a numerical verification method for nonlinear functional equations based on infinite dimensional newton like iteration this paper describes a numerical verification of solutions for infinite dimensional functional equations based on residual forms and newton like iteration the method is based upon a verification method previously developed by the authors several computer assisted proofs for differential equations including nonlinear partial differential equations are presented

numerical verification of stationary solutions for navier stokes problems we present a numerical method to enclose stationary solutions of the navier stokes equations especially 2 d driven cavity problem with regularized boundary condition our method is based on the infinite dimensional newton s method by estimating the inverse of the corresponding linearized operator the method can be applied to the case for high reynolds numbers and we show some numerical examples which confirm us the actual effectiveness

numerical verification of solutions for a simplified signorini problem abstract we proposed some numerical methods for the automatic proof of existence of solutions for variational inequalities i e obstacle problems and elastoplastic torsion problems it is based on nakao s method by using finite element approximation and its explicit error estimates for the problem in this paper we present a numerical verification algorithm of exact solutions for a simplified signorini problem in a computer a numerical example for the signorini problem is presented

computationally efficient algorithms for parameter estimation and uncertainty propagation in numerical models of groundwater flow finite difference and finite element methods are frequently used to study aquifer flow however additional analysis is required when model parameters and hence predicted heads are uncertain computational algorithms are presented for steady and transient models in which aquifer storage coefficients transmissivities distributed inputs and boundary values may all be simultaneously uncertain innovative aspects of these algorithms include a new form of generalized boundary condition a concise discrete derivation of the adjoint problem for transient models with variable time steps an efficient technique for calculating the approximate second derivative during line searches in weighted least squares estimation and a new efficient first order second moment algorithm for calculating the covariance of predicted heads due to a large number of uncertain parameter values the techniques are presented in matrix form and their efficiency depends on the structure of sparse matrices which occur repeatedly throughout the calculations details of matrix structures are provided for a two dimensional linear triangular finite element model

logarithmically improved regularity criterion for the nematic liquid crystal flows in b 1 space abstract in this work we study the regularity criterion of the three dimensional nematic liquid crystal flows it is proved that if the vorticity satisfies 0 t t b 1 2 1 log e t b 1 d t where b 1 denotes the critical besov space then the solution u d becomes a regular solution on 0 t this result extends the recent regularity criterion obtained by fan and ozawa 2012 11

statistical theory of groundwater flow and transport pore to laboratory laboratory to formation and formation to regional scale the statistical approach has been applied increasingly to groundwater flow problems in the last decade as is illustrated in figure 3 by the cumulative number of articles published in this field in water resources reseach this development has been motivated by the recognition of the fact that porous formations are heterogeneous i e with properties which vary in an irregular manner in space flow domains are characterized by the length scale l of their spatial extent and three such scales of a fundamental nature are introduced the laboratory the local and the regional scale heterogeneity is characterized by the spatial correlation scale i of the property of interest the three scales corresponding to the above ones being the pore scale the log hydraulic conductivity and the log transmissivity integral scales the medium properties and related flow variables are regarded as random space functions which satisfy two basic requirements they enjoy some type of stationarity and i l an additional scale d is the measurement or computational scale characterizing the size of the measurement device of a flow variable or the element over which the variable is averaged for computational purposes in both cases the interest resides in the space average of the flow variable over a volume or area of length scale d the primary aim of the theory of flow and transport through porous media is to determine the statistical moments of the space averaged variables given the statistical structure of the spatially variable property the main objective of the study is to show that flow and transport problems at the three fundamental scales can be treated by a unified statistical approach along this line the specific aspects of each scale are examined separately and areas of interest for future research are indicated in the concluding remarks it is submitted that the statistical approach to groundwater flow has become a comprehensive theory beyond the stage of an ad hoc modeling technique

nonstationary stochastic analysis in well capture zone design using first order taylor s series approximation 1 nonstationarity of flow fields due to pumping wells and its impact on advective transport is of particular interest in well capture zone design and wellhead protection however techniques based on monte carlo methods to characterize the associated capture zone uncertainty are time consuming and cumbersome this paper introduces an alternative approach the mean and covariance of system state variables i e head pore water velocity and particle trajectory are approximated using a first order taylor s series with sensitivity coefficients estimated from the adjoint operator for a system of discrete equations the approach allows nonstationarity due to several sources e g transmissivity pumping boundary conditions to be treated by employing numerical solution methods it is able to handle irregular geometry varying boundary conditions complicated sink source terms and different covariance functions all of which are important factors for real world applications a comparison of results for the taylor s series approximation with those from monte carlo analysis showed in general good agreement for most of the tested particles particle trajectory variance calculated using taylor s series approximation is then used to predict well capture zone probabilities under the assumption of normality of the mass transport s state variables verification of this assumption showed that not all particle trajectories depending on their starting location are normally or log normally distributed however the risk of using the first order method to delineate the confidence interval of a well capture zone is minimal since it marginally overestimates the 2 5 probability contour furthermore this should be balanced against its greater computation efficiency over the monte carlo approach

numerical solutions to statistical moment equations of groundwater flow in nonstationary bounded heterogeneous media in this paper we investigate the combined effect of nonstationarity in log hydraulic conductivity and the presence of boundaries on flow in heterogeneous aquifers we derive general equations governing the statistical moments of hydraulic head for steady state flow by perturbation expansions due to their mathematical complexity we solve the moment equations by the numerical technique of finite differences the numerical approach has flexibility in handling moderately irregular geometry different boundary conditions various trends in the mean log hydraulic conductivity spatial variabilities in the magnitude and direction of mean flow and different covariance functions all of which are important factors to consider for real world applications the effect of boundaries on the first two statistical moments involving head is strong and persistent for example in the case of stationary log hydraulic conductivity the head variance is always finite in a bounded domain while the head variance may be infinite in an unbounded domain as in many other stochastic models the statistical moment equations are derived under the assumption that the variance of log hydraulic conductivity is small accounting for a spatially varying large scale trend in the log hydraulic conductivity field reduces the variance of log hydraulic conductivity although this makes the conductivity field nonstationary and significantly increases the mathematical complexity in the problem it justifies the small variance assumption for many aquifers

