regularity of the hamiltonian along optimal trajectories this paper concerns state constrained optimal control problems in which the dynamic constraint takes the form of a differential inclusion if the differential inclusion does not depend on time then the hamiltonian evaluated along the optimal state trajectory and the co state trajectory is independent of time if the differential inclusion is lipschitz continuous then the hamiltonian evaluated along the optimal state trajectory and the co state trajectory is lipschitz continuous these two well known results are examples of the following principle the hamiltonian evaluated along the optimal state trajectory and the co state trajectory inherits the regularity properties of the differential inclusion regarding its time dependence we show that this principle also applies to another kind of regularity if the differential inclusion has bounded variation w r t time then the hamiltonian evaluated along the optimal state trajectory and the co state trajectory has bounded variation two application

communication complexity of the gaussian elimination algorithm on multiprocessors abstract this paper proposes a few lower bounds for communication complexity of the gaussian elimination algorithm on multiprocessors three types of architectures are considered a bus architecture a nearest neighbor ring network and a nearest neighbor grid network it is shown that for the bus and the ring architectures the minimum communication time is o n2 independent of the number of processors while for the grid it is reduced to o k built 1 2 n2 o k built1 2 n for a lock step gaussian elimination algorithm and to o k built 1 2 n2 o k built1 2 for any pipelined gaussian elimination algorithm where k is the total number of processors the practical implications of these bounds are discussed

an a posteriori error estimator for the p and hp versions of the finite element method an a posteriori error estimator is proposed in this paper for the p and hp versions of the finite element method in two dimensional linear elastostatic problems the local error estimator consists in an enhancement of an error indicator proposed by bertoti and szabo int j numer meth engng 1998 42 561 587 which is based on the minimum complementary energy principle in order to obtain the local error estimate this error indicator is corrected by a factor which depends only on the polynomial degree of the element the proposed error estimator shows a good effectivity index in meshes with uniform and non uniform polynomial distributions especially when the global error is estimated furthermore the local error estimator is reliable enough to guide p and hp adaptive refinement strategies

a parallel boundary element formulation for determining effective properties of heterogeneous media this paper presents a parallel implementation of the boundary element method for mimd computer architectures to determine the effective properties of two heterogeneous physical systems the first physical system is comprised of spheres sedimenting in a viscous fluid at low reynolds numbers the effective property is characterized by the hindered settling function which is a measure of the average sedimentation velocity the second physical system is a short fibre reinforced composite the effective property for this system is the composite modulus the determination of effective properties of heterogeneous media requires performing statistical analyses of several realizations of physical systems based on defining characteristics of the media the boundary element method is particularly well suited for studying such systems because of the simplification in the discretization associated with the method however as the number of heterogeneities to be modeled is increased so are the computational demands parallel computation offers the opportunity to model systems of greater complexity we discuss a parallel boundary element formulation based on the torus wrap mapping in this approach blocks of the coefficient matrix associated with the discretized boundary element equations are assigned to processors as opposed to more traditional parallel boundary element implementations where rows or columns are assigned to processors the torus wrap mapping can be shown to minimize communication volume between processors during the lu factorization therefore the present formulation scales well with increases in the number of processors

hamilton jacobi bellman equations for optimal control processes with convex state constraints abstract this work aims at studying some optimal control problems with convex state constraint sets it is known that for state constrained problems and when the state constraint set coincides with the closure of its interior the value function satisfies a hamilton jacobi equation in the constrained viscosity sense this notion of solution has been introduced by h m soner 1986 and provides a characterization of the value functions in many situations where an inward pointing condition ipc is satisfied here we first identify a class of control problems where the constrained viscosity notion is still suitable to characterize the value function without requiring the ipc moreover we generalize the notion of constrained viscosity solutions to some situations where the state constraint set has an empty interior

