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

non parametric test of ordered alternatives in incomplete blocks often in medical studies study subjects become a natural block of observations repeated over a time period some subjects miss observations thus becoming incomplete blocks of observations we are interested in testing an ordered alternative or time trend and propose a non parametric method to detect a trend in incomplete blocks our approach is to estimate the trend by the linear regression method within each block and apply the one sample wilcoxon test to the estimated linear trends the one sample wilcoxon test will be sensitive to the trend if it exists the proposed test statistic is asymptotically normal and consistent we can also estimate the overall magnitude of the linear trend and its confidence interval by a proper non parametric method by monte carlo studies we compare the performance of the proposed test against extended page and jonckheere tests published in 2000 by john wiley sons ltd

longest edge n section algorithms in this paper we survey all known including own recent results properties of the longest edge n section algorithms these algorithms in classical and recently designed conforming form are nowadays used in many applications including finite element simulations computer graphics etc as a reliable tool for controllable mesh generation in addition we present a list of open problems arising in and around this topic

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

teichmuller s problem in space abstract quasiconformal homeomorphisms of the whole space r n onto itself normalized at one or two points are studied in particular the stability theory the case when the maximal dilatation tends to 1 is in the focus our main result provides a spatial analogue of a classical result due to teichmuller unlike teichmuller s result our bounds are explicit explicit bounds are based on two sharp well known distortion results the quasiconformal schwarz lemma and the bound for linear dilatation moreover bernoulli type inequalities and asymptotically sharp bounds for special functions involving complete elliptic integrals are applied to simplify the computations finally we discuss the behavior of the quasihyperbolic metric under quasiconformal maps and prove a sharp result for quasiconformal maps of r n 0 onto itself

nonparametric methods for detecting treatment effects in repeated measures designs abstract sufficient conditions are given that guarantee the limiting distribution of a test proposed by agresti and pendergast 1986 to detect treatment effects in repeated measures designs the test which is appropriate for either the original or aligned data is related to one proposed by koch 1969 and to the rank transformation statistic using pitman asymptotic relative efficiency comparisons situations are presented where these tests are more efficient than their standard parametric and nonparametric competitors the problem of detecting ordered alternatives in repeated measures designs is also considered a rank transformation analog of page s 1963 l is shown to be generally more efficient in the pitman sense than the standard competitors when the number of treatments is not too large

blockwise adaptivity for time dependent problems based on coarse scale adjoint solutions we describe and test an adaptive algorithm for evolution problems that employs a sequence of blocks consisting of fixed though nonuniform space meshes this approach offers the advantages of adaptive mesh refinement but with reduced overhead costs associated with load balancing remeshing matrix reassembly and the solution of adjoint problems used to estimate discretization error and the effects of mesh changes a major issue with a block adaptive approach is determining block discretizations from coarse scale solution information that achieve the desired accuracy we describe several strategies for achieving this goal using adjoint based a posteriori error estimates and we demonstrate the behavior of the proposed algorithms as well as several technical issues in a set of examples

locally adapted tetrahedral meshes using bisection we present an algorithm for the construction of locally adapted conformal tetrahedral meshes the algorithm is based on bisection of tetrahedra a new data structure is introduced which simplifies both the selection of the refinement edge of a tetrahedron and the recursive refinement to conformity of a mesh once some tetrahedra have been bisected we prove that repeated application of the algorithm leads to only finitely many tetrahedral shapes up to similarity and we bound the amount of additional refinement that is needed to achieve conformity numerical examples of the effectiveness of the algorithm are presented

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

two sided estimation of diameters reduction rate for the longest edge n section of triangles with n 4 in this work we study the diameters reduction rate for the iterative application of the longest edge le n section of triangles for n p 4 the maximum diameter d n k of all triangles generated at the kth iteration of the le n section is closely connected with the properties of the triangular mesh generated by this refinement scheme the upper and the lower bounds for d 2 k were proved by kearfott in 9 and for d

self adaptive stress analysis based on stress convergence a new algorithm for self adaptive stress analysis is presented the algorithm is based on h convergence and uses the measured stress convergence rate as an error indicator a theoretical relation between stress convergence rate and pointwise error of stresses is derived for both the 1 d and 2 d case some examples are presented to illustrate the behaviour of the algorithm

a priori and a posterior error estimate of new weak galerkin finite element methods for second order elliptic interface problems on polygonal meshes abstract in this paper we present a posteriori error estimate of a new weak galerkin wg finite element method for the second order elliptic interface problems instead of introducing the local weak derivatives on each element the bilinear form is computed through the inner product of traditional defined derivatives the proposed numerical scheme is symmetric and positive definite and can be applied to polygonal meshes in the a posteriori error estimate we construct a error indicator that can be applied to polygonal meshes or meshes with hanging nodes the reliability and efficiency of the designed error estimator have 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

