convergence in lp space for the homogenization problems of elliptic and parabolic equations in the plane we study the convergence rate of an asymptotic expansion for the elliptic and parabolic operators with rapidly oscillating coefficients first we propose homogenized expansions which are convolution forms of green function and given force term of elliptic equation then using local lp theory the growth rate of the perturbation of green function is found from the representation of elliptic solution by green function we estimate the convergence rate in lp space of the homogenized expansions to the exact solution finally we consider l2 0 t h1 or l 0 t convergence rate of the first order approximation for parabolic homogenization problems

fitted reproducing kernel method for singularly perturbed delay initial value problems in this paper a fitted reproducing kernel is proposed for solving singularly perturbed initial value problems based on it an effective numerical method is proposed for a class of singularly perturbed delay initial value problem numerical results show that the present reproducing kernel is superior to the usual reproducing kernel for solving singularly perturbed initial value problems

error estimates for the combinedh andp versions of the finite element method in theh version of the finite element method convergence is achieved by refining the mesh while keeping the degree of the elements fixed on the other hand thep version keeps the mesh fixed and increases the degree of the elements in this paper we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh in addition it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom even in the presence of corner singularities numerical results comparing theh version p version and combinedh p version for a one dimensional problem are presented

nonlinear equations involving m accretive operators abstract let x be a banach space and t an m accretive operator defined on a subset d t of x and taking values in 2x for the class of spaces whose bounded closed and convex subsets have the fixed point property for nonexpansive self mappings it is shown here that two boundary conditions which imply existence of zeroes for t appear to be equivalent this fact is then used to prove that if there exists x0 d t and a bounded open neighborhood u of x0 such that t x 0 for all x u d t then the open ball b 0 r is contained in the range of t

optimization of dispersive coefficients in the homogenization of the wave equation in periodic structures we study dispersive effects of wave propagation in periodic media which can be modelled by adding a fourth order term in the homogenized equation the corresponding fourth order dispersive tensor is called burnett tensor and we numerically optimize its values in order to minimize or maximize dispersion more precisely we consider the case of a two phase composite medium with an eightfold symmetry assumption of the periodicity cell in two space dimensions we obtain upper and lower bound for the dispersive properties along with optimal microgeometries

singular perturbation analysis of boundary value problems for differential difference equations vi small shifts with rapid oscillations this paper continues the authors study of boundary value problems for singularly perturbed linear second order differential difference equations with small shifts this study was initiated in the companion paper siam j appl math 54 1994 pp 249 272 in this paper the study is extended to problems that have solutions that exhibit rapid oscillations restrictions on the sizes of the shifts in terms of the small parameter are found such that generally the shifted terms cannot be replaced with truncated taylor series in particular it is shown that even when the shifts are small relative to the width of an oscillation they can affect the solution to leading order the conclusion is that oscillatory solutions are more sensitive to small delays than are layer solutions it is shown that a suitably modified version of the standard wkb method can be used to obtain leading order oscillatory solutions of these differential difference equations these preliminary studies of differential difference eq

a right inverse for the divergence operator in spaces of piecewise polynomials in the first part of this paper we study in detail the properties of the divergence operator acting on continuous piecewise polynomials on some fixed triangulation more specifically we characterize the range and prove the existence of a maximal right inverse whose norm grows at most algebraically with the degree of the piecewise polynomials in the last part of this paper we apply these results to thep version of the finite element method for a nearly incompressible material with homogeneous dirichlet boundary conditions we show that thep version maintains optimal convergence rates in the limit as the poisson ratio approaches 1 2 this fact eliminates the need for any reduced integration such as customarily used in connection with the more standardh version of the finite element method

a mixed multiscale finite element method for elliptic problems with oscillating coefficients the recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large scale structure of the solutions without resolving all the fine scale structures motivated by the numerical simulation of flow transport in highly heterogeneous porous media we propose a mixed multiscale finite element method with an over sampling technique for solving second order elliptic equations with rapidly oscillating coefficients the multiscale finite element bases are constructed by locally solving neumann boundary value problems we provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain the asymptotic structure of the solutions numerical experiments are carried out for flow transport in a porous medium with a random log normal relative permeability to demonstrate the efficiency and accuracy of the proposed method

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

first order corrections to the homogenised eigenvalues of a periodic composite medium a convergence proof let e be a dirichlet eigenvalue of the periodically rapidly oscillating elliptic operator a x e and let be a corresponding simple eigenvalue of the homogenised operator a we characterise the possible limit points of the ratio e e as e 0 our characterisation is quite explicit when the underlying domain is a planar convex classical polygon with sides of rational or infinite slopes in particular in this case it implies that there is often a continuum of such limit points

multiscale finite element algorithm of the eigenvalue problems for the elastic equations in composite materials in this paper we discuss the multiscale computation of the eigenvalue problems for the elastic equations in composite materials the multiscale finite element method for solving these problems is presented numerous numerical results validate the proposed method of this paper