infinite horizon problems on stratifiable state constraints sets this paper deals with a state constrained control problem it is well known that unless some compatibility condition between constraints and dynamics holds the value function has not enough regularity or can fail to be the unique constrained viscosity solution of a hamilton jacobi bellman hjb equation here we consider the case of a set of constraints having a stratified structure under this circumstance the interior of this set may be empty or disconnected and the admissible trajectories may have the only option to stay on the boundary without possible approximation in the interior of the constraints in such situations the classical pointing qualification hypothesis are not relevant the discontinuous value function is then characterized by means of a system of hjb equations on each stratum that composes the state constraints this result is obtained under a local controllability assumption which is required only on the strata where some chattering phenomena could occur

error estimation and adaptive meshing in strongly nonlinear dynamic problems we present work aimed at developing a general framework for mesh adaption in strongly nonlinear possibly dynamic problems we begin by showing that the solutions of the incremental boundary value problem for a wide class of materials including nonlinear elastic materials compressible newtonian fluids and viscoplastic solids obey a minimum principle provided that the constitutive updates are formulated appropriately this minimum principle can be taken as a basis for asymptotic error estimation in particular we chose to monitor the error of a lower order projection of the finite element solution the optimal mesh size distribution then follows from a posteriori error indicators which are purely local i e can be computed element by element we demonstrate the robustness and versatility of the computational framework with the aid of convergence studies and selected examples of application

lipschitz regularity of the minimum time function of differential inclusions with state constraints abstract for control systems the local regularity of the minimum time function m i n in the absence of state constraints has been extensively studied and related both to inward pointing conditions and to small time controllability in the neighborhood of a closed target c in the presence of state constraints assessing this regularity is crucial to ensure the existence of solutions when perturbing the initial condition in this paper we prove without imposing the inclusion c int k that for differential inclusions with closed state constraints k and under general assumptions m i n is locally lipschitz continuous on its domain which is open in k we discuss as well extensions to nonautonomous systems and to point targets

l infty l estimates on trajectories confined to a closed subset for control systems with bounded time variation the term distance estimate for state constrained control systems refers to an estimate on the distance of an arbitrary state trajectory from the subset of state trajectories that satisfy a given state constraint distance estimates have found widespread application in state constrained optimal control they have been used to establish regularity properties of the value function to establish the non degeneracy of first order conditions of optimality and to validate the characterization of the value function as a unique solution of the hjb equation the most extensively applied estimates of this nature are so called linear l infty distance estimates the earliest estimates of this nature were derived under hypotheses that required the multifunctions or controlled differential equations describing the dynamic constraint to be locally lipschitz continuous w r t the time variable recently it has been shown that the lipschitz continuity hypothesis can be weakened to a one sided absolute continuity hypothesis this paper provides new less restrictive hypotheses on the time dependence of the dynamic constraint under which linear l infty estimates are valid here one sided absolute continuity is replaced by the requirement of one sided bounded variation this refinement of hypotheses is significant because it makes possible the application of analytical techniques based on distance estimates to important new classes of discontinuous systems including some hybrid control systems a number of examples are investigated showing that for control systems that do not have bounded variation w r t time the desired estimates are not in general valid and thereby illustrating the important role of the bounded variation hypothesis in distance estimate analysis

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

degenerate optimal control problems with state constraints standard necessary conditions for optimal control problems with pathwise state constraints supply no useful information about minimizers in a number of cases of interest e g when the left endpoint of state trajectories is fixed at x0 and x0 lies in the boundary of the state constraint set in these cases a nonzero but nevertheless trivial set of multipliers exists we give conditions for the existence of nontrivial multipliers a feature of these conditions is that they allow nonconvex velocity sets and measurably time dependent data the proof techniques are based on refined estimates of the distance of a given state trajectory from the set of state trajectories satisfying the state constraint originating in the dynamic programming literature

fully reliable error control for first order evolutionary problems abstract this work is focused on the application of functional type a posteriori error estimates and corresponding indicators to a class of time dependent problems we consider the algorithmic part of their derivation and implementation and also discuss the numerical properties of these bounds that comply with obtained numerical results this paper examines two different methods of solution approximation for evolutionary models i e a time marching technique and a space time approach the first part of the study presents an algorithm for global minimisation of the majorant on each of discretisation time cylinders time slabs the effectiveness of this approach to error estimation is confirmed by extensive numerical tests in the second part of the publication the application of functional error estimates is discussed with respect to a space time approach it is followed by a set of extensive numerical tests that demonstrates the efficiency of proposed error control method the numerical results obtained in this paper rely on the implementation carried out using open source software which allows formulating the problem in a weak setting to work in a variational framework appears to be very natural for functional error estimates due to their derivation method the search for the optimal parameters for the majorant is done by a global functional minimisation which to the authors knowledge is the first work using this technique in an evolutionary framework

