the h text p version of the finite element method for domains with curved boundaries this paper analyzes the convergence of the h text p version of the finite element method for elliptic problems on curved domains it proves that when the boundary of the domain is piecewise analytic boundary conditions are piecewise analytic and the sides of the elements are analytic the h text p version has an exponential rate of convergence numerical computations are included the paper generalizes the results previously obtained by the authors the h text p version of finite element method part 1 the basic approximation results comp mech 1 1986 pp 21 41 the h text p version of finite element method part 2 general results and applications comp mech 1 1986 pp 203 220

analyzing the 2d fracture problems via the enriched boundary element free method in this paper the enriched boundary element free method for two dimensional fracture problems is presented an improved moving least squares imls approximation in which the orthogonal function system with a weight function is used as the basis function is used to obtain the shape functions the imls approximation has greater computational efficiency and precision than the existing moving least squares mls approximation and does not lead to an ill conditioned system of equations combining the boundary integral equation bie method and the imls approximation a boundary element free method befm for two dimensional fracture problems is obtained for two dimensional fracture problems the enriched basis function is used at the tip of the crack and then the enriched befm is presented in comparison with other existing meshless boundary integral equation methods the befm is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables and the boundary conditions can be implemented easily which leads to a greater computational precision when the enriched befm is used the singularity of the stresses at the tip of the crack can be shown better than that in the befm for the purposes of demonstration some selected numerical examples are solved using the enriched befm

error analysis for moving least squares approximation in 2d space abstract in this paper we provide a theoretical analysis of the moving least squares mls approximation which belongs to the family of meshless methods first the non matrix form of the mls shape function in two dimensional space is obtained by using consistency conditions the error estimates for mls approximation in sobolev space are presented when u x y c m 1 and u x y w m 1 q respectively we establish the error estimates for interpolating element free galerkin iefg method when it is used for solving poisson s equation the error bound is related to the radii of the weight functions and the bound of the norm of derivatives of shape functions three numerical examples are selected to confirm our analysis

on the sharpness of l 2 error estimates of h 1 0 projections onto subspaces of piecewise high order polynomials in a plane polygonal domain consider a poisson problem u f with homogeneous dirichlet boundary condition and the p version finite element solutions of this we give various upper and lower bounds for the error measured in l 2 in the case of a single element i e a convex domain we reduce the question of sharpness of these estimates to the behavior of a certain inf sup constant which is numerically determined and a likely sharp estimate is then conjectured this is confirmed during a series of numerical experiments also for the case of a reentrant comer for a one dimensional analogue problem of rotational symmetry sharp l 2 error estimates are proven directly and via an extension of the classical duality argument here we give sharp l error estimates in some weighted and unweighted norms also

a weak galerkin finite element method with polynomial reduction the weak galerkin wg is a novel numerical method based on variational principles for weak functions and their weak partial derivatives defined as distributions in the implementation the weak partial derivatives and the weak functions are approximated by polynomials with various degrees of freedom the accuracy and the computational complexity of the corresponding wg scheme is significantly impacted by the selection of such polynomials this paper presents an optimal combination for the polynomial spaces that minimize the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation for illustrative purpose the authors use the second order elliptic equation to demonstrate the basic ideas of polynomial reduction consequently a new weak galerkin finite element method is proposed and analyzed error estimates of optimal order are established for the corresponding wg approximations in both a discrete h 1 norm and the standard l 2 norm in addition the paper presents some numerical results to demonstrate the power of the wg method in dealing with finite element partitions with arbitrary polygons in 2d or polyhedra in 3d the numerical examples include various finite element partitions such as triangular mesh quadrilateral mesh honeycomb mesh in 2d and mesh with deformed cubes in 3d

on the finite element method for singularly perturbed reaction diffusion problems in two and one dimensions second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered the behavior of the ordinary galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed

local and pollution error estimation for finite element approximations of elliptic boundary value problems this paper addresses the issue of local elementwise error estimation of finite element approximations of elliptic boundary value problems the characterization of element error as local and pollution error component is presented and the relationship between energy norms of local errors and that predicted by means of a posteriori error estimators is investigated in addition techniques for calculating element indicators of local error pollution error and other error indicators are presented

theh p version of the finite element method for elliptic equations of order2m the problems of elliptic partial differential equations stemming from engineering problems are usually characterized by piecewise analytic data it has been shown in 3 4 5 that the solutions of the second order and fourth order equations belong to the spacesb 1 where the weighted sobolev norms of thek th derivatives are bounded bycd k l k l k l l 2 wherec andd are constants independent ofk in this case theh p version of the finite element method leads to an exponential rate of convergence measured in the energy norm 6 12 13 theh p version was implemented in the code probe1 18 and has been very successfully used in the industry we will discuss in this paper the generalization of these results for problems of order2m we will show also that the exponential rate can be achieved if the exact solution belongs to the spacesb 1 where the weighted sobolev norm of thek th derivatives is bounded bycd k l k l k l m 1 c andd are independent ofk in addition if the data is piecewise analytic then in fact the exact solution belongs to the spacesb m 1 problems of this type are related obviously to many engineering problems such as problems of plates and shells and are also important in connection with well known locking problems