the asymptotic behavior of the solutions of the cauchy problem generated by ϕ accretive operators abstract the purpose of this paper is to study the asymptotic behavior of the solutions of certain type of differential inclusions posed in banach spaces in particular we obtain the abstract result on the asymptotic behavior of the solution of the boundary value problem u t p u u 1 u f on 0 u u on 0 u 0 x u 0 x in where is a bounded open domain in r n with smooth boundary f t x is a given l 1 function on 0 1 and 1 p p represents the p laplacian operator is the associated neumann boundary operator and a maximal monotone graph in r r with 0 0

modeling the large scale dynamics of saturated groundwater flow using spatial filtering theory 1 theoretical development we develop a model and a framework for understanding the explicitly resolved large scale dynamics of groundwater flow in fully saturated heterogeneous porous media the approach is based upon the large eddy simulation les methodology from geophysical fluid dynamics the idea behind the les approach is to explicitly resolve the larger scales of flow and provide a closure model for the effect of subgrid scales upon resolved scales the technical foundation of the method rests on a spatial filtering operation which allows one to precisely define the scale of the model we spatially filter darcy s law and the continuity equation to produce a model for the resolved scale dynamics we develop les approximations for the resulting flow equations and a closure model for unresolved subgrid scale terms we model the subgrid terms using resolved scale quantities only and thus do not require subgrid scale information the accuracy of these closure models depends upon the scale at which hydraulic conductivity is measured by field tests we characterize the approximations used to model subgrid scale effects asymptotically with a small parameter the dimensionless filter width defined as the ratio of the filter width to the dominant length scale of the explicitly resolved variables

multiscale asymptotic analysis and numerical simulation for the second order helmholtz equations with rapidly oscillating coefficients over general convex domains the multiscale asymptotic analysis and numerical simulation for the second order helmholtz equations with rapidly oscillating coefficients over general convex domains are discussed in this paper a multiscale asymptotic analysis formulation for this problem is presented by constructing properly the boundary layer a multiscale numerical algorithm and a postprocessing technique are given finally numerical results show that the method presented in this paper is effective and reliable

a multiscale reduced basis method for parametrized elliptic partial differential equations with multiple scales we present a technique for solving parametrized elliptic partial differential equations with multiple scales the technique is based on the combination of the reduced basis method c prud homme d rovas k veroy y maday a t patera g turinici reliable real time solution of parametrized partial differential equations reduced basis output bound methods journal of fluids engineering 124 1 2002 70 80 and the multiscale finite element method t y hou x h wu a multiscale finite element method for elliptic problems in composite materials and porous media journal of computational physics 134 1 1997 169 189 to treat problems in which the differential coefficient is characterized by a large number of independent parameters for the multiscale finite element method a large number of cell problems has to be solved at the fine local mesh for each new configuration of the differential coefficient in order to improve the computational efficiency of this method we construct reduced basis spaces that are adapted to the local parameter dependence of the differential operator the approximate solutions of the cell problems are computed accurately and efficiently via performing galekin projection onto the reduced basis spaces and implementing the offline online computational procedure therefore a large number of similar computations at the fine local mesh can be carried out with lower computational cost for each new configuration of the differential coefficient numerical results are provided to demonstrate the accuracy and efficiency of the proposed approach

on mixed finite element methods for first order elliptic systems a physically based duality theory for first order elliptic systems is shown to be of central importance in connection with the galerkin finite element solution of these systems using this theory in conjunction with a certain hypothesis concerning approximation spaces optimal error estimates for galerkin type approximations are demonstrated an example of a grid which satisfies the hypothesis is given and numerical examples which illustrate the theory are provided

existence theorems for m accretive operators in banach spaces abstract in 1985 the second author proved a surjective result for m accretive and expansive mappings for uniformly smooth banach spaces however in this case we have been able to remove the uniform smoothness of the banach space without any additional assumption

convergence of a nonconforming multiscale finite element method the multiscale finite element method msfem t y hou x h wu and z cai math comp 1998 to appear t y hou and x h wu j comput phys 134 1997 pp 169 189 has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coefficients this is accomplished by constructing the multiscale base functions from the local solutions of the elliptic operator our previous study reveals that the leading order error in this approach is caused by the resonant sampling which leads to large error when the mesh size is close to the small scale of the continuous problem similar difficulty also arises in numerical upscaling methods an oversampling technique has been introduced to alleviate this difficulty t y hou and x h wu j comput phys 134 1997 pp 169 189 a consequence of the oversampling method is that the resulting finite element method is no longer conforming here we give a detailed analysis of the nonconforming error our analysis also reveals a new cell resonance error which is caused by the mismatch between the mesh size and the wavelength of the small scale we show that the cell resonance error is of lower order our numerical experiments demonstrate that the cell resonance error is generically small and is difficult to observe in practice

cellular automaton fluids a model for flow in porous media numerical models of fluid flow through porous media can be developed from either microscopic or macroscopic properties the large scale viewpoint is perhaps the most prevalent darcy s law relates the chief macroscopic parameters of interest flow rate permeability viscosity and pressure gradient and may be invoked to solve for any of these parameters when the others are known in practical situations however this solution may not be possible attention is then typically focused on the estimation of permeability and numerous numerical methods based on knowledge of the microscopic pore space geometry have been proposed because the intrinsic inhomogeneity of porous media makes the application of proper boundary conditions difficult microscopic flow calculations have typically been achieved with idealized arrays of geometrically simple pores throats and cracks i propose here an attractive alternative which can freely and accurately model fluid flow in grossly irregular geometries this new method s

