a mixed mol tmol for the numerical solution of the 2d richards equation in layered soils abstract water infiltration into layered soils is studied considering a two dimensional spatial domain the focus is on the treatment of discontinuity at the intersection of non overlapping soils the novelty of this paper is based on a mixed mol tmol which merges desirable features of both method of lines mol and transversal method of lines tmol such a numerical approach allows us the numerical treatment of the solution at the discontinuous interfaces by means of filippov theory for dynamical systems numerical simulations based on implicit and semi implicit schemes of low accuracy are provided for validating this approach

generalized aor method for solving a class of generalized saddle point problems abstract in this paper by constructing appropriate splitting a generalized aor gaor method for solving a class of generalized saddle point problems has been presented and the convergence of the method is studied the numerical experiments results show the efficiency and robustness of the new method

second order expressions for velocity moments in two and three dimensional statistically anisotropic media second order expressions are derived for the mean and covariance of steady state seepage velocity under mean uniform flows in infinite two and three dimensional domains the order of approximation is defined in terms of the variance 2 of a statistically homogeneous and anisotropic natural log hydraulic conductivity field y with a gaussian spatial autocorrelation function results show that second order mean velocity either exceeds or is close to its first order counterpart depending on anisotropy head fluctuations of order larger than affect second order velocity moments to the same extent as do head fluctuations of order in virtually all cases hence neglecting the former renders the results nonasymptotic velocity variances are generally larger when approximated consistently to second than to first order the ratio between second and first order variance approximations is larger in three than in two dimensions larger for transverse than for longitudinal velocity and increases with 2 anisotropy has a significant effect on second order velocity variance second order effects have the greatest influence on longitudinal velocity variance at extreme anisotropy ratios and on transverse velocity variance in isotropic domains

a structure preserving split finite element discretization of the split wave equations abstract we introduce a new finite element fe discretization framework applicable for covariant split equations the introduction of additional differential forms df that form pairs with the original ones permits the splitting of the equations into topological momentum and continuity equations and metric dependent closure equations that apply the hodge star operator our discretization framework conserves this geometrical structure and provides for all dfs proper fe spaces such that the differential operators here gradient and divergence hold in strong form we introduce lowest possible order discretizations of the split 1d wave equations in which the discrete momentum and continuity equations follow by trivial projections onto piecewise constant fe spaces omitting partial integrations approximating the hodge star by nontrivial galerkin projections gp the two discrete metric equations follow by projections onto either the piecewise constant gp0 or piecewise linear gp1 space out of the four possible realizations our framework gives us three schemes with significantly different behavior the split scheme using twice gp1 is unstable and shares the dispersion relation with the p1 p1 fe scheme that approximates both variables by piecewise linear spaces p1 the split schemes that apply a mixture of gp1 and gp0 share the dispersion relation with the stable p1 p0 fe scheme that applies piecewise linear and piecewise constant p0 spaces however the split schemes exhibit second order convergence for both quantities of interest for the split scheme applying twice gp0 we are not aware of a corresponding standard formulation to compare with though it does not provide a satisfactory approximation of the dispersion relation as short waves are propagated much too fast the discovery of the new scheme illustrates the potential of our discretization framework as a toolbox to study and find fe schemes by new combinations of fe spaces

the mimetic finite difference method for the 3d magnetostatic field problems on polyhedral meshes we extend the mimetic finite difference mfd method to the numerical treatment of magnetostatic fields problems in mixed div curl form for the divergence free magnetic vector potential to accomplish this task we introduce three sets of degrees of freedom that are attached to the vertices the edges and the faces of the mesh and two discrete operators mimicking the curl and the gradient operator of the differential setting then we present the construction of two suitable quadrature rules for the numerical discretization of the domain integrals of the div curl variational formulation of the magnetostatic equations this construction is based on an algebraic consistency condition that generalizes the usual construction of the inner products of the mfd method we also discuss the linear algebraic form of the resulting mfd scheme its practical implementation and discuss existence and uniqueness of the numerical solution by generalizing the concept of logically rectangular or cubic meshes by hyman and shashkov to the case of unstructured polyhedral meshes the accuracy of the method is illustrated by solving numerically a set of academic problems and a realistic engineering problem

a generalized shift splitting preconditioner for singular saddle point problems recently some authors cao et al 2014 chen and ma 2015 salkuyeh et al 2014 discussed the generalized shift splitting preconditioner for nonsingular saddle point problems in this paper we further study the generalized shift splitting preconditioner for solving singular saddle point problems with symmetric positive definite 1 1 block theoretical analysis shows that the generalized shift splitting iteration method is unconditionally semi convergent numerical experiments are given to illustrate the efficiency of the proposed preconditioner with appropriate parameters