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

quasiconformal planes with bi lipschitz pieces and extensions of almost affine maps abstract a quasiplane f v is the image of an n dimensional euclidean subspace v of r n 1 n n 1 under a quasiconformal map f r n r n we give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi lipschitz n manifold and for a quasiplane to have big pieces of bi lipschitz images of r n one main novelty of these results is that we analyze quasiplanes in arbitrary codimension n n to establish the big pieces criterion we prove new extension theorems for almost affine maps which are of independent interest this work is related to investigations by tukia and vaisala on extensions of quasisymmetric maps with small distortion

a distribution free rank test for ordered alternatives in randomized complete block designs abstract a distribution free rank test for detecting ordered alternatives in randomized complete block designs and repeated measures designs is proposed for four or fewer treatments the null hypothesis distribution of the statistic is well known and readily available to the user the article establishes the asymptotic normality of an appropriately standardized form of the statistic and the consistency of the test asymptotic relative efficiency comparisons are made with jonckheere s test page s l hollander s test and a normal theory competitor the authors illustrate the good intermediate sample size characteristics of the test via a monte carlo study

a new adaptive local mesh refinement algorithm and its application on fourth order thin film flow problem a new adaptive local mesh refinement method is presented for thin film flow problems containing moving contact lines based on adaptation on an optimal interpolation error estimate in the lp norm 1

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

automatic mesh refinement in limit analysis a strategy for automatic mesh refinement in limit analysis is combined with a recently developed computational method in the absence of estimates of the local error the strategy can be based on the deformations and on slack in the yield condition the approach is tested on standard problems in plane strain including the classical punch problem very accurate results are obtained with the use of moderate computational power

local bisection refinement for n simplicial grids generated by reflection a simple local bisection refinement algorithm for the adaptive refinement of n simplicial grids is presented the algorithm requires that the vertices of each simplex be ordered in a special way re

adaptive mesh refinements for thin shells whose middle surface is not exactly known a strategy concerning mesh refinements for thin shells computation is presented the geometry of the shell is given only by the reduced information consisting in nodes and normals on its middle surface corresponding to a coarse mesh the new point is that the mesh refinements are defined from several criteria including the transverse shear forces which do not appear in the mechanical energy of the applied shell formulation another important point is to be able to construct the unknown middle surface at each step of the refinement for this an interpolation method by edges coupled with a triangle bisection algorithm is applied this strategy is illustrated on various geometries and mechanical problems

hausdorff measure of quasicircles abstract s smirnov 2010 10 proved recently that the hausdorff dimension of any k quasicircle is at most 1 k 2 where k k 1 k 1 in this paper we show that if is such a quasicircle then h 1 k 2 b x r c k r 1 k 2 for all x c r 0 where h s stands for the s hausdorff measure on a related note we derive a sharp weak integrability of the derivative of the riemann map of a quasidisk

on global and local mesh refinements by a generalized conforming bisection algorithm we examine a generalized conforming bisection gcb algorithm which allows both global and local nested refinements of the triangulations without generating hanging nodes it is based on the notion of a mesh density function which prescribes where and how much to refine the mesh some regularity properties of generated sequences of refined triangulations are proved several numerical tests demonstrate the efficiency of the proposed bisection algorithm it is also shown how to modify the gcb algorithm in order to generate anisotropic meshes with high aspect ratios

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

stability of the 8 tetrahedra shortest interior edge partitioning method we consider a tetrahedron partitioning method known in the literature as the 8 tetrahedra shortest interior edge partition for this method which is a variant of freudenthal s algorithm in three space dimensions we prove that the infinite series of refined meshes for any given initial mesh is stable in the sense that the degree of degeneracy of the cells remains bounded we give an explicit estimate in terms of a standard shape quality measure introduced by liu and joe furthermore we show that our estimate is sharp the estimate also holds for freudenthal s algorithm in three space dimensions provided that it is initialized appropriately numerical experiments confirm our result as well as its sharpness

complexity of the bisection method the bisection method is the consecutive bisection of a triangle by the median of the longest side in this paper we prove a subexponential asymptotic upper bound for the number of similarity classes of triangles generated on a mesh obtained by iterative bisection which previously was known only to be finite the relevant parameter is where is the biggest and is the smallest angle of the triangle we get this result by introducing a taxonomy of triangles that precisely captures the behaviour of the bisection method we also prove that the number of directions on the plane given by the sides of the triangles generated is finite additionally we give purely geometrical and intuitive proofs of classical results for the bisection method

mesh adaptations for linear 2d finite element discretizations in structural mechanics especially in thin shell analysis abstract in this paper we describe an automatic refinement of finite element meshes in structural analysis including shell problems at first the adaptation to curved boundaries and surfaces is discussed central point is a topological description which is independent of the discretization the second part delas with local refinements adapted to the solution by error estimation three a posteriori indicators are compared by examples

