reliable and robust a posteriori error estimation for singularly perturbed reaction diffusion problems problems with singular perturbations exhibit solutions with strong boundary layers and other types of local behavior such features lend themselves to adaptive solution methods the quality of any adaptive algorithm ultimately rests on the reliability and robustness of the a posteriori error control an estimator that has proved to be one of the most reliable is the equilibrated residual method the main property of the estimator is that it bounds the true error from above however the method is not robust in the singularly perturbed limit the current work generalizes the error estimator based on the equilibrated residuals and coincides with the standard method in the unperturbed limit it is shown that the new method is robust in the singularly perturbed limit while maintaining reliability yielding a guaranteed upper bound on the true error finally the application of the estimator to the problem of controlling the spatial error in rothe s method for the time discretization of a simple parabolic problem is included

a network design problem for a distribution system with uncertain demands a class of production distribution planning problems with nonstochastic uncertain demands is modeled as a dynamic game between two players who control flows on a network with node and arc capacity constraints simple conditions are derived for determining which player wins the game these conditions are then used to design a minimum cost network with the property that its feasible control strategies are allowed to meet the demand without violating the capacity constraints

on the existence of a feasible flow in a stochastic transportation network many transportation networks e g networks of cooperating power systems and hydrological networks involve a real valued demand function defined on the set of nodes and it is said to be feasible if there exists a flow such that at each node the sum of the incoming flow values is greater than or equal to the demand assigned to this node by the theorem of d gale and a hoffman a system of linear inequalities involving the demand and the arc capacity functions gives necessary and sufficient condition for the feasibility of the demand if the demands and or the arc capacities are random then an important problem is to find the probability that all these inequalities are satisfied this paper proposes a new method to eliminate all redundant inequalities for given lower and upper bounds of the demand function and finds sharp lower and upper bounds for the probability that a feasible flow exists the results can be used to support transportation network analysis and design

explicit upper bounds for dual norms of residuals we derive upper bounds for the dual norms of residuals that are explicit in terms of local poincare constants residuals are continuous linear functionals that are orthogonal to a finite element space and have a singular part supported on the skeleton of the underlying mesh functionals of this type play a key role in a posteriori error estimation our main tools are a discrete partition of unity and suitably weighted trace and poincare inequalities the technique is illustrated for negative first order sobolev norms and a dual norm arising in convection reaction diffusion problems

stabilized and inexact adaptive methods for capturing internal layers in quasilinear pde a method is developed within an adaptive framework to solve quasilinear diffusion problems with internal and possibly boundary layers starting from a coarse mesh the solution process is assumed to start on a mesh where the problem is badly resolved and approximation properties of the exact problem and its corresponding finite element solution do not hold a sequence of stabilized and inexact partial solves allows the mesh to be refined to capture internal layers while an approximate solution is built eventually leading to an accurate approximation of both the problem and its solution the innovations in the current work include a closed form definition for the numerical dissipation and inexact scaling parameters on each mesh refinement as well as a convergence result for the residual of the discrete problem numerical experiments demonstrate the method on a range of problems featuring steep internal layers and high solution dependent frequencies of the diffusion coefficients

practical stability and exponential estimates of trajectory bounds for retarded systems with bounded disturbances the properties of practical stability and exponential estimates of trajectory bounds for retarded systems with uncertain external disturbances are investigated our approach is adaptable to general regarded functional differential equations rfdes with bounded disturbances by applying our results to the lq regulator problem with uncertain input delay and bounded external disturbances it can be found that our results provide an effective method for quantitative stability analysis of retarded systems

a new perspective on hierarchical spline spaces for adaptivity abstract we introduce a framework for spline spaces of hierarchical type based on a parent children relation which is very convenient for the analysis as well as the implementation of adaptive isogeometric methods this framework exploits the innate refinement by functions in the b splines context rather than by elements or cells which is more natural in the finite element context furthermore it entails a new language to handle hierarchical spline spaces which allows to tackle fundamental questions in a very simple manner for example it makes it simple to create hierarchical bases with several desired properties with a refinement procedure which has linear complexity i e the resulting bases have cardinality bounded by the number of initially marked functions

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

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

a posteriori error estimates of finite element methods by preconditioning abstract we present a framework that relates preconditioning with a posteriori error estimates in finite element methods in particular we use standard tools in subspace correction methods to obtain reliable and efficient error estimators as a simple example we recover the classical residual error estimators for the second order elliptic equations

elliptic reconstruction and a posteriori error estimates for parabolic problems it is known that the energy technique for a posteriori error analysis of finite element discretizations of parabolic problems yields suboptimal rates in the norm l infty 0 t l 2 omega in this paper we combine energy techniques with an appropriate pointwise representation of the error based on an elliptic reconstruction operator which restores the optimal order and regularity for piecewise polynomials of degree higher than one this technique may be regarded as the dual a posteriori counterpart of wheeler s elliptic projection method in the a priori error analysis

a priori and a posteriori estimates of stabilized mixed finite volume methods for the incompressible flow arising in arteriosclerosis abstract the flow arising in arteriosclerosis is modeled by the incompressible flow with a slip boundary condition of friction type in this paper whose weak solution satisfies a variational inequality we develop and analyze the stabilized mixed finite volume methods for the resulting model we first achieve the same optimal order results of the finite volume methods as those of the corresponding finite element methods under the same regularity on the exact solution and a slightly additional regularity on the source term also a super close result between the respective solutions of the finite element methods and finite volume methods is derived for the first time for adaptivity we further derive a posterior estimate for the present problem satisfying the variational inequality which reduces the computational cost substantially and provides efficient error control for the solution finally numerical results are shown to support the developed convergence theory

