stochastic upscaling in solid mechanics an excercise in machine learning this paper presents a consistent theoretical and computational framework for upscaling in random microstructures we adopt an information theoretic approach in order to quantify the informational content of the microstructural details and find ways to condense it while assessing quantitatively the approximation introduced in particular we substitute the high dimensional microscale description by a lower dimensional representation corresponding for example to an equivalent homogeneous medium the probabilistic characteristics of the latter are determined by minimizing the distortion between actual macroscale predictions and the predictions made using the coarse model a machine learning framework is essentially adopted in which a vector quantizer is trained using data generated computationally or collected experimentally several parallels and differences with similar problems in source coding theory are pointed out and an efficient computational tool is employed various applications in linear and non linear problems in solid mechanics are examined

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

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

maximum norm stability and error estimates in galerkin methods for parabolic equations in one space variable maximum norm stability and error estimates of best approximation and nonsmooth data types are derived for the approximate solution of a parabolic equation in one space variable using the continuous in time galerkin method based on piecewise polynomial approximating functions on a quasi uniform mesh

sharp l 2 error estimates and superconvergence of mixed finite element methods for non fickian flows in porous media a sharper l2 error estimate is obtained for the non fickian flow of fluid in porous media by means of a mixed ritz volterra projection instead of the mixed ritz projection used in r e ewing y lin and j wang acta math univ comenian n s 70 2001 pp 75 84 moreover local l2 superconvergence for the velocity along the gauss lines and for the pressure at the gauss points is derived for the mixed finite element method via the ritz volterra projection and global l2 superconvergence for the velocity and the pressure is also investigated by virtue of an interpolation postprocessing technique on the basis of the superconvergence estimates some useful a posteriori error estimators are presented for this mixed finite element method

two splitting positive definite mixed finite element methods for parabolic integro differential equations abstract two novel mixed finite element procedures are established for parabolic integro differential equations which can be split into two independent symmetric positive definite sub schemes and do not need to solve a coupled system of equations without requiring the lbb consistency condition the convergence analysis shows that both methods lead to the optimal order l 2 norm error estimate for u and optimal h div norm error estimate for a numerical example is presented to illustrate the theoretical analysis

finite element methods of the two nonlinear integro differential equations abstract transformation of dependent variables such as the kirchhoff transformation is a classical tool for solving nonlinear partial differential equations in larsson et al this approach was used in connection with the finite element method zlamal used it in galerkin approximation in space the euler method in time and applied it to the solution of nonlinear heat equations here we give justification of the methods for the two nonlinear integro heat equations in the case that the discretization is carried through by piecewise linear polynomials in space and by the implicit euler method in time with quadrature rules which require the storage of relatively few values of the solution the estimate of the discretization error in the maximum norm is introduced the results depend on superconvergence in the gradient for a related elliptic problem which is shown to hold only for very special meshes

finite difference heterogeneous multi scale method for homogenization problems in this paper we propose a numerical method the finite difference heterogeneous multi scale method fd hmm for solving multi scale parabolic problems based on the framework introduced in commun math sci 1 1 87 the numerical method relies on the use of two different schemes for the original equation at different grid level which allows to give numerical results at a much lower cost than solving the original equations we describe the strategy for constructing such a method discuss generalization for cases with time dependency random correlated coefficients nonconservative form and implementation issues finally the new method is illustrated with several test examples

numerical upscaling for the eddy current model with stochastic magnetic materials this paper deals with the upscaling of the time harmonic maxwell equations for heterogeneous media we analyze the eddy current approximation of maxwell s equations to describe the electric field for heterogeneous isotropic magnetic materials the magnetic permeability of the materials is assumed to have random heterogeneities described by a gaussian random field we apply the so called coarse graining method to develop a numerical upscaling of the eddy current model the upscaling uses filtering and averaging procedures in fourier space which results in a formulation of the eddy current model on coarser resolution scales where the influence of sub scale fluctuations is modeled by effective scale and space dependent reluctivity tensors the effective reluctivity tensors can be obtained by solving local partial differential equations which contain a laplacian as well as a curl curl operator we present a computational method how the equation of the combined operators can be discretized and solved numerically using an extended variational formulation compared to standard discretizations we compare the results of the numerical upscaling of the eddy current model with theoretical results of eberhard j p eberhard upscaling for the time harmonic maxwell equations with heterogeneous magnetic materials physical review e 72 3 2005 and obtain a very good agreement

heterogeneous multiscale method for optimal control problem governed by parabolic equation with highly oscillatory coefficients abstract in this paper we investigate the heterogeneous multiscale method hmm for the optimal control problem governed by the parabolic equation with highly oscillatory coefficients the state variable and adjoint state variable are approximated by the multiscale discretization scheme that relies on coupled macro and micro finite elements while the control variable is discretized by the piecewise constants by applying the well known lions lemma to the discretized optimal control problem we obtain the necessary and sufficient optimality conditions a priori error estimates are derived for the state co state and the control with uniform bounded constants finally numerical results are presented to illustrate our theoretical findings