convergence rates of neumann problems for stokes systems abstract in quantitative homogenization of the neumann problems for stokes systems with rapidly oscillating periodic coefficients this paper studies the convergence rates of the velocity in l 2 and h 1 as well as those of the pressure term in l 2 without any smoothness assumptions on the coefficients

convergence theorems for ψ expansive and accretive mappings abstract let e be a real banach space and let a d a e e be a lipschitz expansive and accretive mapping such that c o d a 0 r i a suppose that there exists x 0 d a where one of the following holds i there exists r 0 such that r 2 a x 0 or ii there exists a bounded neighborhood u of x 0 such that t x x 0 a x for x u d a and t 0 an iterative sequence x n is constructed to converge strongly to a zero of a related results deal with the strong convergence of this iteration process to fixed points of expansive and pseudocontractive mappings in real banach spaces the convergence results established in this paper are new for this more general class of expansive and accretive or pseudocontractive mappings

non local homogenized limits for composite media with highly anisotropic periodic fibres we consider a homogenization problem for highly anisotropic conducting fibres embedded into an isotropic matrix for a double porosity type scaling in the expression of high contrast between the conductivity along the fibres and the conductivities in the transverse directions we prove the homogenization theorem and derive two scale homogenized equations using a version of the method of two scale convergence supplemented in the case when the spectral parameter 0 by a newly derived variant of high contrast poincare type inequality further elimination of the rapid component from the two scale limit equations results in a non local convolution type integro differential equation for the slowly varying part in the matrix with the non local kernel explicitly related to the green function on the fibre the regularity of the solution to the non local homogenized equation is proved

multiscale analysis and numerical algorithm for the schrodinger equations in heterogeneous media abstract in solid state physics the most widely used techniques to calculate the electronic levels in nanostructures are the effective masses approximation ema and its extension the multiband k p method see 9 they have been particularly successful in the case of heterostructures see e g 4 9 11 this paper discusses the multiscale analysis of the schrodinger equation with rapidly oscillating coefficients the new contributions obtained in this paper are the determination of the convergence rate for the approximate solutions the definition of boundary layer solutions and higher order correctors consequently a multiscale finite element method and some numerical results are presented as one of the main results of this paper we give a reasonable interpretation why the effective mass approximation is very accurate for calculating the band structures in semiconductor in the vicinity of point from the viewpoint of mathematics

asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the dirichlet problem for second order elliptic equations in perforated domains in this paper we study the spectral properties of dirichlet problems for second order elliptic equation with rapidly oscillating coefficients in a perforated domain the asymptotic expansions of eigenvalues and eigenfunctions for this kind of problem are obtained and the multiscale finite element algorithms and numerical results are proposed

an asymptotic formula for the voltage potential in a perturbed e periodic composite medium containing misplaced inclusions of size e we consider composite media made of homogeneous inclusions with c1 boundaries our goal is to compare the potential ue in a perfectly periodic composite with the potential ue d of a perturbed periodic medium where the periodicity defects consist of misplaced inclusions we give an asymptotic expansion of the difference ue d ue away from the defects and show that to first order a misplaced inclusion manifests itself via a polarization tensor which is characterized

a multiscale mortar mixed space based on homogenization for heterogeneous elliptic problems we consider a second order elliptic problem with a heterogeneous coefficient written in mixed form the nonoverlapping mortar domain decomposition method is efficient in parallel if the mortar interface coupling space has a restricted number of degrees of freedom in the heterogeneous case we define a new multiscale mortar space that incorporates purely local information from homogenization theory to better approximate the solution along the interfaces with just a few degrees of freedom in the case of a locally periodic heterogeneous coefficient of period varepsilon we prove that the new method achieves both optimal order error estimates in the discretization parameters and good approximation when varepsilon is small moreover we present three numerical examples to assess its performance when the coefficient is not obviously locally periodic we show that the new mortar method works well and better than polynomial mortar spaces

higher order convergence rates in theory of homogenization ii oscillatory initial data abstract we establish higher order convergence rates in periodic homogenization of fully nonlinear uniformly parabolic cauchy problems accompanied with rapidly oscillating initial data such result is new even for linear problems here we construct higher order initial layer and interior correctors which describe the oscillatory behavior near the initial and interior time zone of the domain to construct higher order correctors we develop a regularity theory in macroscopic scales and prove an exponential decay estimate for initial layer correctors the higher order expansion requires an iteration process successively correcting the initial layer then the interior this leads to a more complicated asymptotic expansion as compared to the non oscillating data case and this complexity is present even in the linear case a notable observation for fully nonlinear operators is that even if the given operator is space time periodic the interior correctors become aperiodic in the time variable as we proceed with the iteration process moreover each interior corrector of higher order is paired with a space time periodic version and the difference between the two decays exponentially fast with time

accretive operators which are always single valued in normed spaces abstract set valued accretive operators in banach spaces have been extensively studied for several decades our main purpose in this paper is to establish a quite revealing result that says that every set valued lower semi continuous accretive mapping defined on a normed space is indeed single valued on the interior of its domain no reference to the well known michael s selection theorem is needed this result is used to extend known theorems concerning the existence of zeros for such operators as well as showing existence of solutions for variational inclusions

