a numerical upscaling method for an elliptic equation with heterogeneous tensorial coefficients the aim of this paper is to develop a new numerical method to determine the effective permeability on a coarse scale level of problems with strongly heterogeneous tensorial coefficients defined on a fine scale this method stems from a primal dual mixed formulation and the upscaled permeability is obtained by imposing the continuity of the numerical flux across the lines of discontinuities moreover a finite volume scheme is built to numerically solve elliptic equations with this kind of coefficients finally the method is illustrated by several numerical experiments copyright 2002 john wiley sons ltd

numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media a numerical procedure for the determination of equivalent grid block permeability tensors for heterogeneous porous media is presented the method entails solution of the fine scale pressure equation subject to periodic boundary conditions to yield upon appropriate averaging of the fine scale velocity field the coarse scale or equivalent grid block permeability when the region over which this coarse scale permeability is computed constitutes a representative elementary volume rev the resulting equivalent permeability may be interpreted as the effective permeability of the region solution of the pressure equation on the fine scale is accomplished through the application of an accurate triangle based finite element numerical procedure which allows for the modeling of geometrically complex features the specification of periodic boundary conditions is shown to yield symmetric positive definite equivalent permeability tensors in all cases the method is verified through application to a periodic model problem and is then applied to the scale up of areal and cross sections with fractally generated permeability fields the applicability and limitations of the method for these more general heterogeneity fields are discussed

properties of some weighted sobolev spaces and application to spectral approximations the weighted sobolev spaces on a square whose weight is a given power of the product of the distances to the edges are introduced trace theorems related to these spaces are proved then a regularity result for the dirichlet problem for the laplace operator is proved next several projection operators with polynomial values are considered for which approximation results in weighted norms are stated finally a collocation spectral method for the dirichlet problem for the laplace operator with inhomogeneous boundary conditions is analyzed this work includes new results on chebyshev approximation

effective properties of composite materials with periodic microstructure a computational approach abstract this study reviews several problems which are specific of composites with periodic microstructure composed of linear or nonlinear constituents the theoretical background of the method is recalled first two different families of numerical methods are considered to solve the problem the first is based on the finite element method the concept of macroscopic degrees of freedom is presented the implementation of periodicity conditions is discussed a general framework permitting either a strain or stress control is proposed the second numerical method is based on fast fourier transforms it considers first the problem of a homogeneous linear reference material undergoing a nonhomogeneous periodic eigenstrain the solution of this problem is based on the explicit form of the periodic green s function of the reference medium the relative merits of the two methods are compared and several examples are discussed both methods give very comparable results on test examples and their domains of applications appear to be complementary

classifying c algebras via ordered mod p k theory we introduce an order structure on k 0 9 k0 z p this group may also be thought of as ko 7z z p we exhibit new examples of real rank zero c algebras that are inductive limits of finite dimensional and dimension drop algebras have the same ordered graded k theory with order unit and yet are not isomorphic in fact they are not even stably shape equivalent the order structure on k0 z 3 z p naturally distinguishes these algebras the same invariant is used to give an isomorphism theorem for such realrank zero inductive limits as a corollary we obtain an isomorphism theorem for all real rank zero approximately homogeneous c algebras that arise from systems of bounded dimension growth and torsion free k0 group at the 1980 kingston conference effros posed the problem of finding suitable invariants for use in studying c algebras that are limits of sequences of homogeneous c algebras these are now called almost homogeneous ah c algebras the classification of ah algebras is a rapidly developing field and we will not attempt to summarize all this activity instead we will focus on the growth of the invariants used specifically we consider an ah algebra a that is the direct limit o f a system of the form

extensions of pure states and projections of norm one abstract we show that the extension property for pure states of a c subalgebra b of a c algebra a leads to the existence of a projection of norm one r a b in the case where b is liminal with hausdorff primitive ideal space furthermore r is given by a dixmier process in which the averaging is effected by a group of unitary elements in the centre of the multiplier algebra m b these results generalize earlier work of j anderson and the author for the case when b is a masa of a various applications are given in the context of inductive limit algebras such as af algebras and more generally kumjian s ultraliminary c algebras