a posteriori error estimation for generalized finite element methods in this paper we address the problem of a posteriori error estimation for generalized finite element methods based on the partition of unity method the computational results focus on the question of the reliability of the error estimators and its assessment

decay rates of adaptive finite elements with dorfler marking we investigate the decay rate for an adaptive finite element discretization of a second order linear symmetric elliptic pde we allow for any kind of estimator that is locally equivalent to the standard residual estimator this includes in particular hierarchical estimators estimators based on the solution of local problems estimators based on local averaging equilibrated residual estimators the zz estimator etc the adaptive method selects elements for refinement with dorfler marking and performs a minimal refinement in that no interior node property is needed based on the local equivalence to the residual estimator we prove an error reduction property in combination with minimal dorfler marking this yields an optimal decay rate in terms of degrees of freedom

a posteriori error estimates for nonlinear problems finite element discretizations of elliptic equations we give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems in a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual in a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite dimensional approximation of the residual this quantity can easily be evaluated and for many practical applications sharp explicit upper and lower bounds are readily obtained the general results are then applied to finite element discretizations of scalar quasi linear elliptic partial differential equations of 2nd order the eigenvalue problem for scalar linear elliptic operators of 2nd order and the stationary incompressible navier stokes equations they immediately yield a posteriori error estimates which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution

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

finite element discretizations of nonlocal minimal graphs convergence abstract in this paper we propose and analyze a finite element discretization for the computation of fractional minimal graphs of order s 0 1 2 on a bounded domain such a plateau problem of order s can be reinterpreted as a dirichlet problem for a nonlocal nonlinear degenerate operator of order s 1 2 we prove that our numerical scheme converges in w 1 2 r for all r s where w 1 2 s is closely related to the natural energy space moreover we introduce a geometric notion of error that for any pair of h 1 functions in the limit s 1 2 recovers a weighted l 2 discrepancy between the normal vectors to their graphs we derive error bounds with respect to this novel geometric quantity as well in spite of performing approximations with continuous piecewise linear lagrangian finite elements the so called stickiness phenomenon becomes apparent in the numerical experiments we present

p version fem for structural acoustics with a posteriori error estimation we demonstrate the advantages of using p version finite element approximations for structural acoustics problems in the mid to high frequency regime we then present a sub domain based a posteriori error estimation procedure to quantify the errors in the setting of a 3d interior acoustics problem with resonances and give numerical results effectivity indices show robust behavior of the error estimator away from the resonant frequencies

delay independent stability conditions and decay estimates for time varying functional differential equations this paper presents sufficient delay independent conditions that guarantee stability of nonlinear time varying functional differential equations fde s estimates on the rate of decay of solutions are also obtained

a variational multiscale a posteriori error estimation method for mixed form of nearly incompressible elasticity this paper presents an error estimation framework for a mixed displacement pressure finite element method for nearly incompressible elasticity the proposed method is based on variational multiscale vms concepts wherein the displacement field is decomposed into coarse scales that can be resolved by a given finite element mesh and fine scales that are beyond the resolution capacity of the mesh variational projection of fine scales onto the coarse scale space via variational embedding of the fine scale solution into the coarse scale formulation leads to the stabilized method with two major attributes first it is free of volumetric locking and second it accommodates arbitrary combinations of interpolation functions for the displacement and pressure fields this vms based stabilized method is equipped with naturally derived error estimators and offers various options for numerical computation of the error specifically two error estimators are explored the first method employs an element based strategy and a representation of error via a fine scale error equation defined over element interiors which is evaluated by a direct post solution evaluation this quantity when combined with the global pollution error results in a simple explicit error estimator the second method involves solving the fine scale error equation through localization to overlapping patches spread across the domain thereby leading to an implicit calculation of the local error this implicit calculation when combined with the global pollution error results in an implicit error estimator the performance of the stabilized method and the error estimators is investigated through numerical convergence tests conducted for two model problems on uniform and distorted meshes the sharpness and robustness of the estimators is shown to be consistent across the test cases employed

on estimators for eigenvalue eigenvector approximations we consider a large class of residuum based a posteriori eigen value eigenvector estimates and present an abstract framework for proving their asymptotic exactness equivalence of the estimator and the error is also established to demonstrate the strength of our abstract approach we present a detailed study of hierarchical error estimators for laplace eigenvalue problems in planar polygonal regions to this end we develop new error analysis for the galerkin approximation which avoids the use of the strengthened cauchy schwarz inequality and the saturation assumption and gives reasonable and explicitly computable upper bounds on the discretization error a brief discussion is also given concerning the design of estimators which are in the same spirit but are based on different a posteriori techniques notably those of gradient recovery type

a unified approach to a posteriori error estimation using element residual methods this paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations it is shown that by solving appropriate local residual type problems one can obtain upper bounds on the error in the energy norm moreover in the special case of adaptiveh p finite element analysis the estimator will also give a realistic estimate of the error a key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems the work extends and combines several existing methods to the case of fullh p finite element approximation on possibly irregular meshes with elements of non uniform degree as a special case the analysis proves a conjecture made by bank and weiser some a posteriori error estimators for elliptic partial differential equations math comput 44 283 301 1985

criteria for quantitative stability for a class of razumikhin type retarded functional differential equations the quantitative stability properties and trajectory bound estimates for a class of retarded functional differential equations rfdes which is characterized via razumikhin type conditions are investigated in this paper resulting in more generalized and improved criteria for rfdes