analytics and numerics of drug release tracking the study of the dependence of fluxes concentrations and response times on the characteristic properties of drug delivery polymeric devices plays an important role in the design of drug release platforms the aim of this paper is to develop mathematical tools for an in depth understanding of drug release tracking the mathematical model presented takes into account the viscoelastic properties of the polymer and the state of the dispersed drug free or chemically bound to the matrix for nonlinear chemical bounds the process is described by a nonlinear integro differential system and the drug release tracking is treated numerically for linear chemical bounds closed formulas for the fluxes and response times are established in terms of the parameters that characterize the drug and the platform these formulas provide a set of a priori estimations for the variables of the model numerical examples which show the effectiveness of the approach are included

wavelet based numerical homogenization a numerical homogenization procedure for elliptic differential equations is presented it is based on wavelet decompositions of discrete operators in fine and coarse scale components followed by the elimination of the fine scale contributions if the operator is in divergence form this is preserved by the homogenization procedure for periodic problems results similar to classical effective coefficient theory are proved the procedure can be applied to problems that are not cell periodic

finite difference approximation of a nonlinear integro differential system finite difference approximation of the nonlinear integro differential system associated with the penetration of a magnetic field into a substance is studied the convergence of the finite difference scheme is proved the rate of convergence of the discrete scheme is given the decay of the numerical solution is compared with the analytical results proven earlier

time stepping schemes for systems of volterra integro differential equations linear volterra type systems of integro differential equations sides with kernels in the form of sums of complex exponentials are considered such systems arise for example in the semi discrete finite element model of dynamic viscoelasticity and of some dispersive wave equations and in the representation of dynamic subsystems on interfaces several new families of time stepping schemes are proposed here for the solution of this class of sides the accuracy stability and efficiency properties of these schemes are demonstrated via both theoretical analysis using a scalar model problem and numerical experiments applied to first and second order sides

finite element approximation of an evolution problem modeling shear band formation abstract an error analysis is provided for the spatial approximation by the finite element method of a nonlinear time dependent problem which models the antiplane shear deformations of a thermoplastic material this problem consists of a system of two coupled partial differential equations with velocity and temperature unknowns optimal order error estimates in the l 2 norm are given for each of these

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

a new wavelet multigrid method the standard multigrid procedure performs poorly or may break down when used to solve certain problems such as elliptic problems with discontinuous or highly oscillatory coefficients the method discussed in this paper solves this problem by using a wavelet transform and schur complements to obtain the necessary coarse grid interpolation and restriction operators a factorized sparse approximate inverse is used to improve the efficiency of the resulting method numerical examples are presented to demonstrate the versatility of the method

generalized p fem in homogenization summary a new finite element method for elliptic problems with locally periodic microstructure of length varepsilon 0 is developed and analyzed it is shown that the method converges as varepsilon rightarrow 0 to the solution of the homogenized problem with optimal order in varepsilon and exponentially in the number of degrees of freedom independent of varepsilon 0 the computational work of the method is bounded independently of varepsilon numerical experiments demonstrate the feasibility and confirm the theoretical results

finite element methods for partial volterra integro differential equations on two dimensional unbounded spatial domains abstract the precise and effective way for solving a partial volterra integro differential equation on unbounded spatial domain is to derive artificial boundary conditions for some feasible and prescribed computational domain thus the unsolvable problem on unbounded domain is converted to the computational problem on bounded domain which is often called reduced problem a feasible numerical method will then be applied to solve the reduced problem with the artificial boundary conditions which often exhibit nonlocal properties in this paper the finite element method for the spatial space and the recently developed discontinuous galerkin time stepping method for the temporal space are considered to solve the reduced problem the convergence rate explicitly with those interesting parameters spatial grid size h temporal mesh size k the location of the artificial boundary d and the truncation number m is obtained this paper is the first theoretical analysis of the numerical methods for volterra integro differential equations on unbounded spatial domains

wavelet galerkin method in multi scale homogenization of heterogeneous media the hierarchical properties of scaling functions and wavelets can be utilized as effective means for multi scale homogenization of heterogeneous materials under galerkin framework it is shown in this work however when the scaling functions are used as the shape functions in the multi scale wavelet galerkin approximation the linear dependency in the scaling functions renders improper zero energy modes in the discrete differential operator stiffness matrix if integration by parts is invoked in the galerkin weak form an effort is made to obtain the analytical expression of the improper zero energy modes in the wavelet galerkin differential operator and the improper nullity of the discrete differential operator is then removed by an eigenvalue shifting approach a unique property of multi scale wavelet galerkin approximation is that the discrete differential operator at any scale can be effectively obtained this property is particularly useful in problems where the multi scale solution cannot be obtained simply by a wavelet projection of the finest scale solution without utilizing the multi scale discrete differential operator for example the multi scale analysis of an eigenvalue problem with oscillating coefficients copyright 2005 john wiley sons ltd