an asymptotically exact pointwise a posteriori error estimator for linear elliptic one dimensional problems with possible singularities in the solution abstract when the finite element solution of a variational problem possesses certain superconvergence properties it is possible very inexpensively to obtain a correction term providing an additional order of approximation of the solution the correction can be used for error estimation locally or globally in whatever norm is preferred or if no error estimation is wanted it can be used for postprocessing of the solution to improve the quality in this paper such a correction term is found and its convergence properties proved in the special case of one dimensional linear elliptic problems special attention is given to the effects of the appearance of singularities and zeros of derivatives in the exact solution

assignment and scheduling in parallel matrix factorization abstract we consider the problem of factoring a dense n n matrix on a network consisting of p mimd processors with no shared memory when the network is smaller than the number of elements in the matrix p n 2 the specific example analyzed is a computational network that arises in computing the lu qr or cholesky factorizations we prove that if the nodes of the network are evenly distributed among processors and if computations are scheduled by a round robin or a least recently executed scheduling algorithm then optimal order of speedup is achieved however such speedup is not necessarily achieved for other scheduling algorithms or if the computation for the nodes is inappropriately split across processors and we give examples of these phenomena lower bounds on execution time for the algorithm are established for two important node assignment strategies

element by element a posteriori error estimation and improvement of stress solutions for two dimensional elastic problems a quantitative method of a posteriori error estimation based on finite element solutions is developed the proposed method can estimate the error of solutions obtained by the finite element method for two dimensional elastic problems with 4 node elements since the error is estimated element by element the proposed method does not require a large memory or long computing time not only rectangular elements but also arbitrarily shaped 4 node elements can be used in this method for estimating the error in the finite element computation with high accuracy the finite element solutions are improved by adding the estimated errors onto the original solutions the proposed method can be utilized for any type of linear problem if an isoparametric finite element method is used

error estimation and adaptivity in elastodynamics abstract the paper discusses adaptive procedures for elastodynamic problems using semi discrete finite elements the adaptivity in space may be done by mesh refinement h version by increase of order of the approximation polynomials p version or a combination of these hp version in time the integration is either made by mode superposition of characteristic functions or by direct integration the adaptation in time may either be made by change of the global time step h version or increase of the order of approximation polynomials p version crucial parts of the analysis are the error estimator and the error indicator which are based either on interpolation theory or on total energy obtained from postprocessed stresses for the mesh generation and regeneration an isoline technique is utilized by which the mesh can be smoothed by interpolation theory the isolines are created for some characteristic scalar function for time discretization a new simple a posteriori local error estimator for the newmark scheme is described it is derived by a post processing technique without solving new equations so the additional computational cost is small and the implementation is convenient the paper also describes a sequence of nested time integration methods for dynamic problems formulated as two first order equations in time these nested integration methods in time based on a hierarchical formulation in time are a stable schemes of order 2k and l stable schemes of order 2k 1 a considerable improvement in terms of accuracy as well as effectiveness is obtained compared to currently available methods

modeling of impact problems using an h adaptive explicit lagrangian finite element method in three dimensions abstract an h adaptive algorithm was developed to enhance modeling of 3d impact problems the algorithm utilizes a recovery based error indicator involving the root mean square error of the green lagrange strain this indicator proved effective for identifying refinements in the contact region which lead to improved solution accuracy the effectiveness of the algorithm is demonstrated for both a reverse taylor impact test and a ballistic penetration problem both the algorithm and implementation issues are discussed herein

