selective refinement derefinement algorithms for sequences of nested triangulations the construction of changing sequences of irregular and nested triangulations based on the use of conforming refinement derefinement algorithms for triangulations is presented and discussed this strategy is particularly appropriate to combine adaptivity and full multigrid algorithms for dealing with moving fronts or fluid dynamics problems it is shown that the quality of all the triangulations iteratively generated depends only on the geometric characteristics of the initial grid a data structure suitable to create manage and modify series of nested triangulations as well as the main features of the deref prototype package are described and numerical examples are given

mesh refinement processes based on the generalized bisection of simplices mesh refinement processes based on the generalized bisection of simplices are discussed and characterized in the context of the general theory of rheinboldt siam j numer anal 17 1980 pp 766 778 then it is proved that such processes allow the construction of sequences of naturally smooth conforming and nested nonuniform meshes in fact it is shown that there exists a generalized mesh refinement operator that for any conforming triangulation delta and for any v subset delta produces a nested smooth conforming triangulation delta containing successors of all elements of v and such that the minimum cell size of delta is bounded from below by half the minimum cell size of delta two conforming mesh refinement algorithms that allow the selective refinement of computational triangulations are explicitly given

on generalized bisection of n simplices a generalized procedure of bisection of n simplices is introduced where the bisection point can be an almost arbitrary point at one of the longest edges it is shown that nested sequences of simplices generated by successive generalized bisection converge to a singleton and an exact bound of the convergence speed in terms of diameter reduction is given for regular simplices which mark the worst case the edge lengths of each worst and best simplex generated by successive bisection are given up to depth n for n 2 and 3 the sequence of worst case diameters is provided until it is halved

coldice a parallel vlasov poisson solver using moving adaptive simplicial tessellation abstract resolving numerically vlasov poisson equations for initially cold systems can be reduced to following the evolution of a three dimensional sheet evolving in six dimensional phase space we describe a public parallel numerical algorithm consisting in representing the phase space sheet with a conforming self adaptive simplicial tessellation of which the vertices follow the lagrangian equations of motion the algorithm is implemented both in six and four dimensional phase space refinement of the tessellation mesh is performed using the bisection method and a local representation of the phase space sheet at second order relying on additional tracers created when needed at runtime in order to preserve in the best way the hamiltonian nature of the system refinement is anisotropic and constrained by measurements of local poincare invariants resolution of poisson equation is performed using the fast fourier method on a regular rectangular grid similarly to particle in cells codes to compute the density projected onto this grid the intersection of the tessellation and the grid is calculated using the method of franklin and kankanhalli 65 66 67 generalised to linear order as preliminary tests of the code we study in four dimensional phase space the evolution of an initially small patch in a chaotic potential and the cosmological collapse of a fluctuation composed of two sinusoidal waves we also perform a warm dark matter simulation in six dimensional phase space that we use to check the parallel scaling of the code

using longest side bisection techniques for the automatic refinement of delaunay triangulations in this paper we discuss study and compare two linear algorithms for the triangulation refinement problem the known longest side triangle bisection refinement algorithm as well as a new algorithm that uses longest side bisection techniques for refining delaunay triangulations we show that the automatic point insertion criterion taken from the fractal property of optimal linear longest side bisection algorithms assures the construction of good quality delaunay triangulations in linear time numerical evidence showing that the practical behaviour of the new algorithm is in complete agreement with the theory is included 1997 by john wiley sons ltd

average adjacencies for tetrahedral skeleton regular partitions for any conforming mesh the application of a skeleton regular partition over each element in the mesh produces a conforming mesh such that all the topological elements of the same dimension are subdivided into the same number of child elements every skeleton regular partition has associated special constitutive recurrence equations in this paper the average adjacencies associated with the skeleton regular partitions in 3d are studied in three dimensions different values for the asymptotic number of average adjacencies are obtained depending on the considered partition in contrast with the two dimensional case j comput appl math 140 2002 673 in addition a priori formulae for the average asymptotic adjacency relations for any skeleton regular partition in 3d are provided

maximum norm error estimators for three dimensional elliptic problems in this paper we define an a posteriori error estimator for finite element approximations of 3 d elliptic problems we prove that the estimator is equivalent up to logarithmic factors of the meshsize to the maximum norm of the error the results are valid for an arbitrary polyhedral domain and rather general meshes we also obtain analogous results for the nonconforming method of crouzeix raviart finally we present some numerical results comparing adaptive procedures based on controlling the error in different norms