numerical verification of solutions for some unilateral boundary value problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for some unilateral boundary value problems using the finite element approximations and explicit a priori error estimates we construct in a computer a set of solutions which satisfies the hypothesis of schauder s fixed point theorem for a compact map on a certain sobolev space further the conditions of verifiability by this method are considered and some numerical examples are presented

explicit h2 estimates and pointwise bounds for solutions of second order elliptic boundary value problems abstract it is well known that the h 2 norm and the c 0 norm of a function u h 2 where r n is a bounded domain n 3 can be estimated in terms of a given uniformly elliptic second order differential operator l and some boundary operator b applied to u if certain regularity assumptions are satisfied if these bounds shall be used for numerical purposes the constants occurring in the estimates must be known explicitly the main goal of the present article is the computation of such explicit constants for simplicity of presentation we restrict ourselves to the case where l u u c x u as an application we prove an existence and inclusion result for nonlinear boundary value problems

nonparabolic dissipative systems modeling the flow of liquid crystals we study a simplified system which retains most of the interesting mathematical properties of the original ericksen leslie equations for the flow of liquid crystals this is a coupled nonparabolic dissipative dynamic system we derive several energy laws which enable us to prove the global existence of the weak solutions and the classical solutions we also discuss uniqueness and some stability properties of the system 1995 john wiley sons inc

nonlinear theory of defects in nematic liquid crystals phase transition and flow phenomena this is a summary of some recent mathematical advances in the theory of defects in nematic liquid crystals it also includes some new results concerning disclinations line defects which have not been published elsewhere

estimating the variance and integral scale of the transmissivity field using head residual increments a modification of previously published solutions regarding the spatial variation of hydraulic heads is discussed whereby the semivariogram of increments of head residuals termed head residual increments hris are related to the variance and integral scale of the transmissivity field a first order solution is developed for the case of a transmissivity field which is isotropic and whose second order behavior can be characterized by an exponential covariance structure the estimates of the variance y2 and the integral scale of the log transmissivity field are then obtained via fitting a theoretical semivariogram for the hri to its sample semivariogram this approach is applied to head data sampled from a series of two dimensional simulated aquifers with isotropic exponential covariance structures and varying degrees of heterogeneity y2 0 25 0 5 1 0 2 0 and 5 0 the results show that this method provided reliable estimates for both and y2 in aquifers with the value of y2 up to 2 0 but the errors in those estimates were higher for y2 equal to 5 0 it is also demonstrated through numerical experiments and theoretical arguments that the head residual increments will provide a sample semivariogram with a lower variance than will the use of the head residuals without calculation of increments

an alternative approach to norm bound computation for inverses of linear operators in hilbert spaces abstract in the present paper we propose a computer assisted procedure to prove the invertibility of a linear operator in a hilbert space and to compute a verified norm bound of its inverse a number of the authors have previously proposed two verification approaches that are based on projection and constructive a priori error estimates the approach of the present paper is expected to bridge the gap between the two previous procedures in actual numerical verifications several verification examples that confirm the actual effectiveness of the proposed procedure are reported

stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers 2 impervious boundary the spatial variability of the head h in a two dimensional flow in the vicinity of an impervious boundary is investigated this study is a continuation of a previous one rubin and dagan 1988 which dealt with the influence of a given head boundary by representing h and the log transmissivity y as space random function and by solving the flow equation first order closed form analytical expressions for the head variogram and the head log transmissivity y covariance are developed these expressions offer the means to evaluate the influence of the boundary on the joint y h random field the assumption of unbounded domain is examined and it is found to be a good approximation at distances larger than three log transmissivity integral scales from the boundary by using conditional probabilities a simple method to estimate the statistical moments close to the boundary is presented it requires the knowledge of the moments of y and h in the unbounded domain only the method is compared with the exact analytical solution showing favorable results

stochastic analysis of head spatial variability in bounded rectangular heterogeneous aquifers groundwater flow through bounded rectangular aquifers is analyzed in a stochastic framework new analytical expressions of the head covariance and variance and the log transmissivity head cross covariance are derived for the simplified first order problem effects of different boundary conditions the aquifer size and the log transmissivity correlation scale upon head moments are studied the results are compared to analytical half plane solutions demonstrating significant differences in several cases finally the problem is solved numerically using the monte carlo simulation method the realizations of the log transmissivity field are generated in two different ways through comparisons with the analytical results the accuracy of the numerical methods is shown to be excellent

stochastic modeling of groundwater flow by unconditional and conditional probabilities the inverse problem the inverse problem is defined here as follows determine the transmissivity at varius points given the shape and boundary of the aquifer and recharge intensity and given a set of measured log transmissivity y and head h values at a few points the log transmissivity distribution is regarded as a realization of a random function of normal and stationary unconditional probability density function pdf the solution of the inverse problem is the conditional normal pdf of y conditioned on measured h and y which is expressed in terms of the unconditional joint pdf of y and h the problem is reduced to determining the unconditional head log transmissivity covariance and head variogram for a selected y covariance which depends on a few unknown parameters this is achieved by solving a first order approximation of the flow equations the method is illustrated for an exponential y covariance and the effect of head and transmissivity measurements upon the reduction of uncertainty of y is investigated systematically it is shown that measurement of h has a lesser impact than those of y but a judicious combination may lead to significant reduction of the predicted variance of y possible applications to real aquifers are outlined