on the techniques for constructing admissible stress fields in model verification performances on engineering examples robust global goal oriented error estimation is used nowadays to control the approximate finite element solutions obtained from simulation in the context of computational mechanics the construction of admissible stress fields ie stress tensors which verify the equilibrium equations is required to set up strict and guaranteed error bounds using residual based error estimators and plays an important role in the quality of the error estimates this work focuses on the different procedures used in the calculation of admissible stress fields which is a crucial and technically complicated point the three main techniques that currently exist called the element equilibration technique eet the star patch equilibration technique spet and the element equilibration star patch technique eespt are investigated and compared with respect to three different criteria namely the quality of associated error estimators computational cost and easiness of practical implementation into commercial finite element codes the numerical results which are presented focus on industrial problems they highlight the main advantages and drawbacks of the different methods and show that the behavior of the three estimators which have the same convergence rate as the exact global error is consistent two and three dimensional experiments have been carried out in order to compare the performance and the computational cost of the three different approaches the analysis of the results reveals that the spet is more accurate than eet and eespt methods but the corresponding computational cost is higher overall the numerical tests prove the interest of the hybrid method eespt and show that it is a correct compromise between quality of the error estimate practical implementation and computational cost furthermore the influence of the cost function involved in the eet and the eespt is studied in order to optimize the estimators

measurable viability theorems and the hamilton jacobi bellman equation abstract we prove viability and invariance theorems for systems with dynamics depending on time in a measurable way and having time dependent state constraints x t f t x t x t p t in the above t p t is an absolutely continuous set valued map and t x f t x is a set valued map which is measurable with respect to t and upper semicontinuous or continuous or locally lipschitz with respect to x for this aim we investigate infinitesimal generators of reachable maps and the lebesgue points of set valued maps the results are applied to define and to study lower semicontinuous solutions of the hamilton jacobi bellman equation u t h t x u x 0 with the hamiltonian h measurable with respect to time locally lipschitz with respect to x and convex in the last variable

filippov s and filippov wazewski s theorems on closed domains abstract the celebrated filippov s theorem implies that given a trajectory x1 0 r n of a differential inclusion x f t x with the set valued map f measurable in t and k lipschitz in x for any initial condition x2 0 r n there exists a trajectory x2 starting from x2 0 such that x1 t x2 t ekt x1 0 x2 0 filippov wazewski s theorem establishes the possibility of approximating any trajectory of the convexified differential inclusion x co f t x by a trajectory of the original inclusion x f t x starting from the same initial condition in the present paper we extend both theorems to the case when the state variable x is constrained to the closure of an open subset r n the latter is allowed to be non smooth we impose a generalized soner type condition on f and yielding extensions of the above classical results to infinite horizon constrained problems applications to the study of regularity of value functions of optimal control problems with state constraints are discussed as well

a recovery type error estimator for the extended finite element method based on singular smooth stress field splitting a new stress recovery procedure that provides accurate estimations of the discretization error for linear elastic fracture mechanic problems analyzed with the extended finite element method xfem is presented the procedure is an adaptation of the superconvergent patch recovery spr technique for the xfem framework it is based on three fundamental aspects a the use of a singular smooth stress field decomposition technique involving the use of different recovery methods for each field standard spr for the smooth field and reconstruction of the recovered singular field using the stress intensity factor k for the singular field b direct calculation of smoothed stresses at integration points using conjoint polynomial enhancement and c assembly of patches with elements intersected by the crack using different stress interpolation polynomials at each side of the crack the method was validated by testing it on problems with an exact solution in mode i mode ii and mixed mode and on a problem without analytical solution the results obtained showed the accuracy of the proposed error estimator copyright 2008 john wiley sons ltd

a unified approach to remeshing strategies for finite element h adaptivity in h adaptivity a remeshing strategy is needed to compute the distribution of required element size using the estimated error distribution several authors have introduced different remeshing strategies yielding very different results in this work these methods are included in a unified framework emphasizing the role of the underlying hypotheses moreover an objective tool to evaluate the accuracy of the resulting finite element solution is presented thus a new remeshing strategy is introduced to optimize the accuracy of the adapted solutions the different remeshing strategies are compared with well known numerical examples

