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

accurate multiscale finite element method for numerical simulation of two phase flow in fractured media using discrete fracture model numerical simulation in naturally fractured media is challenging because of the coexistence of porous media and fractures on multiple scales that need to be coupled we present a new approach to reservoir simulation that gives accurate resolution of both large scale and fine scale flow patterns multiscale methods are suitable for this type of modeling because it enables capturing the large scale behavior of the solution without solving all the small features dual porosity models in view of their strength and simplicity can be mainly used for sugar cube representation of fractured media in such a representation the transfer function between the fracture and the matrix block can be readily calculated for water wet media for a mixed wet system the evaluation of the transfer function becomes complicated due to the effect of gravity in this work we use a multiscale finite element method msfem for two phase flow in fractured media using the discrete fracture model by combining msfem with the discrete fracture model we aim towards a numerical scheme that facilitates fractured reservoir simulation without upscaling msfem uses a standard darcy model to approximate the pressure and saturation on a coarse grid whereas fine scale effects are captured through basis functions constructed by solving local flow problems using the discrete fracture model the accuracy and the robustness of msfem are shown through several examples in the first example we consider several small fractures in a matrix and then compare the results solved by the finite element method then we use the msfem in more complex models the results indicate that the msfem is a promising path toward direct simulation of highly resolution geomodels

hilbert space compression and exactness of discrete groups we show that the hilbert space compression of any unbounded finite dimensional cat 0 cube complex is 1 and deduce that any finitely generated group acting properly co compactly on a cat 0 cube complex is exact and hence has yu s property a the class of groups covered by this theorem includes free groups finitely generated coxeter groups finitely generated right angled artin groups finitely presented groups satisfying the b 4 t 4 small cancellation condition and all those word hyperbolic groups satisfying the b 6 condition another family of examples is provided by certain canonical surgeries defined by link diagrams

a global jacobian method for mortar discretizations of nonlinear porous media flows we describe a nonoverlapping domain decomposition algorithm for nonlinear porous media flows discretized with the multiscale mortar mixed finite element method there are two main ideas 1 linearize the global system in both subdomain and interface variables simultaneously to yield a single newton iteration and 2 algebraically eliminate subdomain velocities and optionally subdomain pressures to solve linear systems for the 1st or the 2nd schur complements solving the 1st schur complement system gives the multiscale solution without the need to solve an interface iteration solving the 2nd schur complement system gives a linear interface problem for a nonlinear model the methods are less complex than a previously developed nonlinear mortar algorithm which requires two nested newton iterations and a forward difference approximation furthermore efficient linear preconditioners can be applied to speed up the iteration the methods are implemented in parallel and a numerical study is performed

a local error estimate of the method of multi scale asymptotic expansions for elliptic problems with rapidly oscillatory coefficients in this paper on basis of o a oleinik a s shamaev g a yosifian mathematical problems in elasticity and homogenization north holland amsterdam 1992 we present a local error estimate of the method of multi scale asymptotic expansions for second order elliptic problems with rapidly oscillatory coefficients

random martingales and localization of maximal inequalities abstract let x d be a metric measure space for r 0 consider the hardy littlewood maximal operator m r f x def sup r r 1 b x r b x r f d we show that if there is an n 1 such that one has the microdoubling condition b x 1 1 n r b x r for all x x and r 0 then the weak 1 1 norm of m r has the following localization property m r l 1 x l 1 x sup r 0 m r r n r l 1 x l 1 x an immediate consequence is that if x d is ahlfors david n regular then the weak 1 1 norm of m r is n log n generalizing a result of stein and stromberg 1983 47 we show that this bound is sharp by constructing a metric measure space x d that is ahlfors david n regular for which the weak 1 1 norm of m 0 is n log n the localization property of m r is proved by assigning to each f l 1 x a distribution over random martingales for which the associated random doob maximal inequality controls the weak 1 1 inequality for m r

implementation of a mortar mixed finite element method using a multiscale flux basis this paper provides a new implementation of a multiscale mortar mixed finite element method for second order elliptic problems the algorithm uses non overlapping domain decomposition to reformulate a fine scale problem as a coarse scale mortar interface problem which is then solved using an iterative method the original implementation by arbogast pencheva wheeler and yotov multiscale model simul 2007 required solving one local dirichlet problem on each subdomain per interface iteration we alter this implementation by forming a multiscale flux basis this basis consists of mortar functions representing the individual flux responses for each mortar degree of freedom on each subdomain independently the computation of these basis functions requires solving a fixed number of dirichlet subdomain problems taking linear combinations of the multiscale flux basis functions replaces the need to solve any dirichlet subdomain problems during the interface iteration this new implementation yields the same solution as the original implementation and is computationally more efficient in cases where the number of interface iterations is greater than the number of mortar degrees of freedom per subdomain the gain in computational efficiency increases with the number of subdomains

a new approximate method for second order elliptic problems with rapidly oscillating coefficients based on the method of multiscale asymptotic expansions abstract in this paper we consider second order elliptic problems with rapidly oscillating coefficients on basis of o a oleinik a s shamaev g a yosifian mathematical problems in elasticity and homogenization north holland amsterdam 1992 wen ming he jun zhi cui a pointwise estimate on the 1 order approximation of g x 0 e ima j appl math 70 2005 241 269 we propose a new approximate method to solve these problems of course we present its error estimate