numerical methods for treating problems of viscoelastic isotropic solid deformation mathematical models for treating problems of linear viscoelasticity involving hereditary constitutive relations for compressible solids are discussed and their discretization using finite element methods in space together with quadrature rules in time to treat the hereditary integrals is described the range of applicability of this type of formulation is reviewed in the context of geometric and constitutive linearity nonlinearity and the limitations imposed by the availability of physical data are discussed one of the above models is a volterra integral equation of the second kind in this when the kernel is separable an established technique due to goursat 1933 can be exploited to reformulate the problem as a system of ordinary differential equations this approach will be described for the special case of a linear viscoelastic isotropic homogeneous synchronous constant poisson s ratio solid this method results in a complete decoupling of the space and time dependence in this case the problem can be solved at each time level by solving first a problem of linear elasticity and then a system of ordinary differential equations for each point in the spatial mesh at which the viscoelastic displacements are required the advantages disadvantages and limitations offered by this and various other schemes for solving problems of viscoelasticity as outlined below are discussed

numerical solution of parabolic integro differential equations by the discontinuous galerkin method the numerical solution of a parabolic equation with memory is considered the equation is first discretized in time by means of the discontinuous galerkin method with piecewise constant or piecewise linear approximating functions the analysis presented allows variable time steps which as will be shown call then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data the combination with finite element discretizat in space is also studied

numerical methods for multiscale elliptic problems we present an overview of the recent development on numerical methods for elliptic problems with multiscale coefficients we carry out a thorough study of two representative techniques the heterogeneous multiscale method hmm and the multiscale finite element method msfem for problems with scale separation but without specific assumptions on the particular form of the coefficients analytical and numerical results show that hmm gives comparable accuracy as msfem with much less cost for problems without scale separation our numerical results suggest that hmm performs at least as well as msfem in terms of accuracy and cost even though in this case both methods may fail to converge since the cost of msfem is comparable to that of solving the full fine scale problem one might expect that it does not need scale separation and still retains some accuracy we show that this is not the case specifically we give an example showing that if there exists an intermediate scale comparable to h the size of the macroscale mesh then msfem commits a finite error even with overlapping

homogenization of parabolic equations with a continuum of space and time scales this paper addresses the issue of the homogenization of linear divergence form parabolic operators in situations where no ergodicity and no scale separation in time or space are available namely we consider divergence form linear parabolic operators in omega subset mathbb r n with l infty omega times 0 t coefficients it appears that the inverse operator maps the unit ball of l 2 omega times 0 t into a space of functions which at small time and space scales are close in h 1 norm to a functional space of dimension n it follows that once one has solved these equations at least n times it is possible to homogenize them both in space and in time reducing the number of operation counts necessary to obtain further solutions in practice we show under a cordes type condition that the first order time derivatives and second order space derivatives of the solution of these operators with respect to caloric coordinates are in l 2 instead of h 1 with euclidean coordinates if the medium is time independent then it is sufficient to solve n times the associated elliptic equation in order to homogenize the parabolic equation

adaptive space time finite element solution for volterra equations arising in viscoelasticity problems abstract we give a short overview of our recent efforts towards constructing adaptive space time finite element solvers for some partial differential volterra equations arising in viscoelasticity theory

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

a fuzzy stochastic multiscale model for fiber composites a one dimensional study abstract we study mathematical and computational models for computing the deformation of fiber reinforced polymers due to external forces a thorough study requires an understanding of both micro structural effects and uncertainty variability in the manufacturing process such as uncertainty in the size and distribution of fibers and variability in material properties and fracture parameters we first show that the present uncertainties variabilities which are of both random and non random types cannot be accurately characterized by current stochastic multiscale models based on precise probability theory such as stationary random fields next we present a new hybrid fuzzy stochastic model which can more accurately describe uncertainties variabilities in fiber composites the new model which is referred to as a fuzzy stationary random field consists of a random field with fuzzy moments we then construct a global local multiscale algorithm in a fuzzy stochastic framework for efficiently computing output quantities of interest such as displacements and stresses in regions of relatively small size e g hot spots the algorithm utilizes the concept of representative volume elements and homogenization and constructs a global solution to compute a local approximation that captures the microscale features of the problem the results are based on and backed by real experimental data through a calibration validation approach

