weak efficiency for multiobjective variational problems abstract sometimes to locate efficient solutions for multiobjective variational problems mvps is quite costly so in this paper we tackle the study of weakly efficient solutions for mvps a new concept of weak vector critical point which generalizes other ones already existent and a new class of pseudoinvex functions are introduced we will apply a new approach to prove that the new class of pseudoinvex functions is equivalent to the class of functions whose weak vector critical points are weakly efficient solution for mvps

efficient mesh optimization using the gradient flow of the mean volume the signed volume function for polyhedra can be generalized to a mean volume function for volume elements by averaging over the triangulations of the underlying polyhedron if we consider these up to translation and scaling the resulting quotient space is diffeomorphic to a sphere the mean volume function restricted to this sphere is a quality measure for volume elements we show that the gradient ascent of this map regularizes the building blocks of hybrid meshes consisting of tetrahedra hexahedra prisms pyramids and octahedra that is the optimization process converges to regular polyhedra we show that the normalized gradient flow of the mean volume yields a fast and efficient optimization scheme for the finite element method known as the geometric element transformation method furthermore we shed some light on the dynamics of this method and the resulting smoothing procedure both theoretically and experimentally

a new paradigm for parallel adaptive meshing algorithms we present a new approach to the use of parallel computers with adaptive finite element methods this approach addresses the load balancing problem in a new way requiring far less communication than current approaches it also allows existing sequential adaptive pde codes such as pltmg and mc to run in a parallel environment without a large investment in recoding in this new approach the load balancing problem is reduced to the numerical solution of a small elliptic problem on a single processor using a sequential adaptive solver without requiring any modifications to the sequential solver the small elliptic problem is used to produce a posteriori error estimates to predict future element densities in the mesh which are then used in a weighted recursive spectral bisection of the initial mesh the bulk of the calculation then takes place independently on each processor with no communication using possibly the same sequential adaptive solver each processor adapts its region of the mesh independently and a nearly load balanced mesh distribution is usually obtained as a result of the initial weighted spectral bisection only the initial fan out of the mesh decomposition to the processors requires communication two additional steps requiring boundary exchange communication may be employed after the individual processors reach an adapted solution namely the construction of a global conforming mesh from the independent subproblems followed by a final smoothing phase using the subdomain solutions as an initial guess we present a series of convincing numerical experiments which illustrate the effectiveness of this approach the justification of the initial refinement prediction step as well as the justification of skipping the two communication intensive steps is supported by some recent j xu and a zhou math comp to appear and not so recent j a nitsche and a h schatz math comp 28 1974 pp 937 958 a h schatz and l b wahlbin math comp 31 1977 pp 414 442 a h schatz and l b wahlbin math comp 64 1995 pp 907 928 results on local a priori and a posteriori error estimation

a moving finite element method with error estimation and refinement for one dimensional time dependent partial differential equations we discuss a moving finite element method for solving vector systems of time dependent partial differential equations in one space dimension the mesh is moved so as to equidistribute the spatial component of the discretization error in h 1 we present a method of estimating this error by using p hierarchic finite elements the error estimate is also used in an adaptive mesh refinement procedure to give an algorithm that combines mesh movement and refinement we discretize the partial differential equations in space using a galerkin procedure with piecewise linear elements to approximate the solution and quadratic elements to estimate the error a system of ordinary differential equations for mesh velocities are used to control element motions we use existing software for stiff ordinary differential equations for the temporal integration of the solution the error estimate and the mesh motion computational results using a code based on our method are presented for several examples

numerical study of the turbulent flow past an airfoil with trailing edge separation presentation d une methode numerique par volume fini pour la resolution des equations de navier stokes bidimensionnelles incompressibles et stationnaires en coordonnees generales curvilignes application de la methode aux ecoulements turbulents sur des profils avec et sans separation au bord de sortie posterieur comparaison des calculs avec des donnees experimentales

local optimization based simplicial mesh untangling and improvement we present an optimization based approach for mesh untangling that maximizes the minimum area or volume of simplicial elements in a local submesh these functions are linear with respect to the free vertex position thus the problem can be formulated as a linear program that is solved by using the computationally inexpensive simplex method we prove that the function level sets are convex regardless of the position of the free vertex and hence the local subproblem is guaranteed to converge maximizing the minimum area or volume of mesh elements although well suited for mesh untangling is not ideal for mesh improvement and its use often results in poor quality meshes we therefore combine the mesh untangling technique with optimization based mesh improvement techniques and expand previous results to show that a commonly used two dimensional mesh quality criterion can be guaranteed to converge when starting with a valid mesh typical results showing the effectiveness of the combined untangling and smoothing techniques are given for both two and three dimensional simplicial meshes

analysis of triangle quality measures several of the more commonly used triangle quality measures are analyzed and compared proofs are provided to verify that they do exhibit the expected extremal properties the asymptotic behavior of these measures is investigated and a number of useful results are derived it is shown that some of the quality measures are equivalent in the sense of displaying the same extremal and asymptotic behavior and that it is therefore possible to achieve a concise classification of triangle quality measures

coordination capacity we develop elements of a theory of cooperation and coordination in networks rather than considering a communication network as a means of distributing information or of reconstructing random processes at remote nodes we ask what dependence can be established among the nodes given the communication constraints specifically in a network with communication rates ri j between the nodes we ask what is the set of all achievable joint distributions p x1 xm of actions at the nodes of the network several networks are solved including arbitrarily large cascade networks distributed cooperation can be the solution to many problems such as distributed games distributed control and establishing mutual information bounds on the influence of one part of a physical system on another

a lagrange dual lower bound to the error exponent of the typical random code a lagrange dual gallager style lower bound is derived for the error exponent function of the typical random code trc pertaining to the i i d random coding ensemble and mismatched stochastic likelihood decoding while the original expression derived from the method of types the csiszar style expression involves minimization over probability distributions defined on the channel input output alphabets the new lagrange dual formula involves optimization of five parameters independently of the alphabet sizes for both stochastic and deterministic mismatched decoding including maximum likelihood decoding as a special case we provide a rather comprehensive discussion on the insight behind the various ingredients of this formula and describe how its behavior varies as the coding rate exhausts the relevant range among other things it is demonstrated that this expression simultaneously generalizes both the expurgated error exponent function at zero rate and the classical random coding exponent function at high rates where it also meets the sphere packing bound