a posteriori error estimators gradient recovery by averaging and superconvergence for the linear finite element solution to a linear elliptic model problem we derive an error estimator based upon appropriate gradient recovery by local averaging in contrast to popular variants like the zz estimator our estimator contains some additional terms that ensure reliability also on coarse meshes moreover the enhanced estimator is proved to be locally efficient and asymptotically exact whenever the recovered gradient is superconvergent we formulate an adaptive algorithm that is directed by this estimator and illustrate its aforementioned properties as well as their importance in numerical tests

on the a posteriori error analysis for equations of prescribed mean curvature we present two approaches to the a posteriori error analysis for prescribed mean curvature equations the main difference between them concerns the estimation of the residual without or with computable weights in the second case the weights are related to the eigenvalues of the underlying operator and thus provide local and computable information about the conditioning we analyze the two approaches from a theoretical viewpoint moreover we investigate and compare the performance of the derived indicators in an adaptive procedure our theoretical and practical results show that it is advantageous to estimate the residual in a weighted way

a posteriori error estimation based on numerical realization of the variational multiscale method this paper presents a numerical realization of the variational multiscale method with the objective of providing a reliable and easy to implement local error estimation technique the variational multiscale framework provides a systematic approach for solution scale decomposition into coarse scales captured by the mesh and fine or subgrid scales in the proposed work the coarse scale errors in the finite element solution are neglected in comparison to the fine scale errors the fine scale variational equation is then localized using a general localization function over an element or a patch of elements to develop a local error estimation technique based on the proposed framework a consistent formulation of a new subdomain error estimator for elliptic problems is derived without the necessity of introducing an error locality assumption the new subdomain error estimator is evaluated numerically within a mesh adaptivity algorithm and it is shown to produce very sharp error estimates that outperform the element residual method estimates

a posteriori error estimators for regularized total variation of characteristic functions we consider a nonuniformly elliptic double obstacle problem arising from convexifying and regularizing a minimization of a functional with total variation in the set of characteristic functions we derive a posteriori estimators for the discretization error with linear finite elements which are uniform in the regularization incorporate computable and local information on the conditioning vanish in the intersection of discrete and exact contact sets and are not affected by possible nonuniqueness moreover we integrate these estimators in an adaptive algorithm and illustrate their properties by various numerical experiments

an improved method for solving quasi linear convection diffusion problems on a coarse mesh a method is developed for solving quasi linear convection diffusion problems starting on a coarse mesh where the data and solution dependent coefficients are unresolved the problem is unstable and approximation properties do not hold the newton like iterations of the solver are based on the framework of regularized pseudotransient continuation where the proposed time integrator is a variation on the newmark strategy designed to introduce controllable numerical dissipation and to reduce the fluctuation between the iterates in the coarse mesh regime where the data is rough and the linearized problems are badly conditioned and possibly indefinite an algorithm and updated marking strategy is presented to produce a stable sequence of iterates as boundary and internal layers in the data are captured by adaptive mesh partitioning the method is suitable for use in an adaptive framework making use of local error indicators to determine mesh refinement and targeted regularization derivation and q linear loc

partition of unity for localization in implicit a posteriori finite element error control for linear elasticity the partition of unity for localization in adaptive finite element method fem for elliptic partial differential equations has been proposed in carstensen and funken siam j sci comput 2000 21 1465 1484 and is applied therein to the laplace problem a direct adaptation to linear elasticity in this paper yields a first estimator l based on patch oriented local weighted interface problems the global korn inequality with a constant c korn yields reliability u u h korn l for any finite element approximation u h to the exact displacement u in order to localize this inequality further and so to involve the global constant c korn directly in the local computations we deduce a new error estimator l the latter estimator is based on local weighted interface problems with rigid body motions rbm as a kernel and so leads to effective estimates only if rbm are included in the local fe test functions therefore the excluded first order fem has to be enlarged by rbm which leads to a partition of unit method pum with rbm called p 1 rbm or to second order fems called p 2 fem for p 1 rbm and p 2 fem or even higher order schemes one obtains the sharper reliability estimate u u h l efficiency holds in the strict sense of l u u h the local weighted interface problems behind the implicit error estimators l and l are usually not exactly solvable and are rather approximated by some fem on a refined mesh and or with a higher order fem the computable approximations l l and l are shown to be reliable in the sense of u u h min l l osc the oscillations osc are known functions of the given data and higher order terms if the data are smooth for first order fem the mathematical proofs are based on weighted korn inequalities and inverse estimates combined with standard arguments the numerical experiments for uniform and adapted fem on benchmarks such as an l shape problem cook s membrane or a slit problem validate the theoretical estimates and also concern numerical bounds for c korn and the locking phenomena

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

a convergent adaptive finite element method for the primal problem of elastoplasticity summary the boundary value problem representing one time step of the primal formulation of elastoplasticity with positive hardening leads to a variational inequality of the second kind with some nondifferentiable functional this paper establishes an adaptive finite element algorithm for the solution of this variational inequality that yields the energy reduction and up to higher order terms the r linear convergence of the stresses with respect to the number of loops applications include several plasticity models linear isotropic kinematic hardening linear kinematic hardening and multisurface plasticity as model for nonlinear hardening laws for perfect plasticity the adaptive algorithm yields strong convergence of the stresses numerical examples confirm an improved linear convergence rate and study the performance of the algorithm in comparison with the more frequently applied maximum refinement rule copyright q 2006 john wiley sons ltd

on an estimate of the decay rate for applications of razumikhin type theorems a decay estimate is established for solutions of a class of razumikhin type functional differential equations fdes the fdes investigated here are general enough to satisfy the requirement of applications of razumikhin type theorems with the help of an application of our result it is illustrated that the method presented in this paper is adaptable to quantitative stability analysis of the retarded functional differential equations encountered in engineering