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

higher order effects on flow and transport in randomly heterogeneous porous media advective solute transport in nonuniform geologic media is generally nonlocal and non fickian neuman 1993 cushman and ginn 1993 in statistically homogeneous log conductivity fields under uniform mean flow the transport is expected to become asymptotically local and fickian at late time during the earlier preasymptotic regime macrodispersivity a measure of the rate at which a plume spreads is expected to vary with solute residence time a first order linear in the natural log hydraulic conductivity variance 2 analysis of this variation has been performed by dagan 1984 1987 1988 he found that when local dispersion is neglected the longitudinal macrodispersivity increases monotonically from zero toward a constant asymptote however the transverse macrodispersivity first increases from zero to a peak value then decreases monotonically toward zero the first order asymptotic analyses by winter 1982 gelhar and axness 1983 and winter et al 1984 also yield zero transverse macrodispersivity when local dispersion is disregarded

on the augmented system approach to sparse least squares problems we study the augmented system approach for the solution of sparse linear least squares problems it is well known that this method has better numerical properties than the method based on the normal equations we use recent work by arioli et al 1988 to introduce error bounds and estimates for the components of the solution of the augmented system in particular we find that using iterative refinement we obtain a very robust algorithm and our estimates of the error are accurate and cheap to compute the final error and all our error estimates are much better than the classical or skeel s error analysis 1979 indicates moreover we prove that our error estimates are independent of the row scaling of the augmented system and we analyze the influence of the bjorck scaling 1967 on these estimates we illustrate this with runs both on large scale practical problems and contrived examples comparing the numerical behaviour of the augmented systems approach with a code using the normal equations these experiments show that while the augmented system approach with iterative refinement can sometimes be less efficient than the normal equations approach it is comparable or better when the least squares matrix has a full row and is in any case much more stable and robust

convergence analysis of the high order mimetic finite difference method we prove second order convergence of the conservative variable and its flux in the high order mfd method the convergence results are proved for unstructured polyhedral meshes and full tensor diffusion coefficients for the case of non constant coefficients we also develop a new family of high order mfd methods theoretical result are confirmed through numerical experiments

symplectic hamiltonian hdg methods for wave propagation phenomena abstract we devise the first symplectic hamiltonian hybridizable discontinuous galerkin hdg methods for the acoustic wave equation we discretize in space by using a hamiltonian hdg scheme that is an hdg method which preserves the hamiltonian structure of the wave equation and in time by using symplectic diagonally implicit and explicit partitioned runge kutta methods the fundamental feature of the resulting scheme is that the conservation of a discrete energy which is nothing but a discrete version of the original hamiltonian is guaranteed we present numerical experiments which indicate that the method achieves optimal approximations of order k 1 in the l 2 norm when polynomials of degree k 0 and runge kutta time marching methods of order k 1 are used in addition by means of post processing techniques and by increasing the order of the runge kutta method to k 2 we obtain superconvergent approximations of order k 2 in the l 2 norm for the displacement and the velocity we also present numerical examples that corroborate that the methods conserve energy and that they compare favorably with dissipative hdg schemes of similar accuracy properties for long time simulations

effective hydraulic conductivity of bounded strongly heterogeneous porous media we develop analytical expressions for the effective hydraulic conductivity ke of a three dimensional heterogeneous porous medium in the presence of randomly prescribed head and flux boundaries the log hydraulic conductivity y forms a gaussian statistically homogeneous and anisotropic random field with an exponential autocovariance by effective hydraulic conductivity of a finite volume in such a field we mean the ensemble mean expected value of all random equivalent conductivities that one could associate with a similar volume under uniform mean flow we start by deriving a first order approximation of an exact expression developed in 1993 by neuman and orr we then generalize this to strongly heterogeneous media by invoking the landau lifshitz conjecture upon evaluating our expressions we find that ke decreases rapidly from the arithmetic mean ka toward an asymptotic value as distance between the prescribed head boundaries increases from zero to about eight integral scales of y the more heterogeneous is the medium the larger is ke relative to its asymptote at any given separation distance our theory compares well with published results of spatially power averaged expressions and with a first order expression developed intuitively by kitanidis in 1990

virtual element methods for hyperbolic problems on polygonal meshes in the present paper we develop the virtual element method for hyperbolic problems on polygonal meshes considering the linear wave equations as our model problem after presenting the semi discrete scheme we derive the convergence estimates in h 1 semi norm and l 2 norm moreover we develop a theoretical analysis on the stability for the fully discrete problem by comparing the newmark method and the bathe method finally we show the practical behaviour of the proposed method through a large set of numerical tests

eigenvalue estimates of an indefinite block triangular preconditioner for saddle point problems in this paper an indefinite block triangular preconditioner for symmetric saddle point problems is discussed the bounds for all the real eigenvalues of the preconditioned matrix are provided and the bounds for the real and imaginary parts of all the complex eigenvalues of the preconditioned matrix are also provided the corresponding theoretical results of paper by bergamaschi 2012 are improved numerical experiments are reported to confirm our theoretical results

symmetric sor method for augmented systems in this paper we develop symmetric successive overrelaxation ssor method to find solution of augmented systems these systems have appeared in many different applications of scientific computing e g the finite element approximation to solve the navier stokes equation and the constrained optimization the convergence of ssor method are studied finally we apply the new method to solve augmented systems the iteration number of this method to receive the solution is very better than sor like method

on local hermitian and skew hermitian splitting iteration methods for generalized saddle point problems in this paper we first present a local hermitian and skew hermitian splitting lhss iteration method for solving a class of generalized saddle point problems the new method converges to the solution under suitable restrictions on the preconditioning matrix then we give a modified lhss mlhss iteration method and further extend it to the generalized saddle point problems obtaining the so called generalized mlhss gmlhss iteration method numerical experiments for a model navier stokes problem are given and the results show that the new methods outperform the classical uzawa method and the inexact parameterized uzawa method

a relaxed dimensional factorization preconditioner for the incompressible navier stokes equations in this paper we introduce a relaxed dimensional factorization rdf preconditioner for saddle point problems properties of the preconditioned matrix are analyzed and compared with those of the closely related dimensional splitting ds preconditioner recently introduced by benzi and guo 7 numerical results for a variety of finite element discretizations of both steady and unsteady incompressible flow problems indicate very good behavior of the rdf preconditioner with respect to both mesh size and viscosity