application of the finite difference heterogeneous multiscale method to the richards equation 1 this paper extends the finite difference heterogeneous multiscale method fdhmm to simulate transient unsaturated water flow problems in random porous media the numerical method is based on the use of two different schemes for the original equation at different grid levels which allows numerical results at a lower cost than solving the original equations the main feature of fdhmm is that the necessary data for the macroscopic model are supplied by solving the microscopic model on a sparse spatial domain the generated code is verified by applying the linearization model of the richards equation considering two different constitutive relationships this method is applied to several test examples with different soil textures and boundary conditions both the dirichlet and the periodic boundary conditions are considered for solving the local microscopic model when the water flow in heterogeneous unsaturated soils is simulated by fdhmm the numerical experiments demonstrate that fdhmm can effectively simulate the transient unsaturated water flow in the specific soils the numerical experiments also demonstrate that fdhmm can achieve accurate global mass balance and is a globally convergent algorithm and the spatial correlation length of random coefficients under the specific standard deviation has relatively little influence on the accuracy of the method

adaptive algebraic multiscale solver for compressible flow in heterogeneous porous media this paper presents the development of an adaptive algebraic multiscale solver for compressible flow c ams in heterogeneous porous media similar to the recently developed ams for incompressible linear flows wang et al 2014 19 c ams operates by defining primal and dual coarse blocks on top of the fine scale grid these coarse grids facilitate the construction of a conservative finite volume coarse scale system and the computation of local basis functions respectively however unlike the incompressible elliptic case the choice of equations to solve for basis functions in compressible problems is not trivial therefore several basis function formulations incompressible and compressible with and without accumulation are considered in order to construct an efficient multiscale prolongation operator as for the restriction operator c ams allows for both multiscale finite volume msfv and finite element msfe methods finally in order to resolve high frequency errors fine scale pre and post smoother stages are employed in order to reduce computational expense the c ams operators prolongation restriction and smoothers are updated adaptively in addition to this the linear system in the newton raphson loop is infrequently updated systematic numerical experiments are performed to determine the effect of the various options outlined above on the c ams convergence behaviour an efficient c ams strategy for heterogeneous 3d compressible problems is developed based on overall cpu times finally c ams is compared against an industrial grade algebraic multigrid amg solver results of this comparison illustrate that the c ams is quite efficient as a nonlinear solver even when iterated to machine accuracy

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

a reduced basis localized orthogonal decomposition in this work we combine the framework of the reduced basis method rb with the framework of the localized orthogonal decomposition lod in order to solve parametrized elliptic multiscale problems the idea of the lod is to split a high dimensional finite element space into a low dimensional space with comparably good approximation properties and a remainder space with negligible information the low dimensional space is spanned by locally supported basis functions associated with the node of a coarse mesh obtained by solving decoupled local problems however for parameter dependent multiscale problems the local basis has to be computed repeatedly for each choice of the parameter to overcome this issue we propose an rb approach to compute in an offline stage lod for suitable representative parameters the online solution of the multiscale problems can then be obtained in a coarse space thanks to the lod decomposition and for an arbitrary value of the parameters thanks to a suitable interpolation of the selected rb the online rb lod has a basis with local support and leads to sparse systems applications of the strategy to both linear and nonlinear problems are given

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 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

adaptive iterative multiscale finite volume method the multiscale finite volume msfv method is a computationally efficient numerical method for the solution of elliptic and parabolic problems with heterogeneous coefficients it has been shown for a wide range of test cases that the msfv results are in close agreement with those obtained with a classical computationally expensive technique the method however fails to give accurate results for highly anisotropic heterogeneous problems due to weak localization assumptions recently a convergent iterative msfv i msfv method was developed to enhance the quality of the multiscale results by improving the localization conditions although the i msfv method proved to be efficient for most practical problems it is still favorable to improve the localization condition adaptively i e only for a sub domain where the original msfv localization conditions are not acceptable e g near shale layers and long coherent structures with high permeability contrasts in this paper a space time adaptive i msfv ai msfv method is introduced it is shown how to improve the msfv results adaptively in space and simulation time the fine scale smoother which is necessary for convergence of the i msfv method is also applied locally finally for multiphase flow problems two criteria are investigated for adaptively updating the msfv interpolation functions 1 a criterion based on the total mobility change for the transient coefficients and 2 a criterion based on the pressure equation residual for the accuracy of the results for various challenging test cases it is demonstrated that iterations in order to obtain accurate results even for highly anisotropic heterogeneous problems are required only in small sub domains and not everywhere the findings show that the error introduced in the msfv framework can be controlled and improved very efficiently with very little additional computational cost compared to the original non iterative msfv method

blueprint for perturbative solution of flow and transport in strongly heterogeneous composite media using fractal and variational multiscale decomposition 1 this paper offers a blueprint for extending perturbative ensemble moment and potentially other e g gaussian or perturbative karhunen loeve solutions of stochastic flow and transport problems in composite randomly heterogeneous porous media to cases in which medium heterogeneity is arbitrarily large the proposed approach is based on a novel idea of combined fractal and variational multiscale decomposition it considers log hydraulic conductivity to be a composite random field with piecewise statistically homogeneous increments characterized by a truncated power variogram i e a truncated fractal this consideration is supported either exactly or approximately by a growing amount of observational data it allows decomposing the field into two or more mutually uncorrelated components having relatively small variances associated with a hierarchy of spatial correlation scales combining this fractal decomposition with a variational multiscale decomposition formalism motivated by the multiscale nature of turbulence allows decomposing flow and transport problems into two or more coupled sets of variational problems each of which is amenable to perturbative solution regardless of how heterogeneous the original field was the problems are defined on a hierarchy of computational grids having discretization intervals proportional to the correlation scales the proposed approach does not require introducing effective parameters on any scale instead it allows resolving flow and transport ensemble moments fully on multiple scales of spatial resolution the approach nevertheless allows subgrid closure another idea inspired by the literature on turbulence through subgrid refinement or the use of effective subgrid parameters it additionally offers the critically important capability of conditioning flow and transport analyses on multiscale measurements details are presented in the context of steady state single phase flow in a composite medium