on the discretization in time for a parabolic integrodifferential equation with a weakly singular kernel ii nonsmooth initial data abstract backward euler methods are considered for a parabolic integrodifferential equation with a memory term containing a weakly singular kernel the integral term is treated by means of a convolution quadrature suggested by lubich some results from lubich s discretized fractional calculus play a crucial role in error analyses the stability and convergence properties of the time discretizations are derived and are applied to the semidiscrete equations built by the galerkin finite element methods in the space variables

spatio spectral concentration of convolutions differential equations may possess coefficients that vary on a spectrum of scales because coefficients are typically multiplicative in real space they turn into convolution operators in spectral space mixing all wavenumbers however in many applications only the largest scales of the solution are of interest and so the question turns to whether it is possible to build effective coarse scale models of the coefficients in such a manner that the large scales of the solution are left intact here we apply the method of numerical homogenisation to deterministic linear equations to generate sub grid scale models of coefficients at desired frequency cutoffs we use the fourier basis to project filter and compute correctors for the coefficients the method is tested in 1d and 2d scenarios and found to reproduce the coarse scales of the solution to varying degrees of accuracy depending on the cutoff we relate this method to mode elimination renormalisation group rg and discuss the connection between accuracy and the cutoff wavenumber the tradeoff is governed by a form of the uncertainty principle for convolutions which states that as the convolution operator is squeezed in the spectral domain it broadens in real space as a consequence basis sparsity is a high virtue and the choice of the basis can be critical

simulation of a waveguide filter using wavelet based numerical homogenization we apply wavelet based numerical homogenization to the simulation of an optical waveguide filter we use the method to derive approximate one dimensional models and subgrid models of the filter numerical examples of the technique are presented and the computational gains are investigated

some new error estimates of a semidiscrete finite volume element method for a parabolic integro differential equation with nonsmooth initial data a semidiscrete finite volume element fve approximation to a parabolic integro differential equation pide is analyzed in a two dimensional convex polygonal domain an optimal order l 2 error estimate for smooth initial data and nearly the same optimal order l 2 error estimate for nonsmooth initial data are obtained more precisely for homogeneous equations an elementary energy technique and a duality argument are used to derive an error estimate of order o left t 1 h 2 ln h right in the l 2 norm for positive time when the given initial function is only in l 2

subgrid modeling for convection diffusion reaction in one space dimension using a haar multiresolution analysis in this paper we propose and study a subgrid model for linear convection diffusion reaction equations with fractal rough coefficients the subgrid model is based on scale extrapolation of a modeling residual from coarser scales using a computed solution on a finest scale as reference we show in experiments that a solution with subgrid model on a scale h in most cases corresponds to a solution without subgrid model on a scale less than h 4 we also present error estimates for the modeling error in terms of modeling residuals

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

diffusion viscoelasticity and erosion in this paper diffusion through a viscoelastic biodegradable material is studied the phenomenon is described by a set of three coupled partial differential equations that take into account passive diffusion stress driven diffusion and the degradation of the material the stability properties of the model are studied erodible viscoelastic materials as biodegradable polymers have a huge range of applications in medicine to make drug eluting implants using the mathematical model the behavior of a particular ocular drug eluting implant which describes drug delivery into the vitreous chamber of the eye is presented the model consists of coupled systems of partial differential equations linked by interface conditions the chemical structure the viscoelastic properties and the diffusion in the implant as well as the transport in the vitreous are taken into account to simulate the evolution in vivo of released drug the dependence of the delivery profile on the properties of the material is addressed numerical simulations that illustrate the interplay between these phenomena are included

non fickian convection diffusion models in porous media in this paper we propose a numerical scheme to approximate the solution of a non fickian coupled model that describes e g miscible transport in porous media the model is defined by a system of a quasilinear elliptic equation which governs the fluid pressure and a quasilinear integro differential equation which models the convection diffusion transport process the numerical scheme is based on a conforming piecewise linear finite element method for the discretization in space the fully discrete approximations is obtained with an implicit explicit method estimates for the continuous in time and the fully discrete methods are derived showing that the numerical approximation for the concentrations and the pressure are second order convergent in a discrete l 2 norm and in a discrete h 1 norm respectively

a parallel method for the numerical solution of integro differential equation with positive memory abstract an efficient parallel numerical method is proposed for an integro differential equation with positive memory instead of solving the equation in classical time marching methods which require massive storage of solutions of previous time steps in order to advance to a next time step the fourier laplace transformation in time is applied to obtain a set of complex valued elliptic problems parameterized by points on a contour in the complex plane using the independence of an elliptic problem corresponding to one contour point is independent of those elliptic problems corresponding to other contour points all elliptic problems can be solved in parallel essentially without data communications then the time domain solution can be obtained by the fourier laplace inversion formula an error analysis and the numerical implementation of this parallel method is presented