cartan subalgebras in c algebras existence and uniqueness we initiate the study of cartan subalgebras in c algebras with a particular focus on existence and uniqueness questions for homogeneous c algebras these questions can be analyzed systematically using the theory of fiber bundles for group c algebras while we are able to find cartan subalgebras in c algebras of many connected lie groups there are classes of discrete groups for instance non abelian free groups whose reduced group c algebras do not have any cartan subalgebras moreover we show that uniqueness of cartan subalgebras usually fails for classifiable c algebras however distinguished cartan subalgebras exist in some cases for instance in nuclear uniform roe algebras

symmetries of the car algebra a automorphism of period two of the uhf c algebra of type 2 is constructed for which the fixed point algebra is not approximately finite dimensional the construction also yields a diagonal maximal commutative c subalgebra whose spectrum is not totally disconnected variations and consequences of the construction are discussed

period two automorphisms of uhf c algebras recently 7 fack and marechal classified the product symmetries of a uhf c algebra among these product symmetries there is a distinguished element si whose fixed point algebra is uhf since they dealt with product symmetries all fixed point algebras are automatically af the symmetry s is characterized among product symmetries by the property pm this had been noted earlier by kishimoto 151

when is the cuntz krieger algebra of a higher rank graph approximately finite dimensional evans d g and sims a when is the cuntz krieger algebra of a higher rank graph approximately finite dimensional preprint arxiv 1112 4549v1

analysis of spectral projectors in one dimensional domains in this paper we analyze a class of projection operators with values in a subspace of polynomials these projection operators are related to the hubert spaces involved in the numerical analysis of spectral methods they are in the first part of the paper the standard sobolev spaces and in the second part some weighted sobolev spaces the weight of which is related to the orthogonality relation satisfied by the chebyshev polynomials these results are used to study the approximation of a model fourth order problem

on the t method for delay differential equations with infinite lag abstract this paper discusses the method for the numerical solution of delay differential equations with infinite lag our analysis is based on the test equation y t ay t by t where a b c and 0 1 in order to solve the storage problem involved in studying the long time behaviour of the solution we use a grid with increased stepsizes monotonicity uniform boundedness asymptotic stability algebraic decay and global discretisation error of the numerical solution are investigated

an approximate solution for a neutral functional differential equation with proportional delays in this paper a novel algorithm based on reproducing kernel theory for neutral functional differential equation with proportional delays is proposed the advantages of the presented method are the establishment of complete approximate solution theory and high precision of absolute error some examples are given to demonstrate the validity and applicability of the new method and some comparisons are made with the existing results

spectral tau approximation of the two dimensional stokes problem we analyse the spectral tau method for the approximation of the stokes system on a square with dirichlet boundary conditions we provide an error estimate in the norm of the sobolev spaceh s for the approximation of a divergence free vector field with polynomial divergence free vector fields we apply this result to prove some convergence estimates for the solution of the discrete stokes problem

covering dimension for nuclear c algebras we introduce the completely positive rank a notion of covering dimension for nuclear c algebras and analyze some of its properties the completely positive rank behaves nicely with respect to direct sums quotients ideals and inductive limits for abelian c algebras it coincides with covering dimension of the spectrum and there are similar results for continuous trace algebras as it turns out a c algebra is zero dimensional precisely if it is af we consider various examples particularly of one dimensional c algebras like the irrational rotation algebras the bunce deddens algebras or blackadar s simple unital projectionless c algebra finally we compare the completely positive rank to other concepts of noncommutative covering dimension such as stable or real rank