variational multiscale stabilized fem formulations for transport equations stochastic advection diffusion and incompressible stochastic navier stokes equations an extension of the deterministic variational multiscale vms approach with algebraic subgrid scale sgs modeling is considered for developing stabilized finite element formulations for the stochastic advection and the incompressible stochastic navier stokes equations the stabilized formulations are numerically implemented using the spectral stochastic formulation of the finite element method ssfem generalized polynomial chaos and karhunen loeve expansion techniques are used for representation of uncertain quantities the proposed stabilized method is then applied to various standard advection diffusion and fluid flow examples with uncertainty in essential boundary conditions comparisons are drawn between the numerical solutions and monte carlo analytical solutions wherever possible

upscaling of permeability of anisotropic heterogeneous formations 1 the general framework in numerical simulations of flow through heterogeneous formations the domain is partitioned into numerical elements the solution requires assigning physical properties to each numerical block the process of transferring information from the scale of actual heterogeneity to that of the numerical elements is known as upscaling we consider steady one phase flow and the only property of interest is the permeability which is regarded as a random space function and the same is true for the upscaled conductivity in this paper we establish the necessary and sufficient conditions to be satisfied by upscaling the necessary conditions are expressed with the aid of the global response of the formation e g the total flux caused by a constant pressure head drop applied on the boundaries the requirement is that the expected value and the variance of the total flux are the same in the actual and upscaled domains these are supplemented by local conditions which stipulate that space averages of flow variables over the numerical blocks in the upscaled domain tend to those in the actual formation when the elements size tends to 0 the result of this analysis is that the upscaled permeability must be such as to ensure the equality in a statistical sense between the space averaged dissipation in the two media by assuming an unbounded domain average uniform flow and stationarity simplified relationships between the moments of the dissipation in the upscaled and actual media are derived it is shown that the effective permeability of the two media is the same the approach is illustrated for stratified formations for which exact composition rules are possible even in this simple case the upscaled permeability is a tensor characterized at second order by two mean components and three covariance functions

conserved quantities of some hamiltonian wave equations after full discretization hamiltonian pdes have some invariant quantities which would be good to conserve with the numerical integration in this paper we concentrate on the nonlinear wave and schrodinger equations under hypotheses of regularity and periodicity we study how a symmetric space discretization makes that the space discretized system also has some invariants or nearly invariants which well approximate the continuous ones we conjecture some facts which would explain the good numerical approximation of them after time integration when using symplectic runge kutta methods or symmetric linear multistep methods for second order systems

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

upscaling of permeability of anisotropic heterogeneous formations 3 applications the general results of indelman and dagan this issue a b are applied to some cases of interest in applications the exact solution of upscaling of stratified formations is used in order to assess the range of applicability of the first order analysis the explicit expressions of the two first moments of the upscaled permeability are derived for media of two dimensional isotropic and three dimensional axisymmetric actual log permeability of gaussian covariance and for parallelepipedic rectangular for two dimensional averaging elements the effects of the actual log permeability covariance and of the anisotropy of partition elements on the upscaled permeability statistics are analyzed the asymptotic behavior of the upscaled permeability for large and small blocks is discussed for general heterogeneity the strong dependence of the small element limit on the properties of actual heterogeneity and on the shape of the elements is illustrated by typical examples in contrast the homogenization of the upscaled media by large blocks depends primarily on the volume area for two dimensions of the partition element relative to the one based on the correlation scale of the log permeability

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

dual homogenization of immiscible steady two phase flows in random porous media 1 we develop a dual conductivity resistivity homogenization for immiscible steady two phase flow in heterogeneous spatially correlated random porous media the objective is to obtain macroscale equations for an equivalent homogeneous medium we assume that the flow is governed locally by the generalized darcy equations two formulations are developed one in terms of capillary pressure pc and the other in terms of saturation s in each case the perturbation approach is used to express average and fluctuation equations and the closure problems are solved using the wiener kinchine spectral theory the macroscale permeability kea and resistivity rea tensors are determined for each fluid a which also yield the relative permeabilities these dual results provide a physically consistent correction of the perturbation based tensors the corrected macroscale tensors are then always positive symmetric and reciprocal to each other we also develop similarly the homogenization of the capillary pressure curve pc x and of its reciprocal x pc where is the wetting fluid content finally these general analytical results are applied to the special case where pc 0 and or 0 which can be interpreted as a mean capillary equilibrium

probabilistic characterization of transport in heterogeneous media abstract the mechanics of transport and flow in a random porous medium are addressed in this paper the hydraulic properties of the porous medium are modeled as spatial random processes the random aspect of the problem is treated by introducing a new dimension along which spectral approximations are implemented thus the hydraulic processes are discretized using the spectral karhunen loeve expansion this expansion represents the random spatial functions as deterministic modes of fluctuation with random amplitudes these amplitudes form a basis in the manifold associated with the random processes the concentrations over the whole domain are also random processes with unknown probabilistic structure these processes are represented using the polynomial chaos basis this is a basis in the functional space described by all second order random variables the deterministic coefficients in this expansion are calculated via a weighted residual procedure with respect to the random measure and the inner product specified by the expectation operator once the spatio temporal variation of the concentrations has been specified in terms of the polynomial chaos expansion individual realizations can be readily computed