a partition of unity enriched dual boundary element method for accurate computations in fracture mechanics we introduce a novel enriched boundary element method bem and dual boundary element method dbem approach for accurate evaluation of stress intensity factors sifs in crack problems the formulation makes use of the partition of unity method pum such that functions obtained from a priori knowledge of the solution space can be incorporated in the element formulation an enrichment strategy is described in which boundary integral equations formed at additional collocation points are used to provide auxiliary equations in order to accommodate the extra introduced unknowns in addition an efficient numerical quadrature method is outlined for the evaluation of strongly singular and hypersingular enriched boundary integrals finally results are shown for mixed mode crack problems these illustrate that the introduction of pum enrichment provides for an improvement in accuracy of approximately one order of magnitude in comparison to the conventional unenriched dbem

error estimates for the interpolating moving least squares method in this paper the interpolating moving least squares imls method presented by lancaster and salkauskas is discussed in details the advantage of the imls method is that the meshless method which is constructed based on the imls method can apply the essential boundary conditions directly and easily a simpler expression of the approximation function of the imls method is obtained then the error estimate of the approximation function and its first and second order derivatives of the imls method are presented in one dimensional case in this paper the theoretical results show that if the order of the polynomial basis functions is big enough and the original function is sufficiently smooth then the approximation function and its partial derivatives are convergent to the exact values in terms of the maximum radius of the domains of influence of nodes for the purpose of demonstration some selected numerical examples are given to prove the theories in this paper

probability inequalities for empirical processes and a law of the iterated logarithm given a class f of functions on x we can view vn as a stochastic process indexed by f and consider limit theorems for this process to prove such theorems it is often helpful to have bounds on the tail of the r v sup w i n f i see for example dudley 1978 kuelbs and dudley 1980 dudley and phillip 1983 or gine and zinn 1983 our main question of interest is for what classes f can best possible bounds be obtained so first we must ask what are these best possible bounds of course

weighted analytic regularity in polyhedra we explain a simple strategy to establish analytic regularity for solutions of second order linear elliptic boundary value problems the abstract framework presented here helps to understand the proof of analytic regularity in polyhedral domains given in the authors paper in m costabel m dauge s nicaise analytic regularity for linear elliptic systems in polygons and polyhedra math models methods appl sci 22 8 2012 we illustrate this strategy by considering problems set in smooth domains in corner domains and in polyhedra

boundary element free method befm for two dimensional elastodynamic analysis using laplace transform in this paper we present a direct meshless method of boundary integral equation bie known as the boundary element free method befm for two dimensional 2d elastodynamic problems that combines the bie method for 2d elastodynamics in the laplace transformed domain and the improved moving least squares imls approximation the formulae for the befm for 2d elastodynamic problems are given and the numerical procedures are also shown the befm is a direct numerical method in which the basic unknown quantities are the real solutions of the nodal variables and the boundary conditions can be implemented directly and easily that leads to a greater computational precision for the purpose of demonstration some selected numerical examples are solved using the befm copyright 2005 john wiley sons ltd

postbuckling analysis of laminated composite plates using the mesh free kp ritz method abstract a ritz method based on kernel particle approximation for the field variables is proposed for the postbuckling analysis of laminated composite plates the first order shear deformation plate theory fsdt is employed to model the plate flexure the ritz method is used to obtain the discretized non linear equations a geometrically non linear analysis is used to trace the postbuckling paths of the plate typical numerical examples including isotropic plates and cross ply and angle ply laminated composite plates have been solved using the proposed method the results are in close agreement with the series solution as well as previous finite element results available in the literature

regularity and numerical solution of eigenvalue problems with piecewise analytic data this paper discusses the regularity of the eigenfunctions of eigenvalue problems with piecewise analytic data and the approximation of the eigenvalues and eigenfunctions of such problems a detailed and systematic numerical study of these approximations is presented together with an analysis of the numerical results in light of the theoretical results the specific aim is to assess the reliability of the theoretical results which are of an asymptotic nature as a guide to practical computations which may take place in the pre asymptotic phase and to look for characteristic features of the numerical results that are not completely explained by known theoretical results

numerical simulation of thermomechanical behaviours of shape memory alloys via a non linear mesh free galerkin formulation a new computational approach is developed based on the mesh free method for the numerical simulation of shape memory alloys a thermomechanical constitutive model is used to describe the complex material response the incremental displacement based mesh free formulation is constructed in a total lagrangian description by employing the moving least square shape function and rate form of the shape memory alloy constitutive model in the variational equations this mesh free formulation is not only applicable to large deformation problems but also accounts for both stress and thermally induced phase transformation moreover the newly developed mesh free codes can deal with complex loading paths and multiple temperature cycles in numerical simulation the performance of the proposed mesh free scheme for the simulation of shape memory alloys is demonstrated in several numerical examples copyright 2005 john wiley sons ltd

