constrained centroidal voronoi tessellations for surfaces centroidal voronoi tessellations are useful for subdividing a region in euclidean space into voronoi regions whose generators are also the centers of mass with respect to a prescribed density function of the regions their extensions to general spaces and sets are also available for example tessellations of surfaces in a euclidean space may be considered in this paper a precise definition of such constrained centroidal voronoi tessellations ccvts is given and a number of their properties are derived including their characterization as minimizers of an energy deterministic and probabilistic algorithms for the construction of ccvts are presented and some analytical results for one of the algorithms are given computational examples are provided which serve to illustrate the high quality of ccvt point sets finally ccvt point sets are applied to polynomial interpolation and numerical integration on the sphere

small area estimation with uncertain random effects random effects models play an important role in model based small area estimation random effects account for any lack of fit of a regression model for the population means of small areas on a set of explanatory variables in a recent article datta hall and mandal showed that if the random effects can be dispensed with via a suitable test then the model parameters and the small area means may be estimated with substantially higher accuracy the work of datta hall and mandal is most useful when the number of small areas m is moderately large for large m the null hypothesis of no random effects will likely be rejected rejection of the null hypothesis is usually caused by a few large residuals signifying a departure of the direct estimator from the synthetic regression estimator as a flexible alternative to the fay herriot random effects model and the approach in datta hall and mandal in this article we consider a mixture model for random effects it is reasonably expected that small areas wit

adaptive finite element methods with convergence rates adaptive finite element methods for numerically solving elliptic equations are used often in practice only recently 12 17 have these methods been shown to converge however this convergence analysis says nothing about the rates of convergence of these methods and therefore does in principle not guarantee yet any numerical advantages of adaptive strategies versus non adaptive strategies the present paper modifies the adaptive method of morin nochetto and siebert 17 for solving the laplace equation with piecewise linear elements on domains in 2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm which is equivalent to the h1 norm namely it is shown that whenever s 0 and the solution u is such that for each n 1 it can be approximated to accuracy o n s in the energy norm by a continuous piecewise linear function on a triangulation with n cells using complete knowledge of u then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process moreover the number of arithmetic computations in the proposed method is also of order o n for each n 1 the construction and analysis of this adaptive method relies on the theory of nonlinear approximation

a posteriori error analysis of finite element solutions for one dimensional problems the paper develops a theory of a posteriors error estimates under the l p energy norm for 2 leqq p leqq infty the theory is based on a general concept of error indicators and error estimators several specific examples of these quantities are introduced and analyzed in detail the results provide a variety of easily computable error estimates under all these important norms for finite element solutions of one dimensional 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

anisotropic centroidal voronoi tessellations and their applications in this paper we introduce a novel definition of the anisotropic centroidal voronoi tessellation acvt corresponding to a given riemann metric tensor a directional distance function is used in the definition to simplify the computation we provide algorithms to approximate the acvt using the lloyd iteration and the construction of anisotropic delaunay triangulation under the given riemannian metric the acvt is applied to the optimization of two dimensional anisotropic delaunay triangulation to the generation of surface cvt and high quality triangular mesh on general surfaces various numerical examples demonstrating the effectiveness of the proposed method are presented

adaptive finite element methods for elliptic equations over hierarchical t meshes isogeometric analysis using nurbs non uniform rational b splines as basis functions gives accurate representation of the geometry and the solution but it is not well suited for local refinement in this paper we use the polynomial splines over hierarchical t meshes pht splines to construct basis functions which not only share the nice smoothness properties as the b splines but also allow us to effectively refine meshes locally we develop a residual based a posteriori error estimate for the finite element discretization of elliptic equations using pht splines basis functions and study their approximation properties in addition we conduct numerical experiments to verify the theory and to demonstrate the effectiveness of the error estimate and the high order approximations provided by the numerical solution

multigrid methods for the computation of singular solutions and stress intensity factors iii interface singularities it is shown in this paper that two dimensional interface problems with large jumps in the coefficients can be solved effectively by the piecewise linear finite element method on quasi uniform grids this is achieved by combining the full multigrid methodology the singular function representation of the solution and the extraction formulas for stress intensity factors the robustness of the algorithm is demonstrated by numerical experiments

