a eulerian lagrangian numerical scheme for the dispersion convection equation using conjugate space time grids abstract a new numerical scheme is proposed for the dispersion convection equation which combines the utility of a fixed grid in eulerian coordinates with the computational power of the lagrangian method convection is formally decoupled from dispersion in a manner which does not leave room for ambiguity the resulting convection problem is solved by the method of characteristics on a grid fixed in space dispersion is handled by finite elements on a separate grid which may but need not coincide wit the former at selected points in spacetime information is projected from one grid to another by local interpolation the conjugate grid method is implemented by linear finite elements in conjunction with piecewise linear interpolation functions and applied to five problems ranging from predominant dispersion to pure convection the results are entirely free of oscillations numerical dispersion exists but can be brought under control either by reducing the spatial increment or by increasing the time step size of the grid used to solve the convection problem contrary to many other methods best results are often obtained when the courant number exceeds 1

on some product type operators from hardy orlicz and bergman orlicz spaces to weighted type spaces abstract the boundedness and compactness of weighted composition operators and a class of integral type operators recently introduced by s stevic from hardy orlicz and weighted bergman orlicz spaces to a class of weighted type spaces on the unit ball of c n are characterized

sobolev hardy and clr inequalities associated with pauli operators in ℝ3 in a previous article the first two authors have proved that the existence of zero modes of pauli operators is a rare phenomenon inter alia it is proved that for b inl3 2 r3 the set of magnetic fields b which do not yield zero modes contains an open dense subset of l3 2 r3 3 here the analysis is taken further and it is shown that sobolev hardy and cwikel lieb rosenbljum clr inequalities hold for pauli operators for all magnetic fields in the aforementioned open dense set

a note on in situ estimates of sorption using push pull tests 1 a sensitivity analysis reveals that large uncertainty exists in estimates of solute retardation factor r made from sorption push pull tests a simplified push pull data interpretation method for estimating instantaneous linear sorption in confined aquifers is developed the new method is less restrictive in its application than the conventional method that uses an approximate analytical solution to the push pull test problem

time stepping along characteristics with incomplete iteration for a galerkin approximation of miscible displacement in porous media miscible displacement of one incompressible fluid by another in a porous medium is modeled by a nonlinear coupled system of two partial differential equations the pressure equation is elliptic while the concentration equation is parabolic but normally convection dominated a sequential backward difference time stepping scheme is defined it approximates the pressure by a standard galerkin procedure and the concentration by a combination of a galerkin method and the method of characteristics optimal convergence rates in l 2 and h 1 are demonstrated for this scheme and for a modified version in which the algebraic equations at each time step are solved approximately by a limited number of preconditioned conjugate gradient iterations time stepping along the characteristics of the hyperbolic part of the concentration equation is shown to result in smaller time truncation errors than those of standard methods numerical results published elsewhere have confirmed that larger time steps are appropriate

some exact solutions of convection diffusion and diffusion equations we examine solutions of the convection diffusion equation during steady radial flow in two and three dimensions with diffusivity proportional to pn with p the peclet number in two dimensions closed form solutions of the laplace transformed equation occur only for n 1 and n 2 and in three dimensions only for n 5 4 and n 2 the inversions for n 1 two dimensions and 5 4 three dimensions cannot be expressed in closed form on the other hand the value n 2 proposed by de gennes 1983 leads to simple exact solutions both in two dimensions and in three in both cases a change of variable gives a one dimensional equation with constant coefficients exact instantaneous and continuous point source solutions for radial flow in two and three dimensions are thus established in general finite difference methods are needed for arbitrary n relevant instantaneous and continuous point source diffusion solutions are developed for radial two and three dimensional systems these represent small time solutions useful for initiating the finite difference solutions for convection diffusion with arbitrary n

