goal oriented error control in adaptive mixed fem for signorini s problem abstract this paper outlines goal oriented finite element error control for signorini s problem the discretization is based on a mixed formulation proposed by hlavacek et al which is extended to higher order polynomials a duality argument based on a variational inequality is applied which allows for the estimates in h as well as hp adaptivity numerical results confirm the applicability of the theoretical findings

bounds for the first positive zero of a mixed bessel function abstract the present article is concerned with lower and upper bounds of the first positive zero of the function h z j z zj z where j gn z is the ordinary bessel function of order 1 and j z is the derivative of j z a lower bound found here improves and extends the range of validity of the order of a lower bound found in a previous work 8 also two upper bounds given here improve a previously known upper bound 8 in the particular case 0 these bounds lead to lower and upper bounds for the first positive zero j 1 of j z which improve well known bounds in the literature

averaging technique for fe a posteriori error control in elasticity part i conforming fem averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing in this paper the reliability of any averaging estimator is shown for low order finite element methods in elasticity theoretical and numerical evidence supports that the reliability is up to the smoothness of given right hand sides and independent of the structure of a shape regular mesh

each averaging technique yields reliable a posteriori error control in fem on unstructured grids part ii higher order fem averaging techniques are popular tools in adaptive finite element methods since they provide efficient a posteriori error estimates by a simple postprocessing in the second paper of our analysis of their reliability we consider conforming h fem of higher i e not of lowest order in two or three space dimensions in this paper reliablility is shown for conforming higher order finite element methods in a model situation the laplace equation with mixed boundary conditions emphasis is on possibly unstructured grids nonsmoothness of exact solutions and a wide class of local averaging techniques theoretical and numerical evidence supports that the reliability is up to the smoothness of given right hand sides

on the green s function for the helmholtz operator in an impedance circular cylindrical waveguide this paper addresses the problem of finding a series representation for the green s function of the helmholtz operator in an infinite circular cylindrical waveguide with impedance boundary condition resorting to the fourier transform complex analysis techniques and the limiting absorption principle when the undamped case is analyzed a detailed deduction of the green s function is performed generalizing the results available in the literature for the case of a complex impedance parameter procedures to obtain numerical values of the green s function are also developed in this article

accurate computation of the zeros of the generalized bessel polynomials a general method for approximating polynomial solutions of second order linear homogeneous differential equations with polynomial coefficients is applied to the case of the families of differential equations defining the generalized bessel polynomials and an algorithm is derived for simultaneously finding their zeros then a comparison with several alternative algorithms is carried out it shows that the computational problem of approximating the zeros of the generalized bessel polynomials is not an easy matter at all and that the only algorithm able to give an accurate solution seems to be the one presented in this paper

averaging techniques yield reliable a posteriori finite element error control for obstacle problems the reliability of frequently applied averaging techniques for a posteriori error control has recently been established for a series of finite element methods in the context of second order partial differential equations this paper establishes related reliable and efficient a posteriori error estimates for the energy norm error of an obstacle problem on unstructured grids as a model example for variational inequalities the surprising main result asserts that the distance of the piecewise constant discrete gradient to any continuous piecewise affine approximation is a reliable upper error bound up to known higher order terms consistency terms and a multiplicative constant

competing subcommunities of mutualist and a generalized kamke theorem a model of a biological community of species is considered the community consists of two competing subcommunities each of which has the property that each pair of species of the subcommunity interact in a mutually beneficial or noninterfering manner sufficient conditions are given that the two subcommunities can coexist the coexistence state being a stable equilibrium the domains of attraction of stable equilibria are considered the sharpest results are for the special case of lotka volterra systems mathematically the fundamental tool is a generalized kamke theorem

a posteriori error control in low order finite element discretisations of incompressible stationary flow problems computable a posteriori error bounds and related adaptive meshrefining algorithms are provided for the numerical treatment of monotone stationary flow problems with a quite general class of conforming and non conforming finite element methods a refined residual based error estimate generalises the works of verfurth dari duran and padra bao and barrett as a consequence reliable and efficient averaging estimates can be established on unstructured grids the symmetric formulation of the incompressible flow problem models certain nonnewtonian flow problems and the stokes problem with mixed boundary conditions a helmholtz decomposition avoids any regularity or saturation assumption in the mathematical error analysis numerical experiments for the partly nonconforming method analysed by kouhia and stenberg indicate efficiency of related adaptive mesh refining algorithms

on the zeros of acv x xc v x where cv x is a cylinder function abstract let ck a v be the kth positive zero of the function acv x xc v x where cv x cos jv x sin yv x is the general cylinder function and 0 0 in particular we establish lower and upper bounds for c1 a v 0 as a consequence we obtain lower and upper bounds for c1 0 v 0 j v 1 the first positive zero of the j v x which are sharper than previously known ones

upper bounds for the first zeros of bessel functions abstract an upper bound for the first positive zero of the bessel functions of first kind j z for 1 is given this upper bound is better than a number of upper bounds found recently by several authors the upper bound given in this paper follows from a step of the ritz s approximation method applied to the eigenvalue problem of a compact self adjoint operator defined on an abstract separable hilbert space some advantages of this method in comparison with other approximation methods are presented

averaging techniques for reliable a posteriori fe error control in elastoplasticity with hardening averaging techniques are popular tools in adaptive finite element methods for numerical simulation in continuum mechanics since they provide efficient a posteriori error control in this paper the reliability of any averaging estimator is shown for low order finite element methods in one time step of elastoplasticity with hardening the constants and higher order terms are effected by the hardening and the smoothness of given right hand sides but are independent of the structure of a shape regular mesh since it involves a different functional analytical framework the case of perfect plasticity is excluded from this paper

