part of the beam the seventh order plate theory captures well the boundary layer effects near the clamped and the free edges and adjacent to the top and the bottom surfaces of a thick orthotropic cantilever beam with the span to the thickness ratio of two also through the thickness variation of the transverse shear and the transverse normal stresses agree well with those computed from the analytical solution of the three dimensional elasticity equations the governing partial differential equations are second order so that lagrange basis functions can be used to solve the problem by the e nite element method

a note on lack of fit tests for linear models without replication a class of three tests overall lack of fit test between cluster lack of fit test and within cluster lack of fit test are proposed for testing the lack of fit of a linear regression model applied to experiments without replicates the power of the proposed tests is significantly higher than those of the known tests under the situations considered here the proposed tests are capable of detecting which type of lack of fit is dominant when both between cluster and within cluster lack of fit are present

a test for lack of fit of a linear multiresponse model this article introduces a multivariate test for lack of fit of a linear multiresponse model the test requires that replicated observations be taken at some points in the experimental region a procedure is presented for the purpose of detecting which responses are responsible for lack of fit whenever the multivariate test is significant an example is given to illustrate the procedure

efg approximation with discontinuous derivatives a technique for incorporating discontinuities in derivatives into meshless methods is presented the technique enriches the approximation by adding special shape functions that contain discontinuities in derivatives the special shape functions have compact support which results in banded matrix equations the technique is described in element free galerkin context but is easily applicable to other meshless methods and projections numerical results for problems in one and two dimensions are reported 1998 john wiley sons ltd

small sample characterizations of near replicate lack of fit tests abstract several ad hoc tests based on near replicates have been proposed for testing lack of fit in regression analysis christensen characterized lack of fit as existing between clusters of near replicates within clusters or as a combination of these pure types of these the between cluster variety is the type commonly associated with the idea of lack of fit christensen examined a test that was new to the normal theory regression literature and established uniformly most powerful invariant umpi properties of the test in particular the test is umpi for orthogonal lack of fit within clusters in this article a new test is proposed that is umpi for orthogonal lack of fit between clusters the relationship of this optimal test to other proposed tests is examined giving the first small sample theoretical justification for these tests the power of the new test is compared to that of others in the literature and consistency results are discussed

optimal detection of a change in distribution abstract document discusses a stopping rule that is a limit of bayes rules derived then an almost minimax rule is presented

optimal designs for identifying the degree of a polynomial regression if an experimenter wants to determine the degree of a polynomial regression on the basis of a sample of observations anderson showed that the following method is optimal starting with the highest specified degree the procedure is to test in sequence whether the coefficients are 0 in this paper optimal designs for anderson s procedure are determined explicitly the optimal design maximizes the minimum power of a given set of alternatives

a generalized ewma control chart and its comparison with the optimal ewma cusum and glr schemes it is known that both the optimal exponentially weighted moving average ewma and cumulative sum cusum control charts are based on a given reference value delta which for the cusum chart is the magnitude of a shift in the mean to be detected quickly in this paper a generalized ewma control chart gewma which does not depend on delta is proposed for detecting the mean shift we compare theoretically the gewma control chart with the optimal ewma cusum and the generalized likelihood ratio glr control charts the results of the comparison in which the in control average run length approaches infinity show that the gewma control chart is better than the optimal ewma control chart in detecting a mean shift of any size and is also better than the cusum control chart in detecting the mean shift which is not in the interval 0 7842 delta 1 3798 delta moreover the glr control chart has the best performance in detecting mean shift among the four control charts except when detecting a particular mean shift delta when the in control average run length approaches infinity

improved xfem an extra dof free well conditioning and interpolating xfem abstract although its unprecedentedly success in approximating the singular field around a crack tip independent of mesh discretization xfem still has been long hindered in engineering applications by its ill conditioning and the difficulty in mass lumping corresponding to extra dofs an improved xfem i xfem is developed to overcome those difficulties from three aspects 1 to eliminate the linear dependence and the ill conditioning issues of the standard and the corrected xfems 2 to get rid of extra dofs in crack tip enrichment to facilitate optimal mass lumping in dynamic analyses 3 to be interpolating at enriched nodes to enable direct essential contact boundary treatments the core of i xfem is a globally interpolating approximation based on local least squares fitting the new method is numerically examined with comprehensive comparisons with the standard xfem and the corrected xfem fries 2008 the new method s excellent performance in terms of accuracy convergence and conditioning stability are demonstrated