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

on generalized symmetric sor method for augmented systems in this paper we establish the generalized symmetric sor method gssor for solving the large sparse augmented systems of linear equations which is the extension of the ssor iteration method the convergence of the gssor method for augmented systems is studied numerical resume shows that this method is effective

nonlocal properties of nonuniform averaged flows in heterogeneous media the properties of nonuniform average potential flows in media of stationary random conductivity are studied the mathematical model of average flow is derived as a system of governing equations to be satisfied by mean velocity and mean head the averaged darcy s law determines the effective conductivity as an integral operator of the convolution type relating the mean velocity to the mean head gradient in a nonlocal way in fourier domain the mean velocity is proportional to the mean head gradient the coefficient of proportionality is referred to as the effective conductivity tensor and is derived by perturbation methods in terms of functionals of the conductivity spatial moments it is shown that the effective conductivity cannot be defined uniquely for potential flows however this nonuniqueness does not affect the spatial distribution of the mean head and velocity analytical expressions of the effective conductivity tensor are derived for two and three dimensional flows and for exponential and gaussian correlations of isotropic conductivity the fundamental solution of the governing equations in effective media mean green function is calculated for the same cases two new asymptotic models of the averaged darcy s law are developed to be applicable to large and small scales of heterogeneity several asymptotic expansions of tue two dimensional and three dimensional mean green functions are derived for exponential and gaussian correlations

analysis of variance method for the equivalent conductivity of rectangular blocks we use an analysis of variance anova decomposition of the log hydraulic conductivity f x ln k x as the sum of an average value f plus main effects and interactions of various orders to evaluate the equivalent conductivity of rectangular blocks in d dimensional space some of the anova components are dealt with exactly the effect of the other components is approximated through calibration to the effective conductivity keff under ergodic conditions our analysis applies to both isotropic and anisotropic lognormal k fields we evaluate the theoretical findings through analytical and numerical comparisons with exact results and previously proposed approximations we find that the anova formula is nearly unbiased although its performance is generally inferior to numerical renormalization advantages of the anova formula are that it is less computationally demanding and explicitly shows the dependence of the block conductivity kb on the various components of fluctuation of f inside the block such dependence is more complex than previously assumed considering the anova formula to be unbiased we derive the bias of the geometric mean of the cardwell parsons bounds for three dimensional rectangular blocks

pumping tests in heterogeneous aquifers an analytical study of what can be obtained from their interpretation using jacob s method interpretation of pumping tests to estimate hydraulic parameter values is typically based on the assumption of aquifer homogeneity the applicability of the traditional methods of interpretation in real aquifers can be questioned since the evaluation of the drawdown curves observed at different locations in a single test may not result in one consistent set of hydraulic parameters thus most hydrogeologists tend to look at estimated transmissivities t as some average property of the medium while estimated storativities s are disregarded in some cases particularly when they are obtained from data measured at the pumping well an analytical study of drawdown under radially convergent flow toward a single point in heterogeneous aquifers shows that large time drawdown values form a straight line on a drawdown versus log time plot jacob s method consists of obtaining estimates for t and s from the slope and intercept of this line we find that even in a heterogeneous field these estimates provide valuable information about the aquifer estimated t values for different observation points tend to converge to a single value which corresponds to the effective t derived under parallel flow conditions estimated storativities however display higher variability but the geometric mean of the sest values can be used as an unbiased estimator of the actual s thus it appears that although jacob s method was originally derived for homogeneous media it can provide valuable information in real aquifers

convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes the stability and convergence properties of the mimetic finite difference method for diffusion type problems on polyhedral meshes are analyzed the optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved

use of power averaging for quantifying the influence of structure organization on permeability upscaling in on lattice networks under mean parallel flow with the local permeability distribution variance s 2 is nonnegligible but remains small it is equal to 0 09 for sparse networks and 0 14 for dense networks representing 4 5 and 7 respectively of the full possible range of w values power averaging is not strictly valid but gives an estimate of upscaling at a few percent here w depends slightly on the local permeability distribution shape beyond its variance but mostly on the morphological network structures most of the morphological control on upscaling for on lattice structures is local and topological and can be explained by the dependence on the average number of neighbor by points effective coordination number within the flowing structure backbone

discrete variational lie group formulation of geometrically exact beam dynamics the goal of this paper is to derive a structure preserving integrator for geometrically exact beam dynamics by using a lie group variational integrator both spatial and temporal discretization are implemented in a geometry preserving manner the resulting scheme preserves both the discrete momentum maps and symplectic structures and exhibits almost perfect energy conservation comparisons with existing numerical schemes are provided and the convergence behavior is analyzed numerically

conservation of energy momentum and actions in numerical discretizations of non linear wave equations for classes of symplectic and symmetric time stepping methods trigonometric integrators and the stormer verlet or leapfrog method applied to spectral semi discretizations of semilinear wave equations in a weakly non linear setting it is shown that energy momentum and all harmonic actions are approximately preserved over long times for the case of interest where the cfl number is not a small parameter such results are outside the reach of standard backward error analysis here they are instead obtained via a modulated fourier expansion in time

a higher order approximation to effective conductivity in media of anisotropic random structure properties of the effective conductivity tensor keff are studied by deriving the second order terms in its expansion in the variance 2 of normally distributed log conductivity it is shown that for media of anisotropic structure the components of the effective conductivity tensor are expressed by a functional of the log conductivity covariance that is it depends on the shape of the correlation function and not only on anisotropy ratios variance 2 and space dimensions however the trace of keff is independent of the log conductivity autocovariance and for a given mean conductivity depends only on 2 the dependence of the effective conductivity on the correlation structure is illustrated for gaussian and exponential autocovariances of log conductivity and for two and three dimensional flows