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

an investigation of eulerian lagrangian methods for solving heterogeneous advection dominated transport problems numerical simulation of solute transport in heterogeneous porous media is greatly complicated by the large velocity and concentration gradients induced by spatial variations in hydraulic conductivity eulerian lagrangian methods for solving the transport equation can give accurate solutions to heterogeneous problems if their interpolation algorithms are properly selected this paper compares the performance of four eulerian lagrangian solvers that rely on linear quadratic cubic spline and taut spline interpolators in each case a tensor product decomposition is used to reduce the general n dimensional interpolation problem to a sequence of n one dimensional problems comparisons of a set of test problems indicate that the linear and taut spline interpolators are dispersive while the quadratic and cubic spline interpolators are oscillatory the cubic and taut spline interpolators give consistently better accuracy than the more conventional linear and quadratic alternatives simulation experiments in two and three dimensional heterogeneous media indicate that the taut spline interpolator which is applied here for the first time to a solute transport problem is able to yield accurate essentially nonoscillatory solutions for high grid peclet numbers the cubic spline interpolator requires significantly less computational effort to achieve performance comparable to the other methods

a new formula for the analytical solution of the radial dispersion problem a new formula is presented for the analytical solution of the radial dispersion problem which analyzes the dispersive transport of a tracer in radial flow from a recharge well this formula is simpler than a solution presented by previous investigators but the two solutions are shown to be equivalent because the analytical solution consists of an integral of an oscillatory function it is evaluated as an alternating series the terms of which are the integrals over successive half cycles of the integrand an efficient algorithm is used to accelerate convergence of the series the accuracies of previous solutions obtained by finite difference numerical laplace transform and perturbation methods are examined by comparison with the analytical solution numerical values of the solution are tabulated for readers interested in checking the accuracies of solute transport codes

on an integral type operator from logarithmic bloch type and mixed norm spaces to bloch type spaces abstract this paper studies the boundedness and compactness of the following integral type operator recently introduced by this author p g f z 0 1 f t z g t z d t t z b where is a holomorphic self map of the unit ball b in c n and g is a holomorphic function on b such that g 0 0 from logarithmic bloch type and mixed norm spaces to bloch type spaces

validity of the local equilibrium assumption for modeling sorbing solute transport through homogeneous soils sorption processes that occur during reactive solute movement through porous media can be modeled using either an equilibrium or kinetic approach because of the resulting conceptual and mathematical simplification many transport models assume local chemical equilibrium is valid for describing sorption reactions this paper presents quantitative criteria to assess the validity of the local equilibrium assumption for one dimensional steady flow through homogeneous soils a method is described whereby formulas for solute breakthrough curve time moments can be determined without knowledge of the nalytical solution to the mass transport model this method is applied to several commonly used nonequilibrium formulations as well as the standard linear equilibrium model the formulations considered include both the physical nonequilibrium models where the sorption rate is controlled by diffusive solute transfer between mobile and stagnant fluid zones and the chemical nonequilibrium models where the overall sorption rate is governed by the rate of reaction at the soil solution interfaces criteria for local equilibrium to be valid are derived by comparing the time moment formulas for the nonequilibrium and equilibrium models these criteria explicitly show that basic system parameters e g seepage velocity dispersion coefficient distribution coefficient sorption rate boundary conditions have a significant influence on the attainment of local equilibrium

a finite volume eulerian lagrangian localized adjoint method for solution of the contaminant transport equations in two dimensional multiphase flow systems an eulerian lagrangian localized adjoint method ellam numerical solution is developed for the multiphase contaminant transport equations in two dimensions the ellam uses finite volume test functions in the space time domain defined by the characteristics of the hyperbolic part of the governing equation the use of the characteristics results in an approximation that allows large time steps while still maintaining accurate solutions this greatly reduces the computational effort required to find solutions to the advection dispersion equation combination of the finite volume test functions with the conservative form of the governing equation results in a local conservation of mass property for problems with a constant saturation such as that of contaminant transport in the saturated zone this property is highly advantageous as the resulting numerical approximation conserves mass both globally and locally for problems with a variable saturation the requirement of local conservation of mass is too stringent particularly around first type boundaries where oscillations can occur because of errors inherent in the numerical determination of fluid velocities and in the backtracking routine a combined conservative nonconservative ellam is developed with an ellam formulation based on the nonconservative form of the governing equation being applied to subdomains intersecting first type boundaries and a conservative ellam being used for all other subdomains the combined conservative nonconservative ellam is compared to a galerkin finite element scheme and is found to have greatly superior performance requiring far fewer time steps to obtain a solution of equivalent accuracy