on efficiency and duality for multiobjective variational problems abstract the concept of efficiency pareto optimum is used to formulate duality for multiobjective variational problems wolfe and mond weir type duals are formulated under generalized f convexity assumptions on the functions involved weak and strong duality theorems are proved

an adaptive finite element procedure for compressible high speed flows abstract a two step explicit finite element procedure for obtaining time accurate solutions to the compressible euler equations is described the method is coupled with an adaptive mesh refinement process which enables steady state solutions of improved quality to be obtained

nonasymptotic and second order achievability bounds for coding with side information we present a novel nonasymptotic or finite blocklength achievability bounds for three side information problems in network information theory these include 1 the wyner ahlswede korner wak problem of almost lossless source coding with rate limited side information 2 the wyner ziv wz problem of lossy source coding with side information at the decoder and 3 the gel fand pinsker gp problem of channel coding with noncausal state information available at the encoder the bounds are proved using ideas from channel simulation and channel resolvability our bounds for all three problems improve on all previous nonasymptotic bounds on the error probability of the wak wz and gp problems in particular those derived by verdu using our novel nonasymptotic bounds we recover the general formulas for the optimal rates of these side information problems finally we also present achievable second order coding rates by applying the multidimensional berry esseen theorem to our new nonasymptotic bounds numerical results show that the second order coding rates obtained using our nonasymptotic achievability bounds are superior to those obtained using existing finite blocklength bounds

sending a bivariate gaussian over a gaussian mac we study the power versus distortion tradeoff for the distributed transmission of a memoryless bivariate gaussian source over a two to one average power limited gaussian multiple access channel in this problem each of two separate transmitters observes a different component of a memoryless bivariate gaussian source the two transmitters then describe their source component to a common receiver via an average power constrained gaussian multiple access channel from the output of the multiple access channel the receiver wishes to reconstruct each source component with the least possible expected squared error distortion our interest is in characterizing the distortion pairs that are simultaneously achievable on the two source components we focus on the equal bandwidth case where the source rate in source symbols per second is equal to the channel rate in channel uses per second we present sufficient conditions and necessary conditions for the achievability of a distortion pair these conditions are expressed as a function of the channel signal to noise ratio snr and of the source correlation in several cases the necessary conditions and sufficient conditions are shown to agree in particular we show that if the channel snr is below a certain threshold then an uncoded transmission scheme is optimal moreover we introduce a source channel vector quantizer scheme which is asymptotically optimal as the snr tends to infinity

sharp bounds for mutual covering a fundamental tool in network information theory is the covering lemma which lower bounds the probability that there exists a pair of random variables among a given number of independently generated candidates falling within a given set we use a weighted sum trick and talagrand s concentration inequality to prove new mutual covering bounds we identify two interesting applications 1 when the probability of the set under the given joint distribution is bounded away from 0 and 1 the covering probability converges to 1 doubly exponentially fast in the blocklength which implies that the covering lemma does not induce penalties on the error exponents in the applications to coding theorems and 2 using hall s marriage lemma we show that the maximum difference between the probability of the set under the joint distribution and the covering probability equals half the minimum total variation distance between the joint distribution and any distribution that can be simulated by selecting a pair from the candidates thus we use the mutual covering bound to derive the exact error exponent in the joint distribution simulation problem in both applications the determination of the exact exponential or double exponential behavior relies crucially on the sharp concentration inequality used in the proof of the mutual covering lemma

distributed channel synthesis two familiar notions of correlation are rediscovered as the extreme operating points for distributed synthesis of a discrete memoryless channel in which a stochastic channel output is generated based on a compressed description of the channel input wyner s common information is the minimum description rate needed however when common randomness independent of the input is available the necessary description rate reduces to shannon s mutual information this paper characterizes the optimal tradeoff between the amount of common randomness used and the required rate of description we also include a number of related derivations including the effect of limited local randomness rate requirements for secrecy applications to game theory and new insights into common information duality our proof makes use of a soft covering lemma known in the literature for its role in quantifying the resolvability of a channel the direct proof achievability constructs a feasible joint distribution over all parts of the system using a soft covering from which the behavior of the encoder and decoder is inferred with no explicit reference to joint typicality or binning of auxiliary interest this paper also generalizes and strengthens this soft covering tool

improving high order finite element approximation through geometrical warping polynomial basis functions are the ubiquitous workhorse of high order finite element methods but their generality comes at a price of high computational cost and fragility in the face of underresolution in this paper a method is presented for constructing a posteriori tailored generally nonpolynomial basis functions for approximating a solution and computing outputs of a system of equations this method is similar to solution based adaptation in which elements of the computational mesh are sized and oriented based on characteristics of the solution the method takes advantage of existing infrastructure in high order methods the reference to global mapping used in constructing curved elements by optimizing this mapping elements are warped to make them ideally suited for representing a target solution or computing a scalar output from the solution guidelines on generating a good initial guess and choosing a generalized set of optimization parameters are provided to minimize tuning time and to intr

insertion of triangulated surfaces into a meccano tetrahedral discretization by means of mesh refinement and optimization procedures this is the peer reviewed version of the following article ruiz girones e oliver a socorro g cascon j escobar j m montenegro r sarrate j insertion of triangulated surfaces into a meccano tetrahedral discretization by means of mesh refinement and optimization procedures international journal for numerical methods in engineering 2018 which has been published in final form at http onlinelibrary wiley com doi 10 1002 nme 5706 pdf this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving

a continuous time generalization of gordan s transposition theorem abstract a certain infinite dimensional generalization of gordan s transposition theorem is obtained as a consequence of a strong duality principle in continuous time programming some implications and corollaries of the main theorem are also discussed

duality for multiobjective variational problems abstract wolfe and mond weir type duals for multiobjective variational problems are formulated under convexity assumptions on the functions involved weak and converse duality theorems are proved to relate properly efficient solutions of the primal and dual problems a close relationship between these variational problems and nonlinear multiobjective programming problems is also indicated