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

guaranteed computable bounds for the exact error in the finite element solution part ii bounds for the energy norm of the error in two dimensions this paper addresses the computation of guaranteed upper and lower bounds for the energy norm of the exact error in the finite element solution these bounds are constructed in terms of the solutions of local residual problems with equilibrated residual loads and are rather sharp even for coarse meshes he sharpness of the bounds can be further improved by employing few iterations of a relatively inexpensive iterative scheme he main result is that the bounds are guaranteed for the nergy norm of the exact error unlike the bounds which ave been proposed in 13 14 which are guaranteed only for the nergy norm of the error with respect to an enriched truth esh finite element solution copyright 2000 john wiley sons ltd

on multivariate discrete moment problems and their applications to bounding expectations and probabilities the discrete moment problem dmp has been formulated as a methodology to find the minimum and or maximum of a linear functional acting on an unknown probability distribution the support of which is a known discrete usually finite set where some of the moments are known the moments may be binomial power or of more general type the multivariate discrete moment problem mdmp has been initiated by the second named author who developed a linear programming theory and methodology for the solution of the dmps and mdmps under some assumptions that concern the divided differences of the coefficients of the objective function the central results in this respect are those that concern the structure of the dual feasible bases in this paper further results are presented in connection with mdmps for the case of power and binomial moments the main theorem theorem 3 1 and its applications help us to find dual feasible bases under the assumption that the objective coefficient function has nonnegative divided differences of a given total order and further divided differences are nonnegative in each variable any dual feasible basis provides us with a bound for the discrete function that consists of the coefficients of the objective function and also for the linear functional the latter bound is sharp if the basis is primal feasible as well the combination of a dual feasible basis structure theorem and the dual method of linear programming is a powerful tool to find the sharp bound for the true value of the functional i e the optimum value of the objective function the lower and upper bounds are frequently close to each other even if the number of utilized moments is relatively small numerical examples are presented for bounding the expectations of functions of random vectors as well as probabilities of boolean functions of event sequences

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

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

variational localizations of the dual weighted residual estimator the dual weighted residual method dwr and its localization for mesh adaptivity applied to elliptic partial differential equations are investigated the contribution of this paper is twofold first we introduce a novel localization technique based on the introduction of a partition of unity this new technique is very easy to apply as neither strong residuals nor jumps over element edges are required second we compare and analyze theoretically and numerically different localization techniques used for mesh adaptivity with respect to their effectivity here we focus on localizations in variational formulations that do not require the evaluation of the corresponding differential operator in the classical strong formulation in our mathematical analysis we show for different localization techniques established methods and our new approach that the local error indicators used for mesh adaptivity converge with proper order in the error functional several numerical tests substantiate our theoretical investigations

axioms of adaptivity this paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of discrete lower bounds inexact solvers inhomogeneous boundary data or the use of equivalent error estimators solely four axioms guarantee the optimality in terms of the error estimators compared to the state of the art in the temporary literature the improvements of this article can be summarized as follows first a general framework is presented which covers the existing literature on optimality of adaptive schemes the abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method second efficiency of the error estimator is neither needed to prove convergence nor quasi optimal convergence behavior of the error estimator in this paper efficiency exclusively characterizes the approximation classes involved in terms of the best approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant third some general quasi galerkin orthogonality is not only sufficient but also necessary for the r linear convergence of the error estimator which is a fundamental ingredient in the current quasi optimality analysis due to stevenson 2007 finally the general analysis allows for equivalent error estimators and inexact solvers as well as different non homogeneous and mixed boundary conditions

a unilaterally constrained quadratic minimization with adaptive finite elements we consider obstacle problems where a quadratic functional associated with the laplacian is minimized in the set of functions above a possibly discontinuous and thin but piecewise affine obstacle in order to approximate minimum point and value we propose an adaptive algorithm that relies on minima with respect to admissible linear finite element functions and on an a posteriori estimator for the error in the minimum value it is proven that the generated sequence of approximate minima converges to the exact one furthermore our numerical results in two and three dimensions indicate that the convergence rate with respect to the number of degrees of freedom is optimal in that it coincides with the one of nonlinear or adaptive approximation

adaptive computations of a posteriori finite element output bounds a comparison of the hybrid flux approach and the flux free approach two different domain decomposition approaches for the a posteriori finite element output bound method with adaptive subdomain refinement are investigated for the three dimensional convection diffusion equation the bound method provides relevant inexpensive rigorous lower and upper bounds for output linear functionals of partial differential equation pde associated with a fine mesh at a fraction of the computational cost of the fine mesh solution to achieve a desired bound gap i e difference between upper and lower output bounds at the lowest cost an adaptive subdomain refinement technique is used to refine subdomains only where needed the bound method based on a finite element domain decomposition technique requires two global calculations and a multitude of subdomain computations there are two main approaches to decompose a global domain into local subdomains for the subdomain based output bound method a hybrid flux elemental neumann subdomain approach and a flux free nodal neumann subdomain approach in this paper the two different domain decomposition approaches are briefly described and compared linear and quadratic finite element discretizations for the hybrid flux and flux free approaches are developed and validated for a convergence test and an adaptive subdomain refinement example the comparison of the hybrid flux and flux free approaches is presented in terms of numerical accuracy and computational cost