convergence rates in l 2 for elliptic homogenization problems we study rates of convergence of solutions in l2 and h1 2 for a family of elliptic systems mathcal l varepsilon with rapidly oscillating coefficients in lipschitz domains with dirichlet or neumann boundary conditions as a consequence we obtain convergence rates for dirichlet neumann and steklov eigenvalues of mathcal l varepsilon most of our results which rely on the recently established uniform estimates for the l2 dirichlet and neumann problems in kenig and shen math ann 350 867 917 2011 commun pure appl math 64 1 44 2011 are new even for smooth domains

analysis of the heterogeneous multiscale method for elliptic homogenization problems the heterogeneous multiscale method hmm is applied to various parabolic problems with multiscale coefficients these problems can be either linear or nonlinear optimal estimates are proved for the error between the hmm solution and the homogenized solution

a review on singularly perturbed differential equations with turning points and interior layers singular perturbation problems with turning points arise as mathematical models for various physical phenomena the problem with interior turning point represent one dimensional version of stationary convection diffusion problems with a dominant convective term and a speed field that changes its sign in the catch basin boundary turning point problems on the other hand arise in geophysics and in modeling thermal boundary layers in laminar flow in this paper we review some existing literature on asymptotic and numerical analysis of singularly perturbed turning point and interior layer problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye towards the goal of developing general methods to solve such problems since major work in this area started after 1970 so this paper limits its coverage to the work done by numerous researchers between 1970 and 2011

eulerian kinematics of flow through spatially periodic models of porous media for spatially periodic models of porous media and incompressible fluids we consider here in detail thephysical interpretation of averaged velocity fields in terms of a purely continuum level eulerian seepage flux relation involving the mass flow across an arbitrarily oriented macroscale surface element we prove that the appropriately averaged microscale velocities do indeed satisfy a macroscale eulerian flux relation albeit to some approximation provided that the surface element is large compared with the pore lengthscale microscale a quantitative discussion of an intermediate scale lying between the pore scale and a characteristic macroscale is achieved through the use of locally periodic velocity fields within the context of seepage velocity kinematics our developments provide a conceptual foundation that renders continuum theories of flow of incompressible fluids through periodic porous media entirely selfcontained on the macroscale free from any details of the discrete microscale structure inaccessible to macroscopic observers from which this continuum theory sprang the error bounds and scaling laws derived are of special interest when the macroscale microscale disparity is not very wide with reference to current rational mechanical theories of immiscible mixtures our analysis provides aconstructive existence proof at least for periodic models of porous media for phase specific velocities heretofore such velocities were usually accepted in a lagrangian sense on an axiomatic basis thus clarification and quantification of thekinematical aspects of the continuum hypothesis for certain classes of multiphase systems can be effecteda priori by our arguments independently of any subsequent macroscaledynamical modelling our analysis whereby we systematically proceed from a discrete picture of the underlying geometry to a purely continuum picture may be likened to that achieved in classical statistical mechanics as in that case we hope that the general results obtained transcend the underlying discrete i e spatially periodic model of the subcontinuum structure

on full two scale expansion of the solutions of nonlinear periodic rapidly oscillating problems and higher order homogenised variational problems we consider a scalar quasilinear equation in the divergence form with periodic rapid oscillations which may be a model of e g nonlinear conducting dielectric or deforming in a restricted way hardening elastic plastic composites with outer periodicity conditions of a fixed large period under some natural growth assumptions on the stored energy function we construct for uniformly elliptic problems a full two scale asymptotic expansion which has a precise double series structure separating the slow and the fast variables in all orders so that its slowly varying part solves asymptotically an infinite order homogenised equation cf bakhvalov n s panasenko g p homogenisation averaging processes in periodic media nauka moscow 1984 in russian english translation kluwer 1989 and whose higher order terms depend on the higher gradients of the slowly varying part we prove the error bound i e that the truncated asymptotic expansion is higher order close to the actual solution in appropriate norms the approach is extended to a non uniformly elliptic case for two dimensional power law potentials we prove the non degeneracy using topological index methods examples and explicit formulae for the higher order terms are given in particular we prove that the first term in the higher order homogenised equations is related to the first order corrector to the mean flux and has in general the form of a fully nonlinear operator which is quadratic with respect to its highest second derivative being a linear combination of the second minors of the hessian with coefficients depending on the first gradient and in dimension two is of monge ampere type we show that this term is present at least for some examples three phase power law laminates

iterative algorithm for multi valued pseudocontractive mappings in banach spaces abstract let d be nonempty open convex subset of a real banach space e let t d k c e be a continuous pseudocontractive mapping satisfying the weakly inward condition and let u d be fixed then for each t 0 1 there exists y t d satisfying y t t t y t 1 t u if in addition e is reflexive and has a uniformly g teaux differentiable norm and is such that every closed convex bounded subset of d has fixed point property for nonexpansive self mappings then t has a fixed point if and only if y t remains bounded as t 1 in this case y t converges strongly to a fixed point of t as t 1 moreover an explicit iteration process which converges strongly to a fixed point of t is constructed in the case that t is also lipschitzian