the boundary node method for three dimensional linear elasticity the boundary node method bnm is developed in this paper for solving three dimensional problems in linear elasticity the bnm represents a coupling between boundary integral equations bie and moving least squares mls interpolants the main idea is to retain the dimensionality advantage of the former and the meshless attribute of the later this results in decoupling of the mesh and the interpolation procedure for problems in linear elasticity free rigid body modes in traction prescribed problems are typically eliminated by suitably restraining the body however an alternative approach developed recently for the boundary element method bem is extended in this work to the bnm this approach is based on ideas from linear algebra to complete the rank of the singular stiffness matrix also the bnm has been extended in the present work to solve problems with material discontinuities and a new procedure has been developed for obtaining displacements and stresses accurately at internal points close to the boundary of a body copyright 1999 john wiley sons ltd

graded mesh approximation in weighted sobolev spaces and elliptic equations in 2d we study the approximation properties of some general finite element spaces constructed using improved graded meshes in our results either the approximating function or the function to be approximated or both are in a weighted sobolev space we consider also the l p version of these spaces the finite element spaces that we define are obtained from em conformally invariant families of finite elements no affine invariance is used stressing the use of elements that lead to higher regularity finite element spaces we prove that for a suitable grading of the meshes one obtains the usual optimal approximation results we provide a construction of these spaces that does not lead to long skinny triangles our results are then used to obtain l 2 error estimates and h m quasi optimal rates of convergence for the fem approximation of solutions of strongly elliptic interface boundary value problems

analysis of periodic schrodinger operators regularity and approximation of eigenfunctions let v be a real valued potential that is smooth everywhere on r3 except at a periodic discrete set s of points where it has singularities of the coulomb type z r we assume that the potential v is periodic with period lattice l we study the spectrum of the schrodinger operator h v acting on the space of bloch waves with arbitrary but fixed wavevector k let t r3 l let u be an eigenfunction of h with eigenvalue and let 0 be arbitrarily small we show that the classical regularity of the eigenfunction u is u h5 2 t in the usual sobolev spaces and u k3 2 m t s in the weighted sobolev spaces the regularity index m can be as large as desired which is crucial for numerical methods for any choice of the bloch wavevector k we also show that h has compact resolvent and hence a complete eigenfunction expansion the case of the hydrogen atom suggests that our regularity results are optimal we present two applications to the numerical approximation of eigenvalues using wave functions and usi

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

multigrid methods for the computation of singular solutions and stress intensity factors i corner singularities we consider the poisson equation u f with homogeneous dirichlet boundary condition on a two dimensional polygonal domain with cracks multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed the convergence rate for the stress intensity factors is mathcal o h 3 2 in whenfel2 and mathcal o h 2 in whenfeh1 the convergence rate in the energy norm is mathcal o h 1 in in the first case and mathcal o h in the second case the costs of these multigrid methods are proportional to the number of elements in the triangulation the general case wherefehm is also discussed

nonlinear analysis of laminated composite plates using the mesh free kp ritz method based on fsdt a mesh free kp ritz method of solution based on the kernel particle approximation for the field variables is proposed for the large deflection flexural analysis of laminated composite plates the first order shear deformation theory fsdt is used for modeling the flexure the nonlinear solution algorithm is based on the total lagrangian formulation with green s strain measures and von karman assumptions the incremental form of nonlinear equations is obtained by taylor series expansion and newton s method is used to solve these equations test problems involving square and rectangular composite plates with ssss and cccc boundary conditions are solved to assess the efficacy of the proposed method the results are in excellent agreement with the series solution as well as the finite element solution already reported in the literature

weighted sobolev spaces and regularity for polyhedral domains we prove a regularity result for the poisson problem u f u p g on a polyhedral domain p r3 using the babuska kondratiev spaces kam p these are weighted sobolev spaces in which the weight is given by the distance to the set of edges 4 33 in particular we show that there is no loss of kam regularity for solutions of strongly elliptic systems with smooth coefficients we also establish a trace theorem for the restriction to the boundary of the functions in kam p

approximation properties of the h p version of the finite element method abstract the paper surveys the approximation and convergence properties of the h p version of the finite element method the basic theoretical results together with the main ideas of the proofs numerical examples are given