an ellam scheme for advection diffusion equations in two dimensions we develop an eulerian lagrangian localized adjoint method ellam to solve two dimensional advection diffusion equations with all combinations of inflow and outflow dirichlet neumann and flux boundary conditions the ellam formalism provides a systematic framework for implementation of general boundary conditions leading to mass conservative numerical schemes the computational advantages of the ellam approximation have been demonstrated for a number of one dimensional transport systems practical implementations of ellam schemes in multiple spatial dimensions that require careful algorithm development are discussed in detail in this paper extensive numerical results are presented to compare the ellam scheme with many widely used numerical methods and to demonstrate the strength of the ellam scheme

toward an oscillation free mass conservative eulerian lagrangian transport model ten numerical schemes to reduce spurious oscillations in transport simulations were implemented and tested in a eulerian lagrangian control volume finite element model the schemes included both new and existing flux corrected transport fct algorithms and nonlinear filters the ability of the methods to eliminate numerical oscillations while preserving mass and minimizing the introduction of numerical diffusion was compared in 2d tests of varying complexity the application of local mass correction algorithms was shown to be vital to avoid the introduction of mass errors by fct while none of the methods emerged as optimal for all cases a new fct method with a local mass correction scheme and a nonlinear filter can be recommended as the best approaches in general the method of choice depends on the problem being solved in particular on the concentration gradients and on the grid resolution

an evaluation of boundary conditions for one dimensional solute transport 1 mathematical development the laplace transform method is employed to obtain the solution to several boundary value problems in which mixing occurs in reservoirs attached to a porous medium flow is assumed to be uniformly one dimensional throughout the porous medium solutions are obtained for both continuous and discontinuous concentrations at the reservoir medium boundary and for resident and flux averaged concentrations a mass balance conducted on each solution shows that neither continuity condition can be proved generally superior an evaluation of the parametric sensitivity of the solutions is also conducted and shows that the distinction between the continuity conditions and the flux and resident concentrations is only important at small peclet numbers the only exception to this is the distinction between flux averaged versus resident continuous concentrations in a finite solution domain in this case the flux concentrations were found to be insensitive to the volume of the downstream reservoir and consequently the flux and resident concentrations can differ substantially for larger reservoirs in consideration of the general behavior of the solutions three conceptual problems were identified which must be investigated by physical modeling 1 determination of the appropriate solution at small peclet number 2 the possibility that dispersive mass flux occurs from the upstream reservoir under conditions of continuous concentration and 3 the use of the flux transformation for continuous concentration in a finite solution domain the physical modeling is discussed in a companion paper

on a product type operator from weighted bergman orlicz space to some weighted type spaces let d z c z 1 be the open unit disk an analytic self map of d and an analytic function on d let d be the differentiation operator and w the weighted composition operator the boundedness and compactness of the product type operator dw from the weighted bergman orlicz space to the bers type space weighted bloch space and weighted zygmund space on d are characterized

push pull tracer tests their information content and use for characterizing non fickian mobile immobile behavior path reversibility and radial symmetry are often assumed in push pull tracer test analysis in reality heterogeneous flow fields mean that both assumptions are idealizations to understand their impact we perform a parametric study which quantifies the scattering effects of ambient flow local scale dispersion and velocity field heterogeneity on push pull breakthrough curves and compares them to the effects of mobile immobile mass transfer mimt processes including sorption and diffusion into secondary porosity we identify specific circumstances in which mimt overwhelmingly determines the breakthrough curve which may then be considered uninformative about drift and local scale dispersion assuming path reversibility we develop a continuous time random walk based interpretation framework which is flow field agnostic and well suited to quantifying mimt adopting this perspective we show that the radial flow assumption is often harmless to the extent that solute paths are reversible the breakthrough curve is uninformative about velocity field heterogeneity our interpretation method determines a mapping function i e subordinator from travel time in the absence of mimt to travel time in its presence a mathematical theory allowing this function to be directly plugged into an existing laplace domain transport model to incorporate mimt is presented and demonstrated algorithms implementing the calibration are presented and applied to interpretation of data from a push pull test performed in a heterogeneous environment a successful four parameter fit is obtained of comparable fidelity to one obtained using a million node 3d numerical model finally we demonstrate analytically and numerically how push pull tests quantifying mimt are sensitive to remobilization but not immobilization kinetics this article is protected by copyright all rights reserved