error estimators for nonconforming finite element approximations of the stokes problem in this paper we define and analyze a posteriori error estimators for nonconforming approximations of the stokes equations we prove that these estimators are equivalent to an appropriate norm of the error for the case of piecewise linear elements we define two estimators both of them are easy to compute but the second is simpler because it can be computed using only the right hand side and the approximate velocity we show how the first estimator can be generalized to higher order elements finally we present several numerical examples in which one of our estimators is used for adaptive refinement

an a posteriori error estimator for anisotropic refinement besides an algorithm for local refinement an a posteriori error estimator is the basic tool of every adaptive finite element method using information generated by such an error estimator the refinement of the grid is controlled for 2nd order elliptic problems we present an error estimator for anisotropically refined grids like n d cuboidal and 3 d prismatic grids that gives correct information about the size of the error additionally it generates information about the direction into which some element has to be refined to reduce the error in a proper way numerical examples are presented for 2 d rectangular and 3 d prismatic grids

estimating the null distribution to adjust observed confidence levels for genome scale screening summary in a novel approach to the multiple testing problem efron 2004 journal of the american statistical association 99 96 104 2007a journal of the american statistical association 102 93 103 2007b annals of statistics 35 1351 1377 formulated estimators of the distribution of test statistics or nominal p values under a null distribution suitable for modeling the data of thousands of unaffected genes nonassociated single nucleotide polymorphisms or other biological features estimators of the null distribution can improve not only the empirical bayes procedure for which it was originally intended but also many other multiple comparison procedures such estimators in some cases improve the proposed multiple comparison procedure mcp based on a recent non bayesian framework of minimizing expected loss with respect to a confidence posterior a probability distribution of confidence levels the flexibility of that mcp is illustrated with a nonadditive loss function designed for genomic screening rather than for validation the merit of estimating the null distribution is examined from the vantage point of the confidence posterior mcp cpmcp in a generic simulation study of genome scale multiple testing conditioning the observed confidence level on the estimated null distribution as an approximate ancillary statistic markedly improved conditional inference specifically simulating gene expression data however indicates that estimation of the null distribution tends to exacerbate the conservative bias that results from modeling heavy tailed data distributions with the normal family to enable researchers to determine whether to rely on a particular estimated null distribution for inference or decision making an information theoretic score is provided as the sum of the degree of ancillarity and the degree of inferential relevance the score reflects the balance conditioning would strike between the two conflicting terms the cpmcp and other methods introduced are applied to gene expression microarray data

mesh refinement strategies for solving singularly perturbed reaction diffusion problems abstract we consider the numerical approximation of a singularly perturbed reaction diffusion problem over a square two different approaches are compared namely adaptive isotropic mesh refinement and anisotropic mesh refinement thus we compare the h refinement and the shishkin mesh approaches numerically with pltmg software 1 it is shown how isotropic elements lead to over refinement and how anisotropic mesh refinement is much more efficient in thin boundary layers

the finite element approximation of the nonlinear poisson boltzmann equation a widely used electrostatics model in the biomolecular modeling community the nonlinear poisson boltzmann equation along with its finite element approximation are analyzed in this paper a regularized poisson boltzmann equation is introduced as an auxiliary problem making it possible to study the original nonlinear equation with delta distribution sources a priori error estimates for the finite element approximation are obtained for the regularized poisson boltzmann equation based on certain quasi uniform grids in two and three dimensions adaptive finite element approximation through local refinement driven by an a posteriori error estimate is shown to converge the poisson boltzmann equation does not appear to have been previously studied in detail theoretically and it is hoped that this paper will help provide molecular modelers with a better foundation for their analytical and computational work with the poisson boltzmann equation note that this article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear poisson boltzmann equation with delta distribution sources and it also introduces the first provably convergent adaptive method for the equation this last result is currently one of only a handful of existing convergence results of this type for nonlinear problems

on the adjacencies of triangular meshes based on skeleton regular partitions for any 2d triangulation the 1 skeleton mesh of is the wireframe mesh defined by the edges of while that for any 3d triangulation the 1 skeleton and the 2 skeleton meshes respectively correspond to the wireframe mesh formed by the edges of and the surface mesh defined by the triangular faces of a skeleton regular partition of a triangle or a tetrahedra is a partition that globally applied over each element of a conforming mesh where the intersection of adjacent elements is a vertex or a common face or a common edge produce both a refined conforming mesh and refined and conforming skeleton meshes such a partition divides all the edges and all the faces of an individual element in the same number of edges faces we prove that sequences of meshes constructed by applying a skeleton regular partition over each element of the preceding mesh have an associated set of difference equations which relate the number of elements faces edges and vertices of the nth and n 1 th meshes by using these constitutive difference equations we prove that asymptotically the average number of adjacencies over these meshes number of triangles by node and number of tetrahedra by vertex is constant when n goes to infinity we relate these results with the non degeneracy properties of longest edge based partitions in 2d and include empirical results which support the conjecture that analogous results hold in 3d