linear estimates for trajectories of state constrained differential inclusions and normality conditions in optimal control abstract we consider solutions to a differential inclusion x f x constrained to a compact convex set here f is a compact possibly non convex valued lipschitz continuous multifunction whose convex closure co f satisfies a strict inward pointing condition at every boundary point x given a reference trajectory x taking values in an e neighborhood of we prove the existence of a second approximating trajectory x 0 t which satisfies the linear estimate x x ac 0 t k e if one of the following two cases occurs i the initial datum x 0 x 0 is given but lies in a compact set containing only points where the boundary is smooth ii the initial point x 0 of the approximating trajectory x can be chosen arbitrarily subsequently we employ these linear ac estimates to establish conditions for normality of the generalized euler lagrange condition for optimal control problems with state constraints in which we have an integral term in the cost we finally provide an illustrative example which underlines the fact that if conditions i and ii above are not satisfied then we can find a degenerate minimizer

on weak residual error estimation a general framework for weak residual error estimators applying to various types of boundary value problems in connection with finite element and finite volume approximations is developed basic ideas commonly shared by various applications in error estimation and adaptive computation are presented and illustrated some numerical results are given to show the effectiveness and efficiency of the estimators

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

an adaptive algorithm for the crank nicolson scheme applied to a time dependent convection diffusion problem an a posteriori upper bound is derived for the nonstationary convection diffusion problem using the crank nicolson scheme and continuous piecewise linear stabilized finite elements with large aspect ratio following lozinski et al 2009 13 a quadratic time reconstruction is used a space and time adaptive algorithm is developed to ensure the control of the relative error in the l 2 h 1 norm numerical experiments illustrating the efficiency of this approach are reported it is shown that the error indicator is of optimal order with respect to both the mesh size and the time step even in the convection dominated regime and in the presence of boundary layers

on the adaptive finite element method of steady state rolling contact for hyperelasticity in finite deformations in this paper we present an adaptive finite element method for steady state rolling contact in finite deformations along with a residual based a posteriori error estimator for rolling contact problem with coulomb friction a general formulation of rolling contact geometry is derived from the point of view of differential geometry solvability conditions for the rolling contact problems are discussed we use newton s method to solve variational equations derived from a penalty regularization of the finite element approximation of the rolling contact problem we provide a numerical example to illustrate the method

mid node admissible spaces for quadratic triangular 2d finite elements with one edge curved a robust element distortion metric based on the new concept of mid node admissible spaces for two dimensional quadratic triangular finite elements is developed the metric is based on the jacobian determinant over the entire element without requiring that it actually be computed everywhere on the element the metric is relatively inexpensive to compute especially for mildly distoted elements the metric is able to detect elementsof poor quality that other distortion metrics fall to detect it also has the ability to approve elements of good quality regardless of the extent to which they may appear geometrically distorted copyright 2001 john wiley sons ltd

time and space adaptive methods applied to localization phenomena in empty and saturated micropolar and standard porous materials localization phenomena as e g shear bands occur as a result of local concentrations of plastic strains in small bands of finite width the reason for this behaviour lies in the basic properties of elasto plastic frictional materials where variations of the lode angle together with the non associated plastic dilatation result in local softening effects proceeding from non viscous empty porous materials it is well known that the computation of shear bands reveals an ill posed problem to overcome this behaviour two different regularization strategies are considered which are 1 the inclusion of independent rotational degrees of freedom micropolar formulation and 2 the inclusion of viscosity effects standard formulation by taking into consideration viscoplastic properties of the solid matrix or a viscous pore fluid respectively the numerical treatment furthermore proceeds from time and space adaptive strategies to refine and to coarsen both the time step and the mesh size considering quasi static problems the space discretization yields a dae system of index 1 in the time domain which can be successfully treated by sdirk methods thus allowing for an efficient estimation of the time error and as a consequence for an adaptive time step control in the space domain the present investigation proceeds from an error indicator of zienkiewicz zhu type where smoothened values of the l2 norms of characteristic quantities are compared to the respective discrete values the efficiency of this procedure is demonstrated by the computation of the biaxial experiment on an empty and a liquid saturated porous soil material and of the slope failure problem of a fluid saturated soil copyright 2001 john wiley sons ltd