a modified dimensional split preconditioner for generalized saddle point problems abstract for large and sparse saddle point linear systems arising from 2d linearized navier stokes equations benzi and guo recently studied a dimensional split ds preconditioner appl numer math 61 2011 66 76 by further applying it to generalized saddle point problems in this paper we present a modified dimensional split mds preconditioner this new preconditioner is based on a splitting of the generalized saddle point matrix resulting in an unconditional convergent fixed point iteration the basic iteration is accelerated by a krylov subspace method like restarted gmres the implementation of the mds preconditioner is discussed and a similar case is also analyzed finally numerical experiments of a model navier stokes problem are presented to illustrate the effectiveness of the mds preconditioner

numerical methods for hamiltonian pdes the paper provides an introduction and survey of conservative discretization methods for hamiltonian partial differential equations the emphasis is on variational symplectic and multi symplectic methods the derivation of methods as well as some of their fundamental geometric properties are discussed basic principles are illustrated by means of examples from wave and fluid dynamics

upscaling of permeability of anisotropic heterogeneous formations 2 general structure and small perturbation analysis the general methodology developed in part 1 indelman and dagan this issue of this study is applied to the detailed analysis of upscaled permeability first the structure of the dissipation function for the general conductivity a tensor of stationary random components is examined with the aid of dimensional analysis it is shown that for arbitrary shapes of the numerical elements the upscaled permeability has also this general structure and the numbers of unknown parameters and equations match this result suggests that the upscaling problem has an unique solution in the particular case of scalar permeability of isotropic covariance of the actual formation it is shown that a similar upscaled permeability is possible only for spherical circular numerical elements otherwise the upscaled permeability has to be a tensor of anisotropic covariance if the actual formation has a scalar permeability of axisymmetric covariance upscaling preserves the last property only for axisymmetric partition elements i e for a sphere cylinder and ellipsoid explicit expressions for the first moments of the upscaled permeability mean covariance are derived at first order in the log permeability variance the detailed computations for a scalar permeability of axisymmetric covariance and for axisymmetric numerical elements lead to simple results the upscaled permeability expected values are components of an axisymmetric tensor whereas the fluctuations are determined by a scalar random space function of anisotropic covariance the general case of upscaling by a first order approximation is examined in appendix b these results will be applied to a few particular cases in part 3 indelman this issue

fourth order approximation of fluid flow through rough walled rock fracture 1 a stochastic approach for estimating the permeability between rough walled rock fracture surfaces is presented in the present analysis the reynolds lubrication equation which is adopted for analyzing fluid flow through a rough surface of rock fracture is solved as a stochastic partial differential equation using perturbative expansion method the results are compared with the numerical deterministic analysis for randomly generated aperture distributions the analytical results are in a good agreement with numerical results obtained by the monte carlo approach through the present analysis a fourth order accurate approximation to the effective permeability of a single joint is given in terms of the mean the standard deviation and the spatial correlation so that the effect of higher order terms is clarified more accurately than the classical second order approximation

on parameterized inexact uzawa methods for generalized saddle point problems abstract for the large sparse saddle point problems bai et al recently studied a class of parameterized inexact uzawa piu methods z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 for a special case that the 1 1 block is solved exactly they determined the convergence domain and computed the optimal iteration parameters and the corresponding optimal convergence factor for the induced method in this paper we develop these methods to the large sparse generalized saddle point problems for the obtained parameterized inexact uzawa method we prove its convergence under suitable restrictions on the iteration parameters in particular we determine its quasi optimal iteration parameters and the corresponding quasi optimal convergence factor for the saddle point problems furthermore this piu method is generalized to obtain a framework of accelerated variants of the parameterized inexact uzawa methods for solving both symmetric and nonsymmetric generalized saddle point problems by using the techniques of vector extrapolation matrix relaxation and inexact iteration

the spectral properties of the hermitian and skew hermitian splitting preconditioner for generalized saddle point problems in this paper we consider the hermitian and skew hermitian splitting hss preconditioner for generalized saddle point problems with nonzero 2 2 blocks the spectral property of the preconditioned matrix is studied in detail under certain conditions all eigenvalues of the preconditioned matrix with the original system being non hermitian will form two tight clusters one is near 0 0 and the other is near 2 0 as the iteration parameter approaches to zero from above so do all eigenvalues of the preconditioned matrix with the original system being hermitian numerical experiments are given to demonstrate the results

a new discretization methodology for diffusion problems on generalized polyhedral meshes we develop a family of inexpensive discretization schemes for diffusion problems on generalized polyhedral meshes with elements having non planar faces the material properties are described by a full tensor we also prove superconvergence for the scalar pressure variable under very general assumptions the theoretical results are confirmed with numerical experiments in the practically important case of logically cubic meshes with randomly perturbed nodes the mixed finite element with the lowest order raviart thomas elements does not converge while the proposed mimetic method has the optimal convergence rate 2007 elsevier b v all rights reserved

symplectic wavelet collocation method for hamiltonian wave equations this paper introduces a novel symplectic wavelet collocation method for solving nonlinear hamiltonian wave equations based on the autocorrelation functions of daubechies compactly supported scaling functions collocation method is conducted for the spatial discretization which leads to a finite dimensional hamiltonian system then appropriate symplectic scheme is employed for the integration of the hamiltonian system under the hypothesis of periodicity the properties of the resulted space differentiation matrix are analyzed in detail conservation of energy and momentum is also investigated various numerical experiments show the effectiveness of the proposed method