robustness of error estimators for finite element solutions of problems with high orthotropy in this paper we address the accuracy of a posteriori error estimators for finite element solutions of problems with high orthotropy especially for cases where rather coarse meshes are used which are often encountered in engineering computations we present sample computations which indicate lack of robustness of all standard residual estimators with respect to high orthotropy the main culprit is the coarseness of the finite element meshes relative to the thickness of the boundary and interface layers in the solution this is often imposed by the size of the problem domain the capabilities for mesh generation and the employed finite element software and cannot be changed hence there is need for a posteriori error estimation approaches which are robust for coarse meshes and which are constructed using non intrusive approaches with respect to the employed software

convex hull property and maximum principle for finite element minimisers of general convex functionals the convex hull property is the natural generalization of maximum principles from scalar to vector valued functions maximum principles for finite element approximations are often crucial for the preservation of qualitative properties of the respective physical model in this work we develop a convex hull property for mathbb p 1 conforming finite elements on simplicial non obtuse meshes the proof does not resort to linear structures of partial differential equations but directly addresses properties of the minimiser of a convex energy functional therefore the result holds for very general nonlinear partial differential equations including e g the p laplacian and the mean curvature problem in the case of scalar equations the introduce techniques can be used to prove standard discrete maximum principles for nonlinear problems we conclude by proving a strong discrete convex hull property on strictly acute triangulations

convergent adaptive finite elements for the nonlinear laplacian the numerical solution of the homogeneous dirichlet problem for the p laplacian p in 1 infty is considered we propose an adaptive algorithm with continuous piecewise affine finite elements and prove that the approximate solutions converge to the exact one while the algorithm is a rather straight forward generalization of those for the linear case p 2 the proof of its convergence is different in particular it does not rely on a strict error reduction

an adaptive strategy for elliptic problems including a posteriori controlled boundary approximation we derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary the numerical solution is assumed to be defined on a finite element mesh whose boundary vertices are located on the boundary of the continuous problem no assumption is made on a geometrically fitting shape a posteriori error estimates are given in the energy norm and the l 2 norm and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation furthermore a strategy is presented to compute the effect of the non discretized part of the domain on the error starting from a coarse mesh this especially implies that parts of the domain where the measured error is small stay non discretized the presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary the reliable computation of the a posteriori estimates and a mesh adaptation based on delaunay techniques numerical examples illustrate that errors outside the initial discretization will be detected

element wise a posteriori estimates based on hierarchical bases for non linear parabolic problems we show that the issue of a posteriori estimate the errors in the numerical simulation of non linear parabolic equations can be reduced to a posteriori estimate the errors in the approximation of an elliptic problem with the right hand side depending on known data of the problem and the computed numerical solution a procedure to obtain local error estimates for the p version of the finite element method by solving small discrete elliptic problems with right hand side the residual of the p fem solution is introduced the boundary conditions are inherited by those of the space of hierarchical bases to which the error estimator belongs we prove that the error in the numerical solution can be reduced by adding the estimators that behave as a locally defined correction to the computed approximation when the error being estimated is that of a elliptic problem constant free local lower bounds are obtained the local error estimation procedure is applied to non linear parabolic differential equations in several space dimensions some numerical experiments for both the elliptic and the non linear parabolic cases are provided copyright 2005 john wiley sons ltd

two stage robust network flow and design under demand uncertainty we describe a two stage robust optimization approach for solving network flow and design problems with uncertain demand in two stage network optimization one defers a subset of the flow decisions until after the realization of the uncertain demand availability of such a recourse action allows one to come up with less conservative solutions compared to single stage optimization however this advantage often comes at a price two stage optimization is in general significantly harder than single stage optimization for network flow and design under demand uncertainty we give a characterization of the first stage robust decisions with an exponential number of constraints and prove that the corresponding separation problem is np hard even for a network flow problem on a bipartite graph we show however that if the second stage network topology is totally ordered or an arborescence then the separation problem is tractable unlike single stage robust optimization under demand uncertainty two stage robust optimization allows one to control conservatism of the solutions by means of an allowed budget for demand uncertainty using a budget of uncertainty we provide an upper bound on the probability of infeasibility of a robust solution for a random demand vector we generalize the approach to multicommodity network flow and design and give applications to lot sizing and location transportation problems by projecting out second stage flow variables we define an upper bounding problem for the two stage min max min optimization problem finally we present computational results comparing the proposed two stage robust optimization approach with single stage robust optimization as well as scenario based two stage stochastic optimization

single commodity stochastic network design under probabilistic constraint with discrete random variables single commodity networks are considered where demands at the nodes are random the problem is to find minimum cost optimal built in capacities at the nodes and arcs subject to the constraint that all demands should be met on a prescribed probability level reliability constraint and some deterministic constraints should be satisfied the reliability constraint is formulated in terms of the gale hoffman feasibility inequalities but their number is reduced by elimination technique the concept of a p efficient point is used in a smart way to convert and then relax the problem into an lp the p efficient points are simultaneously generated with the solution of the lp the joint distribution of the demands is used to obtain the p efficient points for all stochastic inequalities that were not eliminated and the solution of a multiple choice knapsack problem is used to generate p efficient points the model can be applied to planning in interconnected power systems flood control networks design of shelter

a posteriori error analysis for higher order dissipative methods for evolution problems we prove a posteriori error estimates for time discretizations by the discontinuous galerkin method dg q and the corresponding implicit runge kutta radau method irk r q of arbitrary order q 0 for both linear and nonlinear evolution problems of the form u prime mathfrak f u f with 2 angle bounded operator mathfrak f the key ingredient is a novel higher order reconstruction widehat u of the discrete solution u which restores continuity and leads to the differential equation widehat u prime pi mathfrak f u f for a suitable interpolation operator and piecewise polynomial approximation f of f we discuss applications to linear pde such as the convection diffusion equation 1 2 and the wave equation formally and nonlinear pde corresponding to subgradient operators 1 such as the p laplacian as well as lipschitz operators 1 2 we also derive conditional a posteriori error estimates for the time dependent minimal surface problem

equilibrated residual error estimator for edge elements reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem a cheap computation of such functions via equilibration is well known for scalar equations of second order we simplify and modify the equilibration such that it can be applied to the curl curl equation and edge elements the construction is more involved for edge elements since the equilibration has to be performed on subsets with different dimensions for this reason raviart thomas elements are extended in the spirit of distributions