low frequency asymptotic analysis of a string with rapidly oscillating density we consider the eigenvalue problem associated to the vibrations of a string with a rapidly oscillating bounded periodic density it is well known that when the size of the microstructure epsilon is small enough with respect to the wavelength of the eigenfunctions 1 sqrt lambda epsilon eigenvalues and eigenfunctions can be approximated by those of the limit system where the oscillating density is replaced by its average on the other hand it has been observed that when the size of the microstructure is of the order of the wavelength of the eigenfunctions epsilon sim 1 sqrt lambda epsilon singular phenomena may occur in this paper we study the behavior of the eigenvalues and eigenfunctions when 1 sqrt lambda epsilon approaches the critical size epsilon to do this we use the wkb approximation which allows us to find an explicit formula for eigenvalues and eigenfunctions with respect to epsilon in particular our analysis provides all order correction formulas fo

first order corrections to the homogenized eigenvalues of a periodic composite medium this paper studies the eigenvalue problem associated with the vibration of a com posite medium assumed to have a periodic microstructure the investigation concentrates on the first order correction to the homogenized eigenvalues a major difficulty arises due to the interaction of the periodic microstructure with the boundary of the medium the way in which the boundary intersects the underlying periodic structure strongly influences the value of the correction a rigorous derivation is given of the eigenvalue correction formula the properties of the eigenvalue correction are studied in one and two dimensional numerical calculations

a numerical method for singularly perturbed turning point problems with an interior layer the objective of this paper is to present a numerical method for solving singularly perturbed turning point problems exhibiting an interior layer the method is based on the asymptotic expansion technique and the reproducing kernel method rkm the original problem is reduced to interior layer and regular domain problems the regular domain problems are solved by using the asymptotic expansion method the interior layer problem is treated by the method of stretching variable and the rkm four numerical examples are provided to illustrate the effectiveness of the present method the results of numerical examples show that the present method can provide very accurate approximate solutions

rates of convergence and metastability for abstract cauchy problems generated by accretive operators abstract we extract rates of convergence and rates of metastability in the sense of tao for convergence results regarding abstract cauchy problems generated by accretive at zero operators a d a x 2 x where x is a real banach space proved in 8 by proof theoretic analysis of the proofs in 8 and having assumed a new stronger notion of uniform accretivity at zero yielding a new notion of modulus of accretivity we compute the rate of metastability for the convergence of the solution of the abstract cauchy problem generated by a uniformly accretive at zero operator to the unique zero of a via proof mining based on a result by the first author finally we apply our results to a special class of cauchy problems considered in 8 this work is the first application of proof mining to the theory of partial differential equations

time averaging under fast periodic forcing of parabolic partial differential equations exponential estimates the phases of a large class of parabolic partial differential equations with rapid time periodic forcing can be separated up to exponential small errors the originally nonautonomous equation is transformed such that the nonautonomous terms are exponentially small in the period h of the forcing this is a counterpart for partial differential equations of the theorem by a neishtadt 1984 j appl math mech 48 134 139 for ordinary differential equations in our case the exponential rate depends on time t and the estimates have the form h exp c t h 1 3

the effective conductivity of a periodic array of spheres we consider the problem of determining the effective conductivity k of a composite material consisting of equal sized spheres of conductivity alpha arranged in a cubic array within a homogeneous matrix of unit conductivity we modify zuzovski brenner9s 1977 method and thereby obtain a set of infinite linear equations for the coefficients of the formal solution equivalent to that derived by mckenzie et al 1978 using a method originally devised by rayleigh 1892 on solving these equations we derive expressions for k ast to o c 9 c being the volume fraction of the spheres for simple body centred and face centred cubic arrays and also obtain numerical values for k over the whole range of alpha and c we show that these results for cubic arrays can be used to estimate ast for random arrays of identical spheres for arrays of highly conducting and nearly touching spheres batchelor o9brien 1977 showed that begin equation k ast sim begin cases k 1 ln 1 chi k 2 alpha infty chi c c max frac 1 3 rightarrow 1 2k 1 ln alpha k9 2 quad chi 1 alpha rightarrow infty end cases end equation where c max corresponds to the volume fraction when the spheres are actually touching each other and determined k 1 for the three cubic arrays our numerical results are consistent with the above asymptotic expressions except for the fact that the numerical values for the constants k 2 and k9 2 thereby obtained do not quite satisfy the relation k9 2 k 1 3 9 ln 2 k 2 given by batchelor o9brien we have been unable to find the reason for this slight discrepancy

quantitative analysis of boundary layers in periodic homogenization we prove quantitative estimates on the rate of convergence for the oscillating dirichlet problem in periodic homogenization of divergence form uniformly elliptic systems the estimates are optimal in dimensions larger than three and new in every dimension we also prove a regularity estimate on the homogenized boundary condition

singular perturbation analysis of boundary value problems for differential difference equations v small shifts with layer behavior in this paper an investigation is initiated of boundary value problems for singularly perturbed linear second order differential difference equations with small shifts i e where the second order derivative is multiplied by a small parameter and the shift depends on the small parameter similar boundary value problems are associated with expected first exit times of the membrane potential in models for neurons in particular this paper focuses on problems with solutions that exhibit layer behavior at one or both of the boundaries the analyses of the layer equations using laplace transforms lead to novel results it is shown that the layer behavior can change its character and even be destroyed as the shifts increase but remain small in the companion paper siam j appi math 54 1994 pp 273283 similar boundary value problems with solutions that exhibit rapid oscillations are studied

