a posteriori error analysis for stabilised finite element approximations of transport problems we develop the a posteriori error analysis of stabilised finite element approximations to linear transport problems via duality arguments two alternative dual problems are considered one is based on the formal adjoint of the hyperbolic differential operator the other on the transposition of the bilinear form for the stabilised finite element method we show both analytically and through numerical experiments that the second approach is superior in the sense that it leads to sharper a posteriori error bounds and more economical adaptively refined meshes

bubble stabilization of finite element methods for the linearized incompressible navier stokes equations abstract in this paper we discuss the stabilization via bubble functions of a finite element method for stationary linearized incompressible navier stokes equations it is shown that residual free bubbles can reproduce supg

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

an adaptive finite element method for unsteady convection dominated flows with stiff source terms we consider nonstationary convection dominated flows with stiff source terms as a unified approach to such problems a combined finite element method in space and time with streamline diffusion is examined numerically it has good shock capturing features and is implicit and l stable moreover being a galerkin method it admits residual based weighted a posteriori error estimates of optimal order we avoid the use of global stability constants by actually solving the dual problem this in turn leads to efficient and mathematically rigorous mesh refinement strategies where the streamline diffusion parameter is used for optimizing the resulting adaptive scheme all theoretical results are substantiated by numerical test examples for a physical application we turn to detonation waves at moderate relaxation times in one space dimension the finite element method well reproduces the chapman jouguet speed of detonations and their wave structure our error estimator proves accurate and sensitive to parameters at least on time intervals of moderate length the adaptive scheme based on this estimator produces fairly accurate results which appear to be the best available with the present strategy all ingredients of the numerical scheme can be extended in a natural way to higher space dimensions

variation of the modulus of a foliation abstract the modulus is a generalization of the extremal length of plane curves introduced by l ahlfors we study the variation t mod p f t of the p modulus mod p f of a foliation f on a riemannian manifold in particular we consider the product of moduli of orthogonal foliations

conjectures for optimal block designs summary in the study and construction of optimal block designs over the past 20 years a number of conjectures have been repeatedly stated and used the purpose of this note is to bring together some of these conjectures although known to those actively engaged in research in design it is hoped that this note will stimulate other statisticians and mathematicians to give their attention to these unsolved problems

recursive formulae for the average efficiency factor in block and row column designs interchange algorithms are widely used to construct efficient block and row column designs we provide simple recursive formulae for updating this average efficiency factor so that it is no longer computationally expensive to calculate it after each possible interchange

variational multiscale a posteriori error estimation for multi dimensional transport problems this paper presents an explicit a posteriori error estimator for the multi dimensional transport equation based on an approximation to the variational multiscale theory the technique is adequate for methods with a local error distribution such as stabilized methods yielding efficiencies close to one for both the hyperbolic and diffusive limit regarding the hyperbolic limit previous work is extended to multi dimensions adequate norms are proposed for the computation of the error and proper error intrinsic time scales are calculated for the bilinear quadrilateral and the linear triangle furthermore the model considers the element interface error along the element edges enabling the error prediction in the diffusive limit the success of the method can be explained by the fact that in stabilized methods the element local problem captures most of the error and the proposed error intrinsic time scales are an approximation to the solution of the dual problem the proposed technology which can be implemented straightforwardly in existing codes is extremely economical because it is a simple explicit postprocessing

a posteriori error estimation and dynamic adaptivity for symmetric discontinuous galerkin approximations of reactive transport problems a symmetric discontinuous galerkin scheme is applied for solving reactive transport problems an explicit a posteriori error estimator in l 2 l 2 is derived for the semi discrete scheme applied to transport with general kinetic reactions explicit a posteriori error estimators in terms of linear functionals and negative norms are also obtained for transport problems with simple reactions an adaptive approach based on the a posteriori error estimators is proposed to modify the mesh dynamically numerical examples demonstrate the efficiency and effectivity of the estimators moreover it is shown in this study that the proposed adaptivity eliminates the need of slope limiters 2005 elsevier b v all rights reserved

a posteriori error estimation in least squares stabilized finite element schemes abstract we introduce a new approach to a posteriori error estimation in finite element galerkin schemes with least squares stabilization typical examples are the stabilization of equal order approximation in mixed finite element schemes and the streamline diffusion method for convection dominated problems the underlying framework is the general optimal control approach recently proposed for error control in finite element models in this method the approximate solutions of certain dual problems are used as weighting factors in residual based a posteriori bounds for arbitrary functionals of the error on this basis almost optimally economical meshes as well as reliable and efficient error bounds can be developed in a feed back process on successively adapted meshes the new feature in this method when applied for least squares finite element methods is the use of mesh dependent duality arguments to guarantee optimal order estimation of the stabilization terms

on deriving the inverse of a sum of matrices available expressions are reviewed and new ones derived for the inverse of the sum of two matrices one of them being nonsingular particular attention is given to text bf a text bf ubv 1 where textbf a is nonsingular and textbf u b and textbf v may be rectangular generalized inverses of textbf a textbf ubv are also considered several statistical applications are discussed

approaches for adjoint based a posteriori analysis of stabilized finite element methods this study derives a posteriori error estimates for linear functionals of the solution of systems of partial differential equations discretized using stabilized continuous galerkin methods we investigate a convection diffusion equation the stokes equations and incompressible low reynolds number flow governed by the navier stokes equations we consider three well known stabilization methods and show that only one of the three is adjoint consistent and that even this case is contingent upon a proper treatment of the adjoint data a standard approach for a posteriori error analysis uses the adjoint of the stabilized formulation which inherits the difficulties induced by the lack of adjoint consistency we introduce and analyze two alternative approaches the first is based on the addition of stabilization terms to the adjoint data while the second is based on a stabilized formulation of the formal adjoint problem we show that any of the three approaches can be used to derive a fully computable error r