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

analysis for time discrete approximations of blow up solutions of semilinear parabolic equations we prove a posteriori error estimates for time discrete approximations for semilinear parabolic equations with solutions that might blow up in finite time in particular we consider the backward euler and the crank nicolson methods the main tools that are used in the analysis are the reconstruction technique and energy methods combined with appropriate fixed point arguments the final estimates we derive are conditional and lead to error control near the blow up time

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

adaptive variational multiscale methods based on a posteriori error estimation energy norm estimates for elliptic problems we develop a new adaptive multiscale finite element method using the variational multiscale framework together with a systematic technique for approximation of the fine scale part of the solution the fine scale is approximated by a sum of solutions to decoupled localized problems which are solved numerically on a fine grid partition of a patch of coarse grid elements the sizes of the patches of elements may be increased to control the error caused by localization we derive an a posteriori error estimate in the energy norm which captures the dependency of the crucial discretization parameters the coarse grid mesh size the fine grid mesh size and the sizes of the patches based on the a posteriori error estimate we present an adaptive algorithm that automatically tunes these parameters finally we show how the method works in practice by presenting various numerical examples

exact bounds for linear outputs of the advection diffusion reaction equation using flux free error estimates the paper introduces a methodology to compute strict upper and lower bounds for linear functional outputs of the exact solutions of the advection reaction diffusion equation the proposed approach is an alternative to the standard residual type estimators hybrid flux circumventing the need of flux equilibration following a flux free error estimation strategy the presented estimator provides sharper estimates than the ones provided by both the standard hybrid flux techniques and other flux free techniques

error reduction and convergence for an adaptive mixed finite element method an adaptive mixed finite element method amfem is designed to guarantee an error reduction also known as saturation property after each refinement step the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms this error reduction property is established here for the raviart thomas finite element method with a reduction factor p 1 uniformly for the l 2 norm of the flux errors our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels the adaptive algorithm surprisingly does not require any particular mesh design unlike the conforming finite element method the new arguments are a discrete local efficiency and a quasi orthogonality estimate the proof does not rely on duality or on regularity

robust stabilization for uncertain time delay systems containing saturating actuators a class of uncertain time delay systems containing a saturating actuator is considered these systems are characterized by delayed state equations including a saturating actuator with norm bounded parameter uncertainty possibly time varying in the state and input matrices the delay is assumed to be constant bounded but unknown using a razumikhin approach for the stability of functional differential equations upper bounds on the time delay are given such that the considered uncertain system is robustly stabilizable in the case of constrained input via memoryless state feedback control laws these bounds are given in terms of solutions of appropriate finite dimensional riccati equations

convergence analysis of an adaptive nonconforming finite element method an adaptive nonconforming finite element method is developed and analyzed that provides an error reduction due to the refinement process and thus guarantees convergence of the nonconforming finite element approximations the analysis is carried out for the lowest order crouzeix raviart elements and leads to the linear convergence of an appropriate adaptive nonconforming finite element algorithm with respect to the number of refinement levels important tools in the convergence proof are a discrete local efficiency and a quasi orthogonality property the proof does neither require regularity of the solution nor uses duality arguments as a consequence on the data control no particular mesh design has to be monitored

a posteriori error analysis for the mean curvature flow of graphs we study the equation describing the motion of a nonparametric surface according to its mean curvature flow this is a nonlinear nonuniformly parabolic pde that can be discretized in space via a finite element method we conduct an aposteriori error analysis of the spatial discretization and derive upper bounds on the error in terms of computable estimators based on local residual indicators the reliability of the estimators is illustrated with two numerical simulations one of which treats the case of a singular solution

coarse graining schemes for stochastic lattice systems with short and long range interactions we develop coarse graining schemes for stochastic many particle microscopic models with competing short and long range interactions on a dimensional lattice we focus on the coarse graining of equilibrium gibbs states and by using cluster expansions we analyze the corresponding renormalization group map we quantify the approximation properties of the coarse grained terms arising from different types of interactions and present a hierarchy of correction terms we derive semi analytical numerical coarse graining schemes that are accompanied by a posteriori error estimates for lattice systems with short and long range interactions

an improved a priori error analysis of nitsche s method for robin boundary conditions in a previous paper juntunen and stenberg in math comput 78 1353 1374 2009 we have extended nitsche s method in abhandlungen aus dem mathematischen seminar der universitat hamburg 1970 71 for the poisson equation with general robin boundary conditions the analysis required that the solution is in h s with s 3 2 here we give an improved error analysis using a technique proposed by gudi math comput 79 2169 2189 2010