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

convergence of a multigrid method for elliptic equations with highly oscillatory coefficients standard multigrid methods are not so effective for equations with highly oscillatory coefficients new coarse grid operators based on homogenized operators are introduced to restore the fast convergence rate of multigrid methods finite difference approximations are used for the discretization of the equations convergence analysis is based on the homogenization theory proofs are given for a two level multigrid method with the homogenized coarse grid operator for two classes of two dimensional elliptic equations with dirichlet boundary conditions

numerical treatment of a mathematical model arising from a model of neuronal variability in this paper we describe a numerical approach based on finite difference method to solve a mathematical model arising from a model of neuronal variability the mathematical modelling of the determination of the expected time for generation of action potentials in nerve cells by random synaptic inputs in dendrites includes a general boundary value problem for singularly perturbed differential difference equation with small shifts in the numerical treatment for such type of boundary value problems first we use taylor approximation to tackle the terms containing small shifts which converts it to a boundary value problem for singularly perturbed differential equation a rigorous analysis is carried out to obtain priori estimates on the solution of the problem and its derivatives up to third order then a parameter uniform difference scheme is constructed to solve the boundary value problem so obtained a parameter uniform error estimate for the numerical scheme so constructed is established though the convergence of the difference scheme is almost linear but its beauty is that it converges independently of the singular perturbation parameter i e the numerical scheme converges for each value of the singular perturbation parameter however small it may be but remains positive several test examples are solved to demonstrate the efficiency of the numerical scheme presented in the paper and to show the effect of the small shift on the solution behavior

periodic homogenization of green and neumann functions for a family of second order elliptic operators with rapidly oscillating periodic coefficients we study the asymptotic behavior of the green and neumann functions using dirichlet and neumann correctors as a result we obtain asymptotic expansions of poisson kernels and the dirichlet to neumann maps as well as optimal convergence rates in lp and w1 p for solutions with dirichlet or neumann boundary conditions 2014 wiley periodicals inc

homogenization and boundary layers this paper deals with the homogenization of elliptic systems with dirichlet boundary condition when the coefficients of both the system and the boundary data are e periodic we show that as e 0 the solutions converge in l2 with a power rate in e and identify the homogenized limit system due to a boundary layer phenomenon this homogenized system depends in a non trivial way on the boundary our analysis answers a longstanding open problem raised for instance in 7 it extends substantially previous results obtained for polygonal domains with sides of rational slopes as well as our previous paper 12 where the case of irrational slopes was considered

uniformly convergent non standard finite difference methods for singularly perturbed differential difference equations with delay and advance a new class of fitted operator finite difference methods are constructed via non standard finite difference methods nsfdm s for the numerical solution of singularly perturbed differential difference equations having both delay and advance arguments the main idea behind the construction of our method s is to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures significant properties of the governing differential equation and thus provides the reliable numerical results unlike other fofdms constructed in standard ways the methods that we present in this paper are fairly simple to construct and thus enrich the class of fitted operator methods by adding these new methods these methods are shown to be e uniformly convergent with order two which is the highest possible order of convergence obtained via any fitted operator method for the problems under consideration this paper further clarifies several doubts e g why a particular scheme is not suitable for the whole range of values of the associated parameters and what could be the possible remedies finally we provide some numerical examples which illustrate the theoretical findings copyright 2005 john wiley sons ltd

singular perturbation analysis of boundary value problems for differential difference equations a study is made of a class of boundary value problems for linear second order differential difference equations in which the highest order derivative is multiplied by a small parameter consideration of the exact solution of simple model equations provides insight into the appropriate use of singular perturbation techniques for more general problems the resulting analysis leads to several novel features which are not present in problems without differences

quintic b spline method for solving second order linear and nonlinear singularly perturbed two point boundary value problems in this paper we have studied a numerical scheme to solve second order singularly perturbed two point linear and nonlinear boundary value problems the boundary layer of this type of problems exhibits at one end left or right point of the domain due to the presence of perturbation parameter the quintic b spline method and suitable piecewise uniform shishkin mesh have been used linear and nonlinear second order singularly perturbed boundary value problems have been solved by the present method the convergence analysis is also provided and the method is shown to have uniform convergence of fourth order numerical results have demonstrated the efficiency of the present method

a projection method applied to diffusion in a periodic structure in this paper we analyze a method designed to replace pde s with rapidly varying coefficients by pde s with constant coefficients this method is based on a combination of an asymptotic expansion and a variational principle we show that for smooth data the method has the same approximation properties as the more standard approach based only on an asymptotic expansion more importantly we show that the present method works well also for nonsmooth data this may be seen as a result of the optimal way in which the method treats boundary layers