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

voronoi based finite volume methods optimal voronoi meshes and pdes on the sphere we first develop and analyze a finite volume scheme for the discretization of partial differential equations pdes on the sphere the scheme uses voronoi tessellations of the sphere for a model convection diffusion problem the finite volume scheme is shown to produce first order accurate approximations with respect to a mesh dependent discrete firstderivative norm then we introduce the notion of constrained centroidal voronoi tessellations ccvts of the sphere these are special voronoi tessellation of the sphere for which the generators of the voronoi cells are also the constrained centers of mass with respect to a prescribed density function of the cells after discussing an algorithm for determining ccvt meshes on the sphere we discuss and illustrate several desirable properties possessed by these meshes in particular it is shown that ccvt meshes define very high quality uniform and non uniform meshes on the sphere finally we discuss through some computational experiments the performance of the ccvt meshes used in conjunction with the finite volume scheme for the solution of simple model pdes on the sphere the experiments show for example that the ccvt based finite volume approximations are second order accurate if errors are measured in

on the weight enumerators of duadic and quadratic residue codes in this correspondence we compute the weight enumerators of various quadratic residue codes over f sub 2 and f sub 3 together with certain codes of related families like the duadic and the quadratic double circulant codes we use a parallel algorithm to find the number of codewords of a given not too high weight from which we deduce by usual classical methods for self dual and formally self dual codes over f sub 2 and f sub 3 their associated previously unknown weight enumerators we compute weight enumerators for lengths as high as 152 for binary codes and 96 for ternary codes

mesh smoothing using a posteriori error estimates we develop a simple mesh smoothing algorithm for adaptively improving finite element triangulations the algorithm makes use of a posteriori error estimates which are now widely used in finite element calculations in this paper we derive the method present some numerical illustrations and give a brief analysis of the issue of uniqueness

adaptive finite element methods for elliptic pdes based on conforming centroidal voronoi delaunay triangulations a new triangular mesh adaptivity algorithm for elliptic pdes that combines a posteriori error estimation with centroidal voronoi delaunay tessellations of domains in two dimensions is proposed and tested the ability of the first ingredient to detect local regions of large error and the ability of the second ingredient to generate superior unstructured grids result in a mesh adaptivity algorithm that has several very desirable features including the following errors are very well equidistributed over the triangles at all levels of refinement the triangles remain very well shaped even if the grid size at any particular refinement level when viewed globally varies by several orders of magnitude and the convergence rates achieved are the best obtainable using piecewise linear finite elements this methodology can be easily extended to higher order finite element approximations or mixed finite element formulations although only the linear approximation is considered in this paper

on mesh geometry and stiffness matrix conditioning for general finite element spaces the performance of finite element computation depends strongly on the quality of the geometric mesh and the efficiency of the numerical solution of the linear systems resulting from the discretization of partial differential equation pde models it is common knowledge that mesh geometry affects not only the approximation error of the finite element solution but also the spectral properties of the corresponding stiffness matrix in this paper for typical second order elliptic problems some refined relationships between the spectral condition number of the stiffness matrix and the mesh geometry are established for general finite element spaces defined on simplicial meshes the derivation of such relations for general high order elements is based on a new trace formula for the element stiffness matrix it is shown that a few universal geometric quantities have the same dominant effect on the stiffness matrix conditioning for different finite element spaces these results provide guidance to the studies of both linear algebraic solvers and the unstructured geometric meshing

grid generation and optimization based on centroidal voronoi tessellations centroidal voronoi tessellations cvts are voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding voronoi regions such tessellations are of use in very diverse applications including data compression clustering analysis cell biology territorial behavior of animals and optimal allocation of resources in this paper we explore the use of cvts in grid generation in connection with finite element approximations of partial differential equations we being by describing these tessellations and methods for their determination we then discuss their application to mesh generation and finish with some examples of their use for the solution of partial differential equations

a simple mesh generator in matlab creating a mesh is the first step in a wide range of applications including scientific computing and computer graphics an unstructured simplex mesh requires a choice of meshpoints vertex nodes