secret key generation with limited interaction a basic two terminal secret key generation model is considered where the interactive communication rate between the terminals may be limited and in particular may not be enough to achieve the maximum key rate we first prove a multi letter characterization of the key communication rate region where the number of auxiliary random variables depends on the number of rounds of the communication and then provide an equivalent but simpler characterization in terms of concave envelopes in the case of unlimited number of rounds two extreme cases are given special attention first in the regime of very low communication rates the key bits per interaction bit kbib is expressed with a new symmetric strong data processing constant which has a concave envelope characterization analogous to that of the conventional strong data processing constant the symmetric strong data processing constant can be upper bounded by the supremum of the maximal correlation coefficient over a set of distributions which allows us to determine the kbib for binary symmetric sources and conclude in particular that the interactive scheme is not more efficient than the one way scheme at least in the low communication rate regime second a new characterization of the minimum interaction rate needed for achieving the maximum key rate mimk is given and we resolve a conjecture by tyagi regarding the mimk for possibly nonsymmetric binary sources we also propose a new conjecture for binary symmetric sources that the interactive scheme is not more efficient than the one way scheme at any communication rate

rate distortion theory for secrecy systems secrecy in communication systems is measured herein by the distortion that an adversary incurs the transmitter and receiver share secret key which they use to encrypt communication and ensure distortion at an adversary a model is considered in which an adversary not only intercepts the communication from the transmitter to the receiver but also potentially has side information in particular the adversary may have causal or noncausal access to a signal that is correlated with the source sequence or the receiver s reconstruction sequence the main contribution is the characterization of the optimal tradeoff among communication rate secret key rate distortion at the adversary and distortion at the legitimate receiver it is demonstrated that causal side information at the adversary plays a pivotal role in this tradeoff it is also shown that measures of secrecy based on normalized equivocation are a special case of the framework

adaptive finite element analysis of elliptic problems based on bubble type local mesh generation a new mesh adaptivity algorithm that combines a posteriori error estimation with bubble type local mesh generation blmg strategy for elliptic differential equations is proposed the size function used in the blmg is defined on each vertex during the adaptive process based on the obtained error estimator in order to avoid the excessive coarsening and refining in each iterative step two factor thresholds are introduced in the size function the advantages of the blmg based adaptive finite element method compared with other known methods are given as follows the refining and coarsening are obtained fluently in the same framework the local a posteriori error estimation is easy to implement through the adjacency list of the blmg method at all levels of refinement the updated triangles remain very well shaped even if the mesh size at any particular refinement level varies by several orders of magnitude several numerical examples with singularities for the elliptic problems where the explicit error estimators are used verify the efficiency of the algorithm the analysis for the parameters introduced in the size function shows that the algorithm has good flexibility

universal meshes a method for triangulating planar curved domains immersed in nonconforming meshes summary we introduce a new method to triangulate planar curved domains that transforms a specific collection of triangles in a background mesh to conform to the boundary in the process no new vertices are introduced and connectivities of triangles are left unaltered the method relies on a novel way of parameterizing an immersed boundary over a collection of nearby edges with its closest point projection to guarantee its robustness we require that the domain be c2 regular the background mesh be sufficiently refined near the boundary and that specific angles in triangles near the boundary be strictly acute the method can render both straight edged and curvilinear triangulations for the immersed domain the latter includes curved triangles that conform exactly to the immersed boundary and ones constructed with isoparametric mappings to interpolate the boundary at select points high order finite elements constructed over these curved triangles achieve optimal accuracy which has customarily proven difficult in numerical schemes that adopt nonconforming meshes aside from serving as a quick and simple tool for meshing planar curved domains with complex shapes the method provides significant advantages for simulating problems with moving boundaries and in numerical schemes that require iterating over the geometry of domains with no conformity requirements the same background mesh can be adopted to triangulate a large family of domains immersed in it including ones realized over several updates during the coarse of simulating problems with moving boundaries we term such a background mesh as a universal mesh for the family of domains it can be used to triangulate universal meshes hence facilitate a framework for finite element calculations over evolving domains while using only fixed background meshes furthermore because the evolving geometry can be approximated with any desired order numerical solutions can be computed with high order accuracy we present demonstrative examples using universal meshes to simulate the interaction of rigid bodies with stokesian fluids copyright 2014 john wiley sons ltd

simultaneous untangling and smoothing of tetrahedral meshes abstract the quality improvement in mesh optimisation techniques that preserve its connectivity are obtained by an iterative process in which each node of the mesh is moved to a new position that minimises a certain objective function the objective function is derived from some quality measure of the local submesh that is the set of tetrahedra connected to the adjustable or free node although these objective functions are suitable to improve the quality of a mesh in which there are non inverted elements they are not when the mesh is tangled this is due to the fact that usual objective functions are not defined on all r 3 and they present several discontinuities and local minima that prevent the use of conventional optimisation procedures otherwise when the mesh is tangled there are local submeshes for which the free node is out of the feasible region or this does not exist in this paper we propose the substitution of objective functions having barriers by modified versions that are defined and regular on all r 3 with these modifications the optimisation process is also directly applicable to meshes with inverted elements making a previous untangling procedure unnecessary this simultaneous procedure allows to reduce the number of iterations for reaching a prescribed quality to illustrate the effectiveness of our approach we present several applications where it can be seen that our results clearly improve those obtained by other authors

the likelihood encoder for lossy compression a likelihood encoder is studied in the context of lossy source compression the analysis of the likelihood encoder is based on the soft covering lemma it is demonstrated that the use of a likelihood encoder together with the soft covering lemma yields simple achievability proofs for classical source coding problems the cases of the point to point rate distortion function the rate distortion function with side information at the decoder i e the wyner ziv problem and the multi terminal source coding inner bound i e the berger tung problem are examined in this paper furthermore a non asymptotic analysis is used for the point to point case to examine the upper bound on the excess distortion provided by this method the likelihood encoder is also related to a recent alternative technique using the properties of random binning

function gradient and hessian recovery using quadratic edge bump functions an approximate error function for the discretization error on a given mesh is obtained by projecting via the energy inner product the functional residual onto the space of continuous piecewise quadratic functions which vanish on the vertices of the mesh conditions are given under which one can expect this hierarchical basis error estimator to give efficient and reliable function recovery asymptotically exact gradient recovery and convergent hessian recovery in the square norms one does not find similar function recovery results in the literature the analysis given here is based on a certain superconvergence result which has been used elsewhere in the analysis of gradient recovery methods numerical experiments are provided which demonstrate the effectivity of the approximate error function in practice