a meshless local moving kriging method for two dimensional solids abstract an improved meshless local petrov galerkin method mlpg for stress analysis of two dimensional solids is presented in this paper the mlpg method based on the moving least squares approximation is one of the recent meshless approaches however accurate imposition of essential boundary conditions in the mlpg method often presents difficulties because the mlpg shape functions does not possess the kronecker delta property in order to eliminate this shortcoming this approach uses the moving kriging interpolation instead of the traditional moving least square approximation to construct the mlpg shape functions and then the heaviside step function is used as the test function over a local sub domain in this method the essential boundary conditions can be enforced as the fem no domain integration is needed and only regular boundary integration is involved in addition the sensitivity of several important parameters of the present method is mainly studied and discussed comparing with the original meshless local petrov galerkin method the present method has simpler numerical procedures and lower computation cost the effectiveness of the present method for two dimensional solids problem is investigated by numerical examples in this paper

quasi optimal rates of convergence for the generalized finite element method in polygonal domains we consider a mixed boundary value interface problem for the elliptic operator p i j i a i j j u f on a polygonal domain r 2 with straight sides we endowed the boundary of partially with dirichlet boundary conditions u 0 on d and partially with neumann boundary conditions i j i a i j j u 0 on n the coefficients a i j are piecewise smooth with jump discontinuities across the interface which is allowed to have singularities and cross the boundary of in particular we consider triple junctions and even multiple junctions our main result is to construct a sequence of generalized finite element spaces s n that yield h m quasi optimal rates of convergence m 1 for the galerkin approximations u n s n of the solution u more precisely we prove that u u n c dim s n m 2 f h m 1 where c depends on the data for the problem but not on f u or n and dim s n our construction is quite general and depends on a choice of a good sequence of approximation spaces s n on a certain subdomain w that is at some distance to the vertices in case the spaces s n are generalized finite element spaces then the resulting spaces s n are also generalized finite element spaces

stability and convergence of fully discrete galerkin fems for the nonlinear thermistor equations in a nonconvex polygon in this paper we establish the unconditional stability and optimal error estimates of a linearized backward euler galerkin finite element method fem for the time dependent nonlinear thermistor equations in a two dimensional nonconvex polygon due to the nonlinearity of the equations and the non smoothness of the solution in a nonconvex polygon the analysis is not straightforward while most previous efforts for problems in nonconvex polygons mainly focused on linear models our theoretical analysis is based on an error splitting proposed in 30 31 together with rigorous regularity analysis of the nonlinear thermistor equations and the corresponding iterated time discrete elliptic system in a nonconvex polygon with the proved regularity we establish the stability in l infty l infty l l and the convergence in l infty l 2 l l2 for the fully discrete finite element solution without any restriction on the time step size the approach used in this paper may also be applied to other nonlinear parabolic systems in nonconvex polygons numerical results confirm our theoretical analysis and show clearly that no time step condition is needed

boundary element free method befm and its application to two dimensional elasticity problems in this study we first discuss the moving least square approximation mls method in some cases the mls may form an ill conditioned system of equations so that the solution cannot be correctly obtained hence in this paper we propose an improved moving least square approximation imls method in the imls method the orthogonal function system with a weight function is used as the basis function the imls has higher computational efficiency and precision than the mls and will not lead to an ill conditioned system of equations combining the boundary integral equation bie method and the imls approximation method a direct meshless bie method the boundary element free method befm for two dimensional elasticity is presented compared to other meshless bie methods befm is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables and the boundary conditions can be applied easily hence it has higher computational precision for demonstration purpose selected numerical examples are given

the boundary element free method for 2d interior and exterior helmholtz problems abstract the boundary element free method befm is developed in this paper for numerical solutions of 2d interior and exterior helmholtz problems with mixed boundary conditions of dirichlet and neumann types a unified boundary integral equation is established for both interior and exterior problems by using the improved interpolating moving least squares method to form meshless shape functions mixed boundary conditions in the befm can be satisfied directly and easily detailed computational formulas are derived to compute weakly and strongly singular integrals over linear and higher order integration cells three numerical integration procedures are developed for the computation of strongly singular integrals numerical examples involving acoustic scattering and radiation problems are presented to show the accuracy and efficiency of the meshless method

enriched element free galerkin methods for crack tip fields summary the element free galerkin efg method is a meshless method for solving partial di erential equations which uses only a set of nodal points and a cad like description of the body to formulate the discrete model it has been used extensively for fracture problems and has yielded good results when adequate refinement is used near the crack tip but stresses tend to be oscillatatory near the crack tip unless substantial refinement is used an enriched efg formulation for fracture problems is proposed two methods are used 1 adding the asymptotic fields to the trial function and 2 augmenting the basis by the asymptotic fields a local mapping of the enriched fields for curved cracks is also described results show that both methods greatly reduce stress oscillations and allow the calculation of accurate stress intensity factors with far fewer degrees of freedom 1997 by john wiley sons ltd