error analysis of some galerkin least squares methods for the elasticity equations the recent technique of stabilizing mixed finite element methods by augmenting the galerkin formulation with least squares terms calculated separately on each element is considered the error analysis is performed in a unified manner yielding improved results for some methods introduced earlier in addition a new formulation is introduced and analyzed method is used within two different contexts the first is in connection with the plain strain or three dimensional problems near the incompressible limit for which the poisson ratio equals one half it is well known that for these problems a standard lower order displacement finite element method will break down by now it is also well known how a working method can be obtained a new unknown the pressure is introduced and discretized with a finite element space different from that used for the displacements the finite element method obtained falls into the class of saddlepoint problems for which there is a general theory developed by brezzi and babuska 5 7 the application of this theory shows that the method converges optimally if the discrete spaces for the displacement and pressure satisfy a stability condition for a particular choice of displacement and pressure approximations it can be a difficult task to verify this babuska brezzi or inf sup condition during the last decade this problem has been intensively studied and the performances of many combinations are now known cf 18 and 23 26 for some recent extensions in addition there exists a general technique utilizing bubble functions by which a stable method can

blocking efficiency and weighted optimality optimal blocking is explored for experiments such as those incorporating one or more controls where not all treatment comparisons are of equal interest weighted optimality functions are employed in gaining both analytic and enumerative results a catalogue of smaller optimal designs is provided it is shown how design selection based on functions of variances and on functions of efficiency factors are both subsumed by the weighted approach copyright 2011 oxford university press

miscellanea updating the average efficiency factor in α designs this paper presents a method for updating the average efficiency factor of an design after an exchange or interchange in the array this allows the average efficiency factor to be used directly as the objective function in an design computer search algorithm

variational multiscale a posteriori error estimation for systems the euler and navier stokes equations this paper extends explicit a posteriori error estimators based on the variational multiscale theory to systems of equations in particular the emphasis is placed on flow problems the euler and navier stokes equations three error estimators are proposed the standard the naive and the upper bound numerical results show that with a very economical algorithm the attained global and local efficiencies for the naive approach are reasonably close to unity whereas the standard and upper bound approaches give respectively approximate lower and higher error estimates

virtual bubbles and galerkin least squares type methods ga l s abstract the equivalence between stabilized finite element methods or galerkin least squares tyoe methods ga l s and the standard galerkin method with bubble functions is established in an abstract framework the results are applicable to various finite element spaces including high order elements and applications to the advective diffusive model and to the stokes problem are presented illustrating the potential of the abstract theory introduced here

a new stabilized finite element formulation for scalar convection diffusion problems the streamline and approximate upwind petrov galerkin method abstract a new stabilized and accurate finite element formulation for convection dominated problems is herein developed the basis of the new formulation is the choice of a new upwind function the upwind function chosen for the new method provokes its degeneration into the supg or cau methods depending on the approximate solution s regularity the accuracy and stability of the new formulation for the linear and scalar advection diffusion equation is demonstrated in several numerical examples

how accurate is the streamline diffusion finite element method we investigate the optimal accuracy of the streamline diffusion finite element method applied to convection dominated problems for linear bilinear elements the theoretical order of convergence given in the literature is either o h 3 2 for quasi uniform meshes or o h 2 for some uniform meshes the determination of the optimal order in general was an open problem by studying a special type of meshes it is shown that the streamline diffusion method may actually converge with any order within this range depending on the characterization of the meshes

a posteriori error estimators for the raviart thomas element when error estimators for the rte are developed two difficulties prevent the success of the straightforward application of frequently used arguments the hdiv norm is an anisotropic norm i e it refers to differential operators of different orders moreover the traces of hdiv functions are only in h 1 2 therefore one does not obtain optimal a posteriori error estimates when using natural norms this drawback is overcome by using mesh dependent norms

updating formula in an analysis of variance model summary methods for updating the inverse of the information matrix in a regression analysis after deleting or adding one or more points have been known and used for many years in this paper these methods are extended to the case of interchanging two points in an analysis of variance model the results have important applications in the construction of experimental designs

adaptive finite element methods in computational mechanics we present a general approach to adaptivity for finite element methods and give applications to linear elasticity non linear elasto plasticity and nonlinear conservation laws including numerical results

an adaptive finite element method for linear elliptic problems on propose une methode adaptative d elements finis basee sur une estimation d erreur optimale de norme maxima pour les problemes elliptiques lineaires

conformal metrics and size of the boundary we establish a lower bound for the hausdorff dimension of the boundary associated with a conformal deformation of the euclidean metric on the unit ball in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i the deformations we consider are motivated by quasiconformal maps our abstract approach leads to new results on the boundary behavior of these maps we prove a conjecture by hanson on the compression of sets and obtain an improved version of the so called wall theorem we also establish a riesz privalov theorem in higher dimensions

geometric characterizations of gromov hyperbolicity we prove the equivalence of three different geometric properties of metric measure spaces with controlled geometry the first property is the gromov hyperbolicity of the quasihyperbolic metric the second is a slice condition and the third is a combination of the gehring hayman property and a separation condition