adaptive variational multiscale methods for incompressible flow based on two local gauss integrations we consider variational multiscale vms methods with h adaptive technique for the stationary incompressible navier stokes equations the natural combination of vms with adaptive strategy retains the best features of both methods and overcomes many of their deficits a reliable a posteriori projection error estimator is derived which can be computed by two local gauss integrations at the element level finally some numerical tests are presented to illustrate the method s efficiency

on error estimates and adaptivity in elastoplastic solids applications to the numerical simulation of strain localization in classical and cosserat continua the a posteriori error estimates based on the post processing approach are introduced for elastoplastic solids the standard energy norm error estimate established for linear elliptic problems is generalized here to account for the presence of internal variables through the norm associated with the complementary free energy this is known to represent a natural metric for the class of elastoplastic problems of evolution in addition the intrinsic dissipation functional is utilized as a basis for a complementary a posteriori error estimates a posteriori error estimates and adaptive refinement techniques are applied to the finite element analysis of a strain localization problem as a model problem the constitutive equations describing a generalization of standard j2 elastoplasticity within the cosserat continuum are used to overcome serious limitations exhibited by classical continuum models in the post instability region the proposed a posteriori error estimates are appropriately modified to account for the cosserat continuum model and linked with adaptive techniques in order to simulate strain localization problems superior behaviour of the cosserat continuum model in comparison to the classical continuum model is demonstrated through the finite element simulation of the localization in a plane strain tensile test for an elastopiastic softening material resulting in convergent solutions with an h refinement and almost uniform error distribution in all considered error norms

a posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem the reciprocal theorem of betti and rayleigh or in other notions duality arguments are used to derive error estimators for the finite element approximation of various quantities including local variables like single displacements and stresses the proposed error estimator is evaluated solving the set of equations for an additional right hand side simply applying the energy norm error estimators two times furthermore a general h adaptive algorithm is introduced which allows us to optimize meshes with respect to different user specified variables the efficiency of the current approach is demonstrated for plate and shell examples

automated adaptive two dimensional system for the hp version of the finite element method this paper presents an automated adaptive finite element modelling system based on the hp version of the finite element method error estimation and correction indication procedures with an automatic mesh generator and a p version analysis code are combined to provide a computationally efficient modelling environment error prediction techniques are employed to find the best combination of finite element discretization h and overall polynomial order p needed to achieve the prescribed level of accuracy the automated adaptive modelling results demonstrate the capability of the modelling system of predicting discretizations to perform a p version finite element analysis to achieve prescribed levels of accuracy in the most efficient manner

optimal control with state space constraint i optimal control of piecewise deterministic processes with state space constraint is studied under appropriate assumptions it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain finally the uniform continuity of the value function is obtained under a condition on the deterministic drift

iterative method for constrained and mixed approximation an inexpensive improvement of f e m performance a general iterative procedure for solving mixed lagrangian finite element equations is introduced the method allows a dramatic improvement in accuracy of simple irreducible formulations to be achieved in a few resolutions applications to incompressible flow stress analysis and metal forming problems are given the procedures can be easily implemented into standard programs

sensitivity interpretations of the costate variable for optimal control problems with state constraints in optimal control theory it is well known that the costate arc and the associated maximized hamiltonian function can be interpreted in terms of gradients of the value function evaluated along the optimal state trajectory such relations have been referred to as sensitivity relations in the literature we provide in this paper new sensitivity relations for state constrained optimal control problems for the class of optimal control problems considered there is no guarantee that the costate arc is unique a key feature of the results is that they assert some choice of costate arc can be made for which the sensitivity relations are valid the proof technique is to introduce an auxiliary optimal control problem that possesses a richer set of control variables than the original problem the introduction of the additional control variables in effect enlarges the class of variations with respect to which the state trajectory under consideration is a minimizer the extra information thereby obtained yields the desired set of sensitivity relations