finite element analysis for the axisymmetric laplace operator on polygonal domains abstract let l r 2 r r 2 z 2 we consider the equation l u f on a bounded polygonal domain with suitable boundary conditions derived from the three dimensional axisymmetric poisson s equation we establish the well posedness regularity and fredholm results in weighted sobolev spaces for possible singular solutions caused by the singular coefficient of the operator l as r 0 and by non smooth points on the boundary of the domain in particular our estimates show that there is no loss of regularity of the solution in these weighted sobolev spaces besides by analyzing the convergence property of the finite element solution we provide a construction of improved graded meshes such that the quasi optimal convergence rate can be recovered on piecewise linear functions for singular solutions the introduction of a new projection operator from the weighted space to the finite element subspace certain scaling arguments and a calculation of the index of the fredholm operator together with our regularity results are the ingredients of the finite element estimates

improving the rate of convergence of high order finite elements on polygons and domains with cusps let u and uv v be the solution and respectively the discrete solution of the non homogeneous dirichlet problem u f on u 0 for any m and any bounded polygonal domain we provide a construction of a new sequence of finite dimensional subspaces vn such that where f hm 1 is arbitrary and c is a constant that depends only on and not on n we do not assume u hm 1 the existence of such a sequence of subspaces was first proved in a ground breaking paper by babuska 8 our method is different it is based on the homogeneity properties of sobolev spaces with weights and the well posedness of non homogeneous dirichlet problem in suitable sobolev spaces with weights for which we provide a new proof and which is a substitute of the usual shift theorems for boundary value problems in domains with smooth boundary our results extended right away to domains whose boundaries have conical points we also indicate some of the changes necessary to deal with domains with cusps our numerical computation are in agreement with our theoretical results

domain decomposition method for the h p version finite element method domain decomposition method for the h p version of the finite element method in two and three dimensions are discussed using the framework of additive schwarz method various iterative methods are described with their condition numbers estimated further to reduce the cost for solving the sub problems on element interfaces different inexact interface solvers are proposed the effects on the overall condition number as well as their efficient implementation are analyzed

a posteriori estimation and adaptive control of the pollution error in the h version of the finite element method abstract we studied the pollution error in the h version of the finite element method and its effect on the quality of the local error indicators resp the quality of the derivatives recovered by local postprocessing in the interior of the mesh here we show that it is possible to construct a posteriori estimates of the pollution error in a patch of elements by employing the local error indicators over the entire mesh we also give an adaptive algorithm for the local control of the pollution error in a patch of elements of interest

elasto plasticity revisited numerical analysis via reproducing kernel particle method and parametric quadratic programming aiming to simplify the solution process of elasto plastic problems this paper proposes a reproducing kernel particle algorithm based on principles of parametric quadratic programming for elasto plasticity the parametric quadratic programming theory is useful and effective for the assessment of certain features of structural elasto plastic behaviour and can also be exploited for numerical iteration examples are presented to illustrate the essential aspects of the behaviour of the model proposed and the flexibility of the coupled parametric quadratic programming formulations with the reproducing kernel particle method copyright 2002 john wiley sons ltd

complex variable boundary element free method for two dimensional elastodynamic problems abstract we proposed a new direct meshless boundary integral equation technique the complex variable boundary element free method cvbefm based on the complex variable moving least squares cvmls approximation and the boundary element free method befm to study the two dimensional elastodynamic problems with the cvmls approximation the trial function of a two dimensional problem is formed with a one dimensional basis function the number of unknown coefficients in the trial function of the cvmls approximation is less than that in the trial function of the moving least squares mls approximation therefore it requires fewer nodes in the meshless method which formed from the cvmls approximation than that formed from the mls approximation with no lose of precision the laplace transform is used to formulate the boundary integral equations of the two dimensional elastodynamics and then the formulae of the cvbefm for two dimensional elastodynamic problems are derived the cvbefm is a direct numerical method in which the basic unknown quantities are the real solutions of the nodal variables moreover in the cvbefm the boundary conditions can be applied directly and easily that leads to a greater computational precision in this paper we selected a few numerical examples to illustrate the applicability of the cvbefm

exponential convergence in h 1 of hp fem for gevrey regularity with isotropic singularities for functions u in h 1 omega in a bounded polytope omega subset mathbb r d d 1 2 3 with plane sides for d 2 3 which are gevrey regular in overline omega backslash mathscr s with point singularities concentrated at a set mathscr s subset overline omega consisting of a finite number of points in overline omega we prove exponential rates of convergence of hp version continuous galerkin finite element methods on affine families of regular simplicial meshes in omega the simplicial meshes are geometrically refined towards mathscr s but are otherwise unstructured

a semigroup approach to poisson approximation the aim of this paper is twofold first to show that poisson approximation problems for independent summands can in a natural way be treated in a suitable operator semigroup framework allowing at the same time for an asymptotically precise evaluation of the leading term with respect to the total variation distance second to determine asymptotically those poisson distributions which minimize this distance for given bernoulli summands besides semigroup methods coupling techniques as well as direct computations are used