perturbation based moment equation approach for flow in heterogeneous porous media applicability range and analysis of high order terms we present detailed comparisons between high resolution monte carlo simulation mcs and low order numerical solutions of stochastic moment equations smes for the first and second statistical moments of pressure the objective is to quantify the difference between the predictions obtained from mcs and sme natural formations with high permeability variability and large spatial correlation scales are of special interest for underground resources e g oil and water consequently we focus on such formations we investigated fields with variance of log permeability y2 from 0 1 to 3 0 and correlation scales normalized by domain length of 0 05 to 0 5 in order to avoid issues related to statistical convergence and resolution level we used 9000 highly resolved realizations of permeability for mcs we derive exact discrete forms of the statistical moment equations formulations based on equations written explicitly in terms of permeability k based and log transformed permeability y based are considered the discrete forms are applicable to systems of arbitrary variance and correlation scales however equations governing a particular statistical moment depend on higher moments thus while the moment equations are exact they are not closed in particular the discrete form of the second moment of pressure includes two triplet terms that involve log permeability or permeability and pressure we combined mcs computations with full discrete sme equations to quantify the importance of the various terms that make up the moment equations we show that second moment solutions obtained using a low order y based sme formulation are significantly better than those from k based formulations especially when y2 1 as a result y based formulations are preferred the two triplet terms are complex functions of the variance level and correlation length the importance contribution of these triplet terms increases dramatically as y2 increases above one we also show that one of the triplet terms is much more important than the other when comparing k based mcs with y based sme model differences must be taken into consideration these differences model errors are due to embedded assumptions and differences in implementing the discrete forms of the equations

stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers 1 constant head boundary the spatial variability of log transmissivity y and head h in aquifers assumes a major role in determining the solutions of the direct flow problem as well as of the inverse problem investigation of the joint y h variability has been cast in the past in numerical and analytic frames analytic solutions for the y h statistical moments in two dimensions assumed the flow to occur in an unbounded domain in this study exact analytical expressions for the y h moments are derived which account for the presence of a given head boundary in a half plane this result is achieved by a first order approximation and for an exponential y covariance the result offers the opportunity to examine the effects of boundaries on the semivariogram of h and on the y h cross covariance the assumption of unbounded domain is examined and shown to be accurate at distances larger than two y integral scales from the boundary by using conditional probabilities a general and simple method to estimate the moments close to the boundary is presented it requires the knowledge of the moments of y and h in the unbounded domain only the method is compared with the exact analytic solution showing excellent results

a numerical algorithm for the solution of signorini problems in this paper we propose a new iterative algorithm for the solution of a certain class of signorini problems such problems arise in the modelling of a variety of physical phenomena and usually involve the determination of an unknown free boundary here we describe a way of locating the free boundary directly and provide a proof that the algorithm converges when used with analytic methods the advantage of this algorithm is that it can be used in conjunction with any numerical method with minimal development of extra code we demonstrate its application with the boundary element method to some physical problems in both two and three dimensions

two projection methods for the solution of signorini problems abstract two iterative methods based on projection and boundary element methods are considered for signorini problems the regularized problem with the projection boundary condition is first deduced from the signorini problem by using the equivalence between the signorini boundary condition and the projection fixed point problem our methods formulate the signorini boundary condition into a sequence of robin boundary conditions in the new boundary condition there is a penalty parameter the convergence speed of the first method is greatly influenced by the value of which is difficult to choose for individual problems to improve the performance of this method we present a self adaptive projection method which adjusts the parameter automatically per iteration based on the iterative data the main result of this work is to provide the convergence of the methods under mild assumptions as the iteration process is given by the potential and its derivative on the boundary of the domain the unknowns of the problem are computed explicitly by using the boundary element method both theoretical results and numerical experiments indicate efficiency of the methods proposed

existence and uniqueness theorem on mild solutions to the keller segel system coupled with the navier stokes fluid abstract we consider the keller segel system coupled with the navier stokes fluid in the whole space and prove the existence of global mild solutions with the small initial data in the scaling invariant space our method is based on the implicit function theorem which yields necessarily continuous dependence of solutions for the initial data as a byproduct we show the asymptotic stability of solutions as the time goes to infinity since we may deal with the initial data in the weak l p spaces the existence of self similar solutions provided the initial data are small homogeneous functions

the inverse problem for confined aquifer flow identification and estimation with extensions the contributions of this work are twofold first a methodology for estimating the elements of parameter matrices in the governing equation of flow in a confined aquifer is developed the estimation techniques for the distributed parameter inverse problem pertain to linear least squares and generalized least squares methods the linear relationship among the known heads and unknown parameters of the flow equation provides the background for developing criteria for determining the identifiability status of unknown parameters under conditions of exact or overidentification it is possible to develop statistically consistent parameter estimators and their asymptotic distributions the estimation techniques namely two stage least squares and three stage least squares are applied to a specific groundwater inverse problem and compared between themselves and with an ordinary least squares estimator the three stage estimator provides the closer approximation to the actual parameter values but it also shows relatively large standard errors as compared to the ordinary and two stage estimators the estimation techniques provide the parameter matrices required to simulate the unsteady groundwater flow equation second a nonlinear maximum likelihood estimation approach to the inverse problem is presented the statistical properties of maximum likelihood estimators are derived and a procedure to construct confidence intervals and do hypothesis testing is given the relative merits of the linear and maximum likelihood estimators are analyzed other topics relevant to the identification and estimation methodologies i e a continuous time solution to the flow equation coping with noise corrupted head measurements and extension of the developed theory to nonlinear cases are also discussed a simulation study is used to evaluate the methods developed in this study

characteristics and controlling factors of dispersion in bounded heterogeneous porous media 1 in this study we present first order in terms of the log conductivity variance analytical solutions to displacement covariances xij and macrodispersion coefficients dij for transport of conservative solutes in two dimensional bounded heterogeneous porous media these solutions are presented as infinity series and are explicit functions of the statistics of the log hydraulic conductivity using numerical examples the convergence of these solutions in terms of the number of terms in truncated finite summations is first investigated and the accuracy of these solutions is examined by comparing them with results from monte carlo simulations and the semianalytical solutions of osnes 1998 through several examples the general features of time dependent xij and dij are discussed unlike in unbounded domains the longitudinal macrodispersivity d11 for bounded domains does not approach an asymptotic value at large dimensionless time but instead increases quickly near the downstream constant head boundary in addition d11 for bounded domains is always larger than that in unbounded domains and accordingly the transverse macrodispersivity d22 in bounded domains is smaller than that in unbounded domains these differences stem from the lateral no flow boundaries in our bounded domain models we also investigated the effect of domain sizes on xij and dij our study shows that both xij and dij depend not only on the dimensionless size of the domain but also on its aspect ratio the dependence of xij and dij on the particles initial location has also been investigated at detail our results indicate that while this dependence is very small for x11 and d11 the release location has a significant impact on both x22 and d22 our solutions to both displacement covariances and macrodispersivity are compared against those derived from tracer test data at the borden site the comparison shows that our solutions are quite close to observed data indicating that they may be applicable to predict solute transport at the field scale