particle method approximation of oscillatory solutions to hyperbolic differential equations particle methods approximating hyperbolic partial diferential equations with oscillatory solutions are studied convergence is proved for approximations for which the continuous solution is not well resolved on the computational grid highly oscillatory solutions to the broadwell and variable coefficient carleman models are considered homogenization results are given and the approximations of more general systems are discussed numerical examples are presented

robust multilevel solvers for high contrast anisotropic multiscale problems a robust multilevel method for computing the solution of a scalar elliptic equation with anisotropic highly varying tensor coefficients is presented the method which belongs to the class of nonlinear algebraic multilevel iterations amlis uses an abstract framework for general symmetric positive definite bilinear forms previously presented in another publication by the author the crucial ingredient for obtaining robustness with respect to the variations in the coefficients and the anisotropies is the design of a nested sequence of spaces based on local generalized eigenvalue problems a discussion on how to achieve large coarsening factors in this sequence of spaces which is desirable in terms of computational complexity is included particular emphasis is put on how to handle the situation when the computed generating sets of the nested spaces are not minimal i e do not constitute bases several numerical examples are provided verifying the theoretically established robustness results

an error analysis for the finite element method applied to convection diffusion problems abstract this paper analyzes the finite element method applied to a convection diffusion model problem linear elements are used for the trial space the error is measured in a norm closely related to the l sup p norm when the test space is composed of linear elements with parabolic upwinding the method is shown to be optimal when the input data is piecewise smooth a condition which is usually observed in practice without these smoothness assumptions the method is shown to be non optimal even if the class of test spaces is extended to include any elements which have a shape independent of the mesh size author

on metric ramsey type phenomena the main question studied in this article may be viewed as a nonlinear analogue of dvoretzky s theorem in banach space theory or as part of ramsey theory in combinatorics given a finite metric space on n points we seek its subspace of largest cardinality which can be embedded with a given distortion in hilbert space we provide nearly tight upper and lower bounds on the cardinality of this subspace in terms of n and the desired distortion our main theorem states that for any 0 every n point metric space contains a subset of size at least n1 which is embeddable in hilbert space with o log 1 distortion the bound on the distortion is tight up to the log 1 factor we further include a comprehensive study of various other aspects of this problem

upscaling methods for a class of convection diffusion equations with highly oscillating coefficients this paper investigates the upscaling method to the following parabolic equation tc uc d c f x t which stems from the application of solute transport in porous media because of the highly oscillating permeability of the porous media the darcy velocity u hence the dispersion tensor d has many scales with high contrasts thus how to calculate the macro scale equivalent coefficients of the above equation becomes the target of this paper a new upscaling method is proposed and studied via comparing with another upscaling method which was proposed in z chen w deng h ye discrete contin dyn syst 13 2005 941 960 the two different equivalent coefficients computing formulations are based on the solutions of two different cell local problems which one utilizes the elliptic operator with terms of all orders while the other only uses the second order term error estimates between the equivalent coefficients and the homogenized coefficients are given under the assumption that the oscillating coefficients are periodic which is not required by the method numerical experiments are carried out for the periodic coefficients to demonstrate the accuracy of the proposed method moreover the upscaling method is applied to solve the solute transport in a porous medium with a random log normal relative permeability the results show the efficiency and accuracy of the proposed method

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

robust multilevel methods for general symmetric positive definite operators a general robust multilevel method for solving symmetric positive definite systems resulting from discretizing elliptic partial differential equations is developed the term robust refers to the convergence rate of the method being independent of discretization parameters i e the problem size and problem parameters important instances of such problem parameters are in particular highly varying coefficients the method belongs to the class of nonlinear algebraic multilevel iterations the crucial ingredient for obtaining robustness is the construction of a nested sequence of spaces based on local generalized eigenvalue problems the method is analyzed in a rather general setting applying to the scalar elliptic equation the equations of linear elasticity and equations arising in the solution of maxwell s equations numerical results for the scalar elliptic equation are presented showing its robust convergence behavior

proper orthogonal decomposition method for multiscale elliptic pdes with random coefficients abstract in this paper we develop an efficient multiscale reduced basis method to solve elliptic pdes with multiscale and random coefficients in a multi query setting our method consists of offline and online stages in the offline stage a small number of reduced multiscale basis functions are constructed within each coarse grid block using the proper orthogonal decomposition pod method moreover local tensor spaces are defined to approximate the solution space of the multiscale random pdes in the online stage a weak formulation is derived and discretized using the galerkin method to compute the solution since the multiscale reduced basis functions can efficiently approximate the high dimensional solution space our method is very efficient in solving multiscale elliptic pdes with random coefficients convergence analysis of the proposed method is presented which shows the dependence of the numerical error on the number of snapshots and the truncation threshold in the pod method finally numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale problems with or without scale separation in the physical space

weak containment and tensor products of group representations ii let g be a locally compact group and g the set of equivalence classes of irreducible unitary representations of g equipped with the jacobson topology is called the dual space of g in this paper we continue our study of problems related to supports in g of tensor products of representations of g it is well known that ifn and are finite dimensional representations of g and g denotes the conjugate representation of q then the tensor product n contains the trivial one dimensional representation 1g if and only ifg and q have a common subrepresentation can this be generalized to the case where n and are arbitrary and containment is replaced by weak containment this question was first discussed by fell 6 and has been taken up in 4 and 21 the object of sect 1 is to add some further results concerning this problem our main results are the following we show that for a polynomially growing discrete group g and any two unitary representations n and of g 1 is weakly contained in nqff i e 1 suppn if and only if suppnnsupp04 0 this extends the corresponding result for locally nilpotent discrete groups in 21 if g is a motion group and n e we give a criterion on n to satisfy that laesuppn is equivalent to ff n for all ed

geometric multigrid for darcy and brinkman models of flows in highly heterogeneous porous media we apply geometric multigrid methods for the finite element approximation of flow problems governed by darcy and brinkman systems used in modeling highly heterogeneous porous media the method is based on divergence conforming discontinuous galerkin methods and overlapping patch based domain decomposition smoothers we show in benchmark experiments that the method is robust with respect to mesh size and contrast of permeability for highly heterogeneous media

a multiscale a posteriori error estimate we introduce a hierarchic a posteriori error estimate for singularly perturbed reaction diffusion problems the estimator is based on a petrov galerkin method in which the trial space is enriched with nonpolynomial functions or multiscale functions we study the equivalence between the a posteriori estimate and the exact error in the energy norm moreover we prove a relationship between the hierarchic estimator and an explicit residual estimator the approach provides accurate estimates for the boundary layer regions which is confirmed by numerical experiments 2004 elsevier b v all rights reserved

adaptive finite element heterogeneous multiscale method for homogenization problems in this paper we present an a posteriori error analysis for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method unlike standard finite element methods our discretization scheme relies on macro and microfinite elements the desired macroscopic solution is obtained by a suitable averaging procedure based on microscopic data as the macroscopic data such as the macroscopic diffusion tensor are not available beforehand appropriate error indicators have to be defined for designing adaptive methods we show that such indicators based only on the available macro and microsolutions used to compute the actual macrosolution can be defined allowing for a macroscopic mesh refinement strategy which is both reliable and efficient the corresponding a posteriori estimates for the upper and lower bound are derived in the energy norm in the case of a uniformly oscillating tensor we recover the standard residual based a posteriori error estimate for the finite element method applied to the homogenized problem numerical experiments confirm the efficiency and reliability of the adaptive multiscale method

abstract robust coarse spaces for systems of pdes via generalized eigenproblems in the overlaps coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations pdes however it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the pde coefficients especially for systems of pdes here we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities we achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence we prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous pdes and systems of pdes that confirm this property

enhanced rfb method the residual free bubble method rfb is a parameter free stable finite element method that has been successfully applied to a wide range of boundary value problems exhibiting multiple scale behaviour if some local features of the solution are known a priori the approximation properties of the rfb finite element space can be improved by enriching it on selected edges of the partition by edge bubbles that are supported on pairs of neighbouring elements motivated by this idea we define and analyse an enhanced residual free bubble method for the solution of convection dominated convection diffusion problems in 2 d our a priori analysis highlights the limitations of the rfb method and the improved global approximation properties of the new method the theoretical results are supported by detailed numerical experiments

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

global 2d digital image correlation for motion estimation in a finite element framework a variational formulation and a regularized pyramidal multi grid implementation in this study the inverse problem of reconstructing the in plane 2d displacements of a monitored surface through a sequence of two dimensional digital images severely ill posed in hadamard s sense is deeply investigated a novel variational formulation is presented for the continuum 2d digital image correlation problem and critical issues such as semi coercivity and solution multiplicity are discussed by functional analysis tools in the framework of a galerkin finite element discretization of the displacement field a robust implementation for 2d digital image correlation is outlined aiming to attenuate the spurious oscillations which corrupt the deformation scenario especially when very fine meshes are adopted recourse is made to a hierarchical family of grids linked by suitable restriction and prolongation operators and defined over an image pyramid multi grid cycles are performed ascending and descending along the pyramid with only one newton iteration per level irrespective of the tolerance satisfaction as if the problem were linear at each level the conventional least square matching functional is herein enriched by a tychonoff regularization provision preserving the solution against an unstable response the algorithm is assessed on the basis of both synthetic and truly experimental image pairs copyright 2013 john wiley sons ltd

additive schur complement approximation and application to multilevel preconditioning this paper introduces an algorithm for additive schur complement approximation asca which can be applied in various iterative methods for solving systems of linear algebraic equations arising from finite element fe discretization of partial differential equations pde it is shown how the asca can be used to set up a nonlinear algebraic multilevel iteration amli method this requires the construction of a linear multiplicative two level preconditioner at each level the latter is computed in the course of a simultaneous exact two by two block factorization of local stiffness matrices associated with a covering of the entire domain by overlapping subdomains unlike in schwarz type domain decomposition methods this method does not require a global coarse problem but instead uses local coarse problems to provide global communication a robust condition number bound is proved for a particular partitioning of the nodes of a uniform grid the presented numerical tests demonstrate that the asca whe

zonal multiscale finite volume framework in this work the zonal multiscale finite volume zmsfv framework is presented the framework generalizes previous extensions of the multiscale finite volume msfv method and allows to integrate additional models in a unified way the new approach is based on splitting the domain of interest into a classical msfv domain and additional zones within each zone the solution space is spanned by so called zonal functions coupling of the zonal solutions to the surrounding domain is achieved via extended zonal functions the construction of basis zonal and extended zonal functions can be performed such that the functions form a partition of unity the zmsfv framework allows to include the influence of the zonal functions directly in the coarse scale problem similar to previous msfv method formulations the zmsfv framework can ensure conservative coarse scale solutions and allows for the construction of conservative fine scale solutions the main features of the zmsfv framework are used to derive robust preconditioners for iterative solutions of high contrast problems as encountered in subsurface flow and transport simulations it is shown that the obtained convergence rates are independent of the contrast

euclidean quotients of finite metric spaces this paper is devoted to the study of quotients of finite metric spaces the basic type of question we ask is given a finite metric space m and 1 what is the largest quotient of a subset of m which well embeds into hilbert space we obtain asymptotically tight bounds for these questions and prove that they exhibit phase transitions we also study the analogous problem for embeddings into lp and the particular case of the hypercube

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

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 note on the uniform inf sup condition for the brinkman problem in highly heterogeneous media abstract in this short note we study the interpolation spaces based uniform inf sup condition for brinkman equation in highly heterogeneous domains the result is known to hold in the case of a uniform constant permeability our numerical construction and experiments show that it is not true for the taylor hood and the mini elements applied to problems with highly heterogeneous coefficients

rectifiable sets and the traveling salesman problem let k c c be a bounded set in this paper we shall give a simple necessary and sufficient condit ion for k to lie in a rectifiable curve we say that a set is a rectifiable curve if it is the image of a finite interval under a lipschitz mapping recall that for a connected set f c c f is a rectifiable curve not necessarily simple if and only if l f where l denotes one dimensional hausdorff measure this classical result follows from the fact that on any finite graph there is a tour which covers the entire graph and which crosses each edge but not necessarily each vertex at most twice if k is a finite set then we are essentially reduced to the classical traveling salesman problem tsp compute the length of the shortest hami l ton ian cycle which hits all points of k this is the same up to a constant multiple as asking for the inf imum of l f where f is a curve k c f such a f is called a spanning tree in tsp theory for infinite sets k we cannot hope in general to have k be a subset of a jordan curve what we should therefore look at is connected sets which conta in k let fmi n be the shortest minimal spanning tree then we cannot possibly solve our problem for sets k of infinite cardinality if we cannot find f i f c o fmin for any finite set k here and throughout the paper c co c1 c o etc denote various universal constants while there are several algorithms for computing l fml these algorithms work for finite graphs satisfying the triangle inequality and do not use the euclidean properties of k see 13 for an excellent discussion of some of these algorithms therefore these methods cannot solve our problem for general infinite k we present a method which is a minor modification of a well known algorithm farthest insert ion see 13 which yields a f with i f c o l fmi the farthest insert ion algorithm has been extensively studied with large numerical calculations on computers and is experimentally good in the sense that the f produced satisfy i f c o l f i for all examples which have

spectral amge rho amge we introduce spectral element based algebraic multigrid rho amge a new algebraic multigrid method for solving systems of algebraic equations that arise in ritz type finite element discretizations of partial differential equations the method requires access to the element stiffness matrices which enables accurate approximation of algebraically smooth vectors i e error components that relaxation cannot effectively eliminate most other algebraic multigrid methods are based in some manner on predefined concepts of smoothness coarse grid selection and prolongation for example are often defined assuming that smooth errors vary slowly in the direction of strong connections relatively large coefficients in the operator matrix one aim of rho amge is to broaden the range of problems to which the method can be successfully applied by avoiding any implicit premise about the nature of the smooth error rho amge uses the spectral decomposition of small collections of element stiffness matrices to determine local representations of algebraically smooth error components this provides a foundation for generating the coarse level and for defining effective interpolation this paper presents a theoretical foundation for rho amge along with numerical experiments demonstrating its robustness

homogenization and convergence of the vortex method for 2 d euler equations with oscillatory vorticity fields the 2 d incompressible euler equations with oscillatory vorticity fields are studied a homogenization result for 2 d euler equations in velocity vorticity formulation is obtained and weak continuity of the equations is proved convergence of the vortex method is analyzed in the case when the continuous vorticity is not well resolved by the computational particles numerical results are given comparisons are made with the corresponding finite difference approximation

differentiating maps into l1 and the geometry of bv functions this is one of a series of papers examining the interplay between differentiation theory for lipschitz maps x v and bi lipschitz nonembeddability where x is a metric measure space and v is a banach space here we consider the case v l 1 where differentiability fails we establish another kind of differentiability for certain x including n and h the heisenberg group with its carnot caratheodory metric it follows that does not bi lipschitz embed into l 1 as conjectured by j lee and a naor when combined with their work this provides a natural counterexample to the goemans linial conjecture in theoretical computer science the first such counterexample was found by khot vishnoi kv05 a key ingredient in the proof of our main theorem is a new connection between lipschitz maps to l 1 and functions of bounded variation which permits us to exploit results on the structure of bv functions on the heisenberg group fssc01

modeling multiscale diffusion processes in random heterogeneous media a simple stochastic up scaling strategy is presented for incorporating the effects of micro scale variations of thermal conductivity in the thermal analysis of macro scale systems incomplete characterization of the fine scale variation necessitates posing the thermal conductivity as a stochastic multiscale random field a data driven approach coupled with a non linear dimension reduction algorithm is used to construct a low dimensional stochastic descriptor of the fine scale variation of the stochastic conductivity a representative volume element rve argument is utilized to construct a simple model for up scaling the stochastic thermal conductivity in this model the effective stochastic thermal conductivity is estimated by virtually interrogating the stochastic rv a realistic example of the proposed stochastic multiscale diffusion strategy is illustrated utilizing limited experimental data

markov chains in smooth banach spaces and gromov hyperbolic metric spaces a metric space x has markov type 2 if for any reversible flnite state markov chain fztg with z0 chosen according to the stationary distribution and any map f from the state space to x the distance dt from f z0 to f zt satisfles e d 2 k 2 te d 2 for some k k x 2 has markov type 2 this proves a conjecture of ball we also show that trees hyperbolic groups and simply connected riemannian manifolds of pinched negative curvature have markov type 2 our results are applied to settle several conjectures on lipschitz extensions and embeddings in particular we answer a question posed by johnson and lindenstrauss in 1982 by showing that for 1 q 2 p 1 any lipschitz mapping from a subset of lp to lq has a lipschitz extension deflned on all of lp

smoothed aggregation multigrid for markov chains a smoothed aggregation multigrid method is presented for the numerical calculation of the stationary probability vector of an irreducible sparse markov chain it is shown how smoothing the interpolation and restriction operators can dramatically increase the efficiency of aggregation multigrid methods for markov chains that have been proposed in the literature the proposed smoothing approach is inspired by smoothed aggregation multigrid for linear systems supplemented with a new lumping technique that assures well posedness of the coarse level problems the coarse level operators are singular m matrices on all levels resulting in strictly positive coarse level corrections on all levels numerical results show how these methods lead to nearly optimal multigrid efficiency for an extensive set of test problems both when geometric and algebraic aggregation strategies are used

a stochastic mixed finite element heterogeneous multiscale method for flow in porous media a computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales in order to capture the small scale heterogeneity a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method hmm in the spatial domain this new method ensures both local and global mass conservation starting from a specified covariance function the stochastic log permeability is discretized in the stochastic space using a truncated karhunen loeve expansion with several random variables due to the small correlation length of the covariance function this often results in a high stochastic dimensionality therefore a newly developed adaptive high dimensional stochastic model representation technique hdmr is used in the stochastic space this results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method asgc numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method

harmonic maps on amenable groups and a diffusive lower bound for random walks we prove diffusive lower bounds on the rate of escape of the random walk on infinite transitive graphs similar estimates hold for finite graphs up to the relaxation time of the walk our approach uses nonconstant equivariant harmonic mappings taking values in a hilbert space for the special case of discrete amenable groups we present a more explicit proof of the mok korevaar schoen theorem on the existence of such harmonic maps by constructing them from the heat flow on a folner set

a matrix analysis of operator based upscaling for the wave equation scientists and engineers who wish to understand the earth s subsurface are faced with a daunting challenge features of interest range from the microscale centimeters to the macroscale hundreds of kilometers it is unlikely that computational power limitations will ever allow modeling of this level of detail numerical upscaling is one technique intended to reduce this computational burden the operator based algorithm developed originally for elliptic flow problems is modified for the acoustic wave equation with the wave equation written as a first order system in space we solve for pressure and its gradient acceleration the upscaling technique relies on decomposing the solution space into coarse and fine components operator based upscaling applied to the acoustic wave equation proceeds in two steps step one involves solving for fine grid features internal to coarse blocks this stage can be solved quickly via a well chosen set of coarse grid boundary conditions each coarse problem is solved independently of its neighbors in step two we augment the coarse scale problem via this internal subgrid information unfortunately the complexity of the numerical upscaling algorithm has always obscured the physical meaning of the resulting solution via a detailed matrix analysis the coarse scale acceleration is shown to be the solution of the original constitutive equation with input density field corresponding to an averaged density along coarse block edges the pressure equation corresponds to the standard acoustic wave equation at nodes internal to coarse blocks however along coarse cell boundaries the upscaled solution solves a modified wave equation which includes a mixed second derivative term

two scale finite element method for nonselfadjoint elliptic problems with rapidly oscillatory coefficients the purpose of this paper is to solve nonselfadjoint elliptic problems with rapidly oscillatory coefficients a two scale approximate expression of the exact solution for nonselfadjoint elliptic problems is considered and the error estimate of the two scale approximate solution is derived moreover a two scale finite element approximate solution is given and the optimal error estimate is obtained

an immersed weak galerkin method for elliptic interface problems abstract in this paper we present an immersed weak galerkin method for solving second order elliptic interface problems the proposed method does not require solution meshes to be aligned with the interface consequently uniform cartesian meshes can be used for nontrivial interfacial geometry we show the existence and uniqueness of the numerical algorithm and provide the error analysis in the energy norm numerical results are reported to demonstrate the performance of the method

analysis of the multiscale finite element method for nonlinear and random homogenization problems in this paper we study the convergence of the multiscale finite element method for nonlinear and random homogenization problems error estimates similar to those for linear homogenization problems are obtained here

a new multiscale finite element method for high contrast elliptic interface problems we introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes and without resolving the interfaces a typical application would be the modelling of flow in a porous medium containing a number of inclusions of low or high permeability embedded in a matrix of high respectively low permeability our method is h 1 conforming with degrees of freedom at the nodes of a triangular mesh and requiring the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface but which use standard linear approximation otherwise a key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems under moderate assumptions we prove that our methods have optimal convergence rate of o h in the energy norm and o h 2 in the l 2 norm where h is the coarse mesh diameter and the hidden constants in these estimates are independent of the contrast i e ratio of largest to smallest value of the pde coefficient for standard elements the best estimate in the energy norm would be o h 1 2 e with a hidden constant which in general depends on the contrast the new interior boundary conditions depend not only on the contrast of the coefficients but also on the angles of intersection of the interface with the element edges

a multiscale probabilistic collocation method for subsurface flow in heterogeneous media 1 owing to the spatial variability of the media properties uncertainty quantification for subsurface flow and solute transport usually requires high resolution simulations in this work a multiscale probabilistic collocation method mspcm is developed for solving such problems in a computationally efficient manner the subsurface flow problem is cast in a stochastic framework and probabilistic collocation strategy is used to represent the original stochastic differential equation the resulting equations are a set of decoupled deterministic equations with respect to collocation points a multiscale finite element method is utilized to solve these deterministic problems on a coarse mesh coarse scale basis functions are constructed on a field in which the conductivity varies spatially at each set of stochastic collocation points the coarse scale solution is then obtained by solving a modified coarse formulation that takes into account the fine scale heterogeneity the fine scale solution is reconstructed after the coarse scale solution is available since the pcm and multiscale finite element method are implemented at different levels the mspcm inherits their respective advantages in which a stochastic problem is decomposed by fewer realizations and is solved on a coarser grid the performance of the proposed method is demonstrated with numerical examples the capability of mspcm in reproducing the probability density functions pdfs of head and velocity is investigated the numerical results show that the mspcm with proper coarsening level is able to capture small scale heterogeneity with a coarse mesh and to generate satisfactory probability density functions of head and velocity

a multiscale finite element linearization scheme for the unsaturated flow problems in heterogeneous porous media 1 numerical solution of unsaturated water flow in heterogeneous porous media is difficult not only because of the spatial variability in the parameters used to characterize the relevant physical properties of the natural porous media but also because of the mathematical problems that arise in dealing with the nonlinearities in this study a multiscale finite element linearization scheme is presented for effectively simulating unsaturated flow in heterogeneous porous media spanning over many scales the central goal of this method is to obtain the large scale solution of richards equation with heterogeneous coefficients accurately and efficiently on a coarse grid without resolving all the small scale details the underlying idea is to employ the slodicka linear relaxation approximation scheme to address the nonlinear characteristics of the equation and to use the multiscale finite element base functions to account for the spatial variability in the equation coefficients we describe the principle for constructing such a method and give an algorithm for implementing it numerical experiments are carried out for the unsaturated flow equation with both periodic and randomly generated lognormal hydraulic conductivity to demonstrate the efficiency and accuracy of the proposed method

a probabilistic graphical model approach to stochastic multiscale partial differential equations we develop a probabilistic graphical model based methodology to efficiently perform uncertainty quantification in the presence of both stochastic input and multiple scales both the stochastic input and model responses are treated as random variables in this framework their relationships are modeled by graphical models which give explicit factorization of a high dimensional joint probability distribution the hyperparameters in the probabilistic model are learned using sequential monte carlo smc method which is superior to standard markov chain monte carlo mcmc methods for multi modal distributions finally we make predictions from the probabilistic graphical model using the belief propagation algorithm numerical examples are presented to show the accuracy and efficiency of the predictive capability of the developed graphical model

application of a conservative generalized multiscale finite element method to flow models in this paper we propose a method for the construction of locally conservative flux fields from generalized multiscale finite element method gmsfem pressure solutions the flux values are obtained from an element based postprocessing procedure in which an independent set of 4x4 linear systems need to be solved to test the performance of the method we consider two heterogeneous permeability coefficients and couple the resulting fluxes to a two phase flow model the increase in accuracy associated with the computation of the gmsfem pressure solutions is inherited by the postprocessed flux fields and saturation solutions and is closely correlated to the size of the reduced order systems in particular the addition of more basis functions to the enriched coarse space yields solutions that more accurately capture the behavior of the fine scale model a number of numerical examples are offered to validate the performance of the method

flexible conjugate gradients we analyze the conjugate gradient cg method with preconditioning slightly variable from one iteration to the next to maintain the optimal convergence properties we consider a variant proposed by axelsson that performs an explicit orthogonalization of the search directions vectors for this method which we refer to as flexible cg we develop a theoretical analysis that shows that the convergence rate is essentially independent of the variations in the preconditioner as long as the latter are kept sufficiently small we further discuss the real convergence rate on the basis of some heuristic arguments supported by numerical experiments depending on the eigenvalue distribution corresponding to the fixed reference preconditioner several situations have to be distinguished in some cases the convergence is as fast with truncated versions of the algorithm or even with the standard cg method whereas quite large variations are allowed without too much penalty in other cases the flexible variant effectively outperforms the standard method while the need for truncation limits the size of the variations that can be reasonably allowed

holder curves and parameterizations in the analyst s traveling salesman theorem abstract we investigate the geometry of sets in euclidean and infinite dimensional hilbert spaces we establish sufficient conditions that ensure a set of points is contained in the image of a 1 s holder continuous map f 0 1 l 2 with s 1 our results are motivated by and generalize the sufficient half of the analyst s traveling salesman theorem which characterizes subsets of rectifiable curves in r n or l 2 in terms of a quadratic sum of linear approximation numbers called jones beta numbers the original proof of the analyst s traveling salesman theorem depends on a well known metric characterization of rectifiable curves from the 1920s which is not available for higher dimensional curves such as holder curves to overcome this obstacle we reimagine jones non parametric proof and show how to construct parameterizations of the intermediate approximating curves f k 0 1 we then find conditions in terms of tube approximations that ensure the approximating curves converge to a holder curve as an application to the geometry of measures we identify conditions that guarantee fractional rectifiability of pointwise doubling measures in r n

an analysis of the nlmc upscaling method for high contrast problems abstract in this paper we propose simple multiscale basis functions with constraint energy minimization to solve elliptic problems with high contrast medium our methodology is based on the recently developed non local multicontinuum method nlmc the main ingredient of the method is the construction of suitable local basis functions with the capability of capturing multiscale features and non local effects in our method each coarse block is decomposed into various regions according to the contrast ratio and we require that the contrast ratio should be relatively small within each region the basis functions are constructed by solving a local problem defined on the oversampling domains and they have mean value one on the chosen region and zero mean otherwise numerical analysis shows that the resulting basis functions can be localizable and have a decay property the convergence of the multiscale solution is also proved finally some numerical experiments are carried out to illustrate the performances of the proposed method they show that the proposed method can solve problem with high contrast medium efficiently in particular if the oversampling size is large enough then we can achieve the desired error

a mixed multiscale finite element method for convex optimal control problems with oscillating coefficients abstract we study numerical approximation of convex optimal control problems governed by elliptic partial differential equations with oscillating coefficients since the objective functional contains flux we approximate the problems using the mixed finite element methods we first analyze the standard mixed finite element approximation schemes then motivated by the numerical simulation of the primal variable and the flux in highly heterogeneous porous media we use a multiscale mixed finite element method to solve the state equations the multiscale finite element bases are constructed by locally solving neumann boundary value problems the analysis of the approximate control problems is carried out under the assumption that the oscillating coefficients are locally periodic which allows us to use homogenization theory to obtain the asymptotic structure of the solutions although the numerical schemes are designed for the general case