the discrete continuity equation in primitive variable solutions of incompressible flow abstract the use of a non staggered computational grid for the numerical solutions of the incompressible flow equations has many advantages over the use of a staggered grid a penalty however is inherent in the finite difference approximations of the governing equations on non staggered grids in the primitive variable solutions the penalty is that the discrete continuity equation does not converge to machine accuracy rather it converges to a source term which is proportional to the fourth order derivative of the pressure the time increment and the square of the grid spacing an approach which minimizes the error in the discrete continuity equation is developed numerical results obtained for the driven cavity problem confirm the analytical developments

a class of nondifferentiable continuous programming problems abstract optimality conditions duality and converse duality results are obtained for a class of continuous programming problems with a nondifferentiable term in the integrand of the objective function the proofs are based on a fritz john theorem for constrained optimization in abstract spaces the results generalize various well known results in variational problems with differentiable functions and also give a dynamic analogue of certain nondifferentiable programming problems

obtaining smooth mesh transitions using vertex optimization mesh optimization has proven to be an effective way to improve mesh quality for arbitrary lagrangian eulerian ale simulations to date however most of the focus has been on improving the geometric shape of individual elements and these methods often do not result in smooth transitions in element size or aspect ratio across groups of elements we present an extension to the mean ratio optimization that addresses this problem and yields smooth transitions within regions and across regions in the ale simulation while this method is presented in the context of ale simulations it is applicable to a wider set of applications that require mesh improvement including the mesh generation process published in 2007 by john wiley sons ltd

anisotropic mesh adaptation for 3d flows on structured and unstructured grids abstract this paper presents a mesh optimization methodology in three dimensions mom3d an initial mesh is continually adapted during the solution process without the need for global remeshing the adaptation procedure uses an interpolation error estimate whose magnitude and direction are controlled by the hessian the matrix of second derivatives of the solution this metric error is projected over mesh edges and drives the nodal movement scheme as well as the edge refinement and coarsening strategies these operations yield highly anisotropic grids in which the mesh movement significantly contributes to the stretching and realignment of the edges along unidirectional features of flow problems the results presented have been chosen to illustrate some important points first the method is gauged on problems with exact solutions demonstrating good agreement between the error estimate and the true error as well as an equidistribution of the error the cost effectiveness of grid adaptation is then addressed by determining the size of an anisotropic grid that would be equivalent to that of a given non adapted finer grid for the same error level the capture of sharp discontinuities through highly anisotropic grids is illustrated on a transonic flow flow in a gas turbine combustor demonstrates how automatically generated meshes can sometimes cause convergence difficulties and how mesh adaptation can cure these ills finally the flow over a wing nacelle pylon configuration is studied to further validate the solver mesh adaptation capabilities by comparing the numerical results against experiments

a method of grid optimization for finite element methods a computationally useful criterion for grid optimization is derived based on a measure of the interpolation error associated with the finite element model the result is intended to be used to improve the quality of finite element solutions by changing the location of the nodes within a fixed number of degrees of freedom examples of the application of the criterion are provided the finite element method has become an important tool of engineering analysis and design much of the popularity of the method is due to the freedom that it allows in the construction of the discretized model it has long been realized however that this freedom must be carefully exercised since the quality of the finite element solution greatly depends on how the discretization is performed several methods have been proposed to improve the discretized model in an iterative manner one such method is presented in this paper a natural way of improving the quality of finite element solutions is to increase the number of degrees of freedom the process is normally performed after an initial solution is already available several schemes have been devised to introduce the new degrees of freedom in a selective manner in order to produce the greatest possible improvement of the previous solution this calls for the definition of criteria to identify the regions of the domain where the finite element approximation is poorer the new degrees of freedom are added in these regions by either increasing the order of polynomial approximation inside elements the so called p method or by subdivision of elements or h method the process is continued until a specified accuracy is achieved the quality of the finite element solution may be improved also by optimizing the disposition of the nodes analysts often rely on their experience to construct grids that make an efficient use of the available degrees of freedom it is also possible to improve the quality of existing meshes iteratively using predefined guidelines for the redistribution of the nodes this paper considers the development of such guidelines

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

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

fast delaunay triangulation in three dimensions abstract an efficient algorithm for delaunay triangulation of a given set of points in three dimensions based on the point insertion technique is presented various steps of the triangulation algorithm are reviewed and many acceleration procedures are devised to speed up the triangulation process new features include the search of a neighbouring point by the layering scheme locating the containing tetrahedron by random walk formulas of important geometrical quantities of a new tetrahedron based on those of an existing one a novel approach in establishing the adjacency relationship the use of adjacency table and the management of memory the resulting scheme is one of the fastest triangulation algorithms known to the authors which is able to generate tetrahedra generation rate of 15 000 tetrahedra per second for randomly generated points on a hp 735 machine

simulating curvilinear crack propagation in two dimensions with universal meshes summary we formulate a class of delicately controlled problems to model the kink free evolution of quasistatic cracks in brittle isotropic linearly elastic materials in two dimensions the evolving crack satisfies familiar principles griffith s criterion local symmetry and irreversibility a novel feature of the formulation is that in addition to the crack path the loading is also treated as an unknown specifically a scaling factor for prescribed dirichlet and neumann boundary conditions is computed as part of the solution to yield an always propagating and ostensibly kink free crack and a continuous loading history beyond the initial step a dimensionless statement of the problem depends only on the poisson s ratio of a homogeneous material and is in particular independent of its young s modulus and fracture toughness numerical resolution of the formulated problem relies on two new ideas the first is an algorithm to compute triangulations conforming to cracked domains by locally deforming a given background mesh in the vicinity of evolving cracks the algorithm is robust under mild assumptions on the sizes and angles of triangles near the crack and its smoothness hence a subset if not the entire family of cracked domains realized during the course of a simulation can be discretized with the same background mesh we term the latter a universal mesh for such a family of domains universal meshes facilitate adopting a discrete representation for the crack as splines in our examples preclude the need for local splitting retriangulation operations liberate the crack from following directions prescribed by the mesh and enable the adoption of standard fems to compute the elastic fields in the cracked solid second we employ a method specifically designed to approximate the stress intensity factors for curvilinear cracks we examine the performance of the resulting numerical method with detailed examples including comparisons with an exact solution of a crack propagating along a circular arc which we construct and comparisons with experimental fracture paths in all cases we observe convergence of computed paths their derivatives and loading histories with refinement of the universal mesh copyright 2014 john wiley sons ltd