nonlocal and localized analyses of conditional mean steady state flow in bounded randomly nonuniform domains 2 computational examples in paper 1 guadagnini and neuman this issue of this two part series we presented exact nonlocal equations for first and second conditional ensemble moments of hydraulic head and flux under steady state groundwater flow in bounded randomly nonuniform domains in the presence of uncertain source and boundary terms we derived recursive approximations for these equations based on expansion in powers of a small parameter y which represents the standard estimation error of log hydraulic conductivity developed a two dimensional finite element scheme for their solution under the action of deterministic forcing terms to first order in y2 showed how to localize the conditional mean flow equations so as to yield familiar looking darcian differential equations in which however all quantities are conditional on data and therefore nonunique and proposed that traditional deterministic flow models be reinterpreted in light of these localized equations we stressed that all quantities which enter into our nonlocal and localized conditional moment equations are defined on a consistent support scale thereby obviating the need for upscaling from support to grid scale or the a priori definition of a computational grid that such a grid can therefore be defined a posteriori based on the degree of smoothness one expects the conditional moments to exhibit and that conditioning points are often spaced widely enough to render these moments relatively smooth and hence resolvable on a coarser grid than is necessary for monte carlo simulations this notwithstanding we use in this paper a high resolution grid to explore superimposed mean uniform and convergent flows in unconditional and conditional log hydraulic conductivity domains and to compare the relative accuracies of corresponding nonlocal localized and monte carlo finite element solutions whereas our nonlocal solution is nominally restricted to mildly nonuniform media with y2 1 we find that it actually yields remarkably accurate results for strongly nonuniform media with y2 at least as large as 4 localized solutions deteriorate relative to nonlocal results as one approaches some points of conditioning and singularity or as y2 increases though they require much less computer time than do nonlocal solutions localized solutions have the added disadvantage that they yield no information about predictive uncertainty

analytical solutions to statistical moments for transient flow in two dimensional bounded randomly heterogeneous media 1 in this paper we derive analytical solutions to statistical moments for transient saturated flow in two dimensional bounded randomly heterogeneous porous media by perturbation expansions we first derive partial differential equations governing the zeroth order head h 0 and the first order head term h 1 where orders are in terms of the standard deviation of the log transmissivity we then solve h 0 and h 1 analytically both of which are expressed as infinite series the head perturbation h 1 is then used to derive expressions for autocovariance of the hydraulic head and the cross covariance between the log transmissivity and head the expressions for the mean flux and flux covariance tensor are formulated from the head moments based on darcy s law using numerical examples we demonstrate the convergence of these solutions we also examine the accuracy of these first order solutions by comparing them with solutions from both monte carlo simulations and the numerical moment equation method

estimation of aquifer parameters under transient and steady state conditions 1 maximum likelihood method incorporating prior information in this series of three papers a method is presented to estimate the parameters of groundwater flow models under steady and nonsteady state conditions the parameters include values and directions of principal hydraulic conductivities or transmissivities in anisotropic media specific storage or storativity interior and boundary recharge or leakage rates coefficients of head dependent interior and boundary sources and boundary heads in transient situations the initial head distribution can also be estimated if the system is originally at a steady state paper 1 of the series discusses some of the advantage in treating the inverse problem statistically and in regularizing its solution by means of penalty criteria based on prior estimates of the parameters the inverse problem is posed in the framework of maximum likelihood theory cast in a manner that accounts for prior information about the parameters since not all the factors which contribute to the prior errors can be quantified statistically at the outset the covariance matrices of these errors are expressed in terms of several parameters which if unknown can be estimated jointly with the hydraulic parameters by a stagewise optimization process when transient head data are separated by a fixed time interval the temporal structure of these data is approximated by a lag one autoregressive model with a correlation coefficient that can be treated as another unknown parameter estimation errors are analyzed by examining the lower bound of their covariance matrix in the eigenspace paper 1 concludes by suggesting that certain model identification criteria developed in the time series literature all of which are based on the maximum likelihood concept might be useful for selecting the best groundwater model or the best method of parameterizing a particular model among a number of given alternatives

on transport in porous formations characterized by heterogeneity of evolving scales solute transport in natural formations at the regional scale is influenced by several scales of heterogeneity which correspond to the presence of several geological units called facies as customarily assumed in stochastic theories inside the facies the transport can be characterized by a single scale of heterogeneity at the regional scale several geological units are present such that a hierarchy of relevant scales needs to be defined a possible model for this spatial variability assumes the log conductivity as a random space function of stationary increments characterized by a power law semivariogram with this hypothesis the ergodic dispersion coefficient grows unbounded as time increases leading to the phenomenon called anomalous dispersion an alternative approach considers the plume in nonergodic conditions and assumes the effective dispersion coefficient which is defined through differentiation in time of the expected value of the spatial second order plume moment as representative of macrodispersion large differences have been observed in the resulting plume spreading while approaching the problem using the above alternative definitions in this paper we provide first order analytical solutions for the longitudinal effective dispersion coefficient d l as well as for the expected value of the longitudinal spatial plume moment that complement semianalytical expressions recently proposed in literature furthermore we provide a semianalytical expression for the standard deviation of the longitudinal second order moment which is important in assessing the interval of confidence of the estimation provided by suitable numerical simulations are performed to validate analytical and semianalytical expressions as well as to assess the impact of the cutoff in the log conductivity power spectrum imposed by choosing a finite domain dimension we conclude that according to recently published results the dispersion is anomalous when the hurst coefficients h is larger than 0 5 while it is fickian for h 0 5 this is in contrast with the ergodic analysis which concludes that the dispersion is anomalous irrespective of the hurst coefficient hence the effective dispersion coefficient is more effective than the ergodic dispersion coefficient to represent the plume spreading however the standard deviation of the longitudinal spatial second order moment is of the same order of magnitude as the expected value leading to the conclusion that the estimations provided by d l and are affected by large uncertainties numerical results are in good agreement with the analytical solutions and under some hypotheses they are not influenced by the cutoff this is not the case for the ergodic second order longitudinal moment which strongly depends on the imposed cutoff