fully reliable localized error control in the fem if the first task in numerical analysis is the calculation of an approximate solution the second is to provide a guaranteed error bound and is often of equal importance the standard approaches in the a posteriori error analysis of finite element methods suppose that the exact solution has a certain regularity or the numerical scheme enjoys some saturation property for coarse meshes those asymptotic arguments are difficult to recast into rigorous error bounds the aim of this paper is to provide reliable computable error bounds which are efficient and complete in the sense that constants are estimated as well the main argument is a localization via a partition of unity which leads to problems on small domains two fully reliable estimates are established the sharper one solves an analytical interface problem with residuals following babuska and rheinboldt siam j numer anal 15 1978 pp 736 754 the second estimate is a modification of the standard residual based a posteriori estimate with explicit constants from local analytical eigenvalue problems for some class of triangulations we show that the efficiency constant is smaller than 2 5 according to our numerical experience the overestimation of our computable estimates proved to be reasonably small with an overestimation by a factor between 2 5 and 4 only

small data oscillation implies the saturation assumption the saturation assumption asserts that the best approximation error in h 1 0 with piecewise quadratic finite elements is strictly smaller than that of piecewise linear finite elements we establish a link between this assumption and the oscillation of f delta u and prove that small oscillation relative to the best error with piecewise linears implies the saturation assumption we also show that this condition is necessary and asymptotically valid provided f in l 2

convergence and quasi optimality of adaptive fem with inhomogeneous dirichlet data we consider the solution of a second order elliptic pde with inhomogeneous dirichlet data by means of adaptive lowest order fem as is usually done in practice the given dirichlet data are discretized by nodal interpolation as model example serves the poisson equation with mixed dirichlet neumann boundary conditions for error estimation we use an edge based residual error estimator which replaces the volume residual contributions by edge oscillations for 2d we prove convergence of the adaptive algorithm even with optimal convergence rate for 2d and 3d we show convergence if the nodal interpolation operator is replaced by the l 2 projection or the scott zhang quasi interpolation operator as a byproduct of the proof we show that the scott zhang operator converges pointwise to a limiting operator as the mesh is locally refined this property might be of independent interest besides the current application finally numerical experiments conclude the work

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

guaranteed energy error bounds for the poisson equation using a flux free approach solving the local problems in subdomains this is the pre peer reviewed version of the following article pares n santos h diez p guaranteed energy error bounds for the poisson equation using a flux free approach solving the local problems in subdomains international journal for numerical methods in engineering 23 marc 2009 vol 79 num 10 p 1203 1244 which has been published in final form at http www3 interscience wiley com journal 122269073 abstract

a pde approach to fractional diffusion we derive a computable a posteriori error estimator for the harmonic extension problem which localizes the fractional powers of elliptic operators supplemented with dirichlet boundary conditions our a posteriori error estimator relies on the solution of small discrete problems on anisotropic cylindrical stars it exhibits built in flux equilibration and is equivalent to the energy error up to data oscillation under suitable assumptions we design a simple adaptive algorithm and present numerical experiments which reveal a competitive performance

bounds for quantities of interest and adaptivity in the element free galerkin method a novel approach to implicit residual type error estimation in mesh free methods and an adaptive refinement strategy are presented this allows computing upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest the proposed approach precludes the main drawbacks of standard residual type estimators circumventing the need of flux equilibration and resulting in a simple implementation that avoids integrals on edges sides of a domain decomposition mesh this is especially interesting for mesh free methods the adaptive strategy proposed leads to a fast convergence of the bounds to the desired precision copyright 2008 john wiley sons ltd

bounds of functional outputs for parabolic problems part ii bounds of the exact solution the paper introduces a methodology to compute upper and lower bounds for linear functional outputs of the exact solutions of parabolic problems in this second part the bounds account for the error both in space and time the assumption stating that the error introduced by the time marching scheme is negligible used in the first part is removed here the bounds are computed starting from an approximation of the exact solution associated with a spatial mesh and a time grid nevertheless the bounds are guaranteed with respect to the exact solution with no reference to any mesh or time discretization

closed form two sided bounds for probabilities that at least r and exactly r out of n events occur in two previous papers prekopa prekopa a 1986a boole bonferroni inequalities and linear programming oper res 36 145 162 prekopa a 1986b sharp bounds on probabilities using linear programming to appear in oper res gave algorithms to approximate probabilities that at least r and exactly r out of n events occur 1 r n primal and dual linear programming problems were formulated and solved by dual type algorithms the purpose of the present paper is to give closed forms for the basis inverse and the corresponding dual vector in case of an arbitrary basis furthermore to give closed forms for the lower and upper bounds approximating the probability in question in case of a dual feasible basis in the case when the probability that at least one out of n events occurs is approximated it is shown that the absolute values of the components of any dual vector form a monotonically decreasing sequence the paper improves the method of inclusion exclusion proves new probability inequalities and

localized pointwise a posteriori error estimates for gradients of piecewise linear finite element approximations to second order quasilinear elliptic problems two types of pointwise a posteriori error estimates are presented for gradients of finite element approximations of second order quasilinear elliptic dirichlet boundary value problems over convex polyhedral domains omega in space dimension n ge 2 we first give a residual estimator which is equivalent to nabla u u h l infty omega up to higher order terms the second type of residual estimator is designed to control nabla u u h locally over any subdomain of omega it is a novel a posteriori counterpart to the localized or weighted a priori estimates of sch98 this estimator is shown to be equivalent up to higher order terms to the error measured in a weighted global norm which depends on the subdomain of interest all estimates are proved for general shape regular meshes which may be highly graded and unstructured the constants in the estimates depend on the unknown solution u in the nonlinear case but in a fashion which places minimal restrictions on the regularity of u