the coexistence of a community of species with limited competition abstract this paper is a study of a system modeling a biological community of species with limited competition the community consists of two competing subcommunities all species of which cooperate and some species of one subcommunity can invade the steady state of another subcommunity whereas others cannot sufficient conditions are given that all species can coexist for lotka volterra systems we can improve this result by showing that there is a unique globally asymptotically stable steady state

discrete infinite dimensional type k monotone dynamical systems and time periodic reaction diffusion systems abstract the asymptotic behavior of discrete type k monotone dynamical systems and reaction diffusion equations is investigated the studying content includes the index theory for fixed points permanence global stability convergence everywhere and coexistence it is shown that the system has a globally asymptotically stable fixed point if every fixed point is locally asymptotically stable with respect to the face it belongs to and at this point the principal eigenvalue of the diagonal partial derivative about any component not belonging to the face is not one a nice result presented is the sufficient and necessary conditions for the system to have a globally asymptotically stable positive fixed point it can be used to establish the sufficient conditions for the system to persist uniformly and the convergent result for all orbits applications are made to time periodic lotka volterra systems with diffusion and sufficient conditions for such systems to have a unique positive periodic solution attracting all positive initial value functions are given for more general time periodic type k monotone reaction diffusion systems with spatial homogeneity a simple condition is given to guarantee the convergence of all positive solutions

edge residuals dominate a posteriori error estimates for low order finite element methods we prove that up to higher order perturbation terms edge residuals yield global upper and local lower bounds on the error of linear finite element methods both in h1 and l2 norms we present two proofs one uses the standard l2 projection and the other relies on a new weighted clement type interpolation operator

positivity preserving finite element approximation we consider finite element operators defined on rough functions in a bounded polyhedron in rn insisting on preserving positivity in the approximations we discover an intriguing and basic difference between approximating functions which vanish on the boundary of and approximating general functions which do not we give impossibility results for approximation of general functions to more than first order accuracy at extreme points of we also give impossibility results about invariance of positive operators on finite element functions this is in striking contrast to the well studied case without positivity

stable coexistence and bi stability for competitive systems on ordered banach spaces abstract autonomous and asymptotically autonomous semiflows modeling two species competition are studied which are strongly order preserving on a convex subset of a banach space with competitive order conditions are derived for stable coexistence bi stability and competitive exclusion a complete classification of all possible outcomes is obtained in case that there is at most one positive steady state representing coexistence of both species

residual type a posteriori error estimates for elliptic obstacle problems a posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems an instrumental ingredient is a new interpolation operator which requires minimal regularity exhibits optimal approximation properties and preserves positivity both upper and lower bounds are proved and their optimality is explored with several examples sharp a priori bounds for the a posteriori estimators are given and extensions of the results to double obstacle problems are briefly discussed

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

averaging technique for a posteriori error control in elasticity part iii locking free nonconforming fem abstract in the third part of our investigations on averaging techniques for a posteriori error control in elasticity we focus on nonconforming finite elements in two dimensions kouhia and stenberg comput methods appl mech engrg 124 1995 195 established robust a priori error estimates for a galerkin discretisation where the first component of the discrete displacement function is discretised with conforming and the second with nonconforming p1 finite elements here we study robust i e independent reliability and efficiency estimates for averaging error estimators numerical evidence supports that the reliability depends on the smoothness of given right hand sides and independent of the structure of a shape regular mesh

efficient and reliable a posteriori error estimators for elliptic obstacle problems a posteriori error estimators are derived for linear finite element approximations to elliptic obstacle problems these estimators yield global upper and local lower bounds for the discretization error here discretization error means the sum of two contributions the distance between continuous and discrete solution in the energy norm and some quantity that is related to the distance of continuous and discrete contact set moreover the local error indicators in the interior of the discrete contact set reduce to quantities that measure only data resolution

each averaging technique yields reliable a posteriori error control in fem on unstructured grids part i low order conforming nonconforming and mixed fem averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing in this paper their reliablility is shown for conforming nonconforming and mixed low order finite element methods in a model situation the laplace equation with mixed boundary conditions emphasis is on possibly unstructured grids nonsmoothness of exact solutions and a wide class of averaging techniques theoretical and numerical evidence supports that the reliability is up to the smoothness of given right hand sides

averaging technique for fe a posteriori error control in elasticity part ii λ independent estimates in the second part of our investigation on a posteriori error estimates and a posteriori error control in finite element analysis in elasticity we focus on robust a posteriori error bounds first we establish a residual based a posteriori error estimate which is reliable and efficient up to higher order terms and independent multiplicative constants the lame constant steers the incompressibility second we show the robust efficiency and reliability of averaging techniques in certain norms numerical evidence supports that the reliability of depends on the smoothness of given right hand sides and is independent of the structure of a shape regular mesh

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

inhomogeneous dirichlet conditions in a priori and a posteriori finite element error analysis the numerical solution of elliptic boundary value problems with finite element methods requires the approximation of given dirichlet data ud by functions ud h in the trace space of a finite element space on d in this paper quantitative a priori and a posteriori estimates are presented for two choices of ud h namely the nodal interpolation and the orthogonal projection in l2 d onto the trace space two corresponding extension operators allow for an estimate of the boundary data approximation in global h1 and l2 a priori and a posteriori error estimates the results imply that the orthogonal projection leads to better estimates in the sense that the influence of the approximation error on the estimates is of higher order than for the nodal interpolation