on generalized successive overrelaxation methods for augmented linear systems for the augmented system of linear equations golub wu and yuan recently studied an sor like method bit 41 2001 71 85 by further accelerating it with another parameter in this paper we present a generalized sor gsor method for the augmented linear system we prove its convergence under suitable restrictions on the iteration parameters and determine its optimal iteration parameters and the corresponding optimal convergence factor theoretical analyses show that the gsor method has faster asymptotic convergence rate than the sor like method also numerical results show that the gsor method is more effective than the sor like method when they are applied to solve the augmented linear system this gsor method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation matrix relaxation and inexact iteration besides we also demonstrate a complete version about the convergence theory of the sor like method

mimetic finite difference methods for hamiltonian wave equations in 2d abstract in this paper we consider the numerical solution of the hamiltonian wave equation in two spatial dimensions we construct a two step procedure in which we first discretize the space by the mimetic finite difference mfd method and then we employ a standard symplectic scheme to integrate the semi discrete hamiltonian system derived the main characteristic of the mfd methods when applied to stationary problems is to mimic important properties of the continuous system this approach yields a full numerical procedure suitable to integrate hamiltonian problems a complete theoretical analysis of the method and some numerical simulations are developed in the paper

assessment of the predictive capabilities of stochastic theories in a three dimensional laboratory test aquifer effective hydraulic conductivity and temporal moments of breakthrough curves 1 conservative and sorptive tracer experiments were conducted in a highly heterogeneous lnk2 1 79 and anisotropic h v 3 52 three dimensional test aquifer under well controlled laboratory conditions to evaluate the effective conductivity and temporal moments predicted by stochastic theories the spatial distribution of lnk in the test aquifer conformed to a statistically homogeneous system thus allowing evaluation of results from stochastic theories for stationary random fields effective hydraulic conductivity keff in the mean flow direction for the test aquifer was compared with different stochastic theoretical expressions breakthrough curves btcs at individual deep penetrating observation wells and averaged breakthrough curves at control planes were examined the mean arrival time and the travel time variance estimated using averaged btcs from many observation wells at the same control plane for bromide and lithium were slightly overestimated by stochastic theories yet most of the experimental data ranged within the confidence interval rendered by the uncertainty in the statistical properties of the test aquifer a highlight of our experimental approach is that the continuous sampling procedure also permitted the evaluation of higher order temporal moments that were analyzed to study the asymmetry and peakedness of btcs by means of the coefficients of skewness and kurtosis respectively small perturbation analytical solutions for the coefficient of skewness and kurtosis were provided it is seen that although stochastic theories based upon small perturbations provide reasonably good estimates of the coefficients of skewness and kurtosis associated with averaged btcs at control planes in general they can largely underestimate the peakedness and tailing of btcs observed at individual deep penetrating observation wells

geometric variational discretization of continuum theories this study derives geometric variational discretization of continuum theories arising in fluid dynamics magnetohydrodynamics mhd and the dynamics of complex fluids a central role in these discretizations is played by the geometric formulation of fluid dynamics which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume preserving diffeomorphisms of the fluid domain inspired by this framework we construct a finite dimensional approximation to the diffeomorphism group and its lie algebra thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group the extension to mhd and complex fluid flow is then made through an appeal to the theory of euler poincare systems with advection which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters upon deriving a family of structured integrators for these systems we test their performance via a numerical implementation of the update schemes on a cartesian grid among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries automatic satisfaction of solenoidal constraints on vector fields good long term energy behavior robustness with respect to the spatial and temporal resolution of the discretization and applicability to irregular meshes

analysis of the inexact uzawa algorithm for saddle point problems in this paper we consider the so called inexact uzawa algorithm for iteratively solving linear block saddle point problems such saddle point problems arise for example in finite element and finite difference discretizations of stokes equations the equations of elasticity and mixed finite element discretization of second order problems we consider both the linear and nonlinear variants of the inexact uzawa iteration we show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block in the case of nonlinear iteration the inexact uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear uzawa algorithm applications to the stokes equations and mixed finite element discretization of second order elliptic problems are discussed and finally the results of numerical experiments involving the algorithms are presented

mimetic finite difference method for the stokes problem on polygonal meshes various approaches to extend finite element methods to non traditional elements general polygons pyramids polyhedra etc have been developed over the last decade the construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis the mimetic finite difference mfd method works on general polygonal meshes and has many similarities with low order finite element methods both schemes try to preserve the fundamental properties of the underlying physical and mathematical models the essential difference between the two schemes is that the mfd method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices since no extension of basis functions inside the mesh elements is required practical implementation of the mfd method is simple for polygonal meshes that may include degenerate and non convex elements in this article we present a new mfd method for the stokes problem on arbitrary polygonal meshes and analyze its stability the method is developed for the general case of tensor coefficients which allows us to apply it to a linear elasticity problem as well numerical experiments show for the velocity variable second order convergence in a discrete l 2 norm and first order convergence in a discrete h 1 norm for the pressure variable first order convergence is shown in the l 2 norm

on hermitian and skew hermitian splitting iteration methods for the linear matrix equation axb c in this paper a hermitian and skew hermitian splitting hss iteration method for solving the linear matrix equation axb c is proposed this method is formed by extending the corresponding hss iteration method for solving ax b the analysis shows that the hss iteration method will converge under certain assumptions moreover the optimal parameter of this iteration method is presented numerical results show that this new method is more efficient and robust than existing methods