numerical inclusion methods of solutions for variational inequalities we consider a numerical method that enables us to verify the existence of solutions for variational inequalities this method is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations using the finite element approximations and explicit a priori error estimates we present an effective verification procedure that through numerical computation generates a set which includes the exact solution copyright 2002 john wiley sons ltd

nonlocal and localized analyses of conditional mean steady state flow in bounded randomly nonuniform domains 1 theory and computational approach we consider the effect of measuring randomly varying hydraulic conductivitiesk x on one s ability to predict numerically without resorting to either monte carlo simulation or upscaling steady state flow in bounded domains driven by random source and boundary terms our aim is to allow optimum unbiased prediction of hydraulic heads h x and fluxes q x by means of their ensemble moments h x c and q x c respectively conditioned on measurements of k x these predictors have been shown by neuman and orr 1993a to satisfy exactly an integrodifferential conditional mean flow equation in which q x c is nonlocal and non darcian here we develop complementary integrodifferential equations for second conditional moments of head and flux which serve as measures of predictive uncertainty obtain recursive closure approximations for both the first and second conditional moment equations through expansion in powers of a small parameter y which represents the standard estimation error of ln k x and show how to solve these equations to first order in y2 by finite elements on a rectangular grid in two dimensions in the special case where one treats k x as if it was locally homogeneous and mean flow as if it was locally uniform one obtains a localized darcian approximation q x c kc x h x c in which kc x is a space dependent conditional hydraulic conductivity tensor this leads to the traditional deterministic darcian steady state flow equation which however acquires a nontraditional meaning in that its parameters and state variables are data dependent and therefore inherently nonunique it further explains why parameter estimates obtained by traditional inverse methods tend to vary as one modifies the database localized equations yield no information about predictive uncertainty our stochastic derivation of these otherwise standard deterministic flow equations makes clear that uncertainty measures associated with estimates of head and flux obtained by traditional inverse methods are generally smaller often considerably so than measures of corresponding predictive uncertainty which can be assessed only by means of stochastic models such as ours we present a detailed comparison between finite element solutions of nonlocal and localized moment equations and monte carlo simulations under superimposed mean uniform and convergent flow regimes in two dimensions paper 1 presents the theory and computational approach and paper 2 guadagnini and neuman this issue describes unconditional and conditional computational results

numerical verification of solutions for variational inequalities in this paper we consider a numerical technique that enables us to verify the existence of solutions for variational inequalities this technique is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations using the finite element approximations and explicit a priori error estimates for obstacle problems we present an effective verification procedure that through numerical computation generates a set which includes the exact solution further a numerical example for an obstacle problem is presented

global existence of weak solutions of the nematic liquid crystal flow in dimension three for any bounded smooth domain 3 or 3 we establish the global existence of a weak solution u d 0 3 s2 of the initial boundary value or the cauchy problem of the simplified ericksen leslie system llf modeling the hydrodynamic flow of nematic liquid crystals for any initial and boundary or cauchy data u0 d0 h h1 s2 with d0 s 2 the upper hemisphere furthermore u d satisfies the global energy inequality 2016 wiley periodicals inc

numerical verifications of solutions for elliptic equations in nonconvex polygonal domains in this paper methods for numerical verifications of solutions for elliptic equations in nonconvex polygonal domains are studied in order to verify solutions using computer it is necessary to determine some constants which appear in a priori error estimations we propose some methods for determination of these constants in numerical examples calculating these constants for anl shaped domain we verify the solution of a nonlinear elliptic equation

stochastic analysis of spatial variability in two dimensional steady groundwater flow assuming stationary and nonstationary heads two dimensional steady groundwater flow in a confined aquifer with spatially variable transmissivity t is analyzed stochastically using spectral analysis and the theory of intrinsic random functions conditions that ensure a stationary statistically homogeneous head process are derived and using two convenient forms for the covariance function of the ln t process the head covariance function is studied in addition the head variogram is obtained for a particular nonstationary case and the asymptotic head variogram is derived under very general conditions results are compared to those obtained by gelhar 1976 for one and two dimensional phreatic flow and bakr et al 1978 for one and three dimensional confined flow multidimensional flow analysis results in a significantly reduced head variance the head correlation remains high over much greater distances than the ln t correlation the variogram obtained when stationary heads are assumed is identical to that obtained for nonstationary heads for dimensionless lag distances up to 2 times the correlation scale of the log transmissivity the variogram for nonstationary heads continues to grow logarithmically as lag distance increases independent of the form of the input covariance in the nonstationary case the conditions for stationarity are contrasted with the corresponding results obtained for the one and three dimensional cases of gutjahr and gelhar 1981 the head variance calculated from the stationary theory is found to agree with that of previous monte carlo simulations

a note on the influence of a constant velocity boundary condition on flow and transport in heterogeneous formations fluid flow and solute transport in an anisotropic heterogeneous porous medium with mean flow normal to a constant flux boundary are considered the statistical moments of flow and transport variables are determined at second order in log conductivity fluctuation and they are expressed in terms of the log conductivity variance and integral scales the mean flow velocity and the distance from the boundary the variance of the longitudinal and transverse components of velocity as well as hydraulic head variance and the longitudinal macrodispersivity are analyzed for a bounded medium with axisymmetric gaussian log conductivity covariance structure in this case all of the moments can be solved by means of a single numerical quadrature the constant flow boundary increases the variability of head and of flow transverse to the mean flow direction and causes a reduction of the macrodispersivity in a zone adjacent to the boundary our results should be useful for the design and testing of numerical models and have important implications for surface infiltration of solutes