estimates for trajectories confined to a cone in mathbb r lowercase n we develop estimates on the distance of a trajectory associated with a differential inclusion dot x in f in mathbb r n to a given set of feasible f trajectories where feasible means with values confined to a given cone when the cone is a half space it is known that a linear k h x estimate of the mathbf w 1 1 distance is valid where k is a constant independent of the initial choice of trajectory and h x is a measure of the constraint violation a recent counterexample has unexpectedly demonstrated that linear estimates of the distance are no longer valid when the state constraint set is the intersection of two half spaces this paper addresses fundamental questions concerning the approximation of general f trajectories by f trajectories confined to a cone which is the intersection of two half spaces in this context we establish the validity of a superlinear k h x ln h x estimate of the distance we demonstrate by means of an example that the structure of this estimate is optimal we show furthermore that a linear estimate can be recovered in the case when the velocity set f is strictly convex

trajectories of differential inclusions with state constraints abstract the paper deals with solutions of a differential inclusion x f x constrained to a compact convex set here f is a compact possibly non convex valued lipschitz continuous multifunction whose convex closure co f satisfies a strict inward pointing condition at every boundary point x given a reference trajectory x taking values in an e neighborhood of we prove the existence of a second trajectory x 0 t which satisfies x x w 1 1 c e 1 ln e as shown by an earlier counterexample this bound is sharp

l estimates on trajectories confined to a closed subset this paper concerns the validity of estimates on the distance of an arbitrary state trajectory from the set of state trajectories which lie in a given state constraint set these so called distance estimates have wide spread application in state constrained optimal control including justifying the use of the maximum principle in normal form and establishing regularity properties of value functions we focus on linear l distance estimates which of all the available estimates have so far been the most widely used such estimates are known to be valid for general closed state constraint sets provided the functions defining the dynamic constraint are lipschitz continuous with respect to the time and state variables we ask whether linear l distance estimates remain valid when the lipschitz continuity hypothesis governing t dependence of the data is relaxed we show by counter example that these distance estimates are not valid in general if the hypothesis of lipschitz continuity is replaced by continuity we also provide a new hypothesis absolute continuity from the left for the validity of linear l estimates the new hypothesis is less restrictive than lipschitz continuity and even allows discontinuous time dependence in certain cases it is satisfied in particular by differential inclusions exhibiting non lipschitz t dependence at isolated points governed for example by a fractional power modulus of continuity the relevance of distance estimates for state constrained differential inclusions permitting fractional power time dependence is illustrated by an example in engineering design where we encounter an isolated square root type singularity concerning the t dependence of the data

finite element mesh control by integer programming a novel technique is presented for the control of mapped quadrilateral and hexahedral meshes generated inside a collection of connected subregions target division numbers on the subregion edges are given as input by allowing these numbers to vary meshes can be found which satisfy both the subregion constraints imposed by the mesh patterns being employed and the compatibility constraints on edges shared by two or more adjacent subregions among all such feasible solutions the optimal one is obtained by minimizing the changes to the division numbers using an integer programming technique

on trajectories satisfying a state constraint w 1 1 estimates and counterexamples this paper concerns properties of solutions to a differential inclusion f trajectories satisfying a state constraint estimates on the w 1 1 distance of a given f trajectory to the set of f trajectories satisfying the constraint have an important role in state constrained optimal control theory regarding the derivation of nondegenerate necessary conditions sensitivity analysis characterization of the value function in terms of the hamilton jacobi equation and other applications according to some of the earlier literature estimates in which the w 1 1 distance is related linearly to the degree of constraint violation of the original f trajectory are valid for state constraints defined by a collection of one or more inequality constraint functionals we show by counterexample that linear w 1 1 estimates are not in general valid for multiple state constraints we also identify cases involving several state constraints where not even a weaker linear l infty estimate holds we further show that it is possible to justify linear w 1 1 estimates by means of a modification of earlier constructive techniques when there is only one state constraint in a future companion paper estimates for trajectories confined to a cone in mathbb r n siam j control optim submitted we develop weaker estimates for multiple state constraints and identify additional hypotheses under which w 1 1 estimates are valid in this broader setting

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

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