on the classification of simple inductive limit c algebras ii the isomorphism theorem in this article it is proved that the invariant consisting of the scaled ordered k group and the space of tracial states together with the natural pairing between them is a complete invariant for the class of unital simple c algebras which can be expressed as the inductive limit of a sequence a 1 to a 2 to cdots to a n to cdots with a n bigoplus i 1 t n p n i m n i c x n i p n i where xn i is a compact metrizable space and pn i is a projection in m n i c xn i for each n and i and the spaces xn i are of uniformly bounded finite dimension note that the c algebras in the present class are not assumed to be of real rank zero as they were in eg2

existence uniqueness and approximation for generalized saddle point problems extensions of the well known results of brezzi on saddle point problems are presented the class of problems is generalized to include the unsymmetric case and the known stability and approximation results are strengthened and applied to the generalized problem an application to the solution of certain exterior interface problems is indicated

c algebras of real rank zero abstract the concept of real rank of a c algebra is introduced as a non commutative analogue of dimension it is shown that real rank zero is equivalent to the previously defined conditions fs and hp and that it is invariant under strong morita equivalence in particular under stable isomorphism real rank zero is also invariant under inductive limits and split extensions and the class may well be regarded as the conceptual completion of the af algebras in some cases notably when the algebra is matroid it is shown that the multiplier algebra also has real rank zero although that is not true in general by a result of g j murphy this implies a weyl von neumann type result for self adjoint multiplier elements in these cases

symmetries on the discrete heisenberg group c algebra we consider symmetries on the c algebra of the discrete heisenberg group and their generalization to the generalized discrete heisenberg groups and study the structure of the crossed products by the symmetries and their fixed point algebras and also compute the stable rank of the crossed products and their fixed point algebras

stability analysis of theta methods for neutral functional differential equations this paper deals with the subject of numerical stability for the neutral functional differential equation y t ay t by qt cy pt qquad t 0 it is proved that numerical solutions generated by theta methods are convergent if c b if and only if theta geq 1 2 and that the one leg theta method is lambda stable if theta 1 we also find out that inappropriate stepsize causes spurious solution in the marginal case where rm re a 0 and a b

dimension and stable rank in the k theory of c algebras in topological k theory which can be viewed as the algebraic side of the theory of vector bundles some of the interesting properties which one investigates are for example the conditions under which bundles must possess trivial direct summands or the extent to which the cancellation property for direct whitney sums of bundles holds such properties turn out to be controlled in part by the dimension of the base space and results describing the nature of this control are among what are frequently called stability results 17 during the past few years there has been a sudden flowering of a theory for c algebras see surveys 35 36 37 since c algebras are profitably thought of as non commutative locally compact spaces with the finitely generated projective modules being the appropriate generalization of vector bundles according to swan s theorem 40 it would be natural to look for stability results for c algebras but until now the theory has been focussed on the c groups of c algebras and there has been little discussion of stability properties presumably in part for lack of an appropriate concept of dimension for c algebras one of the main objectives of this paper is to introduce a concept of dimension for c algebras which directly generalizes the classical concept of dimension for compact spaces 16 20 22 and to develop some techniques for calculating it notably for c algebras which are obtained as crossed product algebras for an action of the group of integers in algebraic k theory which is the generalization of topological c theory to rings a substantial number of stability properties have been discovered 2 39 42 48 within that theory the concept which has played a role most analogous to that of dimension is the bass stable rank but despite its many successes the bass stable rank has often been difficult to calculate even for relatively uncomplicated rings another main objective of the present paper is to make accessible for application to c algebras some of the stability results from algebraic c theory this comes about from the fact that the concept of dimension which we introduce dominates the bass stable rank i was led to seek stability results for c algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation c algebras discussed in 30 which as elliott 11 has suggested can appropriately be called 2 dimensional non commutative tori in more concrete terms the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent the present paper provides the general theory which is needed for the affirmative answer which i obtained in a subsequent paper i will derive the

finite group actions on c algebras with the rohlin property i basic properties of finite group actions with the rohlin property on unital c algebras are investigated a characterization of finite group actions with the rohlin property on the cuntz algebra 02 is given in terms of central sequences which may be considered as an equivariant version of e kirchberg and n c phillips s characterization of 02 a large class of symmetries on 02 are classified in terms of the fixed point algebras for conjugacy and the crossed products for cocycle conjugacy model actions of symmetries of 02 are constructed for given k theoretical invariants