space time spectral method for a weakly singular parabolic partial integro differential equation on irregular domains abstract the spectral method is proposed for the partial integro differential equations with a weakly singular kernel on irregular domains the space discretization is based on the nodal spectral element method using the lagrange polynomials basis associated with the gauss lobatto legendre quadrature nodes also the model is discretized in time with the legendre spectral galerkin method the discretization leads to conversion of the problem to a sylvester matrix equation which can be solved efficiently by the qz algorithm gardiner et al 1992 the convergence of the method is proven by providing a priori l 2 error estimate numerical results illustrate the efficiency and spectral accuracy of the proposed method

finite element methods for parabolic and hyperbolic partial integro differential equations on formule plusieurs methodes de galerkin a temps discret pour des equations integrodifferentielles paraboliques

on a linearized backward euler method for the equations of motion of oldroyd fluids of order one in this paper a linearized backward euler method is discussed for the equations of motion arising in the oldroyd model of viscoelastic fluids some new a priori bounds are obtained for the solution under realistically assumed conditions on the data further the exponential decay properties for the exact as well as the discrete solutions are established finally a priori error estimates in bh 1 and bl 2 norms are derived for the the discrete problem which are valid uniformly for all time t 0

an iteratively adaptive multi scale finite element method for elliptic pdes with rough coefficients we propose an iteratively adaptive multi scale finite element method msfem for elliptic pdes with rough coefficients the choice of the local boundary conditions for the multi sale basis functions determines the accuracy of the msfem numerical solution and one needs to incorporate the global information of the elliptic equation into the local boundary conditions of the multi scale basis functions to recover the underlying fine mesh solution of the equation in our proposed iteratively adaptive method we achieve this global to local information transfer through the combination of coarse mesh solving using adaptive multi scale basis functions and fine mesh smoothing operations in each iteration step we first update the multi scale basis functions based on the approximate numerical solutions of the previous iteration steps and obtain the coarse mesh approximate solution using a galerkin projection then we apply several steps of smoothing operations to the coarse mesh approximate solution on the underlying fine mesh to get the updated approximate numerical solution the proposed algorithm can be viewed as a nonlinear two level multi grid method with the restriction and prolongation operators adapted to the approximate numerical solutions of the previous iteration steps convergence analysis of the proposed algorithm is carried out under the framework of two level multi grid method and the harmonic coordinates are employed to establish the approximation property of the adaptive multi scale basis functions we demonstrate the efficiency of our proposed multi scale methods through several numerical examples including a multi scale coefficient problem a high contrast interface problem and a convection dominated diffusion problem

patch dynamics with buffers for homogenization problems an important class of problems exhibits smooth behaviour on macroscopic space and time scales while only a microscopic evolution law is known for such time dependent multi scale problems an equation free framework has been proposed of which patch dynamics is an essential component patch dynamics is designed to perform numerical simulations of an unavailable macroscopic equation on macroscopic time and length scales it uses appropriately initialized simulations of the available microscopic model in a number of small boxes patches which cover only a fraction of the space time domain we show that it is possible to use arbitrary boundary conditions for these patches provided that suitably large buffer regions shield the boundary artefacts from the interior of the patches we analyze the accuracy of this scheme for a diffusion homogenization problem with periodic heterogeneity and illustrate the approach with a set of numerical examples which include a non linear reaction diffusion equation and the kuramoto sivashinsky equation

a new expanded mixed method for parabolic integro differential equations a new expanded mixed scheme is studied and analyzed for linear parabolic integro differential equations the proposed method s gradient belongs to the simple square integrable space replacing the classical h div space the new expanded mixed projection is introduced the existence and uniqueness of solution for semi discrete scheme are proved and the fully discrete error estimates based on both backward euler scheme are obtained moreover the optimal a priori error estimates in l 2 and h 1 norm for the scalar unknown u and the error results in l 2 norm for its gradient and its flux the coefficients times the negative gradient are derived finally some numerical results are calculated to verify our theoretical analysis

a combined finite element and oversampling multiscale petrov galerkin method for the multiscale elliptic problems with singularities in this paper we construct a combined finite element and oversampling multiscale petrov galerkin method fe omspgm to solve the multiscale problems which may have singularities in some special portions of the computational domain for example in the simulation of subsurface flow singularities lie in the porous media with channelized features or in near well regions since the solution behaves like the green function the basic idea of fe omspgm is to utilize the traditional finite element method fem directly on a fine mesh of the problematic part of the domain and using the petrov galerkin version of oversampling multiscale finite element method omspgm on a coarse mesh of the other part the transmission condition across the fe omspg interface is treated by the penalty technique the fe omspgm takes advantages of the fem and omspgm which uses much less dofs than the standard fem and may be more accurate than the omspgm for problems with singularities although the error analysis is carried out under the assumption that the oscillating coefficients are periodic our method is not restrict to the periodic case numerical examples with periodic and random highly oscillating coefficients as well as the multiscale problems on the l shaped domain and multiscale problems with high contrast channels or well singularities are presented to demonstrate the efficiency and accuracy of the proposed method