constructive a priori error estimates for a full discrete approximation of the heat equation in this paper we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation our method is based on the finite element galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space the present estimates play an essential role in the numerical verification method of exact solutions for the nonlinear parabolic equations this implies that by utilizing the present results we could get the guaranteed a posteriori error estimates for various kinds of nonlinear evolutional problems our results can also be considered as an explicit optimal estimate with the limited regularity of solutions which should be unknown up to the present

on error estimation of finite element approximations to the elliptic equations in nonconvex polygonal domains numerical verification methods so called nakao s methods on existence or uniqueness of solutions to pdes have been developed by nakao and his group including the authors they are based on the error estimation of approximate solutions which are mainly computed by fem it is a standard way of the error estimation of fem to estimate the projection errors by elementwise interpolation errors there are some constants in the error estimation which depend on the mesh size parameters h the explicit values of the constants are necessary in order to use nakao s method however there were not so many researches for the computation of the explicit values of the constants then we had to develop the computation by ourselves especially with guaranteed accuracy note that the methods of the computation depend on the dimension the degree of bases and the shape of the domain etc the present paper shows how we have developed the methods to calculate the constants and describes new results for nonconvex domains

numerical verifications for solutions to elliptic equations using residual iterations with a higher order finite element the verifications of solutions to weakly nonlinear elliptic equations by the method described e g by nakao 1988 1989 etc are sometimes hardly accomplished when the right hand sides of the equations are very large to overcome such difficulties a residual iteration technique with approximate solution was introduced by nakao 1993 in the present paper we propose an a posteriori method for the residual iteration and show that a remarkable improvement in efficiency and in accuracy of the verification can be obtained when we use a higher order finite element

a posteriori existence in numerical computations a typical a posteriori error analysis for numerical methods for differential equations consists of two elements the stability of the continuous problem and an a posteriori computed residual estimate if the problem is nonlinear then this is possible only if it is strongly monotone in this paper we will estimate the stability constant a posteriori as well in order to apply the inverse function theorem to derive the existence of solutions from the numerical computation and thus obtain rigorous a posteriori error estimates for nonmonotone equations

parameter estimation by ensemble kalman filters with transformed data approach and application to hydraulic tomography 1 ensemble kalman filters enkfs are a successful tool for estimating state variables in atmospheric and oceanic sciences recent research has prepared the enkf for parameter estimation in groundwater applications enkfs are optimal in the sense of bayesian updating only if all involved variables are multivariate gaussian subsurface flow and transport state variables however generally do not show gaussian dependence on hydraulic log conductivity and among each other even if log conductivity is multi gaussian to improve enkfs in this context we apply nonlinear monotonic transformations to the observed states rendering them gaussian gaussian anamorphosis ga similar ideas have recently been presented by beal et al 2010 in the context of state estimation our work transfers and adapts this methodology to parameter estimation additionally we address the treatment of measurement errors in the transformation and provide several multivariate analysis tools to evaluate the expected usefulness of ga beforehand for illustration we present a first time application of an enkf to parameter estimation from 3 d hydraulic tomography in multi gaussian log conductivity fields results show that 1 ga achieves an implicit pseudolinearization of drawdown data as a function of log conductivity and 2 this makes both parameter identification and prediction of flow and transport more accurate combining enkfs with ga yields a computationally efficient tool for nonlinear inversion of data with improved accuracy this is an attractive benefit given that linearization free methods such as particle filters are computationally extremely demanding

liquid crystal flows in two dimensions this paper is concerned with a simplified hydrodynamic equation proposed by ericksen and leslie modeling the flow of nematic liquid crystals in dimension two we establish both interior and boundary regularity theorems for such a flow under smallness conditions as a consequence we establish the existence of global in time weak solutions on a bounded smooth domain in mathbb r 2 which are smooth everywhere with possible exceptions of finitely many singular times

nonstationary flow and nonergodic transport in random porous media 1 transport processes taking place in natural formations are often characterized by the spatial nonstationarity of flow field and the nonergodic condition of solute spreading the former may originate from statistical inhomogeneity of porous media properties or from the influence of boundary conditions as well as from conditioning procedures on measured data while the latter is generally due to the finite size of the solute source in real world applications these aspects sometimes dominate the transport phenomena and known literature results based on statistical homogeneity and ergodicity assumptions cannot be properly applied in this paper a method is proposed to handle different and concurrent causes of the flow field nonstationarity and to describe nonergodic transport of inert solutes by spatial moments in a domain of finite size the goal is reached by expanding the steady state flow equation in taylor series limited to the first order and by the recursive application of finite element fe method the unknowns are the piezometric head mean values and its derivatives in respect to the fluctuating porous media hydraulic conductivity from the latter the velocity statistics are derived on the basis of darcy s law according to the first order approximation by neglecting the pore scale dispersion the spatial moments of a solute plume are a posteriori obtained by a consistent lagrangian analysis starting from the knowledge of the velocity field covariance matrices without any restriction regarding the statistical homogeneity of flow and or ergodicity conditions the proposed method is here applied to investigate the influence of boundary conditions on the solute transport developing in a limited domain the results obtained from different test cases give a velocity field behavior in agreement with literature findings and allow us to define the nonergodic transport in bounded domains the comparison of the results with some ad hoc developed monte carlo simulations ensures the applicability of the proposed approach at least to the small heterogeneity cases here considered

a numerical approach to the proof of existence of solutions for some generalized obstacle problems in 8 we proposed some numerical verification methods for automatic proof of the existence of solution for obstacle problems in this paper we propose a new iterative algorithm to automatically prove the existence of solutions for some generalized obstacle problems