covering dimension for nuclear c algebras ii the completely positive rank is an analogue of the topological covering dimension defined for nuclear c algebras via completely positive approximations these may be thought of as simplicial approximations of the algebra which leads to the concept of piecewise homogeneous maps and a notion of noncommutative simplicial complexes we introduce a technical variation of completely positive rank and show that the two theories coincide in many important cases furthermore we analyze some of their properties in particular we show that both theories behave nicely with respect to ideals and that they coincide with the covering dimension of the spectrum for certain continuous trace c algebras

c algebras associated with interval maps for each piecewise monotonic map r of 0 1 we associate a pair of c algebras f and or and calculate their k groups the algebra f is an ai algebra we characterize when f and or are simple in those cases f has a unique trace and or is purely infinite with a unique kms state in the case that is markov these algebras include the cuntz krieger algebras o a and the associated af algebras f a other examples for which the k groups are computed include tent maps quadratic maps multimodal maps interval exchange maps and transformations for the case of interval exchange maps and of transformations the c algebra or coincides with the algebras defined by putnam and katayama matsumoto watatani respectively

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

polynomial approximation of divergence free functions etude de la meilleure approximation d une fonction sans divergence par un polynome algebrique ou trigonometrique sans divergence etablissement d une estimation optimale

a non mortar mixed finite element method for elliptic problems on non matching multiblock grids abstract we consider the approximation of second order elliptic equations on domains that can be described as a union of sub domains or blocks we assume that a grid is defined on each block independently so that the resulting grid over the entire domain need not be conforming i e match across the block boundaries several techniques have been developed to approximate elliptic equations on multiblock grids that utilize a mortar finite element space defined on the block boundary interface itself we define a mixed finite element method that does not use such a mortar space the method has an advantage in the case where adaptive local refinement techniques will be used in that there is no mortar grid to refine as is typical of mixed methods our method is locally conservative element by element it is also globally conservative across the block boundaries theoretical results show that the approximate solution converges at the optimal rate to the true solution we present computational results to illustrate and confirm the theory

crossed products by finite group actions with the rokhlin property we prove that a number of classes of separable unital c algebras are closed under crossed products by finite group actions with the rokhlin property including a ai algebras at algebras and related classes characterized by direct limit decompositions using semiprojective building blocks b simple unital ah algebras with slow dimension growth and real rank zero c c algebras with real rank zero or stable rank one d simple c algebras for which the order on projections is determined by traces e c algebras whose quotients all satisfy the universal coefficient theorem f c algebras with a unique tracial state along the way we give a systematic treatment of the derivation of direct limit decompositions from local approximation conditions by homomorphic images which are not necessarily injective

stability of runge kutta methods for the generalized pantograph equation this paper deals with stability properties of runge kutta rk methods applied to a non autonomous delay differential equation dde with a constant delay which is obtained from the so called generalized pantograph equation an autonomous dde with a variable delay by a change of the independent variable it is shown that in the case where the rk matrix is regular stability properties of the rk method for the dde are derived from those for a difference equation which are examined by similar techniques to those in the case of autonomous ddes with a constant delay as a result it is shown that some rk methods based on classical quadrature have a superior stability property with respect to the generalized pantograph equation stability of algebraically stable natural rk methods is also considered

compact group actions on uhf algebras obtained by folding the interval we show that for any integer p greater than one there exists a c algebra which is not af such that mp a mp where mp mp is the uhf algebra of type p using this construction of uhf algebras as inductive limits of subhomogeneous algebras on an interval embedded in each other via a folding operation on the interval we exhibit actions of compact groups on uhf algebras whose fixed point algebras are not af in particular for any compact group g e there exists an action of g on mp whose fixed point algebra is not af