ribbon tableaux hall littlewood functions quantum affine algebras and unipotent varieties we introduce a new family of symmetric functions which are q analogs of products of schur functions defined in terms of ribbon tableaux these functions can be interpreted in terms of the fock space representation fq of uq sln and are related to hall littlewood functions via the geometry of flag varieties we present a series of conjectures and prove them in special cases the essential step in proving that these functions are actually symmetric consists in the calculation of a basis of highest weight vectors of fq using ribbon tableaux

metric based upscaling we consider divergence form elliptic operators in dimension n ge 2 with l coefficients although solutions of these operators are only holder continuous we show that they are differentiable c1 with respect to harmonic coordinates it follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales this new numerical homogenization method is based on the transfer of a new metric in addition to traditional averaged homogenized quantities from subgrid scales into computational scales error bounds can be given and this method can also be used as a compression tool for differential operators 2006 wiley periodicals inc

error estimates for semidiscrete finite element methods for parabolic integro differential equations application des methodes d elements finis aux equations integro differentielles de type parabolique avec un noyau integral forme par un operateur aux derivees partielles d ordre 2

applications and numerical analysis of partial differential volterra equations a brief survey this article contains a concise survey of the numerical analysis of volterra equations and leads up to some recent results on a posteriori error estimation for finite element approximations

evaluation of kriging based surrogate models constructed from mesoscale computations of shock interaction with particles abstract macro scale computations of shocked particulate flows require closure laws that model the exchange of momentum energy between the fluid and particle phases closure laws are constructed in this work in the form of surrogate models derived from highly resolved mesoscale computations of shock particle interactions the mesoscale computations are performed to calculate the drag force on a cluster of particles for different values of mach number and particle volume fraction two kriging based methods viz the dynamic kriging method dkg and the modified bayesian kriging method mbkg are evaluated for their ability to construct surrogate models with sparse data i e using the least number of mesoscale simulations it is shown that if the input data is noise free the dkg method converges monotonically convergence is less robust in the presence of noise the mbkg method converges monotonically even with noisy input data and is therefore more suitable for surrogate model construction from numerical experiments this work is the first step towards a full multiscale modeling of interaction of shocked particle laden flows

a multiscale finite element method for optimal control problems governed by the elliptic homogenization equations in this article we develop and analyze a priori estimates for optimal control problems with multiscale governed by the elliptic homogenization equations the multiscale finite element is applied to capture the effect of microscale through modification of finite element basis functions without resolving all the small scale features the optimal estimate is derived for elliptic homogenization problems without resonance effect o e h by using an over sampling technique and the boundary layer assumption furthermore the a priori estimate is obtained for the optimal control problems governed by the elliptic homogenization equations

quantum quasi symmetric functions and hecke algebras the algebra of quasi symmetric functions is known to describe the characters of the hecke algebra of type at v 0 we present a quantization of this algebra defined in terms of filtrations of induced representations of the 0 hecke algebra we show that this q deformed algebra admits a simple realization in terms of quantum polynomials for generic values of q the algebra of quantum quasi symmetric functions is isomorphic to the one of noncommutative symmetric functions this gives rise to a one parameter family of hilbert space structures on the algebra of noncommutative symmetric functions as well as to new interesting bases

gamblets for opening the complexity bottleneck of implicit schemes for hyperbolic and parabolic odes pdes with rough coefficients implicit schemes are popular methods for the integration of time dependent pdes such as hyperbolic and parabolic pdes however the necessity to solve corresponding linear systems at each time step constitutes a complexity bottleneck in their application to pdes with rough coefficients we present a generalization of gamblets introduced in 62 enabling the resolution of these implicit systems in near linear complexity and provide rigorous a priori error bounds on the resulting numerical approximations of hyperbolic and parabolic pdes these generalized gamblets induce a multiresolution decomposition of the solution space that is adapted to both the underlying hyperbolic and parabolic pde and the system of odes resulting from space discretization and to the time steps of the numerical scheme

optimal long time l_p 0 t stability and semidiscrete error estimates for the volterra formulation of the linear quasistatic viscoelasticity problem the purpose of this article is to show how the solution of the linear quasistatic compressible viscoelasticity problem written in volterra form with fading memory may be sharply bounded in terms of the data if certain physically reasonable assumptions are satisfied the bounds are derived by making precise assumptions on the memory term which then make it possible to avoid the gronwall inequality and use instead a comparison theorem which is more sensitive to the physics of the problem once the data stability estimates are established we apply the technique also to deriving a priori error bounds for semidiscrete finite element approximations our bounds are derived for viscoelastic solids and fluids under the small strain assumption in terms of the eigenvalues of a certain matrix derived from the stress relaxation tensor for isotropic materials we can be explicit about the form of these bounds while for the general case we give a formula for their computation