farkas type conditions of general interval linear systems for ae solvability abstract we present farkas type necessary and sufficient conditions for ae solvability of two specific general interval linear systems we generalize the farkas type theorems of interval linear systems to a unified framework moreover each particular previously published result of farkas type condition of interval linear systems is a special case of our general approach

a posterior error estimator and lower bound of a nonconforming finite element method in this paper we present an a posteriori error estimator and the lower bound for a nonconforming finite element approximation i e the extended crouzeix raviart element of the laplace eigenvalue problem under the guideline of the analysis to the laplace source problem we first give out an error indicator and prove it as the global upper and local lower bounds of the approximation error we also give the lower bound analysis for this type of nonconforming element on the adaptive meshes some numerical experiments are presented to verify our theoretical results

an adaptive finite element strategy for the three dimensional time dependent navier stokes equations abstract an adaptive strategy for three dimensional time dependent problems in the context of the fem is presented the basic tools are a mechanism for local refinement and coarsening of simplicial meshes and an unexpensive error estimator the algorithm for local grid modification is based on bisecting tetrahedra the method is applied to the navier stokes equations

adaptive refinement for arbitrary finite element spaces with hierarchical bases abstract an adaptive refinement algorithm is presented and interpreted as the selective enrichment of a finite element space through the hierarchical basis each elemental division corresponds exactly to the inclusion of a small number of new basis functions while existing basis functions remain unchanged which bases to add i e which division to perform are determined so that those that make the largest contribution to the function are included first the space of c p th degree piecewise polynomials over triangles are used for illustration but the techniques apply to any family of function spaces that can be represented with a hierarchical basis numerical examples are presented to show that the technique can regain the optimal smooth order of convergence for the solution of partial differential equations with nonsmooth solutions

local error estimation and adaptive remeshing scheme for least squares mixed finite elements following the theme of our previous work on least squares finite elements 10 28 we describe an adaptive remeshing scheme using local residuals as the error indicator this choice of indicator is natural and exact at the element level in the norm associated with the corresponding least squares statement the remeshing strategy applied here involves mesh enrichment by point insertion in a delaunay scheme several refined grids and error plots are included for a representative model elliptic boundary value problem

a new adaptive mesh refinement strategy for numerically solving evolutionary pde s a graph based implementation of quadtree meshes for dealing with adaptive mesh refinement amr in the numerical solution of evolutionary partial differential equations is discussed using finite volume methods the technique displays a plug in feature that allows replacement of a group of cells in any region of interest for another one with arbitrary refinement and with only local changes occurring in the data structure the data structure is also specially designed to minimize the number of operations needed in the amr implementation of the new scheme allows flexibility in the levels of refinement of adjacent regions moreover storage requirements and computational cost compare competitively with mesh refinement schemes based on hierarchical trees low storage is achieved for only the children nodes are stored when a refinement takes place these nodes become part of a graph structure thus motivating the denomination autonomous leaves graph alg for the new scheme neighbors can then be reached without accessing their parent nodes additionally linear system solvers based on the minimization of functionals can be easily employed alg was not conceived with any particular problem or geometry in mind and can thus be applied to the study of several phenomena some test problems are used to illustrate the effectiveness of the technique

nonlinear equations and optimization abstract interval analysis has led to improved numerical algorithms both for the solution of nonlinear equations and for nonlinear optimization in fact interval analysis has made possible algorithms for solving a problem often predicted to be numerically insoluble the nonlinear global optimization problem in this paper we survey interval methods for solving nonlinear equations and optimization

optimal multilevel methods for graded bisection grids we design and analyze optimal additive and multiplicative multilevel methods for solving h 1 problems on graded grids obtained by bisection we deal with economical local smoothers after a global smoothing in the finest mesh local smoothing for each added node during the refinement needs to be performed only for three vertices the new vertex and its two parent vertices we show that our methods lead to optimal complexity for any dimensions and polynomial degree the theory hinges on a new decomposition of bisection grids in any dimension which is of independent interest and yields a corresponding decomposition of spaces we use the latter to bridge the gap between graded and quasi uniform grids for which the multilevel theory is well established