a new approach to solid modeling with trivariate t splines based on mesh optimization abstract we present a new method to construct a trivariate t spline representation of complex genus zero solids for the application of isogeometric analysis the proposed technique only demands a surface triangulation of the solid as input data the key of this method lies in obtaining a volumetric parameterization between the solid and the parametric domain the unitary cube to do that an adaptive tetrahedral mesh of the parametric domain is isomorphically transformed onto the solid by applying a mesh untangling and smoothing procedure the control points of the trivariate t spline are calculated by imposing the interpolation conditions on points sited both on the inner and on the surface of the solid the distribution of the interpolating points is adapted to the singularities of the domain in order to preserve the features of the surface triangulation

continuous mesh framework part i well posed continuous interpolation error in the context of mesh adaptation riemannian metric spaces have been used to prescribe orientation density and stretching of anisotropic meshes but such structures are only considered to compute lengths in adaptive mesh generators in this article a riemannian metric space is shown to be more than a way to compute a length it is proven to be a reliable continuous mesh model in particular we demonstrate that the linear interpolation error can be evaluated continuously on a riemannian metric space from one hand this new continuous framework proves that prescribing a riemannian metric field is equivalent to the local control in the mathbf l 1 norm of the interpolation error this proves the consistency of classical metric based mesh adaptation procedures on the other hand powerful mathematical tools are available and are well defined on riemannian metric spaces calculus of variations differentiation optimization dots whereas these tools are not defined on discrete meshes

quantum rate distortion reverse shannon theorems and source channel separation we derive quantum counterparts of two key theorems of classical information theory namely the rate distortion theorem and the source channel separation theorem the rate distortion theorem gives the ultimate limits on lossy data compression and the source channel separation theorem implies that a two stage protocol consisting of compression and channel coding is optimal for transmitting a memoryless source over a memoryless channel in spite of their importance in the classical domain there has been surprisingly little work in these areas for quantum information theory in this paper we prove that the quantum rate distortion function is given in terms of the regularized entanglement of purification we also determine a single letter expression for the entanglement assisted quantum rate distortion function and we prove that it serves as a lower bound on the unassisted quantum rate distortion function this implies that the unassisted quantum rate distortion function is nonnegative and generally not equal to the coherent information between the source and distorted output in spite of barnum s conjecture that the coherent information would be relevant here moreover we prove several quantum source channel separation theorems the strongest of these are in the entanglement assisted setting in which we establish a necessary and sufficient condition for transmitting a memoryless source over a memoryless quantum channel up to a given distortion

a distortion measure to validate and generate curved high order meshes on cad surfaces with independence of parameterization the first author was sponsored by cur from diue of the generalitat de catalunya and the european social fund grants fi dgr 11 and be dgr 12 and by the ferran sunyer i balaguer foundation grant fsb 13 the work of the corresponding author was partially supported by the boeing co us air force office of scientific research european commission through the boeing mit alliance the computational math program marie sklodowska curie actions hipermegaflows individual fellowship grant

optimality conditions and duality for multiobjective variational problems with generalized b invexity we extend the concepts of b type i and generalized b type i functions to the continuous case and we use these concepts to establish sufficient optimality conditions and duality results for multiobjective variational programming problems

multiterminal source encoding with one distortion criterion the authors unify earlier investigations concerning the encoding of two correlated sources x sub k y sub k by means of separate encoders decoding is done by a single decoder which receives the outputs from both encoders the reconstruction of x sub k is required to be perfect in the usual shannon sense the authors determine the admissible rate region r d where d is the distortion of the reconstruction of y sub k the binary hamming case is investigated explicitly

numerical solutions for the pressure poisson equation with neumann boundary conditions using a non staggered grid 1 abstract numerical solutions are obtained for the pressure poisson equation with neumann boundary conditions using a non staggered grid the existence of a solution for this equation requires the satisfaction of a compatibility condition which relates the source of the poisson equation and the neumann boundary conditions this compatibility condition is not automatically satisfied on non staggered grids failure to satisfy the compatibility condition leads to non convergent iterative solutions consistent finite difference approximations for the pressure equation with neumann boundary conditions are developed to satisfy the compatibility condition on non staggered grids the method is applied to calculate the pressure coefficient in a driven cavity when given the velocity field the velocity is computed from the stream function vorticity formulation of the navier stokes equations

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

tetrahedral mesh optimisation and adaptivity for steady state and transient finite element calculations abstract a method for optimising a pre existing mesh of tetrahedral finite elements is described it is based on a series of mesh connectivity and node position searches of the landscape defining mesh quality a riemannian metric reflecting the a posteriori error measure is used to calculate element size and shape a functional is defined which embodies both shape and size quality of an element with respect to the metric and is used to gauge mesh quality a heuristic based local search strategy is adopted local in the sense that it has no hill climbing abilities the paper presents applications of the method to complex steady state and time dependent problems which highlight its anisotropic feature capturing abilities numerical evidence is provided which suggests that the computational complexity time of the proposed algorithm varies linearly with the number of elements or nodes of the finite element mesh

an algebraic method for smoothing surface triangulations on a local parametric space this paper presents a new procedure to improve the quality of triangular meshes defined on surfaces the improvement is obtained by an iterative process in which each node of the mesh is moved to a new position that minimizes a certain objective function this objective function is derived from algebraic quality measures of the local mesh the set of triangles connected to the adjustable or free node if we allow the free node to move on the surface without imposing any restriction only guided by the improvement of the quality the optimization procedure can construct a high quality local mesh but with this node in an unacceptable position to avoid this problem the optimization is done in the parametric mesh where the presence of barriers in the objective function maintains the free node inside the feasible region in this way the original problem on the surface is transformed into a two dimensional one on the parametric space in our case the parametric space is a plane chosen in terms of the local mesh in such a way that this mesh can be optimally projected performing a valid mesh that is without inverted elements several examples and applications presented in this work show how this technique is capable of improving the quality of triangular surface meshes copyright 2005 john wiley sons ltd