convergence analysis for the numerical boundary corrector for elliptic equations with rapidly oscillating coefficients we develop the convergence analysis of a numerical scheme for approximating the solution of the elliptic problem l epsilon u epsilon frac partial partial x i a ij x epsilon frac partial partial x j u epsilon f mbox in omega u epsilon 0 mbox on partial omega where a y a ij y is a periodic symmetric positive definite matrix and omega 0 1 2 the major goal of the numerical scheme is to capture the epsilon scale of the oscillations of the solution u epsilon on a mesh size h epsilon mbox or h epsilon the numerical scheme is based on asymptotic expansions constructive boundary corrector and finite element approximations new a priori error estimates are established for the asymptotic expansions and for the constructive boundary correctors under weak assumptions on the regularity of the problem these estimates permit to establish sharp finite element error estimates and to consider composite materials applications depending on the regularity of the problem we establish for the numerical scheme a priori error estimates of o h 2 epsilon 3 2 epsilon h on the l 2 norm and o h epsilon 1 hat delta for the broken h 1 norm where hat delta in frac 1 4 0

a numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations abstract a boundary value problem for second order singularly perturbed delay differential equation is considered when the delay argument is sufficiently small to tackle the delay term the researchers m k kadalbajoo k k sharma numerical analysis of singularly perturbed delay differential equations with layer behavior appl math comput 157 2004 11 28 r e o malley jr singular perturbation methods for ordinary differential equations springer verlag new york 1991 used taylor s series expansion and presented an asymptotic as well as numerical approach to solve such type boundary value problem but the existing methods in the literature fail in the case when the delay argument is bigger one because in this case the use of taylor s series expansion for the term containing delay may lead to a bad approximation in this paper to short out this problem we present a numerical scheme for solving such type of boundary value problems which works nicely in both the cases i e when delay argument is bigger one as well as smaller one to handle the delay argument we construct a special type of mesh so that the term containing delay lies on nodal points after discretization the proposed method is analyzed for stability and convergence to demonstrate the efficiency of the method and how the size of the delay argument and the coefficient of the delay term affects the layer behavior of the solution several test examples are considered

numerical study of singularly perturbed differential difference equation arising in the modeling of neuronal variability the objective of this paper is to present a numerical study of a class of boundary value problems of singularly perturbed differential difference equations spdde which arise in computational neuroscience in particular in the modeling of neuronal variability the mathematical modeling of the determination of the expected time for the generation of action potential in the nerve cells by random synaptic inputs in dendrites includes a general boundary value problem for singularly perturbed differential difference equation with shifts the problem considered in this paper exhibit turning point behavior which add to the complexity in the construction of numerical approximation to the solution of the problem as well as in obtaining theoretical estimates on the solution exponentially fitted finite difference scheme based on il in allen southwell fitting is used on a specially designed mesh some numerical examples are given to validate convergence and computational efficiency of the proposed numerical scheme effect of the shifts on the layer structure is illuminated for the considered examples

schaefer krasnoselskii fixed point theorems using a usual measure of weak noncompactness abstract we present some extension of a well known fixed point theorem due to burton and kirk t a burton c kirk a fixed point theorem of krasnoselskii schaefer type math nachr 189 1998 423 431 for the sum of two nonlinear operators one of them compact and the other one a strict contraction the novelty of our results is that the involved operators need not to be weakly continuous finally an example is given to illustrate our results

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

the operational significance of the continuum hypothesis in the theory of water movement through soils and aquifers the operational meaning of the representative elementary volume rev concept on which current foundational theories of water movement through porous media are based is analyzed critically it is concluded that the rev concept as applied to real porous media is both unnecessarily restrictive and experimentally unverifiable in its place a relativist concept is proposed in which macroscopic physical variables are defined as convolution products of microscopic properties of a porous medium with weighting functions that represent the appropriate measuring instruments the effects of resolution and precision differences among experimental measuring devices now recognized as critical in the assessment of spatial variability in the properties of soils and aquifers can be incorporated naturally within the relativist concept but are excluded a priori in the rev concept the relativist point of view unlike the rev concept has a clear operational meaning and in principle extends theoretical validity to all local measurements made on porous media it is sufficient mathematically to derive conventional macroscopic balance equations for mass and linear momentum as well as a differential equation for the isothermal transport of water through an unsaturated anisotropic deformable soil or aquifer

second order homogenization of boundary and transmission conditions for one dimensional waves in periodic media abstract we consider the homogenized boundary and transmission conditions governing the mean field approximations of 1d waves in finite periodic media within the framework of two scale analysis we establish the homogenization ansatz up to the second order of approximation for both types of problems by obtaining the relevant boundary correctors and exposing the enriched boundary and transmission conditions as those of robin type rigorous asymptotic analysis is performed for boundary conditions while the applicability to transmission conditions is demonstrated via numerical simulations within this framework we also propose an optimized second order model of the homogenized wave equation for 1d periodic media that follows more accurately the exact dispersion relationship and generally enhances the performance of second order approximation the proposed analysis is applied toward the long wavelength approximation of waves in finite periodic bilaminates subject to both boundary and transmission conditions a set of numerical simulations is included to support the mathematical analysis and illustrate the effectiveness of the homogenization scheme