h stability of runge kutta methods with general variable stepsize for pantograph equation this paper deals with h stability of runge kutta methods with variable stepsize for pantograph equations it is shown that the runge kutta methods with a regular matrix a are h stable if and only if the modulus of stability function at infinity is less than 1 furthermore the same conclusion can be obtained if the runge kutta methods are stiffly accurate

on neutral functional differential equations with proportional delays abstract in this paper we develop a comprehensive theory on the well posedness of the initial value problem for the neutral functional differential equation y t ay t i 1 b i y q i t i 1 c i y p i t t 0 y 0 y 0 and the asymptotic behaviour of its solutions we prove that the existence and uniqueness of solutions depend mainly on the coefficientsci i 1 2 and on the smoothness of functions in the solution space as far as the asymptotic behaviour of analytic solutions is concerned thecihave little effect we prove that if re a 0 then the solutiony t either grows exponentially or is polynomial the most interesting result is that if re a 0 anda 0 then the asymptotic behaviour of the solution depends mainly on the characteristic equation a i 1 b i q i 0 these results can be generalized to systems of equations finally we present some examples to illustrate the change of asymptotic behaviour in response to the variation of some parameters the main idea used in this paper is to express the solution in either dirichlet or dirichlet taylor series form

generalized inf sup conditions for chebyshev spectral approximation of the stokes problem a mixed problem and its approximation in an abstract framework are considered they are proved to be well posed if and only if several inf sup conditions are satisfied these results are applied to the stokes equations in a square formulated in chebyshev weighted sobolev spaces and their approximations two kinds of spectral discretizations are analyzed a galerkin method and a collocation method at the chebyshev nodes

stability of a class of runge kutta methods for a family of pantograph equations of neutral type this paper deals with the stability of runge kutta methods for a class of neutral infinite delay differential equations with different proportional delays under suitable conditions the asymptotic stability of some runge kutta methods with variable stepsize are considered by the stability function at infinity it is proved that the even stage gauss legendre methods are not asymptotically stable but the radau ia methods radau iia methods and lobatto iiic methods are all asymptotically stable furthermore some numerical experiments are given to demonstrate the main conclusions

spectral approximation of the periodic nonperiodic navier stokes equations in order to approximate the navier stokes equations with periodic boundary conditions in two directions and a no slip boundary condition in the third direction by spectral methods we justify by theoretical arguments an appropriate choice of discrete spaces for the velocity and the pressure the compatibility between these two spaces is checked via an infsup condition we analyze a spectral and a collocation pseudo spectral method for the stokes problem and a collocation pseudo spectral method for the navier stokes equations we derive error bounds of spectral type i e which behave likem wherem depends on the number of degrees of freedom of the method and represents the regularity of the data

finite group actions on certain stably projectionless c algebras with the rohlin property we introduce the rohlin property and approximate representability for finite group actions on stably projectionless c algebras and study their basic properties we give some examples of finite group actions on the razakjacelon algebra w and show some classification results of these actions this study is based on the work of izumi robert s classification theorem and kirchberg s central sequence c algebras

a new collocation method for approximate solution of the pantograph functional differential equations with proportional delay the paper presents a new numerical method for solving functional differential equations with proportional delays of the first and higher orders the method consists of replacing the initial equation by an approximate equation which has an exact analytic solution with a set of free parameters these free parameters are determined by the use of the collocation procedure some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results the numerical results show that the proposed method is of a high accuracy and is efficient for solving a wide class of functional differential equations with proportional delays including equations of neutral type the method is applicable to both initial and boundary value problems

homogenized behavior of two phase flows in naturally fractured reservoirs with uniform fractures distribution abstract we consider the behavior of two phase flow in a periodically fractured porous medium e denotes the parameter associated to the period size we deal with homogenization methods to find asymptotic flow behavior as e tends to zero the homogenized porosity and permeability depend no longer on heterogeneities this limit gives the behavior of two phase flows in globally equivalent homogeneous media in some special cases we compute the arithmetic and harmonic means which are used classically in simulation codes