structured preconditioners for nonsingular matrices of block two by two structures for the large sparse block two by two real nonsingular matrices we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations for the specific versions such as modified block jacobi type modified block gauss seidel type and modified block unsymmetric symmetric gauss seidel type preconditioners we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices also we show that when these structured preconditioners are employed to precondition the krylov subspace methods such as gmres and restarted gmres fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two by two coefficient matrices in particular these structured preconditioners can lead to efficient and high quality preconditioning matrices for some typical matrices from the real world applications

on positive definite and skew hermitian splitting iteration methods for continuous sylvester equation ax xb c in this paper we present a positive definite and skew hermitian splitting pss iteration method for continuous sylvester equations ax xb c with positive definite semi definite matrices the theoretical analysis shows that the pss iteration method will converge unconditionally and the optimal parameter of the new method is presented moreover to reduce the computing cost an inexact variant of the pss iteration method ipss and the analysis of its convergence property in detail have been established numerical results show that this new method and its inexact invariant are efficient and robust solvers for this class of continuous sylvester equations

shift splitting preconditioners for saddle point problems in this paper we first present a shift splitting preconditioner for saddle point problems the preconditioner is based on a shift splitting of the saddle point matrix resulting in an unconditional convergent fixed point iteration based on the idea of the splitting we further propose a local shift splitting preconditioner some properties of the local shift splitting preconditioned matrix are studied these preconditioners extend those studied by bai yin and su for solving non hermitian positive definite linear systems bai et al 2006 finally numerical experiments of a model stokes problem are presented to show the effectiveness of the proposed preconditioners

new preconditioners for saddle point problems we present new preconditioners based on matrix splittings for the saddle point problems the spectral property of one of the preconditioned matrix is studied in detail numerical examples are used to illustrate the efficiency of the new preconditioners

optimal parameter in hermitian and skew hermitian splitting method for certain two by two block matrices the optimal parameter of the hermitian skew hermitian splitting hss iteration method for a real two by two linear system is obtained the result is used to determine the optimal parameters for linear systems associated with certain two by two block matrices and to estimate the optimal parameters of the hss iteration method for linear systems with n by n real coefficient matrices numerical examples are given to illustrate the results

effective hydraulic conductivity of a randomly heterogeneous porous medium 1 the effective hydraulic conductivity of a randomly heterogeneous isotropic porous medium has been calculated by means of a new perturbative technique which differs greatly from that commonly used in the approach developed we construct the perturbation series for a seepage velocity and then utilize darcy s law for calculating pressure on the basis of the analysis of high order approximations it is shown that in the general case the effective conductivity in the large scale limit does depend on the form of the correlation function thus the widespread landau lifshitz matheron formula which operates on the conjecture that the first two terms of the perturbation series are those for the taylor series expansion of an exponential function proves to be invalid the calculations were carried out in a space of arbitrary dimension the dependence of the effective conductivity on the variance has been obtained up to the third order approximation inclusive in terms of a conductivity logarithm for convenience and simplification of series analysis the feynman diagrammatic technique was developed and utilized in a one dimensional 1d case our result in the large scale limit when a heterogeneity scale is small enough with respect to the characteristic scale of groundwater flow gives an exact formula for the effective conductivity

on second order correction to the effective conductivity of two dimensional anisotropic media it has been shown recently that the effective conductivity keff of anisotropic media depends on the shape of correlations indelman and abramovich 1994 this property results from the analysis of the second order expression of keff in the log conductivity variance 2 which is derived in terms of quadratures and calculated by numerical integration for two and three dimensional flows and for exponential and gaussian covariances the present paper is a continuation of this study and aims at deriving an analytical closed form expression of keff at order 4 for the case of two dimensional flow in lognormal anisotropic media of gaussian log conductivity correlation

spectral properties and conservation laws in mimetic finite difference methods for pdes the mimetic finite difference mfd methods for pdes mimic crucial properties of mathematical systems duality and self adjointness of differential operators conservation laws and properties of the solution on general polytopal meshes in this article the structure and the spectral properties of the linear systems derived by the spatial discretization of diffusion problem are analysed in addition the numerical approximation of parabolic equations is discussed where the mfd approach is used in the space discretization while implicit method and explicit runge kutta chebyshev schemes are used in time discretization moreover we will show how the numerical solution preserves certain conservation laws of the theoretical solution

preconditioned conjugate gradient method for generalized least squares problems abstract a variant of the preconditioned conjugate gradient method to solve generalized least squares problems is presented if the problem is min ax b t w 1 ax b with a r m n and w r m m symmetric and positive definite the method needs only a preconditioner a 1 r n n but not the inverse of matrix w or of any of its submatrices freund s comparison result for regular least squares problems is extended to generalized least squares problems an error bound is also given