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

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

volume conserving smoothing for piecewise linear curves surfaces and triple lines abstract we present smoothing algorithms for piecewise linear curves surfaces and triple lines of intersection of surfaces that are based on the the idea of sequentially relaxing either individual nodes or edges in the mesh each relaxation is designed both to smooth the mesh and to conserve down to round off error the area or volume enclosed by the curve or surface for the case of smoothing surfaces and lines of intersection of surfaces each relaxation consists of a pure smoothing component and a volume conserving correction which is chosen to be of minimum norm since surfaces and triple intersection lines can be conservatively smoothed the algorithms are suitable for improving multimaterial grids used by physics simulations where exactly conserving the volume of each individual material may be a requirement or at least highly desirable the algorithms are also suitable for smoothing piecewise linear functions of one or two variables while simultaneously preserving their integrals we show examples of the application of the more powerful edge based algorithms to curve surface and multimaterial volume grids and to a thin film simulation

tetrahedral mesh improvement via optimization of the element condition number the authors present a new shape measure for tetrahedral elements that is optimal in that it gives the distance of a tetrahedron from the set of inverted elements this measure is constructed from the condition number of the linear transformation between a unit equilateral tetrahedron and any tetrahedron with positive volume using this shape measure they formulate two optimization objective functions that are differentiated by their goal the first seeks to improve the average quality of the tetrahedral mesh the second aims to improve the worst quality element in the mesh they review the optimization techniques used with each objective function and presents experimental results that demonstrate the effectiveness of the mesh improvement methods they show that a combined optimization approach that uses both objective functions obtains the best quality meshes for several complex geometries

variational method for untangling and optimization of spatial meshes a variational method that can provably construct 3d quasi isometric mappings between domains of a complex shape is introduced a local maximum principle for polyconvex mesh element distortion measures is formulated it allows us to control the invertibility and distortion bounds for non simplicial elements in the minimization process a simple and efficient technique for construction of boundary orthogonal meshes suggested in garanzha 2000 is applied to the construction of hexahedral meshes and thick prismatic mesh layers around complex shapes the mesh untangling technique which is a generalization of the penalty method suggested in garanzha and kaporin 1999 is verified on a wide set of challenging test problems another untangling technique based on theoretical ideas from ivanenko 1997 is implemented and tested it provably constructs admissible meshes using a finite number of minimization steps a minimization technique for the mesh distortion functional is described the approach is based on the global gradient search technique with preconditioning and domain decomposition for local mesh optimization and untangling application areas for explicit and implicit minimization methods are evaluated

algebraic mesh quality metrics quality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics the theory based on the jacobian and related matrices provides a means of constructing classifying and evaluating mesh quality metrics the jacobian matrix is factored into geometrically meaningful parts a nodally invariant jacobian matrix can be defined for simplicial elements using a weight matrix derived from the jacobian matrix of an ideal reference element scale and orientation invariant algebraic mesh quality metrics are defined the singular value decomposition is used to study relationships between metrics equivalence of the element condition number and mean ratio metrics is proved the condition number is shown to measure the distance of an element to the set of degenerate elements algebraic measures for skew length ratio shape volume and orientation are defined abstractly with specific examples given two combined metrics shape volume and shape volume orientation are algebraically defined and examples of such metrics are given algebraic mesh quality metrics are extended to nonsimplicial elements a series of numerical tests verifies the theoretical properties of the metrics defined

area control in generating smooth and convex grids over general plane regions a new method adaptive smoothness functional to produce convex and smooth grids over general plane regions has been introduced in a recent work by the authors 10 this method belongs to the variational grid generation approach theoretical results showing that a convex grid over a region is obtained when this method is applied were presented the basic assumption was that at least one convex grid exists a procedure to control large cells bilateral smoothness functional in addition to smoothness and convexity was also presented experimental results showing the effectiveness of these methods were reported however no theoretical results were reported assuring that the area control can be always exerted this article continues the same line of research introducing a new version of the bilateral smoothness functional that improves the control of large areas unlike the former method theoretical results show the effectiveness of the new bilateral smoothness functional to exert such control optimal grids obtained with the new functional are compared with those reached using the older version demostrating the improvement

numerical solutions for the incompressible navier stokes equations in primitive variables using a non staggered grid 11 abstract in part i of this study a method was developed for the solution of the pressure poisson equation with neumann boundary conditions on a non staggered grid this method was used to determine the pressure when given the velocity field from a stream function vorticity solution in part ii the pressure equation is solved iteratively with the momentum navier stokes equations on a non staggered grid in this case the solution of the pressure equation not only provides the pressure but also serves to indirectly satisfy the continuity equation this primitive variables formulation has a major advantage over the stream function vorticity method in its applicability for three dimensional flow numerical results are obtained and compared with the stream function vorticity results for the driven cavity of part i

achieving finite element mesh quality via optimization of the jacobian matrix norm and associated quantities part ii a framework for volume mesh optimization and the condition number of the jacobian matrix three dimensional unstructured tetrahedral and hexahedral finite element mesh optimization is studied from a theoretical perspective and by computer experiments to determine what objective functions are most effective in attaining valid high quality meshes the approach uses matrices and matrix norms to extend the work in part i to build suitable 3d objective functions because certain matrix norm identities which hold for 2 x 2 matrices do not hold for 3 x 3 matrices significant differences arise between surface and volume mesh optimization objective functions it is shown for example that the equivalence in two dimensions of the smoothness and condition number of the jacobian matrix objective functions does not extend to three dimensions and further that the equivalence of the oddy and condition number of the metric tensor objective functions in two dimensions also fails to extend to three dimensions matrix norm identities are used to systematically construct dimensionally homogeneous groups of objective functions the concept of an ideal minimizing matrix is introduced for both hexahedral and tetrahedral elements non dimensional objective functions having barriers are emphasized as the most logical choice for mesh optimization the performance of a number of objective functions in improving mesh quality was assessed on a suite of realistic test problems focusing particularly on all hexahedral whisker weaved meshes performance is investigated on both structured and unstructured meshes and on both hexahedral and tetrahedral meshes although several objective functions are competitive the condition number objective function is particularly attractive the objective functions are closely related to mesh quality measures to illustrate it is shown that the condition number metric can be viewed as a new tetrahedral element quality measure

mesh relaxation a new technique for improving triangulations summary given a list of points defining a domain boundary a three stage process is often used to triangulate a domain first an appropriate distribution of interior points is generated next the points are connected to form triangles and finally the connectivity data are used to reposition the interior points using the laplacian smoothing technique thereby usually improving the shapes of the triangles this paper describes a new technique for mesh improvement adjusting the connection structure during the second stage of this process the new scheme which we call mesh relaxation consists of a procedure for iteratively making the mesh topology more regular by edge swapping for each interior edge a relaxation index is computed that depends on the degrees of its end points and adjacent points any edge for which this index exceeds a prescribed threshold will be swapped i e replaced by a new edge connecting the adjacent points of the original edge after all edge swaps are completed laplacian smoothing is applied to the mesh examples show that when the mesh point density varies smoothly and due care is taken in the vicinity of the boundary mesh relaxation can dramatically increase the regularity of the mesh and produce improved triangle shapes