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

local problems on stars a posteriori error estimators convergence and performance a new computable a posteriori error estimator is introduced which relies on the solution of small discrete problems on stars it exhibits built in flux equilibration and is equivalent to the energy error up to data oscillation without any saturation assumption a simple adaptive strategy is designed which simultaneously reduces error and data oscillation and is shown to converge without mesh pre adaptation nor explicit knowledge of constants numerical experiments reveal a competitive performance show extremely good effectivity indices and yield quasi optimal meshes

tetrahedral mesh generation and optimization based on centroidal voronoi tessellations the centroidal voronoi tessellation based delaunay triangulation cvdt provides an optimal distribution of generating points with respect to a given density function and accordingly generates a high quality mesh in this paper we discuss algorithms for the construction of the constrained cvdt from an initial delaunay tetrahedral mesh of a three dimensional domain by establishing an appropriate relationship between the density function and the specified sizing field and applying the lloyd s iteration the constrained cvdt mesh is obtained as a natural global optimization of the initial mesh simple local operations such as edges faces flippings are also used to further improve the cvdt mesh several complex meshing examples and their element quality statistics are presented to demonstrate the effectiveness and efficiency of the proposed mesh generation and optimization method copyright 2003 john wiley sons ltd

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

adaptive anisotropic meshing for steady convection dominated problems obtaining accurate solutions for convection diffusion equations is challenging due to the presence of layers when convection dominates the diffusion to solve this problem we design an adaptive meshing algorithm which optimizes the alignment of anisotropic meshes with the numerical solution three main ingredients are used first the streamline upwind petrov galerkin method is used to produce a stabilized solution second an adapted metric tensor is computed from the approximate solution third optimized anisotropic meshes are generated from the computed metric tensor by an anisotropic centroidal voronoi tessellation algorithm our algorithm is tested on a variety of two dimensional examples and the results shows that the algorithm is robust in detecting layers and efficient in avoiding non physical oscillations in the numerical approximation

a posteriori error estimates for finite volume approximations of elliptic equations on general surfaces abstract in this paper we present a residual based a posteriori error estimate for the finite volume discretization of steady convection diffusion reaction equations defined on surfaces in r 3 which are often implicitly represented as level sets of smooth functions reliability and efficiency of the proposed a posteriori error estimator are rigorously proved numerical experiments are also conducted to verify the theoretical results and demonstrate the robustness of the error estimator

all first order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable all first order averaging or gradient recovery operators for lowest order finite element methods are shown to allow for an efficient a posteriori error estimation in an isotropic elliptic model problem in a bounded lipschitz domain in r d given a piecewise constant discrete flux p h e p h that is the gradient of a discrete displacement as an approximation to the unknown exact flux p that is the gradient of the exact displacement recent results verify efficiency and reliability of m min p h q h l 2 q h qh in the sense that m is a lower and upper bound of the flux error p p h l 2 up to multiplicative constants and higher order terms the averaging space q h consists of piecewise polynomial and globally continuous finite element functions in d components with carefully designed boundary conditions the minimal value m is frequently replaced by some averaging operator a p h q h applied within a simple post processing to p h the result q h ap h e q h provides a reliable error bound with m a p h ap h l 2 this paper establishes a c eff m and so equivalence of m and a this implies efficiency of a for a large class of patchwise averaging techniques which includes the zz gradient recovery technique the bound c eff 3 88 established for tetrahedral p 1 finite elements appears striking in that the shape of the elements does not enter the equivalence a m is robust with respect to anisotropic meshes the main arguments in the proof are ascoli s lemma a strengthened cauchy inequality and elementary calculations with mass matrices

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments