time discretization of an integro differential equation of parabolic type the stability and convergence properties of two time discretizations of an integro differential equation of parabolic type are studied the methods reduce to the backward euler and crank nicolson methods if the integral term is absent the integral term is approximated in each case by a quadrature rule with relatively high order truncation error so that a relatively large time step can be used for the quadrature in order to reduce the memory and computational requirements of the method

on unifying the concepts of scale instrumentation and stochastics in the development of multiphase transport theory a generalized theory of multiphase transport is presented which combines the concepts of scale instrumentation stochastics and time series with the development of transport equations by defining the filtering process as a convolution of a measure p with a field property we are able to exploit the fourier transform to place plausible restrictions on an instrument in frequency space so as to make its measurement relevant to its physical environment an ideal instrument is defined which filters out high frequency noise corresponding to short distances and yet does not alter the structure of low frequencies using ideal instruments we successively filter out lower frequency noise in a multiscale multiphase environment formulas are developed to relate the autocorrelation of a field property on one scale of motion to that on any other scale while taking into account the types of instruments used in the measuring process an equation relating the integral scale on one scale of motion to the integral scale on any other scale of motion is developed power spectra are developed which relate spectra on different scales to the measuring instrument used by successively applying filtering theorems a hierarchy of multiscale transport equations is developed filtered properties in the transport equations are mass averages different properties are allowed to be measured by different instruments and different instruments are allowed on different scales of motion and for different phases the concept of a wide sense stationary ergodic process is introduced to develop mass average autocorrelations and spectra over scales of motion as a function of measuring devices

a quasilinear parabolic system arising in modelling of catalytic reactors a system of four quasilinear parabolic equations arising in modelling of catalytic reactors is studied the system is coupled in a nonstandard way we prove that the system has unique global solution the asymptotic behavior as t is studied

fixed point and approximation results for multimaps in s kkm class the paper discusses new fixed point and approximation theorems for multimaps in the class s kkm

approachability and fixed points for non convex set valued maps this paper studies the approachability of upper semicontinuous nonconvex set valued maps by single valued continuous functions and its applications to fixed point theory the use of approximate selections was initiated by j von neumann 41 in the proof of his well known minimax principle this remarkable property was extended to the normed space setting by cellina s and cellina and lasota 9 in the context of a degree theory for upper semicontinuous convex compact valued maps it also holds for maps with compact contractible values defined on finite polyhedra of iw ma cole11 40 and more generally defined on compact anrs mclennan 39 it was recently taken up by gbrniewicz granas and kryszewski 22 241 in the context of an index theory for non convex maps defined on compact anrs the content of this paper is divided into three parts the first part is devoted to background material concerning the types of spaces and set valued maps studied herein in the second part we define the abstract class d of approachable set valued maps that is the class of maps a x y

a pseudo parabolic variational inequality and stefan problem abstract we study existence and uniqueness questions concerning non linear variational inequalities of the type of 1 1a 1 1b in the introduction applications to free boundary problems are also given author

numerical low rank approximation of matrix differential equations abstract the efficient numerical integration of large scale matrix differential equations is a topical problem in numerical analysis and of great importance in many applications standard numerical methods applied to such problems require an unduly amount of computing time and memory in general based on a dynamical low rank approximation of the solution a new splitting integrator is proposed for a quite general class of stiff matrix differential equations this class comprises differential lyapunov and differential riccati equations that arise from spatial discretizations of partial differential equations the proposed integrator handles stiffness in an efficient way and it preserves the symmetry and positive semidefiniteness of solutions of differential lyapunov equations numerical examples that illustrate the benefits of this new method are given in particular numerical results for the efficient simulation of the weather phenomenon el nino are presented

a data driven stochastic collocation approach for uncertainty quantification in mems this work presents a data driven stochastic collocation approach to include the effect of uncertain design parameters during complex multi physics simulation of micro electromechanical systems mems the proposed framework comprises of two key steps first probabilistic characterization of the input uncertain parameters based on available experimental information and second propagation of these uncertainties through the predictive model to relevant quantities of interest the uncertain input parameters are modeled as independent random variables for which the distributions are estimated based on available experimental observations using a nonparametric diffusion mixing based estimator botev nonparametric density estimation via diffusion mixing technical report 2007 the diffusion based estimator derives from the analogy between the kernel density estimation kde procedure and the heat dissipation equation and constructs density estimates that are smooth and asymptotically consistent the diffusion model allows for the incorporation of the prior density and leads to an improved density estimate in comparison with the standard kde approach as demonstrated through several numerical examples following the characterization step the uncertainties are propagated to the output variables using the stochastic collocation approach based on sparse grid interpolation smolyak soviet math dokl 1963 4 240 243 the developed framework is used to study the effect of variations in young s modulus induced as a result of variations in manufacturing process parameters or heterogeneous measurements on the performance of a mems switch copyright 2010 john wiley sons ltd

low rank second order splitting of large scale differential riccati equations we apply first and second order splitting schemes to the differential riccati equation such equations are very important in e g linear quadratic regulator lqr problems where they provide a link between the state of the system and the optimal input the methods can also be extended to generalized riccati equations e g arising from lqr problems given in implicit form in contrast to previously proposed schemes such as bdf or rosenbrock methods the splitting schemes exploit the fact that the nonlinear and affine parts of the problem when considered in isolation have closed form solutions we show that if the solution possesses low rank structure which is frequently the case then this is preserved by the method this feature is used to implement the methods efficiently for large scale problems the proposed methods are expected to be competitive as they at most require the solution of a small number of linear equation systems per time step finally we apply our low rank implementations to the riccati equations arising from two lqr problems the results show that the rank of the solutions stay low and the expected orders of convergence are observed

on fully discrete galerkin approximations for partial integro differential equations of parabolic type the subject of this work is the application of fully discrete galerkin finite element methods to initial boundary value problems for linear partial integro differential equations of parabolic type we investigate numerical schemes based on the pade discretization with respect to time and associated with certain quadrature formulas to approximate the integral term a preliminary error estimate is established which contains a term related to the quadrature rule to be specified in particular we consider quadrature rules with sparse quadrature points so as to limit the storage requirements without sacrificing the order of overall convergence for the backward euler scheme the crank nicolson scheme and a third order pade type scheme the specific quadrature rules analyzed are based on the rectangular the trapezoidal and simpson s rule for all the schemes studied optimal order error estimates are obtained in the case that the solution of the problem is smooth enough since this is important for our error analysis we also discuss the regularity of the exact solutions of our equations high order regularity results with respect to both space and time are given for the solution of problems with smooth enough data 17 refs

finite element approximation of diffusion equations with convolution terms approximation of solutions to diffusion equations with memory represented by convolution integral terms is considered such problems arise from modeling of flows in fissured media convergence of the method is proved and results of numerical experiments confirming the theoretical results are presented the advantages of implementation of the algorithm in a multiprocessing environment are discussed

homogenization of richardsʼ equation of van genuchten mualem model abstract this paper is devoted to the homogenization of richards equation of van genuchten mualem model which is a nonlinear degenerate parabolic differential equation it is usually used to model the motion of saturated unsaturated water flow in porous media we firstly apply the kirchhoff transformation to the equation and obtain a simpler equivalent equation with a linear oscillated diffusion term then under the real assumption for van genuchten mualem model we obtain the homogenized equation based on the two scale convergence theory some results on the first order corrector are also presented

discretization with variable time steps of an evolution equation with a positive type memory term we consider the linearized form of an evolution equation used to model viscoelasticity and heat conduction in materials with memory the equation is discretized in time using several schemes based on the backward euler and crank nicolson methods and incorporating appropriate quadratures for the memory term by permitting variable time steps we remove a restriction present in earlier related work by the first two authors

large scale averaging analysis of single phase flow in fractured reservoirs the method of large scale averaging is introduced to derive and analyze the most general form of the dual porosity model of single phase flow in naturally fractured reservoirs the dual porosity model contains the usual equations based on darcy s law and the coupling terms representing the fluid transfer between the matrix and the fractures a transient closure problem is developed to obtain and analyze the fracture and matrix permeability tensors and the fluid transfer terms the spatial averaging theorem is presented from a mathematical point of view and proved rigorously by the distribution theory the problem of wellposedness of the dual porosity model is also considered the techniques developed here are not restricted to either regular geometric fractures or spatially periodic reservoirs

multiscale model reduction for transport and flow problems in perforated domains abstract convection dominated transport phenomenon is important for many applications in these applications the transport velocity is often a solution of heterogeneous flow problems which results to a coupled flow and transport phenomena in this paper we consider a coupled flow stokes problem and transport unsteady convection diffusion problem in perforated domains perforated domains see fig 1 represent void space outside hard inclusions as in porous media filters and so on we construct a coarse scale solver based on generalized multiscale finite element method gmsfem for a coupled flow and transport the main idea of the gmsfem is to develop a systematic approach for computing multiscale basis functions we use a mixed formulation and appropriate multiscale basis functions for both flow and transport to guarantee a mass conservation for the transport problem we use petrov galerkin mixed formulation which provides a stability as a first approach we use the multiscale flow solution in constructing the basis functions for the transport equation in the second approach we construct multiscale basis functions for coupled flow and transport without solving global flow problem the novelty of this approach is to construct a coupled multiscale basis function numerical results are presented for computations using offline basis we also present an algorithm for adaptively adding online multiscale basis functions which are computed using the residual information numerical examples using online gmsfem show the speed up of convergence

probabilistic collocation for period 1 limit cycle oscillations in this paper probabilistic collocation for limit cycle oscillations pclco is proposed probabilistic collocation pc is a non intrusive approach to compute the polynomial chaos description of uncertainty numerically polynomial chaos can require impractical high orders to approximate long term time integration problems due to the fast increase of required polynomial chaos order with time pclco is a pc formulation for modeling the long term stochastic behavior of dynamical systems exhibiting a periodic response i e a limit cycle oscillation lco in the pc method deterministic time series are computed at collocation points in probability space in pclco pc is applied to a time independent parametrization of the periodic response of the deterministic solves instead of to the time dependent functions themselves due to the time independent parametrization the accuracy of pclco is independent of time the approach is applied to period 1 oscillations with one main frequency subject to a random parameter numerical results are presented for the harmonic oscillator a two dof airfoil flutter model and the fluid structure interaction of an elastically mounted cylinder

an upscaling method using coefficient splitting and its applications to elliptic pdes in this paper we develop an upscaling method using coefficient splitting techniques green s function is constructed using the differential operator associated with the first part of the splitting an effective upscaling coefficient is recursively calculated by green s function the computation of the upscaling process involves some independent steps combining the proposed upscaling method with the stochastic collocation method we present a stochastic space reduction collocation method where the stochastic collocation method is performed on a lower dimension stochastic space than the full dimension stochastic space we thoroughly analyze the convergence of the proposed upscaling method for both deterministic and stochastic elliptic pdes computation complexity is also addressed for the stochastic upscaling method a number of numerical tests are presented to confirm the convergence analysis

stability for implicit explicit schemes for non equilibrium kinetic systems in weighted spaces with symmetrization abstract we consider kinetic systems and prove their stability working in weighted spaces in which the systems are symmetric we prove stability for various explicit and implicit semi discrete and fully discrete schemes the applications include advective and diffusive transport coupled to the accumulation of immobile components governed by non equilibrium relationships we also discuss extensions to nonlinear relationships and multiple species

matrix dependent multigrid homogenization for diffusion problems for problems with strongly varying or discontinuous diffusion coefficients we present a method to compute coarse scale operators and to approximately determine the effective diffusion tensor on the coarse scale level the approach is based on techniques that are used in multigrid such as matrix dependent prolongations and the construction of coarse grid operators by means of the galerkin approximation in numerical experiments we compare our multigrid homogenization method with continuous homogenization renormalization and simple averaging approaches

low rank methods for a class of generalized lyapunov equations and related issues in this paper we study possible low rank solution methods for generalized lyapunov equations arising in bilinear and stochastic control we show that under certain assumptions one can expect a strong singular value decay in the solution matrix allowing for low rank approximations since the theoretical tools strongly make use of a connection to the standard linear lyapunov equation we can even extend the result to the d dimensional case described by a tensorized linear system of equations we further provide some reasonable extensions of some of the most frequently used linear low rank solution techniques such as the alternating directions implicit adi iteration and the krylov plus inverted krylov k pik method by means of some standard numerical examples used in the area of bilinear model order reduction we will show the efficiency of the new methods

long time numerical solution of a parabolic equation with memory long time stability and convergence properties of two time discreti zation methods for an integro differential equation of parabolic type are studied the methods are based on the standard backward euler and second order back ward differencing methods the memory term is approximated by a quadrature rule with emphasis on such rules with reduced computational memory require ments discretization of the spatial partial differential operators by the finite element method is also considered

multi component multiphase porous flow an axiomatic foundation for models of multi component multiphase porous flow appearing ubiquitously in the engineering literature is developed this unifies and extends various disparate and empirical formulations appearing in the literature constitutive restrictions are derived from an appropriate statement of the second law of thermodynamics and the corresponding dissipation inequalities establish stability of solutions the convexity properties and variational structure of these models are elucidated

ingredients for a general purpose stochastic finite elements implementation abstract the stochastic finite element method has seen a number of incarnations over the past few years ranging from perturbation based methods and neumann expansion based methods to methods for evaluating bounds on the response variability and more recent functional expansions combined with error minimization techniques of these only the last class of methods provides an explicit expression to the solution of the problem as opposed to evaluating a global norm of this solution and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method the main obstacle in developing a substantial user community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest this paper presents the main ingredients for developing a general purpose version of the spectral stochastic finite element method

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

the existence of weak solutions to single porosity and simple dual porosity models of two phase incompressible flow abstract it is shown that there exists a weak solution to a degenerate and singular elliptic parabolic partial integro differential system of equations these equations model two phase incompressible flow of immiscible fluids in either an ordinary porous medium or in a naturally fractured porous medium the full model is of dual porosity type though the single porosity case is covered by setting the matrixto fracture flow terms to zero this matrix to fracture flow is modeled simply in terms of fracture quantities that is no distinct matrix equations arise the equations are considered in a global pressure formulation that is justified by appealing to a physical relation between the degeneracy of the wetting fluid s mobility and the singularity of the capillary pressure function in this formulation the elliptic and parabolic parts of the system are separated hence it is natural to consider various boundary conditions including mixed nonhomogeneous saturation dependent ones of the first three types a weak solution is obtained as a limit of solutions to discrete time problems the proof makes no use of the corresponding regularized system the hypotheses required for various earlier results on single porosity systems are weakened so that only physically relevant assumptions are made in particular the results cover the cases of a singular capillary pressure function a pure neumann boundary condition and an arbitrary initial condition

on the benefits of the ldlt factorization for large scale differential matrix equation solvers abstract we propose efficient algorithms for solving large scale matrix differential equations in particular we deal with the differential riccati equations dre and state the applicability to the differential lyapunov equations dle we focus on methods based on standard versions of ordinary differential equations in the matrix setting the application of these methods yields algebraic lyapunov equations ales with a certain structure to be solved in every step the alternating direction implicit adi algorithm and krylov subspace based methods allow to exploit this special structure however a direct application of classic low rank formulations requires the use of complex arithmetic using an l d l t type decomposition of both the right hand side and the solution of the equation we avoid this problem thus the proposed methods are a more practical alternative for large scale problems arising in applications also they make the application of higher order methods feasible the numerical results show the better performance of the proposed methods compared to earlier formulations

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

an efficient spde approach for el nino abstract we consider the numerical approximation of stochastic partial differential equations spdes based models for a quasi periodic climate pattern in the tropical pacific ocean known as el nino phenomenon we show that for these models the mean and the covariance are given by a deterministic partial differential equation and by an operator differential equation respectively in this context we provide a numerical framework to approximate these parameters directly we compare this method to stochastic differential equations and spdes based models from the literature solved by taylor methods and stochastic galerkin methods respectively numerical results for different scenarios taking as a reference measured data of the years 2014 and 2015 last nino event validate the efficiency of our approach

a linear homogenization problem with time dependent coefficient on etudie un probleme provenant d une equation a structure differentielle pure decrivant par exemple le comportement viscoelastique d un materiau composite

an approximation scheme for a nonlinear degenerate parabolic equation with a second order differential volterra operator a nonlinear degenerate parabolic initial boundary value problem with a second order differential volterra operator is studied the approximate solution is found due to the semi discretization in time which is based on rothe s method the convergence of the approximation scheme existence and uniqueness of solution as well as some regularity results are proved

a simplified method for upscaling composite materials with high contrast of the conductivity a large class of industrial composite materials such as metal foams fibrous glass materials mineral wools and the like are widely used in insulation and advanced heat exchangers these materials are characterized by a substantial difference between the thermal properties of the highly conductive materials glass or metal and the insulator air as well as low volume fractions and complex network like structures of the highly conductive components in this paper we address the important issue for the engineering practice of developing fast reliable and accurate methods for computing the macroscopic upscaled thermal conductivities of such materials we assume that the materials have constant macroscopic thermal conductivity tensors which can be obtained by upscaling techniques based on the postprocessing of a number of linearly independent solutions of the steady state heat equation on representative elementary volumes revs we propose theoretically justify and computationally study a numerical method for computing the effective conductivities of materials for which the ratio delta of low and high conductivities satisfies delta ll1 we show that in this case one needs to solve the heat equation in the region occupied by the highly conductive media only further we prove that under certain conditions on the microscale geometry the proposed method gives an approximation that is mathcal o delta close to the upscaled conductivity finally we illustrate the accuracy and the limitations of the method on a number of numerical examples

finite elements for elliptic problems with stochastic coefficients abstract we describe a deterministic finite element fe solution algorithm for a stochastic elliptic boundary value problem sbvp whose coefficients are assumed to be random fields with finite second moments and known piecewise smooth two point spatial correlation function separation of random and deterministic variables parametrization of the uncertainty is achieved via a karhunen loeve kl expansion an o n log n algorithm for the computation of the kl eigenvalues is presented based on a kernel independent fast multipole method fmm truncation of the kl expansion gives an m 1 wiener polynomial chaos pc expansion of the stochastic coefficient and is shown to lead to a high dimensional deterministic boundary value problem dbvp analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r r1 rm in an m r wiener pc expansion for the approximate solution pointwise error bounds for the fem approximations of kl eigenpairs the truncation of the kl expansion and the fe solution to the dbvp are given numerical examples show that m depends on the spatial correlation length of the random diffusion coefficient the variable polynomial degree r in pc stochastic galerkin fem allows to handle kl expansions with m up to 30 and r1 up to 10 in moderate time

analysis of large scale averaged models for two phase flow in fractured reservoirs the method of large scale averaging has been recently applied to derive a general dual porosity model of two phase flow in naturally fractured reservoirs the present paper is aimed to develop mathematical theory behind the derivation of this model in particular two volume averaging theorems which are key mathematical tools in the model derivation are proved rigorously using distribution theory existence and uniqueness of a weak solution to a simplied dual porosity model are shown

convolution quadrature and discretized operational calculus ii operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations singular integrals power and logarithmic singularities finite part integrals multiple timescale convolution volterra integral equations wiener hopf integral equations frequency domain conditions which determine the stability of such equations can be carried over to the discretization

additive schwarz algorithms for parabolic convection diffusion equations in this paper we consider the solution of linear systems of algebraic equations that arise from parabolic finite element problems we introduce three additive schwarz type domain decomposition methods for general not necessarily selfadjoint linear second order parabolic partial differential equations and also study the convergence rates of these algorithms the resulting preconditioned linear system of equations is solved by the generalized minimal residual method numerical results are also reported

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

existence and uniqueness of weak solutions for nonlocal parabolic problems via the galerkin method abstract fractional differential equations are becoming increasingly popular as a modeling tool to describe a wide range of non classical phenomena with spatial heterogeneities throughout the applied science and engineering a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal operators which include fractional laplacians on bounded domains in r n we develop the galerkin method to prove existence and uniqueness of weak solutions to nonlocal parabolic problems moreover we study the existence of orthonormal basis of eigenvectors associated to these nonlocal operators

finite difference methods for two phase incompressible flow in porous media two phase incompressible flow in porous media is governed by a system of nonlinear partial differential equations convection physically dominates diffusion and the object of this paper is to develop a finite difference procedure that reflects this dominance the pressure equation which is elliptic in appearance is discretized by a standard five point difference method the concentration equation is treated by an implicit finite difference method that applies a form of the method of characteristics to the transport terms a convergence analysis is given for the method

ritz volterra projections to finite element spaces and applications to integrodifferential and related equations the object of this paper is to investigate the convergence of finite element approximations to solutions of parabolic and hyperbolic integrodifferential equations and also of equations of sobolev and viscoelasticity type the concept of ritz volterra projection will be seen to unify much of the analysis for the different types of problems optimal order error estimates are obtained in l p for 2 leqq p infty and almost optimal order pointwise results given

numerical solution of a nonlinear integro differential equation abstract galerkin s method with appropriate discretization in time is considered for approximating the solution of the nonlinear integro differential equation u t x t 0 t a t x u x x d f x t 0 an error estimate in a suitable norm will be derived for the difference u uh between the exact solution u and the approximant uh it turns out that the rate of convergence of uh to u as h 0 is optimal this result was confirmed by the numerical experiments

finite element approximation of a parabolic integro differential equation with a weakly singular kernel we give error estimates for the numerical solution by means of the galerkin finite element method of an integro differential equation of parabolic type with a memory term containing a weakly singular kernel optimal order estimates are shown for spatially semidiscrete and completely discrete methods special attention is paid to the regularity of the exact solution

analysis of the simulation of single phase flow through a naturally fractured reservoir a general form of the double porosity model for single phase flow through a naturally fractured reservoir is derived by explicitly considering fluid flow in individual matrix blocks the warren and root model is shown to be a crude approximation to this model the general model consists of a parabolic equation coupled to a series of parabolic equations it is shown that the coupling term can be viewed as a positive semidefinite perturbation of the time derivative and hence it is verified that the model is well posed a finite element method is presented to approximate the solution and optimal order l 2 error estimates are derived

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

two scale homogenization of non linear degenerate evolution equations using the notion of two scale convergence developed by allaire the homogenization of a degenerate non linear evolution equation with periodically oscillating coefficients is presented a two scale homogenized system is obtained as the limit of the periodic problem monotone operator methods and two scale convergence are employed to show that the solutions of the periodic problem converge to the unique solution of the homogenized system homogenized initial conditions are also obtained and the sense in which they hold for the homogenized initial value problem is made specific

perturbation of the mapping and solvability theorems in banach space abstract here we consider perturbations of continuous mappings on banach spaces and investigate their images under various conditions consequently we study the solvability of some classes of equations and inclusions for these we start by the investigation of local properties of the considered mapping and local comparisons of this mapping with certain smooth mappings moreover we study different mixed problems

micro structure models of diffusion in fissured media abstract two diffusion models are developed which recognize the local geometry of the individual cells or storage sites and the exchange of flux on the micro scale of these cells the cauchy problems for these model systems are shown to be resolved by holomorphic semigroups and various classical models are obtained as limits of these disributed micro structure models

convergence rate estimate for a domain decomposition method we provide a convergence rate analysis for a variant of the domain decomposition method introduced by gropp and keyes for solving the algebraic equations that arise from finite element discretization of nonsymmetric and indefinite elliptic problems with dirichlet boundary conditions in 2 we show that the convergence rate of the preconditioned gmres method is nearly optimal in the sense that the rate of convergence depends only logarithmically on the mesh size and the number of substructures if the global coarse mesh is fine enough

flow of oil and water in a porous medium abstract the flow of two immiscible and incompressible fluids in a porous medium is described by a system of quasilinear degenerate partial differential equations in this paper the existence of a weak solution by regularization is shown

a transport model with micro and macro structure abstract in this paper we consider solute transport through porous media where the chemical species undergoes a chemical process through the surface of the porous skeleton the problem is modeled by a system of differential equations for the macro concentration u u x t and the micro concentration u u x x t we prove existence and uniqueness and some properties of the set with positive concentration

a free boundary problem for a degenerate parabolic system abstract the degenerate parabolic system 1 1 in the introduction serves as a model for heat conduction in a heterogeneous medium consisting of two components the first component is made up of small pieces suspended in the second component and the second component undergoes a change of phase at a prescribed temperature this phenomenon occurs in a mixture of gravel and wet soil for example melting of frozen soil existence and uniqueness results of weak solutions of the degenerate parabolic problem are shown by employing monotone operator theory local regularity such as continuity and boundedness of the solution is studied a discussion is provided about the mutual interplay of the thermodynamic temperature the temperature in the first component and the conductive temperature the temperature in the second component author

numerical homogenization of the richards equation for unsaturated water flow through heterogeneous soils homogenized equations and the corresponding effective constitutive relations are generally necessary for numerically modeling large scale unsaturated flow processes in soils recently based on the kirchhoff transformation and the two scale convergence theory a homogenization method for the richards equation with the mualem van genuchten model has been proposed with a constant model parameter relating to the inverse of the air entry pressure and the soil pore size distribution the homogenized model is computationally efficient and convenient to use because of its explicit expression in this study we generalize this method allowing to be a spatially distributed random field and proposing a homogenized richards equation in the mixed form h under the condition that the effective hydraulic conductivity tensor is diagonal this generalization eliminates the limitation of a constant in practical applications the proposed homogenized model is meaningful in most situations because the flow problems are influenced mainly by the diagonal terms of conductivity and the off diagonal terms are often neglected two dimensional numerical tests are conducted in soil profiles with different degrees of spatial heterogeneity structure to illustrate that the homogenized model can capture the fine scale flow behaviors on coarse grids effectively homogenization for the richards equation with other two commonly used constitutive relations the brooks corey model and the gardner russo model is also illustrated in this study