a weak galerkin mixed finite element method for second order elliptic problems abstract a new weak galerkin wg method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations this method called wg mfem is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons polyhedra the wg mfem is capable of providing very accurate numerical approximations for both the primary and flux variables allowing the use of discontinuous approximating functions on arbitrary shape of polygons polyhedra makes the method highly flexible in practical computation optimal order error estimates in both discrete h and l norms are established for the corresponding weak galerkin mixed finite element solutions

the hp adaptive fem based on continuous sobolev embeddings isotropic refinements the aim of this paper is to present a new class of smoothness testing strategies in the context of hp adaptive refinements based on continuous sobolev embeddings in addition to deriving a modified form of the 1d smoothness indicators introduced in 26 they will be extended and applied to a higher dimensional framework a few numerical experiments in the context of the hp adaptive fem for a linear elliptic pde will be performed

a note on the l_1 consistency of variable kernel estimates 1 introduction most consistent nonparametric density estimates have a built in smoothing parameter numerous schemes have been proposed see e g references found in rudemo 1982 or devroye and penrod 1984 for selecting the smoothing parameter as a function of the data only a process called automatization and for introducing locally adaptable smoothing parameters in this note we give conditions which insure that estimators of the form

a two level finite element discretization of the streamfunction formulation of the stationary quasi geostrophic equations of the ocean in this paper we proposed a two level finite element discretization of the nonlinear stationary quasi geostrophic equations which model the wind driven large scale ocean circulation optimal error estimates for the two level finite element discretization were derived numerical experiments for the two level algorithm with the argyris finite element were also carried out the numerical results verified the theoretical error estimates and showed that for the appropriate scaling between the coarse and fine mesh sizes the two level algorithm significantly decreases the computational time of the standard one level algorithm

two dimensional linear elasticity by the boundary node method abstract this paper presents a further development of the boundary node method bnm for 2 d linear elasticity in this work the boundary integral equations bie for linear elasticity have been coupled with moving least square mls interpolants this procedure exploits the mesh less attributes of the mls and the dimensionality advantages of the bie as a result the bnm requires only a nodal data structure on the bounding surface of a body a cell structure is employed only on the boundary in order to carry out numerical integration in addition the mls interpolants have been suitably truncated at corners in order to avoid some of the oscillations observed while solving potential problems by the bnm mukherjee and mukherjee 1997a numerical results presented in this paper including those for the solution of the lame and kirsch problems show good agreement with analytical solutions

the optimal convergence rate of a c1 finite element method for non smooth domains we establish optimal up to arbitrary e 0 convergence rates for a finite element formulation of a model second order elliptic boundary value problem in a weightedh 2 sobolev space with 5th degree argyris elements this formulation arises while generalizing to the case of non smooth domains an unconditionally stable scheme developed by liu et al j g liu j liu r l pego stability and convergence of efficient navier stokes solvers via a commutator estimate comm pure appl math 60 2007 pp 1443 1487 for the navier stokes equations we prove the optimality for both quasiuniform and graded mesh refinements and provide numerical results that agree with our theoretical predictions

regularity of the solutions for elliptic problems on nonsmooth domains in ℝ3 part i countably normed spaces on polyhedral domains this is the first of a series of three papers devoted to the regularity of solutions of elliptic problems on nonsmooth domains in 3 the present paper introduces various weighted spaces and countably weighted spaces in the neighbourhood of edges and vertices of polyhedral domains and it concentrates on exploring the structure of these spaces such as the embeddings of weighted sobolev spaces the relation between weighted sobolev spaces and weighted continuous function spaces and the relations between the weighted sobolev spaces and countably weighted sobolev spaces in cartesian coordinates and in the spherical and cylindrical coordinates these well defined spaces are the foundation for the comprehensive study of the regularity theory of elliptic problems with piecewise analytic data in 3 which are essential for the design of effective computation and the analysis of the h p version of the finite element method for solving elliptic problems in three dimensional nonsmooth domains arising from mechanics and engineering

some guidelines and guarantees for common random numbers common random numbers crn is a widely used technique for reducing variance in comparing stochastic systems through simulation its popularity derives from its intuitive appeal and ease of implementation however though crn has been observed to work well with a broad range of models the class of systems for which it is provably advantageous has remained rather limited this paper has two purposes we first discuss the effectiveness and optimality of crn in a general setting stressing the roles played by monotonicity and continuity properties we then present specific new classes of systems and comparisons for which crn is beneficial and even optimal our conclusions for these systems are largely consistent with simulation practice and lend further theoretical support to folklore our results differ from those of previous analyses primarily because we put conditions on the timing of events rather than the sequence of states in a discrete event simulation we formulate our results in three settings corresponding to three applications of crn distributional comparisons structural comparisons and sensitivity analysis in each case we make use of conditions that simultaneously ensure monotonicity and continuity in the timing of events these properties are established through explicit recursions for event epochs in terms of increasing continuous functions

on the sharpness of certain local estimates for projections into finite element spaces influence of a re entrant corner in a plane polygonal domain with a reentrant corner consider a homogeneous dirichlet problem for poisson s equation au f with f smooth and the corresponding galerkin finite element solutions in a family of piecewise polynomial spaces based on quasi uniform uniformly regular triangulations with the diameter of each element compara