conservation laws of mixed type describing three phase flow in porous media in this paper we examine the mathematical structure of a model for three phase incompressible flow in a porous medium we show that in the absence of diffusive forces the system of conservation laws describing the flow is not necessarily hyperbolic we present an example in which there is an elliptic region in saturation space for reasonable relative permeability data a linearized analysis shows that in nonhyperbolic regions solutions grow exponentially however the nonhyperbolic region if present will be of limited extent which inherently limits the exponential growth to examine these nonlinear effects we resort to fine grid numerical experiments with a suitably dissipative numerical method these experiments indicate that the solutions of riemann problems remain well behaved in spite of the presence of a linearly unstable elliptic region in saturation space in particular when initial states are outside the elliptic region the riemann problem solution appears to stay outside the region further

univalent σ harmonic mappings we study mappings from 2 into 2 whose components are weak solutions to the elliptic equation in divergence form div u 0 which we call harmonic mappings we prove sufficient conditions for the univalence i e injectivity of such mappings moreover we prove local bounds in bmo on the logarithm of the jacobian determinant of such univalent mappings thus obtaining the a e nonvanishing of their jacobian in particular our results apply to harmonic mapping associated with any periodic structure and therefore they play an important role in homogenization

divergence free finite elements on tetrahedral grids for k 6 it was shown two decades ago that the p k p k 1 mixed element on triangular grids approximating the velocity by the continuous p k piecewise polynomials and the pressure by the discontinuous p k 1 piecewise polynomials is stable for all k 4 provided the grids are free of a nearly singular vertex the problem with the method in 3d was posted then and remains open the problem is solved partially in this work it is shown that the p k p k 1 element is stable and of optimal order in approximation on a family of uniform tetrahedral grids for all k 6 the analysis is to be generalized to non uniform grids when we can deal with the complicity of 3d geometry for the divergence free elements the finite element spaces for the pressure can be avoided in computation if a classic iterated penalty method is applied the finite element solutions for the pressure are computed as byproducts from the iterate solutions for the velocity numerical tests are provided

an analysis of thep version of the finite element method for nearly incompressible materials in this paper we analyze the behavior of the so calledp version of the finite element method when applied to the equations of plane strain linear elasticity we establish optimal rate error estimates that are uniformly valid independent of the value of the poisson ratio v in the interval 0 1 2 this shows that thep versiondoes not exhibit the degeneracy phenomenon which has led to the use of various only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material

a corpuscular approach to continuum mechanics basic considerations the macroscopic behaviour of a material derives ultimately from its microstructure which at the molecular level and below is of a fundamentally discrete nature as much is often known of the microstructure molecular interactions and thermal motions associated with specific materials it is of interest to ascertain whether and to what extent such information can usefully be incorporated into continuum models the first step in this direction is to treat discrete fundamental entities for example molecules ions or atoms as particles point masses and relate continuum theories to corpuscular considerations this is of course an important feature of the kinetic theory of gases lattice dynamics and statistical mechanics the first two theories take account of particle interactions and thermal motions as these pertain to general motions of moderately rarefied gases and vibrations in ordered solids respectively but such considerations play no explicit part in standard formulations of statistical mechanics many discussions which bear upon the relations between discrete descriptions of materials and continuum mechanics have been given motivated in the main by considerations of microstructure such as the contributions of ericksen 1960 1961 dahler scriven 1963 eringen suhubi 1964 green rivlin 1964 krumhansl 1965 eringen kroner krumhansl kunin mindlin and rivlin in kroner 1968 rivlin 1968 1976 alblas 1976 and capriz podio guidugli 1976 1977 this work is intended to complement the aforementioned by emphasising the roles played in general by corpuscular interactions and thermal motions in the formulation of continuum theories

numerical methods on shishkin mesh for singularly perturbed delay differential equations with a grid adaptation strategy abstract a numerical study is made for a class of boundary value problems of second order differential equations in which the highest order derivative is multiplied by a small parameter and both the differentiated convection and undifferentiated reaction terms are with negative shift we analyze three difference operators l k h k 1 2 3 a simple upwind scheme midpoint upwind scheme and a hybrid scheme respectively on a shishkin mesh to approximate the solution of the problem theoretical error bounds are established the accuracy by one of the grid adaptation strategies namely grid redistribution is examined by solving the considered problem using hybrid method for some values of and n as a result the accuracy gets improved by this grid adaptation strategy with almost the same computational cost a few numerical results exhibiting the performance of these three schemes are presented

sufficient conditions for the uniform convergence of a difference scheme for a singularly perturbed turning point problem a number of results exist in the literature for singularly perturbed differential equations without turning points in particular a number of difference schemes have been proposed that satisfy a stronger than normal convergence criteria known as uniform convergence this guarantees that the schemes model the boundary layers well we wish to examine whether these schemes will also be uniformly convergent if the equation has turning points to this end we derive sufficient conditions for uniform convergence which are satisfied not only by these schemes but by a more general class of schemes we show that the rate of convergence is determined by a characteristic parameter of the problem which may be less than one we confirm these theoretical results by numerical calculations

a priori estimates and analysis of a numerical method for a turning point problem bounds are obtained for the derivatives of the solution of a turning point problem these results suggest a modification of the el mistikawy werle finite difference scheme at the turning point a uniform error estimate is obtained for the resulting method and illustrative numerical results are given i introduction we will examine the following two point boundary value problem with dirichlet data at the endpoints l la ly eyx x p x yx x q x y x f x for i x 1