prediction of steady state flow in nonuniform geologic media by conditional moments exact nonlocal formalism effective conductivities and weak approximation we consider the effect of measuring randomly varying local hydraulic conductivities k x on one s ability to predict steady state flow within a bounded domain driven by random source and boundary functions more precisely we consider the prediction of local hydraulic head h x and darcy flux q x by means of their unbiased ensemble moments h x and q x conditioned on measurements of k x these predictors satisfy a deterministic flow equation in which q x x h x r x where x is a relatively smooth unbiased estimate of k x and r x is a residual flux we derive a compact integral expression for r x which is rigorously valid for a broad class of k x fields including fractals it demonstrates that q x is nonlocal and non darcian so that an effective hydraulic conductivity does not generally exist we show analytically that under uniform mean flow the effective conductivity may be a scalar a symmetric or a nonsymmetric tensor or a set of directional scalars which do not form a tensor we demonstrate numerically that in two dimensional mean radial flow it may increase from the harmonic mean of k x near interior and boundary sources to the geometric mean far from such sources for cases where r x can neither be expressed nor approximated by a local expression we propose a weak integral approximation closure which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of k x measurements increase the nonlocal deterministic flow equation can be solved numerically by standard methods our theory shows clearly how the scale of grid discretization should relate to the scale quantity and quality of available data after providing explicit approximations for the second moments of head and flux prediction errors we conclude by discussing practical methods to compute x from noisy measurements of k x and to calculate required second moments of the associated estimation errors when k x is lognormal

equivalent projectors for virtual element methods in the original virtual element space with degree of accuracy k projector operators in the h 1 seminorm onto polynomials of degree k can be easily computed on the other hand projections in the l 2 norm are available only on polynomials of degree k 2 directly from the degrees of freedom here we present a variant of the virtual element method that allows the exact computations of the l 2 projections on all polynomials of degree k the interest of this construction is illustrated with some simple examples including the construction of three dimensional virtual elements the treatment of lower order terms the treatment of the right hand side and the l 2 error estimates

uzawa low method and preconditioned uzawa low method for three order block saddle point problem in this paper uzawa low method for three order block saddle point problem is presented and the corresponding convergence conditions are established furthermore by introducing a preconditioner we propose centered preconditioned uzawa low method cpu low method and obtain its convergence conditions as well experimental results show that the two proposed methods are more efficient than uzawa low method for the original saddle point problem and the cpu low method is superior to the uzawa low method for three order block saddle point problem

scales of fluctuation and the propagation of uncertainty in random porous media the mechanics of transport and flow in a random porous medium are addressed in this paper the hydraulic properties of the porous medium are modeled as spatial random processes the heterogeneity of the medium is modeled as a superposition of scales of heterogeneity that are statistically uncorrelated the stochastic process making up a hydraulic property is thus represented as a linear combination of normalized deterministic shapes multiplied by a set of uncorrelated random variables the karhunen loeve expansion is used to formalize this representation the uncertainty contained in each of these scales is then propagated through the system using the governing differential equation as a constraint the solution process will thus consist of the original scales used to represent the porous medium as well as various interaction between these scales the objective of this paper is to develop a procedure for quantifying the uncertainty in the field variables using the scales of fluctuation of the porous medium as the measuring stick

sympletic finite difference approximations of the nonlinear klein gordon equation we analyze three finite difference approximations of the nonlinear klein gordon equation and show that they are directly related to symplectic mappings two of the schemes the perring skyrme and ablowitz kruskal ladik are long established and the third is a new higher order accurate scheme we test the schemes on traveling wave and periodic breather problems over long time intervals and compare their accuracy and computational costs with those of symplectic and nonsymplectic method of lines approximations and a nonsymplectic energy conserving method

new choices of preconditioning matrices for generalized inexact parameterized iterative methods for large sparse saddle point problems chen and jiang recently studied a class of generalized inexact parameterized iterative methods see f chen y l jiang a generalization of the inexact parameterized uzawa methods for saddle point problems appl math comput 206 2008 765 771 in this paper the methods are modified and some choices of preconditioning matrices are given these preconditioning matrices have advantages in solving large sparse linear system numerical experiments of a model stokes problem are presented

high resolution monte carlo simulation of flow and conservative transport in heterogeneous porous media 1 methodology and flow results in this the first of two papers concerned with the use of numerical simulation to examine flow and transport parameters in heterogeneous porous media via monte carlo methods various aspects of the modelling effort are examined in particular the need to save on core memory causes one to use only specific realizations that have certain initial characteristics in effect these transport simulations are conditioned by these characteristics also the need to independently estimate length scales for the generated fields is discussed the statistical uniformity of the flow field is investigated by plotting the variance of the seepage velocity for vector components in the x y and z directions finally specific features of the velocity field itself are illuminated in this first paper in particular these data give one the opportunity to investigate the effective hydraulic conductivity in a flow field which is approximately statistically uniform comparisons are made with first and second order perturbation analyses the mean cloud velocity is examined to ascertain whether it is identical to the mean seepage velocity of the model finally the variance in the cloud centroid velocity is examined for the effect of source size and differing strengths of local transverse dispersion

a note on spectrum analysis of augmentation block preconditioned generalized saddle point matrices in this note we consider a set of augmentation block preconditioners for solving generalized saddle point systems whose coefficient matrices have singular 1 1 blocks results concerning the eigenvalue distribution and forms of the eigenvectors of the augmentation block preconditioned generalized saddle point matrix and its minimal polynomial are given and an optimal augmentation block preconditioner of the set is derived these results extend previous ones in the literature numerical experiments that show the very effective performance of the optimal augmentation block preconditioner are reported

effective medium approximation and exact formulae for electrokinetic phenomena in porous media electrokinetic phenomena in porous media are studied by application of the effective medium theory and the theory of duality transformation we deduce new exact relations and analytical formulae for the effective constants of the macroscopic tensor we also prove that the effective tensors obtained by these approaches coincide for 2d problems the obtained results for the electrokinetic processes are closely related to similar results derived for piezoelectric composites because of a common mathematical background