the multi level hp method for three dimensional problems dynamically changing high order mesh refinement with arbitrary hanging nodes abstract one main challenge of the h p version of the finite element method is the high implementational complexity of the method resulting from the added need of handling hanging nodes appropriately the multi level h p formulation recently introduced for two dimensional applications aims at alleviating these difficulties without compromising the approximation quality this is achieved by changing from the conventional refine by replacement approach to a refine by superposition idea the current work shows that the multi level h p approach can be extended naturally to three dimensional refinement without increasing the complexity of the rule set ensuring linear independence and compatibility of the shape functions in this way a three dimensional h p refinement scheme is formulated in which hanging nodes are avoided by definition this ease of complexity allows for a highly flexible discretization kernel featuring arbitrary irregular meshes and a continuous refinement and coarsening throughout the simulation runtime different numerical examples demonstrate that even in the presence of singularities this novel refinement scheme yields exponential convergence with respect to both the number of unknowns and the computational time it is further shown that the refinement scheme is able to capture complex solution features that demand for three dimensional refinement patterns the dynamic discretization properties of the approach are demonstrated by continuously refining and coarsening the mesh during the simulation runtime to keep the refinement zone local to a moving singularity finally it is shown that the high approximation power of the multi level h p scheme also carries over to curved geometries common in engineering practice without a significant detrimental effect on the conditioning of the stiffness matrix

optimal anisotropic meshes for minimizing interpolation errors in l p norm in this paper we present a new optimal interpolation error estimate in l p norm 1 p for finite element simplicial meshes in any spatial dimension a sufficient condition for a mesh to be nearly optimal is that it is quasi uniform under a new metric defined by a modified hessian matrix of the function to be interpolated we also give new functionals for the global moving mesh method and obtain optimal monitor functions from the viewpoint of minimizing interpolation error in the l p norm some numerical examples are also given to support the theoretical estimates

algorithms for refining triangular grids suitable for adaptive and multigrid techniques two general algorithms for refining triangular computational meshes based on the bisection of triangles by the longest side are presented and discussed the algorithms can be applied globally or locally for selective refinement of any conforming triangulation and always generate a new conforming triangulation after a finite number of interactions even when locally used the algorithms also ensure that all angles in subsequent refined triangulations are greater than or equal to half the smallest angle in the original triangulation the shape regularity of all triangles is maintained and the transition between small and large triangles is smooth in a natural way proofs of the above properties are presented the second algorithm is a simpler improved version of the first which retains most of the properties of the latter the algorithms can be used either for constructing irregular computational meshes or for locally refining any given triangulation in this sense they can be adequately combined with adaptive and or multigrid techniques for solving finite element systems examples of the application of the algorithms are given and two possible generalizations are pointed out

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

a grid generator based on 4 triangles conforming mesh refinement algorithms a procedure to generate 2d meshes of triangles in a quite general fashion allowing for local and selective mesh refinement is presented and discussed the grid generator is based on the iterative application of 4 triangles conforming mesh refinement algorithms for triangulations which are also introduced in this paper these algorithms are modified versions of those proposed in int j numer methods eng 20 745 756 1984 and they can be used for global refinement of a grid as well as for local refinement the grid generator works in the following way given any initial coarse triangulation which properly defines the geometry of the problem a set of user defined refinement subregions ri i 1 2 n and an associated set of tolerance parameters hi an irregular and conforming final triangulation is automatically constructed in such a way that the diameter of all the triangles contained in ri is smaller than hi i 1 2 n moreover all angles in the final triangulation are greater than or equal to half the smallest angle in the initial coarse one the refinement is propagated only to assure the conformity and smoothness of the grid and consequently the number of involved nodes will be minimized the refinement of the final mesh will be determined by the subregions ri and the parameters ni and will be essentially independent of the initial coarse grid the 4 triangles conforming mesh refinement algorithms are presented and their properties discussed the implementation of these techniques is discussed and examples of the application of the grid generator are given

on the shape of tetrahedra from bisection we present a procedure for bisecting a tetrahedron t successively into an infinite sequence of tetrahedral meshes j 0 91 g2 which has the following properties 1 each mesh sfn is conforming 2 there are a finite number of classes of similar tetrahedra in all the 3n n 0 3 for any tetrahedron t7 in s7n jj t cl1 t where ti is a tetrahedron shape measure and cl is a constant 4 3 j c2 1 2 ni3j t where d t denotes the diameter of tetrahedron t1 and c2 is a constant estimates of cl and c2 are provided properties 2 and 3 extend similar results of stynes and adler and of rosenberg and stenger respectively for the 2 d case the diameter bound in property 4 is better than one given by kearfott

weak galerkin based a posteriori error estimates for second order elliptic interface problems on polygonal meshes abstract in this paper we present a posteriori error estimate of weak galerkin wg finite element methods based on the second order elliptic interface problems this estimate can be applied to polygonal meshes or meshes with hanging nodes the reliability and efficiency of the designed error estimator has been proved in this work extensive numerical tests are performed to validate our algorithm these results demonstrate the effectiveness of the adaptive mesh refinement guided by the proposed error estimator