treatment of material discontinuity in two meshless local petrov galerkin mlpg formulations of axisymmetric transient heat conduction summary we use two meshless local petrov galerkin mlpg formulations to analyse heat conduction in a bimetallic circular disk the continuity of the normal component of the heat flux at the interface between two materials is satisfied either by the method of lagrange multipliers or by using a jump function the convergence of the h 0 and h 1 error norms for the four numerical solutions with an increase in the number of equally spaced nodes and in the number of quadrature points is scrutinized with an increase in the number of uniformly spaced nodes the two error norms decrease monotonically for the mlpg5 formulation but are essentially unchanged for the mlpg1 formulation to our knowledge this is the first study comparing the performance of the two methods of modelling a discontinuity in the gradient of a field variable at the interface between two different materials copyright 2004 john wiley sons ltd

measures of lack of fit for response surface designs and predictor variable transformations some first order 2 k p two level factorials and fractional factorials plus center points and second order cube plus star plus center point composite response surface designs are discussed from the point of view of their ability to detect certain likely kinds of lack of fit of degree one higher than has been fitted this leads to consideration of conditions for representationa adequacy of first and second order models in transformed predictor variables it is shown how to use the estimated regression coefficients from the higher degree model to check if power transformations of the predictor variables could eliminate the lack of fit and also actually to estimate the transformations

on optimality of the shiryaev roberts procedure for detecting a change in distribution in 1985 for detecting a change in distribution pollak introduced a specific minimax performance metric and a randomized version of the shiryaev roberts procedure where the zero initial condition is replaced by a random variable sampled from the quasi stationary distribution of the shiryaev roberts statistic pollak proved that this procedure is third order asymptotically optimal as the mean time to false alarm becomes large the question of whether pollak s procedure is strictly minimax for any false alarm rate has been open for more than two decades and there were several attempts to prove this strict optimality in this paper we provide a counterexample which shows that pollak s procedure is not optimal and that there is a strictly optimal procedure which is nothing but the shiryaev roberts procedure that starts with a specially designed deterministic point

mls moving least square based finite elements for three dimensional nonmatching meshes and adaptive mesh refinement abstract with the aid of moving least square mls approximation a new class of three dimensional finite elements are proposed for treating nonmatching meshes and adaptive mesh refinement for which the existing finite elements are hardly efficient with a special choice of the weight function supports and the base functions the method results in useful elements with the polynomial shape function for which the c 1 continuity breaks down on the boundaries between the neighboring subdomains comprising one element the effectiveness of the new elements in handling the discontinuities due to nonmatching interfaces and automatic mesh refinement is demonstrated via three dimensional examples

numerical methods for modal analysis of stepped piezoelectric beams abstract this paper analyzes different numerical methods for modal analysis of stepped piezoelectric beams modeled by the euler bernoulli beam theory results from standard numerical approaches that rely on the discretization of the stepped beam assumed modes and finite element methods are compared with the solution of the exact transcendental eigenvalue problem for the infinite dimensional system an accurate and manageable novel method that enriches the assumed modes basis functions with special jump functions is presented numerical results are compared with experimental data and the accuracy of the adopted beam model is validated

combination of the meshless finite difference approach with the monte carlo random walk technique for solution of elliptic problems abstract this paper proposes a stochastic approach for the fast and effective numerical analysis of the second order elliptic differential equations it is based upon the well known monte carlo mc method with a random walk rw technique carried out on the grid of points this method allows for accurate estimation of the solution of the differential equation at selected point s of the domain and or its boundary it extends the standard formulation of the monte carlo random walk mc rw approach by means of its appropriate combination with the meshless version of the finite difference method in this manner the proposed approach may deal with elliptic equations in more general non homogeneous form as well as boundary conditions of both essential and natural types moreover arbitrarily irregular clouds of nodes may be used with no a priori imposed nodes structure therefore the meshless mc rw approach may be applied to the significantly wider class of problems with more complex geometry this concept was examined on variety of 2d boundary value problems selected numerical results are presented and discussed a simple matlab code is included as well