a weak galerkin finite element method for second order elliptic problems this paper introduces a finite element method by using a weakly defined gradient operator over generalized functions the use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play an important role in numerical methods for partial differential equations this article intends to provide a general framework for managing differential operators on generalized functions as a demonstrative example the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem in which the classical gradient operator is replaced by the discrete weak gradient the resulting numerical scheme is called a weak galerkin wg finite element method it can be seen that the weak galerkin method allows the use of totally discontinuous functions in the finite element procedure for the second order elliptic problem an optimal order error estimate in both a discrete h 1 and l 2 norms are established for the corresponding weak galerkin finite element solutions a superconvergence is also observed for the weak galerkin approximation

analytic regularity of stokes flow on polygonal domains in countably weighted sobolev spaces we investigate the analytic regularity of the stokes problem in a polygonal domain r2 with straight sides and piecewise analytic data we establish a shift theorem in weighted sobolev spaces of arbitrary order with explicit control of the order dependence of the constants the shift theorem in the framework of countably weighted sobolev spaces implies in particular interior analyticity and gevrey type analytic regularity near the corners

the boundary node method for potential problems the element free galerkin efg method allows one to use a nodal data structure usually with an underlying cell structure within the domain of a body of arbitrary shape the usual efg combines moving least squares mls interpolants with a variational principle weak form and has been used to solve two dimensional 2 d boundary value problems in mechanics such as in potential theory elasticity and fracture this paper proposes a combination of mls interpolants with boundary integral equations bie in order to retain both the meshless attribute of the former and the dimensionality advantage of the latter this new method called the boundary node method bnm only requires a nodal data structure on the bounding surface of a body whose dimension is one less than that of the domain itself an underlying cell structure is again used for numerical integration in principle the bnm for 3 d problems should be extremely powerful since one would only need to put nodes points on the surface of a solid model for an object numerical results are presented in this paper for the solution of laplace s equation in 2 d dirichlet neumann and mixed problems have been solved some on bodies with piecewise straight and others with curved boundaries results from these numerical examples are extremely encouraging 1997 by john wiley sons ltd

interior energy error estimates for the weak galerkin finite element method consider the poisson equation in a polytopal domain omega subset mathbb r d d 2 3 as the model problem we study interior energy error estimates for the weak galerkin finite element approximation to elliptic boundary value problems in particular we show that the interior error in the energy norm is bounded by three components the best local approximation error the error in negative norms and the trace error on the element boundaries this implies that the interior convergence rate can be polluted by solution singularities on the domain boundary even when the solution is smooth in the interior region numerical results are reported to support the theoretical findings to the best of our knowledge this is the first local energy error analysis that applies to general meshes consisting of polytopal elements and hanging nodes

on besov regularity of solutions to nonlinear elliptic partial differential equations abstract in this paper we study the regularity of the solutions of some nonlinear elliptic equations in kondratiev spaces on certain domains of polyhedral type general embedding theorems between kondratiev spaces and besov spaces will allow to avoid drawbacks to the standard sobolev regularity theory for those nonsmooth domains this will give us the opportunity to derive optimal rates for certain nonlinear approximation schemes

coupled samples in simulation assume that we wish to generate two samples of n independent identically distributed random variables x1 xn and y1 yn where x1 and y1 have densities f and g respectively if these samples are used in a simulation and f is close to g it is sometimes desirable to have close simulation results this can be achieved by insisting that both samples agree in most of their components that is xi yi for as many i as possible under the given distributional constraints samples with this property are said to be optimally coupled in this paper we propose and study various methods of coupling two samples a sequence of samples and an infinite family of samples

a novel complex variable element free galerkin method for two dimensional large deformation problems abstract based on complex variable theory and moving least squares mls approximation the improved complex variable moving least squares icvmls approximation is discussed in this paper compared with complex variable moving least squares cvmls approximation the function in the icvmls approximation has an explicit physics meaning by using a new basis function the icvmls approximation can obtain greater precision and computational efficiency based on the icvmls approximation an improved complex variable element free galerkin icvefg method which belongs to a novel element free galerkin efg method is presented for two dimensional large deformation problems the galerkin weak form is employed to obtain the equations while the penalty method is used to apply the essential boundary conditions then the corresponding formulae of the icvefg method for two dimensional large deformation problems are obtained compared with the efg method the icvefg method has greater precision and efficiency