the stability of stars of triangular equilibrium plate elements equilibrium models for finite element analyses are becoming increasingly important in complementary roles to those from conventional conforming models but when formulating equilibrium models questions of stability or admissibility of loads are of major concern this paper addresses these questions in the context of flat plates modelled with triangular hybrid elements involving membrane and or flexural actions patches of elements that share a common vertex are considered and such patches are termed stars stars may be used in global analyses as assemblies of elements forming macro elements or in local analyses the conditions for stability or the existence and number of spurious kinematic modes are determined in a general algebraic procedure for any degree of the interpolation polynomials and for any geometric configuration the procedure involves the determination of the rank of a compatibility matrix by its transformation to row echelon form examples are presented to illustrate some of the characteristics of spurious kinematic modes when they exist in stars with open or closed links copyright 2008 john wiley sons ltd

boole bonferroni inequalities and linear programming we present a method to obtain sharp lower and upper bounds for the probability that at least one out of a number of events in an arbitrary probability space will occur the input data are some of the binomial moments of the occurrences such as the sum of the probabilities of the individual events or the sum of the joint probabilities of all pairs of events we develop a special very simple linear programming algorithm to obtain these bounds the method allows us to compute good bounds in an optimal way utilizing only the first few terms in the inclusion exclusion formula possible applications include obtaining bounds for the reliability of a stochastic system solving algorithmically some stochastic programming problems and approximating multivariate probabilities in statistics in a numerical example we approximate the probability that a gaussian process runs below a given level in a number of consecutive epochs

afem for the laplace beltrami operator on graphs design and conditional contraction property we present an adaptive finite element method afem of any polynomial degree for the laplace beltrami operator on c 1 graphs in r d d 2 we first derive residual type a posteriori error estimates that account for the interaction of both the energy error in h 1 and the surface error in w 1 1 due to approximation of we devise a marking strategy to reduce the total error estimator namely a suitably scaled sum of the energy geometric and inconsistency error estimators we prove a conditional contraction property for the sum of the energy error and the total estimator the conditional statement encodes resolution of in w 1 1 we conclude with one numerical experiment that illustrates the theory

a posteriori error estimates and maximal regularity for approximations of fully nonlinear parabolic problems in banach spaces a posteriori error estimates are provided for discretizations in time of abstract nonlinear parabolic problems u f u by the backward euler method in the maximal regularity framework of banach spaces the estimates are of conditional type i e are valid under assumptions on the approximate solution and the proofs are based on appropriate fixed point arguments

subdomain based flux free a posteriori error estimators a new residual type flux free error estimator is presented it estimates upper and lower bounds of the error in energy norm the proposed approach precludes the main drawbacks of standard residual type estimators circumvents the need of flux equilibration and results in a simple implementation that uses standard resources available in finite element codes this is specially interesting for 3d applications where the implementation of this technique is as simple as in 2d recall that on the contrary the complexity of the flux equilibration techniques increases drastically in the 3d case the bounds for the energy norm of the error are used to produce upper and lower bounds of linear functional outputs representing quantities of engineering interest the presented estimators demonstrate their efficiency in numerical tests producing sharp estimates both for the energy and the quantities of interest

sharp bounds on probabilities using linear programming in a previous paper 1988 the author proposed methods to obtain sharp lower and upper bounds for probabilities that at least one out of n events occurs based on the knowledge of some of the binomial moments of the number of events which occur and linear programming formulations this paper presents further results concerning other and more general logical functions of events we give sharp lower and upper bounds for the probabilities that a exactly r events b at least r events occur using linear programming the basic facts are expressed by the dual feasible basis characterization theorems which are interpreted in terms of the vertices of the dual problems we mention some linear inequalities among the binomial moments generalize the theory for the case of nonconsecutive binomial moments and present numerical examples

ten remarks on nonconvex minimisation for phase transition simulations nonconvex minimisation problems are encountered in many applications such as phase transitions in solids or liquids but also in optimal design tasks or micromagnetism in contrast to rubber type elastic materials and many other variational problems in continuum mechanics the minimal energy may be not attained in the sense of sobolev functions the nonrank 1 convex minimisation problem m is ill posed the gradients of infimising sequences are enforced to develop finer and finer oscillations called microstructures their limit is a measure called gradient young measure gym and describes the effective energy density w qc the quasiconvex envelope of the original energy density w this gives rise to a relaxed minimisation problem r which is well posed in the sense that the minimum is attained the paper compares computational aspects of the two problems m and r neither adaptive finite element methods nor effective solvers may work for discrete versions of m but theoretical and numerical evidence supports that they work for r the drawback of r is that the relaxed energy density w qc is not always given by a known closed form formula instead a numerical relaxation has to be involved which is of the form m and hence yields a computational challenge other semiconvexity notions are requested and give rise to a new category of algorithms the demand for quasiconvexification algorithms is highly motivated by recent models in finite plasticity as the time discretisation in the latter models typically leads to nonrank 1 convex minimisation problems 2004 published by elsevier b v

the facets of the polyhedral set determined by the gale hoffman inequalities the gale hoffman inequalities characterize feasible external flow in a capacitated network among these inequalities those that are redundant can be identified through a simple arc connectedness criterion

recovery of equilibrium on star patches using a partition of unity technique a technique for recovering equilibrated element stresses is developed for finite element models of structural mechanics problems the data for the method consist of the prescribed loading and the stress and displacement fields resulting from a conventional compatible finite element model local problems are defined for each star of elements via the introduction of fictitious body forces and strains these problems can be solved independently for equilibrium in a process that can be easily parallelized the quality of the solutions is assessed for two dimensional linear elastic problems by using them to compute bounds of the error of the finite element solutions in terms of both global and local quantities of interest copyright 2009 john wiley sons ltd