a coupled interpolating meshfree method for computing sound radiation in infinite domain summary in this paper the coupling of the improved interpolating element free galerkin iiefg method and the variable order infinite acoustic wave envelope element wee method is studied a coupled iiefg wee method for computing sound radiation is proposed to make use of their advantages while evading their disadvantages the coupling is achieved by constructing the hybrid shape function of continuity and compatibility on the interface between the iiefg and wee domains in the iiefg domain the improved interpolating moving least squares iimls method is employed to form the shape functions satisfying the kronecker delta condition while nonsingular weight functions can be used the impacts of the size of the influence domain and the shape parameter on the performance of this coupled method are investigated the numerical results show that the coupled iiefg wee method can take full advantage of both the iiefg and wee methods and that it not only can achieve higher accuracy but also has a faster convergence speed than the conventional method of the finite element coupled with the wee the experimental results show that the method is very flexible for acoustic radiation prediction in the infinite domain this article is protected by copyright all rights reserved

a novel meshless scheme for solving surface integral equations with flat integral domains numerical solutions for electromagnetic em integral equations rely on the discretization of integral domains and the use of meshes for geometric description meshing geometries is very tedious especially for complicated structures with many details tiny parts and geometric discontinuities corners or edges and remeshing could be required in many scenarios to reduce the costs of generating quality meshes meshless or mesh free methods were developed and they have been extensively studied in mechanical engineering though there are less obvious interests in em community the meshless methods employ discrete nodes to replace meshes in the description of geometries but the background meshes for integrations are still needed traditionally in this work we first address the traditional meshless scheme for solving em integral equations based on the moving least square mls approximation for unknown currents and the use of background meshes for integrations and then develop a novel meshless scheme by applying the green s lemma to the em surface integral equations with flat domains the novel scheme transforms a surface integral over a flat domain into a line integral along its boundaries when excluding a singular patch in the domain since only the domain boundaries are discretized and no background meshes are needed the scheme is truly meshless numerical examples for em scattering by flat surface objects are presented to demonstrate the effectiveness of the novel scheme

meshless local petrov galerkin mlpg approximation to the two dimensional sine gordon equation during the past few years the idea of using meshless methods for numerical solution of partial differential equations pdes has received much attention throughout the scientific community and remarkable progress has been achieved on meshless methods the meshless local petrov galerkin mlpg method is one of the truly meshless methods since it does not require any background integration cells the integrations are carried out locally over small sub domains of regular shapes such as circles or squares in two dimensions and spheres or cubes in three dimensions in this paper the mlpg method for numerically solving the non linear two dimensional sine gordon sg equation is developed a time stepping method is employed to deal with the time derivative and a simple predictor corrector scheme is performed to eliminate the non linearity a brief discussion is outlined for numerical integrations in the proposed algorithm some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped sg equation is investigated the final numerical results confirm the ability of proposed method to deal with the unsteady non linear problems in large domains

reproducing kernel hierarchical partition of unity part i formulation and theory this work is concerned with developing the hierarchical basis for meshless methods a reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart to form such hierarchical partition a class of basic wavelet functions are introduced based upon the built in consistency conditions the differential consistency conditions for the hierarchical kernel functions are derived it serves as an indispensable instrument in establishing the interpolation error estimate which is theoretically proven and numerically validated for a special interpolant with different combinations of the hierarchical kernels a synchronized convergence effect may be observed being different from the conventional legendre function based p type hierarchical basis the new hierarchical basis is an intrinsic pseudo spectral basis which can remain as a partition of unity in a local region because the discrete wavelet kernels form a partition of nullity these newly developed kernels can be used as the multi scale basis to solve partial differential equations in numerical computation as a p type refinement copyright 1999 john wiley sons ltd

interpolating element free galerkin method for the regularized long wave equation and its error analysis abstract combining the simplified interpolating moving least squares imls method with the galerkin weak form an interpolating element free galerkin iefg method is presented for the numerical study of the regularized long wave rlw equation the trial function in the imls method has one less unknown coefficient than in the traditional mls approximation and the shape functions satisfy the interpolating property at nodes then the iefg method can apply the essential boundary conditions naturally and directly and has high computational accuracy and then this paper studies the error estimates of the iefg method for rlw equation the error estimate of the semi discrete iefg method for the rlw equation on the influence domain radius is first studied and then the convergence rate of the full discrete scheme of the method on the influence domain radius and time step is studied the theoretical results show that the iefg method for rlw equation has good convergence rate for the purpose of demonstration three selected numerical examples are given to demonstrate the method and the mathematical theory of this paper