boundary value problems in spaces of distributions on smooth and polygonal domains we study boundary value problems of the form email protected f on w and bu g on the boundary email protected with either dirichlet or neumann boundary conditions where w is a smooth bounded domain in r n and the data f g are distributions this problem has to be first properly reformulated and for practical applications it is of crucial importance to obtain the continuity of the solution u in terms of f and g for f 0 taking advantage of the fact that u is harmonic on w we provide four formulations of this boundary value problem one using nontangential limits of harmonic functions one using green functions one using the dirichlet to neumann map and a variational one we show that these four formulations are equivalent we provide a similar analysis for f 0 and discuss the roles of f and g which turn to be somewhat interchangeable in the low regularity case the weak formulation is more convenient for numerical approximation whereas the nontangential limits definition is closer to the intuition and easier to check in concrete situations we extend the weak formulation to polygonal domains using weighted sobolev spaces we also point out some new phenomena for the concentrated loads at the vertices in the polygonal case

weighted regularization of maxwell equations in polyhedral domains we present a new method of regularizing time harmonic maxwell equations by a bf grad div term adapted to the geometry of the domain this method applies to polygonal domains in two dimensions as well as to polyhedral domains in three dimensions in the presence of reentrant corners or edges the usual regularization is known to produce wrong solutions due the non density of smooth fields in the variational space we get rid of this undesirable effect by the introduction of special weights inside the divergence integral standard finite elements can then be used for the approximation of the solution this method proves to be numerically efficient

regularity estimates for elliptic boundary value problems in besov spaces we consider the dirichlet problem for poisson s equation on a nonconvex plane polygonal domain new regularity estimates for its solution in terms of besov and sobolev norms of fractional order are proved the analysis is based on new interpolation results and multilevel representations of norms on sobolev and besov spaces the results can be extended to a large class of elliptic boundary value problems some new sharp finite element error estimates are deduced

h p spectral element methods for three dimensional elliptic problems on non smooth domains this paper is devoted to the study of non conforming h p spectral element methods for three dimensional elliptic problems on non smooth domains using parallel computers to overcome the singularities which arise in the neighbourhoods of vertices edges and vertex edges we use local systems of coordinates together with geometric meshes which become finer near corners and edges we provide estimates for the error and the computational complexity of our method and present results of numerical simulations that have been performed to validate the theoretical estimates obtained in 22

h p spectral element methods for three dimensional elliptic problems on non smooth domains part iii error estimates preconditioners computational techniques and numerical results abstract the present paper is the third of a series of papers devoted to the study of h p spectral element methods for three dimensional elliptic problems on non smooth domains using parallel computers in this paper we provide error estimates preconditioners and numerical results the spectral element functions are fully non conforming we propose preconditioners on non smooth domains which can be diagonalized using separation of variables technique optimal error estimates in terms of number of layers in the geometrical mesh and in terms of number of degrees of freedom are obtained the method is easy to implement on a parallel computer and we briefly outline computational techniques we give results of numerical simulations to confirm the theoretical estimates theoretical results have been also validated by computational experiments which are published independently in dutt et al 2014

well posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks we prove a regularity result for the anisotropic linear elasticity equation p u rm div left boldmath mathsf c cdot nabla u right f with mixed displacement and traction boundary conditions on a curved polyhedral domain omega subset mathbb r 3 in weighted sobolev spaces mathcal k m 1 a 1 omega for which the weight is given by the distance to the set of edges in particular we show that there is no loss of mathcal k m a regularity our curved polyhedral domains are allowed to have cracks we establish a well posedness result when there are no neighboring traction boundary conditions and a 0 that depends on p on the boundary conditions and on the domain our results extend to other strongly elliptic systems and higher dimensions

asymptotic exactness of an a posteriori error estimator based on the equilibrated residual method in this paper we analyze an a posteriori error estimator based on the equilibrated residual method we prove that this estimator is asymptotically exact in the energy norm for regular solutions and o h 1 alpha alpha 0 meshes numerical examples are included to illustrate the theoretical results

an improved moving least squares ritz method for two dimensional elasticity problems we propose an improved moving least squares ritz imls ritz method with its element free framework developed for studying two dimensional elasticity problems using the imls approximation for the field variables the discretized governing equations of the problem are derived via the ritz procedure in the imls an orthogonal function system with a weight function is employed as the basis for construction of its displacement field by using the element free imls ritz method solutions of the two dimensional elasticity problems are obtained the applicability of the element free imls ritz method is illustrated through three selected example problems the convergence characteristics of the method are examined by varying the number of nodes and geometric parameters of these examples the accuracy of the method is validated by comparing the computed results with the efg and exact solutions

moving least squares differential quadrature method and its application to the analysis of shear deformable plates a moving least squares differential quadrature mlsdq method is developed and employed for the analysis of moderately thick plates based on the first order shear deformation theory fsdt to carry out the analysis the governing equations in terms of the generalized displacements transverse deflection and two rotations of the plate are formulated by employing the moving least squares approximation the weighting coefficients used in the mlsdq approximation are computed through a fast computation of shape functions and their derivatives numerical examples illustrating the accuracy stability and convergence of the mlsdq method are presented effects of support size order of completeness and node irregularity on the numerical accuracy are investigated copyright 2003 john wiley sons ltd