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

bounded flatness in q triangulated regular n simplexes bounded flatness of simplexes is crucial for simplicial algorithms to provide solutions of satisfactory accuracy by using combinatorial arguments we show that iterative q refinement with arbitrary mesh of euclidean simplexes a crucial step in the implementation of simplicial algorithms yields subsimplexes of bounded flatness the flatness is bounded by n 1 email protected n 2 f n r where f n r is the flatness of the regular unit n simplex

a new proof of the degeneracy property of the longest edge n section refinement scheme for triangular meshes in this note by using complex variable functions we present a new simpler proof of the degeneracy property of the longest edge n section of triangles for n p 4 this means that the longest edge n section of triangles for n p 4 produces a sequence of triangles with minimum interior angle converging to zero

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

dimension of quasicircles we introduce canonical antisymmetric quasiconformal maps which minimize the quasiconformality constant among maps sending the unit circle to a given quasicircle as an application we prove astala s conjecture that the hausdorff dimension of a k quasicircle is at most 1 k 2

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

convergence and optimality of adaptive mixed finite element methods the convergence and optimality of adaptive mixed finite element methods for the poisson equation are established in this paper the main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality a quasi orthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result this discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation

planar mesh refinement cannot be both local and regular we show that two desirable properties for planar mesh refinement techniques are incompatible mesh refinement is a common technique for adaptive error control in generating unstructured planar triangular meshes for piecewise polynomial representations of data local refinements are modifications of the mesh that involve a fixed maximum amount of computation independent of the number of triangles in the mesh regular meshes are meshes for which every interior vertex has degree 6 at least for some simple model meshing problems optimal meshes are known to be regular hence it would be desirable to have a refinement technique that if applied to a regular mesh produced a larger regular mesh we call such a technique a regular refinement in this paper we prove that no refinement technique can be both local and regular our results also have implications for non local refinement techniques such as delaunay insertion or rivara s refinement

the 4 triangles longest side partition of triangles and linear refinement algorithms in this paper we study geometrical properties of the iterative 4 triangles longest side partition of triangles and of a 3 triangles partition as well as practical algorithms based on these partitions used both directly for the triangulation refinement problem and as a basis for point insertion strategies in delaunay refinement algorithms the 4 triangles partition is obtained by joining the midpoint of the longest side with the opposite vertex and the midpoints of the two remaining sides by means of simple geometrical arguments we show that the iterative partition of obtuse triangles systematically improves the triangles while they remain obtuse in the following sense the sequence of smallest angles monotonically increases while the sequence of largest angles monotonically decreases in an amount at least equal to the smallest angle of each iteration this allows us to improve the known bound on the smallest angle without making use of previous results and to obtain a better a priori bound on the number of similarly distinct triangles as a function of the geometry of the initial triangle numerical evidence showing that the practical behavior of the 4 triangles partition is in complete agreement with this theory is included a 4 triangles refinement algorithm is also discussed and illustrated furthermore we show that the time cost of the algorithm is linear independently of the size of the triangulation

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

a posteriori error estimation for the laplace beltrami equation on spheres with spherical splines abstract we prove a posteriori upper and lower bounds for the error estimates when solving the laplace beltrami equation on the unit sphere by using the galerkin method with spherical splines adaptive mesh refinements based on these a posteriori error estimates are used to reduce complexity and computational cost of the corresponding discrete problems the theoretical results are corroborated by numerical experiments