variational mesh adaptation ii error estimates and monitor functions the key to the success of a variational mesh adaptation method is to define a proper monitor function which controls mesh adaptation in this paper we study the choice of the monitor function for the variational adaptive mesh method developed in the previous work j comput phys 174 2001 924 two types of monitor functions scalar matrix and non scalar matrix ones are defined based on asymptotic estimates of interpolation error obtained using the interpolation theory of finite element methods the choice of the adaptation intensity parameter is also discussed for each of these monitor functions asymptotic bounds on interpolation error are obtained for adaptive meshes that satisfy the regularity and equidistribution conditions two dimensional numerical results are given to verify the theoretical findings

spline parameterization method for 2d and 3d geometries based on t mesh optimization abstract we present a method to obtain high quality spline parameterization of 2d and 3d geometries for their use in isogeometric analysis as input data the proposed method demands a boundary representation of the domain and it constructs automatically a spline transformation between the physical and parametric domains parameterization of the interior of the object is obtained by deforming isomorphically an adapted parametric t mesh onto the physical domain by applying a t mesh untangling and smoothing procedure which is the key of the method mesh optimization is based on the mean ratio shape quality measure the spline representation of the geometry is calculated by imposing interpolation conditions using the data provided by one to one correspondence between the meshes of the parametric and physical domains we give a detailed description of the proposed technique and show some examples also we present some examples of the application of isogeometric analysis in geometries parameterized with our method

strong functional representation lemma and applications to coding theorems this paper shows that for any random variables x and y it is possible to represent y as a function of x z such that z is independent of x and i x z y le log i x y 1 4 bits we use this strong functional representation lemma sfrl to establish a bound on the rate needed for one shot exact channel simulation for general discrete or continuous random variables strengthening the results by harsha et al and braverman and garg and to establish new and simple achievability results for one shot variable length lossy source coding multiple description coding and gray wyner system we also show that the sfrl can be used to reduce the channel with state noncausally known at the encoder to a point to point channel which provides a simple achievability proof of the gelfand pinsker theorem

an automatic adaptive meshing technique for delaunay triangulations abstract a new algorithm is described for adaptivity of delaunay triangulations it distinguishes between regions of mesh according to whether they are to be refined unaltered or coarsened the method automatically identifies nodes of the mesh which are candidates for deletion for a region to be coarsened a node deletion process is proposed and it is proved that the resulting triangulation is delaunay after coarsening the mesh retains a high quality for two reasons the first is because a set of nodes have been used to obtain the coarse mesh which are equi distant apart in some sense defined by the original mesh the second is because the triangulation is delaunay the mesh refinement procedure sub divides elements into a number of similar elements in the regions to be refined two dimensional examples demonstrate the quality of the mesh after coarsening refinement

simultaneous untangling and smoothing of moving grids in this work a technique for simultaneous untangling and smoothing of meshes is presented it is based on an extension of an earlier mesh smoothing strategy developed to solve the computational mesh dynamics stage in fluid structure interaction problems in moving grid problems mesh untangling is necessary when element inversion happens as a result of a moving domain boundary the smoothing strategy formerly published by the authors is defined in terms of the minimization of a functional associated with the mesh distortion by using a geometric indicator of the element quality this functional becomes discontinuous when an element has null volume making it impossible to obtain a valid mesh from an invalid one to circumvent this drawback the functional proposed is transformed in order to guarantee its continuity for the whole space of nodal coordinates thus achieving the untangling technique this regularization depends on one parameter making the recovery of the original functional possible as this parameter tends to 0 this feature is very important consequently it is necessary to regularize the functional in order to make the mesh valid then it is advisable to use the original functional to make the smoothing optimal finally the simultaneous untangling and smoothing technique is applied to several test cases including 2d and 3d meshes with simplicial elements as an additional example the application of this technique to a mesh generation case is presented copyright 2008 john wiley sons ltd

a posteriori error estimation of hierarchical type for the schrodinger operator with inverse square potential we develop an a posteriori error estimate for mixed boundary value problems of the form delta fancyscript v u f v u f where the potential fancyscript v v may possess inverse square singularities at finitely many points in the domain we prove that our error estimate can be efficiently computed and is asymptotically identical to the actual error in the energy norm on a family of geometrically graded meshes appropriate for singular solutions of such problems therefore our estimate can be used for a practical stopping criterion a variety of numerical experiments support our theoretical results we also offer a direct convergence and effectivity comparison between the geometrically graded meshes which are based on a priori knowledge of possible singularities in the solution and adaptively refined meshes driven by local error indicators associated with our a posteriori error estimate

a geometrical approach to mesh smoothing we study the problem of smoothing finite element meshes of triangles and tetrahedra where vertices are recursively moved to improve the overall quality of the elements with respect to a given shape quality metric we propose a geometric approach to solving the local optimization problem level sets of the given metric are used to characterize the set of optimal point s we also introduce a new mesh quality metric for tetrahedra

second order finite element approximations and a posteriori error estimation for two dimensional parabolic systems we discuss an adaptive local refinement finite element method of lines for solving vector systems of parabolic partial differential equations on two dimensional rectangular regions the partial differential system is discretized in space using a galerkin approach with piecewise eight node serendipity functions an a posteriori estimate of the spatial discretization error of the finite element solution is obtained using piecewise fifth degree polynomials that vanish on the edges of the rectangular elements of a grid ordinary differential equations for the finite element solution and error estimate are integrated in time using software for stiff differential systems the error estimate is used to control a local spatial mesh refinement procedure in an attempt to keep a global measure of the error within prescribed limits examples appraising the accuracy of the solution and error estimate and the computational efficiency of the procedure relative to one using bilinear finite elements are presented

a posteriori error estimate techniques for coupled navier stokes equations and energy equation abstract in this paper we discuss how to enhance the adaptive finite element simulation of compressible navier stokes flow via a posteriori error estimate analysis for the coupled navier stokes equations and energy equation the efficiency of this study is shown through numerical experiments corresponding to a supersonic flow around naca0012 airfoil