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

block spectral mapping for multi scale solution in this paper an efficient multi scale method is presented discrete block domains with high mesh resolution are used to resolve fine scale features and the number of the blocks to be solved is kept as small as being sufficient for constructing a block block spatial spectrum a pointwise spectrum for the blockwise variation is generated for each mesh point on the corresponding block boundaries these block to block spectra provide the required formation for all the block interfaces and hence enable the fine scale solutions in these solved blocks to be mapped to a large domain simultaneously the structured blocking also provides a convenient domain division for parallel processing the method has been implemented in a 3 d time marching finite volume solver for the reynolds averaged compressible navier stokes equations the double fourier series is adopted to construct the pointwise spatial spectrum which is well suited for wall bounded problems where a domain layer with fine scale resolution is required several test cases of mass and heat transfer problems have been examined and the results show consistently that the accurate solutions for fine scale domains can be obtained very efficiently with a reduction in degrees of freedom by two orders of magnitude illustrating the potential of solving complex problems which might currently be intractable

automatic spectral coarse spaces for robust finite element tearing and interconnecting and balanced domain decomposition algorithms we introduce spectral coarse spaces for the balanced domain decomposition and the finite element tearing and interconnecting methods these coarse spaces are specifically designed for the two level methods to be scalable and robust with respect to the coefficients in the equation and the choice of the decomposition we achieve this by solving generalized eigenvalue problems on the interfaces between subdomains to identify the modes that slow down convergence theoretical bounds for the condition numbers of the preconditioned operators which depend only on a chosen threshold and the maximal number of neighbors of a subdomain are presented and proved for the finite element tearing and interconnecting method there are two versions of the two level method one based on the full dirichlet preconditioner and the other on the cheaper lumped preconditioner some numerical tests confirm these results copyright 2013 john wiley sons ltd

multiscale asymptotic expansion and finite element methods for the mixed boundary value problems of second order elliptic equation in perforated domains in this paper we will discuss the mixed boundary value problems for the second order elliptic equation with rapidly oscillating coefficients in perforated domains and will present the higher order multiscale asymptotic expansion of the solution for the problem which will play an important role in the numerical computation the convergence theorems and their rigorous proofs will be given finally a multiscale finite element method and some numerical results will be presented

proper actions of wreath products and generalizations we study stability properties of the haagerup property and of coarse embeddability in a hilbert space under certain semidirect products in particular we prove that they are stable under taking standard wreath products our construction also allows for a characterization of subsets with relative property t in a standard wreath product

domain decomposition for multiscale pdes we consider additive schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic pdes with highly variable coefficients in contrast to standard analyses we do not assume that the coefficients can be resolved by a coarse mesh this situation arises often in practice for example in the computation of flows in heterogeneous porous media in both the deterministic and monte carlo simulated stochastic cases we consider preconditioners which combine local solves on general overlapping subdomains together with a global solve on a general coarse space of functions on a coarse grid we perform a new analysis of the preconditioned matrix which shows rather explicitly how its condition number depends on the variable coefficient in the pde as well as on the coarse mesh and overlap parameters the classical estimates for this preconditioner with linear coarsening guarantee good conditioning only when the coefficient varies mildly inside the coarse grid elements by contrast our new results show that with a good choice of subdomains and coarse space basis functions the preconditioner can still be robust even for large coefficient variation inside domains when the classical method fails to be robust in particular our estimates prove very precisely the previously made empirical observation that the use of low energy coarse spaces can lead to robust preconditioners we go on to consider coarse spaces constructed from multiscale finite elements and prove that preconditioners using this type of coarsening lead to robust preconditioners for a variety of binary i e two scale media model problems moreover numerical experiments show that the new preconditioner has greatly improved performance over standard preconditioners even in the random coefficient case we show also how the analysis extends in a straightforward way to multiplicative versions of the schwarz method

compression functions of uniform embeddings of groups into hilbert and banach spaces we construct finitely generated groups with arbitrary prescribed hilbert space compression alpha from the interval 0 1 for a large class of banach spaces e including all uniformly convex banach spaces the e compression of these groups coincides with their hilbert space compression moreover the groups that we construct have asymptotic dimension at most 3 hence they are exact in particular the first examples of groups that are uniformly embeddable into a hilbert space respectively exact of finite asymptotic dimension with hilbert space compression 0 are given these groups are also the first examples of groups with uniformly convex banach space compression 0

lp compression traveling salesmen and stable walks we show that if h is a group of polynomial growth whose growth rate is at least quadratic then the lp compression of the wreath product z oh equals max 1 p 1 2 we also show that the lp compression of z oz equals max p 2p 1 2 3 and the lp compression of z oz 0 the zero section of z oz equipped with the metric induced from z o z equals max p 1 2p 3 4 the fact that the hilbert compression exponent of z o z equals 3 while the hilbert compression exponent of z o z 0 equals 3 4 is used to show that there exists a lipschitz function f z o z 0 l2 which cannot be extended to a lipschitz function defined on all of z o z