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

discretization error analysis and adaptive meshing algorithms for fluorescence diffuse optical tomography part ii for imaging problems in which numerical solutions need to be computed for both the inverse and the underlying forward problems discretization can be a major factor that determines the accuracy of imaging in this work we analyze the effect of discretization on the accuracy of fluorescence diffuse optical tomography we model the forward problem by a pair of diffusion equations at the excitation and emission wavelengths and consider a finite element discretization method for the numerical solution of the forward problem for the inverse problem we use an optimization framework which allows incorporation of a priori information in the form of zeroth and first order tikhonov regularization terms next we convert the inverse problem into a variational problem and use galerkin projection to discretize the inverse problem following the discretization we analyze the error in reconstructed images due to the discretization of the forward and inverse problems and present two theorems which point out the factors that may lead to high error such as the mutual dependence of the forward and inverse problems the number of sources and detectors their configuration and their positions with respect to fluorophore concentration and the formulation of the inverse problem finally we demonstrate the results and implications of our error analysis by numerical experiments in the second part of the paper we apply our results to design novel adaptive discretization algorithms

adaptive mesh refinement processes for finite element solutions the nature of adaptive processes is reviewed using as an example a specific finite element problem several possible formulations for the objective of mesh refinement processes are given for the most natural of these objectives the a algorithms of artificial intelligence turn out to provide a solution process which is known to be optimal in a certain sense but practically there is insufficient information and these processes tend to be inefficient on the other hand if the objective is replaced by a strictly local objective based on pareto optimality the error estimates of the resulting mesh sequence are shown to exhibit the order of convergence which is known to be best possible for the type of elements used

on the non degeneracy property of the longest edge trisection of triangles the longest edge le trisection of a triangle t is obtained by joining the two equally spaced points of the longest edge of t with the opposite vertex in this paper we prove that for any given triangle t with smallest interior angle t 0 if the minimum interior angle of the three triangles obtained by the le trisection of t into three new triangles is denoted by t 1 then t 1 email protected c 1 where c email protected 3arctan 3 5 3 1403 moreover we show empirical evidence on the non degeneracy property of the triangular meshes obtained by iterative application of the le trisection of triangles if t n denotes the minimum angle of the triangles obtained after n iterative applications of the le trisection then t n t c where c is a positive constant independent of n an experimental estimate of c 6 7052025350 is provided

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

proving the non degeneracy of the longest edge trisection by a space of triangular shapes with hyperbolic metric from an initial triangle three triangles are obtained joining the two equally spaced points of the longest edge with the opposite vertex this construction is the base of the longest edge trisection method let be an arbitrary triangle with minimum angle let be any triangle generated in the iterated application of the longest edge trisection let be the minimum angle of thus c with c 3 arctan 3 11 is proved in this paper a region of the complex half plane endowed with the poincare hyperbolic metric is used as the space of triangular shapes the metric properties of the piecewise smooth complex dynamic defined by the longest edge trisection are studied this allows us to obtain the value c

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

snowballs are quasiballs we introduce snowballs which are compact sets in r 3 homeomorphic to the unit ball they are 3 dimensional analogs of domains in the plane bounded by snowflake curves for each snowball b a quasiconformal map f colon r 3 to r 3 is constructed that maps b to the unit ball

on the bisection method for triangles let uvw be a triangle with vertices u v and w it is bisected as follows choose a longest edge say vw of uvw and let a be the midpoint of vw the uvw gives birth to two daughter triangles uva and uwa continue this bisection process forever we prove that the infinite family of triangles so obtained falls into finitely many similarity classes and we obtain sharp estimates for the longest jth generation edge

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

rectifiable sets and the traveling salesman problem let k c c be a bounded set in this paper we shall give a simple necessary and sufficient condit ion for k to lie in a rectifiable curve we say that a set is a rectifiable curve if it is the image of a finite interval under a lipschitz mapping recall that for a connected set f c c f is a rectifiable curve not necessarily simple if and only if l f where l denotes one dimensional hausdorff measure this classical result follows from the fact that on any finite graph there is a tour which covers the entire graph and which crosses each edge but not necessarily each vertex at most twice if k is a finite set then we are essentially reduced to the classical traveling salesman problem tsp compute the length of the shortest hami l ton ian cycle which hits all points of k this is the same up to a constant multiple as asking for the inf imum of l f where f is a curve k c f such a f is called a spanning tree in tsp theory for infinite sets k we cannot hope in general to have k be a subset of a jordan curve what we should therefore look at is connected sets which conta in k let fmi n be the shortest minimal spanning tree then we cannot possibly solve our problem for sets k of infinite cardinality if we cannot find f i f c o fmin for any finite set k here and throughout the paper c co c1 c o etc denote various universal constants while there are several algorithms for computing l fml these algorithms work for finite graphs satisfying the triangle inequality and do not use the euclidean properties of k see 13 for an excellent discussion of some of these algorithms therefore these methods cannot solve our problem for general infinite k we present a method which is a minor modification of a well known algorithm farthest insert ion see 13 which yields a f with i f c o l fmi the farthest insert ion algorithm has been extensively studied with large numerical calculations on computers and is experimentally good in the sense that the f produced satisfy i f c o l f i for all examples which have