the hopf algebra of odd symmetric functions abstract we consider a q analogue of the standard bilinear form on the commutative ring of symmetric functions the q 1 case leads to a z graded hopf superalgebra which we call the algebra of odd symmetric functions in the odd setting we describe counterparts of the elementary and complete symmetric functions power sums schur functions and combinatorial interpretations of associated change of basis relations

multiscale modeling using goal oriented adaptivity and numerical homogenization part i mathematical formulation and numerical results abstract this paper is the first in this series to develop a numerical homogenization method for heterogeneous media and integrate it with goal oriented finite element mesh adaptivity we describe the physical application step and flash imprint lithography in brief and present the mathematical ideas and numerical verification the method requires the moore penrose pseudoinverse of element stiffness matrices algorithms for efficiently computing the pseudoinverse of sparse matrices will be presented in the second paper the purpose of numerical homogenization is to reduce the number of degrees of freedom find locally optimal effective material properties and perform goal oriented mesh refinement traditionally a finite element mesh is designed after obtaining material properties in different regions the mesh has to resolve material discontinuities and rapid variations in the solution in our approach however we generate a sequence of coarse meshes possibly 1 irregular and homogenize material properties on each coarse mesh element using a locally posed constrained convex quadratic optimization problem this upscaling is done using the moore penrose pseudoinverse of the linearized fine scale element stiffness matrices and a material independent interpolation operator numerical verification is done using a two dimensional conductivity problem with known analytical limit finally we present results for two and three dimensional geometries the results show that this method uses orders of magnitude fewer degrees of freedom to give fast and approximate solutions of the original fine scale problem

maximum norm stability and maximal l p lp regularity of fems for parabolic equations with lipschitz continuous coefficients in this paper we study the semi discrete galerkin finite element method for parabolic equations with lipschitz continuous coefficients we prove the maximum norm stability of the semigroup generated by the corresponding elliptic finite element operator and prove the space time stability of the parabolic projection onto the finite element space in l infty q t l qt and l p 0 t l q omega lp 0 t lq 1

long time behavior of the two grid finite element method for fully discrete semilinear evolution equations with positive memory abstract based on two grid discretizations two fully discrete finite element algorithms for semilinear parabolic integro differential equations with positive memory are proposed with the backward euler scheme for the temporal discretization the basic idea of the space two grid finite element algorithms is to approximate the semilinear equations on a coarse space grid and to solve the linearized equations on a finer space grid at each time step to further decreases the amount of computational work a space time two grid algorithm based on a coarse space grid with large time stepsize t and a finer space grid with small time stepsize t for the evolutional equations is proposed in this paper the sharp long time stability and error estimates for the standard finite element method the space two grid finite element method and the space time two grid finite element method are derived it is showed that the two grid algorithms long time stability and error estimates are similar to those of the direct resolution of the semilinear problem on a fine grid

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

on the exponential decay of waves with memory in this paper we consider general linear damped wave equations with memory we establish energy estimates that under the assumption of exponentially bounded kernels induce exponential decaying solutions numerical waves that mimic their continuous counterpart are also introduced using a finite element approach

a multi resolution non parametric bayesian framework for identification of spatially varying model parameters this paper proposes a hierarchical multi resolution framework for the identification of model parameters and their spatially variability from noisy measurements of the response or output such parameters are frequently encountered in pde based models and correspond to quantities such as density or pressure fields elasto plastic moduli and internal variables in solid mechanics conductivity fields in heat diffusion problems permeability fields in fluid flow through porous media etc the proposed model has all the advantages of traditional bayesian formulations such as the ability to produce measures of confidence for the inferences made and providing not only predictive estimates but also quantitative measures of the predictive uncertainty in contrast to existing approaches it utilizes a parsimonious non parametric formulation that favors sparse representations and whose complexity can be determined from the data the proposed framework in non intrusive and makes use of a sequence of forward solvers operating at various resolutions as a result inexpensive coarse solvers are used to identify the most salient features of the unknown field s which are subsequently enriched by invoking solvers operating at finer resolutions this leads to significant computational savings particularly in problems involving computationally demanding forward models but also improvements in accuracy it is based on a novel adaptive scheme based on sequential monte carlo sampling which is embarrassingly parallelizable and circumvents issues with slow mixing encountered in markov chain monte carlo schemes the capabilities of the proposed methodology are illustrated in problems from nonlinear solid mechanics with special attention to cases where the data is contaminated with random noise and the scale of variability of the unknown field is smaller than the scale of the grid where observations are collected

on the numerical approximation of an evolution problem in nonlinear viscoelasticity abstract an error analysis is furnished for the spatial approximation via the finite element method of certain nonlinear time dependent problems emanating in the theory of viscoelastic solids numerical examples are given first for a model equation and then for numerical equilibrium solutions in cases of a nonconvex double well potential and a convex potential the latter derived from the nonconvex one via relaxation by an equal area construction