mixed type duality for multiobjective variational problems the concept of mixed type duality has been extended to the class of multiobjective variational problems a number of duality relations are proved to relate the efficient solutions of the primal and its mixed type dual problems the results are obtained for convex generalized convex functions these studies have been generalized to the case of invex generalized invex functions our results apparently generalize a fairly large number of duality results previously obtained for finite dimensional nonlinear programming problems under various convexity assumptions

a posteriori error estimates based on hierarchical bases the authors present an analysis of an a posteriors error estimator based on the use of hierarchical basis functions the authors analyze nonlinear nonselfadjoint and indefinite problems as well as the selfadjoint positive definite case because both the analysis and the estimator itself are quite simple it is easy to see how various approximations affect the quality of the estimator as examples the authors apply the theory to some scalar elliptic equations and the stokes system of equations

adaptive triangular quadrilateral mesh generation in this paper we begin by recalling an adaptive mesh generation method governed by isotropic and anisotropic discrete metric maps by means of the generation of a unit mesh with respect to a riemannian structure we propose then an automatic triangular to quadrilateral mesh conversion scheme which generalizes the standard case to the anisotropic context in addition we introduce an optimal vertex smoothing procedure application test examples in particular a cfd test are given to demonstrate the eciency of the proposed method 1998 john wiley sons ltd int j numer meth engng 41 915 934 1998

on optimal triangular meshes for minimizing the gradient error construction of optimal triangular meshes for controlling the errors in gradient estimation for piecewise linear interpolation of data functions in the plane is discussed using an appropriate linear coordinate transformation rigorously optimal meshes for controlling the error in quadratic data functions are constructed it is shown that the transformation can be generated as a curvilinear coordinate transformation for anyc data function with nonsingular hessian matrix using this transformation a construction of nearly optimal meshes for general data functions is described and the error equilibration properties of these meshes discussed in particular it is shown that equilibration of errors is not a sufficient condition for optimality a comparison of meshes generated under several different criteria is made and their equilibrating properties illustrated

an analysis of simplex shape measures for anisotropic meshes this paper analyses and generalises several simplex shape measures documented in the literature and currently used for mesh adaptation and mesh optimisation it first summarises important properties of simplices and their degeneration in euclidean space different simplex shape measures are then defined and validated according to a validity criterion the shape measures are generalised to riemannian spaces in order to extend their use to anisotropic meshes they are then analysed and compared using complementary approaches a visualisation method helping to show their regularity some theoretical relations establishing their equivalence and a discussion on the evaluation of the global quality of a mesh conclusions are drawn on the choice of a simplex shape measure to guide mesh optimisation

optimization of a regularized distortion measure to generate curved high order unstructured tetrahedral meshes we present a robust method for generating high order nodal tetrahedral curved meshes the approach consists of modifying an initial linear mesh by first introducing high order nodes second displacing the boundary nodes to ensure that they are on the cad surface and third smoothing and untangling the mesh obtained after the displacement of the boundary nodes to produce a valid curved high order mesh the smoothing algorithm is based on the optimization of a regularized measure of the mesh distortion relative to the original linear mesh this means that whenever possible the resulting mesh preserves the geometrical features of the initial linear mesh such as shape stretching and size we present several examples to illustrate the performance of the proposed algorithm furthermore the examples show that the implementation of the optimization problem is robust and capable of handling situations in which the mesh before optimization contains a large number of invalid elements we consider cases with polynomial approximations up to degree ten large deformations of the curved boundaries concave boundaries and highly stretched boundary layer elements the meshes obtained are suitable for high order finite element analyses

local and parallel finite element algorithms based on two grid discretizations a number of new local and parallel discretization and adaptive nite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems these algorithms are motivated by the observation that for a solution to some elliptic problems low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a ne grid by some local and parallel procedure the theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for nite element solutions on general shape regular grids some numerical experiments are also presented to support the theory in this paper we will propose some new parallel techniques for nite element computation these techniques are based on our understanding of the local and global properties of a nite element solution to some elliptic problems simply speaking the global behavior of a solution is mostly governed by low frequency components while the local behavior is mostly governed by high frequency compo nents the main idea of our new algorithms is to use a coarse grid to approximate the low frequencies and then to use a ne grid to correct the resulted residue which contains mostly high frequencies by some local parallel procedures let us now give a somewhat more detailed but informal and hopefully infor mative description of the main ideas and results in this paper we consider the following very simple model problem posed on a convex polygonal domain r 2 u bru f in

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

asymptotically exact a posteriori error estimators part ii general unstructured grids in part i of this work siam j numer anal 41 2003 pp 2294 2312 we analyzed superconvergence for piecewise linear finite element approximations on triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms in this work we consider superconvergence for general unstructured but shape regular meshes we develop a postprocessing gradient recovery scheme for the finite element solution uh inspired in part by the smoothing iteration of the multigrid method this recovered gradient superconverges to the gradient of the true solution and becomes the basis of a global a posteriori error estimate that is often asymptotically exact next we use the superconvergent gradient to approximate the hessian matrix of the true solution and form local error indicators for adaptive meshing algorithms we provide several numerical examples illustrating the effectiveness of our procedures

error exponents of typical random codes we define the error exponent of the typical random code trc as the long block limit of the negative normalized expectation of the logarithm of the error probability of the random code as opposed to the traditional random coding error exponent which is the limit of the negative normalized logarithm of the expectation of the error probability for the ensemble of uniformly randomly drawn fixed composition codes we provide exact error exponents of trcs for a general discrete memoryless channel and a wide class of stochastic decoders collectively referred to as the generalized likelihood decoder gld this ensemble of fixed composition codes is shown to be no worse than any other ensemble of independent codewords that are drawn under a permutation invariant distribution e g independent identically distributed codewords we also present relationships between the error exponent of the trc and the ordinary random coding error exponent as well as the expurgated exponent for the gld finally we demonstrate that our analysis technique is applicable also to more general communication scenarios such as list decoding for fixed size lists as well as the decoding with an erasure list option in forney s sense

adaptive remeshing for compressible flow computations an adaptive mesh procedure for improving the quality of steady state solutions of the euler equations in two dimensions is described the procedure is implemented in conjunction with a finite element solution algorithm using linear triangular elements and an explicit time stepping scheme the mesh adaptation is accomplished by regeneration using information provided by the computed solution on the current mesh the meshes produced are characterised by the appearance of stretched elements in the vicinity of one dimensional flow features and can exhibit a large variation in element size this allows solutions of enhanced quality to be produced in a computationally efficient manner