numerical verification of solutions for elasto plastic torsion problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for the elasto plastic torsion problems governed by the variational inequality based upon the finite element approximations and the explicit a priori error estimates for a simple problem we present an effective verification procedure that through numerical computation generates a set which includes the exact solution this paper is an extension of the previous paper 1 in which we mainly dealt with the obstacle problems but some special techniques are utilized to verify the solutions for nondifferentiable nonlinear equations concerned with the present problem a numerical example is illustrated

stochastic models of subsurface flow infinite versus finite domains and stationarity stochastic solutions of the differential equation describing one dimensional flow through a porous medium with spatially variable hydraulic conductivity which is represented by a stationary statistically homogeneous random process are developed using several techniques the analytical approximations using first order analysis propagation of error covariance differential equations and variogram analysis all yield consistent results which demonstrate the important effects of boundary conditions and conditioning using the theory of intrinsic random functions stochastic solutions are developed for the case when the logarithm of the hydraulic conductivity is a three dimensional stationary random field in contrast to the one dimensional case it is found that the resulting three dimensional head perturbation will be locally stationary under very general conditions results from the one dimensional analytical solutions are found to be in agreement with previous monte carlo simulations for a flow system of finite length the solution based on linearization in the logarithm of hydraulic conductivity proved to be very robust showing reasonable agreement with monte carlo results even for the largest input standard deviation of lnk 4 6

existence and uniqueness of mild solutions to the incompressible nematic liquid crystal flow abstract in this paper we consider the incompressible nematic liquid crystal flow in the whole space and obtain the existence of global mild solutions with small initial data in the scaling invariant space the main tool we have used is the implicit function theorem furthermore we establish the asymptotic stability of solutions when the time goes to infinity and in view of our construction of solutions in the weak l p spaces we finally derive the existence of self similar solutions provided the initial data are small homogeneous functions

first order analysis of uncertainty in numerical models of groundwater flow part 1 mathematical development in part 1 of this paper first and second order analysis of uncertainty is applied to numerical models of groundwater flow the models are cast in state space form with boundary conditions and inputs that are functionally dependent but statistically independent of time using a compact matrix calculus notation first and second order taylor series expansions of the model equations are derived and used to estimate the mean and variance covariance properties of piezometric head predictions given corresponding statistics for aquifer parameters material properties initial conditions boundary conditions and inputs the mathematical results demonstrate that the prediction uncertainty is a function of the magnitude of the parameter uncertainty and sensitivity of the predictions to the parameters furthermore the first order estimate of the piezometric head is identical to the dete ministic result part 2 of this paper to be presented later will illustrate these and other results through numerous applications of the methodology

numerical verification of solutions for signorini problems using newton like method we proposed a numerical method to verify the existence of solutions for a simplified signorini problem comput math appl 2000 40 1003 1013 using sequential iteration method we numerically constructed a set containing solutions that satisfies the hypothesis of schauder s fixed point theorem in a certain sobolev space it is difficult to apply this method to the problem of which associated operator is not retractive in a neighborhood of the solution in this paper in order to overcome such a difficulty we describe an alternative approach to this problem numerical examples are presented copyright 2007 john wiley sons ltd

role of boundary conditions in convergence and nonlocality of solutions to stochastic flow problems in bounded domains steady flow through a heterogeneous porous medium in a bounded domain is investigated using a recursive perturbation scheme the effect of boundary conditions on a two dimensional flow with arbitrary variation of the mean flow allowing large gradients is investigated using analytical expressions for the head and velocity covariance functions boundary conditions were decomposed into deterministic and stochastic components two flow cases with the same zero order and different first order boundary conditions were analyzed boundary conditions are deterministic for the first case and random for the second significant differences between the two cases indicate random or absence of randomness processes must be modeled at boundaries first order solutions for the head and velocity covariance functions for a bounded rectangular domain are derived the resulting integral kernels involve greens functions and they are evaluated by numerical integration boundary conditions for higher order problems influence the absolute value and shape of the kernels and therefore of the head and velocity variance and covariance functions it is found that the validity of the perturbation scheme is dependent on the magnitude of the kernels and not only on the condition f2 1 where f is the variance of the log hydraulic conductivity assumed gaussian and weakly homogeneous in space the type of boundary conditions affects the values of the kernels and therefore determine the convergence limits for the problem milder head gradients also allow larger values of f2 nonlocality is also contingent on boundary type weakly homogeneous log fluctuating conductivity fields give rise to head and velocity covariances which are not weakly homogeneous the inhomogeneous fields are obtained from a linear filter of the solution to the problem without stochasticity higher gradient regions induce higher head and velocity variances in the presence of space varying gradients nonlocal effects are most important away from the boundaries lpar center of the domain small local head gradients result in small head and velocity gradients and therefore where observations are made in stagnation regions data should be analyzed taking into account this effect the results may be used to interpret experimental data for columns or data taken where conditions do not fit the average uniform flow assumption and when processes at the boundaries influence the flow and therefore the mixing of contaminants

stochastic analysis of flow in a heterogeneous unsaturated saturated system 1 in this study we develop a stochastic model for transient unsaturated saturated flow in randomly heterogeneous media with the method of moment equations we first derive partial differential equations governing the statistical moments of the flow quantities by perturbation expansions and then implement these equations under general conditions with the method of finite differences the stochastic model developed is applicable to the entire domain of a bounded multidimensional unsaturated saturated system in the presence of random or deterministic recharge and sink source and in the presence of multiscale nonstationary medium features the unsaturated and saturated zones are coupled through the water table whose position is random in randomly heterogeneous porous media the presence of the water table renders the flow moments strongly nonstationary even in the absence of medium nonstationary features this finding is confirmed with monte carlo simulations the resulting first two moments of flow may be used to approximate confidence intervals for the flow quantities in addition the integrated stochastic flow model provides the prerequisite quantities for realistically studying contaminant transport in unsaturated saturated systems with a stochastic approach