a remark on split least squares mixed element procedures for pseudo parabolic equations abstract in this paper we introduce two novel split least squares mixed element procedures for pseudo parabolic equations by selecting the least squares functional properly each procedure can be split into two independent symmetric positive definite sub procedures one of sub procedures is for the primitive unknown variable u which is the same as the standard galerkin finite element procedure and the other is for the introduced flux variable optimal order error estimates are developed a numerical example is given to show the efficiency of the introduced schemes

moving average fields macro scale response measures and homogenizing micro scale variation there are two critical issues when deriving a macro scale prediction model starting from a more complete underlying model the first is the precise relationship of the fields predicted by the more complete model and the fields predicted by the macro scale model the second is the manner of solving a closure problem that is invariably encountered in all such derivations the understanding that moving averages of the fields predicted by the more complete model are the fields predicted by the macro scale model is challenged on the grounds that accomplishing a moving average does not eliminate micro scale variation it only appears to do so in one representation of the moving average field the solution of a closure problem by assumption is challenged on the grounds that the most common assumptions are demonstrably invalid even while leading to prediction models that can provide reasonable estimates of the macro scale response in some scenarios in presenting the challenges it is further shown how a multiresolution analysis by an orthogonal wavelet system provides a framework for both precisely defining macro scale response fields i e fields from which all micro scale variation has been eliminated and presenting a formally exact solution for a precisely described closure problem

homogenization based finite element analysis of unidirectional composites by classical and multiresolutional techniques computational studies of unidirectional transient problems in multiresolutional periodic heterogeneous media using homogenization technique and the finite element method are documented below this homogenization method being the extension of the classical two scale asymptotic approach and based on the wavelet representation for composite parameters is used here to calculate effective material parameters of a composite efficiency of this method is compared against asymptotic homogenization technique and simple spatial averaging of material properties using the fem solution for some transient heat transfer and free vibration problems on the other hand a comparison with the solutions obtained for a multiresolutional decomposition of material parameters for composites is also made numerical results show that the homogenized parameters resulting from the new approach are bounded by spatial averages and by asymptotic method results this result is valid also for the eigenvalues of a periodic and simply supported composite beam the multiresolutional homogenization proposed appears to be especially efficient in case of smaller numbers of periodicity cells in the structure while increased number of cells needs application of an asymptotic homogenization method further computational studies are necessary in this area but the application of the multiresolutional technique is natural because of multiscale character of composites moreover this method may be very efficient in common symbolic fem programs implementation

discontinuous galerkin finite element approximation of nonlinear non fickian diffusion in viscoelastic polymers we consider discrete schemes for a nonlinear model of non fickian diffusion in viscoelastic polymers the model is motivated by but not the same as that proposed by cohen et al in siam j appl math 55 pp 348 368 1995 the spatial discretisation is effected with both the symmetric and non symmetric interior penalty discontinuous galerkin finite element method and the time discretisation is of crank nicolson type we also discuss two means of handling the nonlinearity either implicitly which requires the solution of nonlinear equations at each time level or through a linearisation based on extrapolating from previous time levels the same optimal orders of convergence are proven in both cases and to verify this some numerical results are also given for the linearised scheme

error estimates with sharp constants for a fading memory volterra problem in linear solid viscoelasticity the problem characterizing nonageing linear isothermal quasi static isotropic compressible solid viscoelasticity in the time interval 0 t is described this is essentially a volterra equation of the second kind arrived at by adding smooth fading memory to the elliptic linear elasticity equations we analyze the errors resulting from replacing the relaxation functions with practical approximations in a semidiscrete finite element approximation and in a fully discrete scheme derived by replacing the hereditary integral with the trapezoidal rule for numerical integration the error estimates are sharp in the sense that if certain bounds on the data are independent of t then so also are the constants involved in them this is a consequence of bypassing the usual gronwall lemmas with arguments that are more sensitive to the fading memory of the physical problem

a multiscale stochastic finite element method on elliptic problems involving uncertainties in this study a multiscale stochastic finite element method mssfem is developed to resolve scale coupling stochastic elliptic problems based on formulations of a stochastic variational approach and scale bridging multiscale shape functions by employing polynomial chaos expansions and lagrange polynomials stochastic galerkin method and stochastic collocation method are introduced and combined with deterministic multiscale methods i e variational multiscale method and multiscale finite element method the resulting spectral and pseudo spectral stochastic finite element methods are incorporated into the fine and coarse mesh computation of the mssfem a benchmark multiscale model and the numerical experiment based on the formulated mssfem are provided for illustration it is expected that the proposed mssfem can act as a paradigm for solving of general stochastic partial differential equations involving multiscale stochastic data