a numerical verification method for the existence of weak solutions for nonlinear boundary value problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for nonlinear two point boundary value problems bvp we formulate the problem as a fixed point of a newton like operator and present a verification algorithm by computer based on sadovskii s fixed point theorem in order to compute the newton type iterative sequences in infinite dimensional space two concepts the rounding and the rounding error are introduced also some numerical examples are illustrated we emphasize that since the present method uses the weak formulation of bvp and the finite element approximations instead of the existing validation techniques it is aslo applicable in principle to the bvp for partial differential equations as it stands

stochastic modeling of groundwater flow by unconditional and conditional probabilities 1 conditional simulation and the direct problem mathematical modeling of groundwater flow in heterogeneous porous formations of large extent is investigated the formation properties hydraulic conductivity transmissivity as well as flow variables head specific discharge solute concentration are regarded as random variables subjected to uncertainty the main aim of the study is to analyze the influence of conditional probability of input variables upon the statistical structure of the dependent variables the unconditional probability density functions are supposed to be stationary multivariate normal while conditioning accounts for the measured values at a few points of the formation two problems of groundwater flow are discussed in part 1 conditional simulation and the direct problem for steady flow analytical results are obtained by using perturbation approximations average uniform head gradient flows as well as recharge and flow to wells are discussed in unconditional modeling the input variable the conductivity or transmissivity log is regarded as stationary and is represented by its constant mean and variogram the ensemble of formations on which statistical calculations are carried out represents all aquifers with same probability density distributions in conditional modeling the measured values at a few points are kept fixed and uncertainty prevails only at other points consequently statistical computations are performed for the subensemble of aquifers which preserve the measured values and as a result both input and output variables are nonstationary the main effect of conditioning is to reduce the variance i e the uncertainty of these variables this effect is particularly important in the case of flow toward wells application of the modeling method to field problems is outlined

a note on higher order corrections of the head covariances in steady aquifer flow average uniform flow takes place in a heterogeneous aquifer of infinite extent the input to the problem is the hydraulic conductivity which is regarded as a random space function that is lognormal stationary and statistically isotropic the output is the water head field which is a random function satisfying the equation of steady flow the first order approximation of the head in an asymptotic expansion for small log conductivity variance is a normal function characterized completely by the head log conductivity cross covariance and the head covariance or variogram these covariances are proportional to the log conductivity variance by using spectral methods second order corrections of the head covariances proportional to the log conductivity variance squared are derived explicitly detailed calculations are carried out for an exponential log conductivity covariance the main finding of the note is that the first order approximation is very robust and even for a log conductivity variance equal to unity the second order correction of the head variances is smaller than 10 of the first order approximation

stochastic subsurface hydrology from theory to applications research on stochastic analysis of subsurface flow has developed rapidly in the last decade but applications of this approach have been very limited the purpose of this paper is to illustrate how currently available techniques and results can be used to answer important questions about the large scale behavior of naturally heterogeneous aquifers perturbation based spectral theory which presumes local statistical homogeneity provides generic theoretical results for the head variance effective conductivity tensor and macrodispersivity tensor in a field these results emphasize the key role of the variance and spatial correlation scales of the log hydraulic conductivity field field information of variances and correlation scales of natural materials is summarized the validity of some of the generic stochastic results is evaluated through comparisons with monte carlo simulations and field observations a specific field application example is developed to illustrate how the stochastic results are used to estimate large scale parameters and determine the reliability of three dimensional numerical simulations using typical log conductivity covariance parameters the effective hydraulic conductivity tensor and the macrodispersivity tensor are estimated the calculated head variance based on the simulated mean hydraulic gradient is used as a measure of the adequacy of the calculation of the steady state flow model discussion emphasizes limitations and extensions of this approach and ongoing field evaluations of the results

stochastic identification of transmissivity and effective recharge in steady groundwater flow 2 case study the study is a continuation and extension of a previous work dagan 1985a whose aim was to identify the values of the log transmissivity y for steady flow the common basic assumptions are that y is a normal and stationary random space function the aquifer is unbounded and a first order approximation of the flow equation is adopted the expected value of the water head h as well as the y unconditional autocovariance are supposed to have analytical expressions which depend on a parameters vector the proposed solution of the inverse problem consists of identifying with the aid of the model and of the measurements of y and h and subsequently computing the statistical moments of y conditioned on the same data the additional features of the present study are 1 incorporation of a constant but random effective recharge and its identification and 2 accounting for the fact that estimation is associated with some uncertainty whereas before was assumed to be identified with certainty analytical expressions are derived for the y and h covariances for an exponential autocovariance of y paper 2 rubin and dagan this issue of the study illustrates the applications of the method to a real life case

meshless projection iterative analysis of signorini problems using a boundary element free method this paper presents a meshless method based on coupling a boundary element free method befm with a new projection iterative scheme for signorini problems using the new iterative scheme the nonlinear complementary signorini boundary conditions are transformed into neumann boundary conditions then mixed elliptic problems are formed and are solved by the befm in this method only a nodal data structure on the boundary is required and the coefficient matrices of the discrete system equations only have to be computed once in the whole iterative process convergence of the iterative scheme is discussed theoretically numerical examples involving beach percolation and an electropaint process are given to illustrate the convergence rate and high computational efficiency of the method

projection and self adaptive projection methods for the signorini problem with the bem abstract we propose two new projection methods for solving the signorini problem and study the relationship between them since the signorini boundary condition is equivalent to a projection fixed point problem the methods formulate the signorini boundary condition into a sequence of robin boundary conditions and only require solving a variational problem with simple boundary conditions at each iterative step the convergence speed of the first method depends on the parameter heavily and it is difficult to choose a proper for individual problems to improve the efficiency of the method we propose an adaptive projection method which adjusts the parameter automatically per iteration we establish the convergence analysis of the methods because the signorini boundary condition is given in an iterative form by the potential and its derivative on the boundary of the domain we can easily provide the boundary element approximation of the problem finally we present some numerical simulations which illustrate the performance of the methods