an explicit unconditionally stable numerical solution of the advection problem in irrotational flow fields 1 a new methodology for the eulerian numerical solution of the advection problem is proposed the methodology is based on the conservation of both the zero and the first order spatial moments inside each element of the computational domain and leads to the solution of several small systems of ordinary differential equations since the systems are solved sequentially one element after the other the method can be classified as explicit the proposed methodology has the following properties 1 it guarantees local and global mass conservation 2 it is unconditionally stable and 3 it applies second order approximation of the concentration and its fluxes inside each element limitation of the procedure to irrotational flow fields for the 2 d and 3 d cases is discussed the results of three 1 d and 2 d literature tests are compared with those obtained using other techniques a new 2 d test with radially symmetric flow is also carried out

source terms in eulerian lagrangian contaminant transport simulation standard eulerian treatment of source terms in eulerian lagrangian numerical simulations results in poor performance at higher courant numbers to regain the customary high accuracy of eulerian lagrangian methods under these conditions a lagrangian treatment of source terms is needed it is also important to include the effects of fluid sources as well as contaminant sources a new lagrangian source formulation is presented which has been implemented in a finite element simulator for contaminant transport in rivers and estuaries test problems demonstrate the high accuracy of the technique under a range of conditions and its applicability to general multi dimensional problems and unstructured grids copyright 2004 john wiley sons ltd

modeling of scale dependent dispersion in hydrogeologic systems an approach for handling scale dependent dispersion using a finite element linear triangular solute transport model was developed dispersivity is allowed to vary temporally as a function of the mean travel distance of the solute from an input source various scale dependent dispersivity relationships were presented including types which approach continuously to a maximum or asymptotic value the finite element model results were verified for a one dimensional case by comparison of the theoretical variances of the solute distribution for a given dispersivity travel distance function with the variances calculated from the finite element model concentration distributions the model was applied successfully to the results of several tracer tests reported in the literature that have exhibited a scale effect for systems that exhibit a constant asymptotic dispersivity at large times or mean travel distances the importance of scale dependent dispersion at early times or short travel distances was shown to be minimal in long term predictions of solute transport

the mast fv fe scheme for the simulation of two dimensional thermohaline processes in variable density saturated porous media a novel methodology for the simulation of 2d thermohaline double diffusive processes driven by heterogeneous temperature and concentration fields in variable density saturated porous media is presented the stream function is used to describe the flow field and it is defined in terms of mass flux the partial differential equations governing system is given by the mass conservation equation of the fluid phase written in terms of the mass based stream function as well as by the advection diffusion transport equations of the contaminant concentration and of the heat the unknown variables are the stream function the contaminant concentration and the temperature the governing equations system is solved using a fractional time step procedure splitting the convective components from the diffusive ones in the case of existing scalar potential of the flow field the convective components are solved using a finite volume marching in space and time mast procedure this solves a sequence of small systems of ordinary differential equations one for each computational cell according to the decreasing value of the scalar potential in the case of variable density groundwater transport problem where a scalar potential of the flow field does not exist a second mast procedure has to be applied to solve again the odes according to the increasing value of a new function called approximated potential the diffusive components are solved using a standard galerkin finite element method the numerical scheme is validated using literature tests

variable rate pumping tests for radially symmetric nonuniform aquifers conventional pumping test methodology is of limited effectiveness for defining the spatial distribution of aquifer properties because of the nonuniqueness of the parameter estimates sensitivity analysis can be used to develop a pumping test procedure that significantly decreases the uncertainty associated with the estimated parameters this approach employs systematic variations in pumpage rates to achieve reductions in parameter uncertainty these reductions are obtained by increasing the sensitivity of drawdown to flow properties while simultaneously constraining the growth in the correlation between the effects of different flow properties on observation well drawdown numerical experiments demonstrate the importance of the magnitude and frequency of the rate variations the spatial and temporal pattern of data collection as well as the dependence of the technique on the total duration of the pumping test significant decreases in parameter uncertainty can be expected in any flow system in which the primary component of flow is in the radial direction this study demonstrates that sensitivity analysis can be an important tool in the development of methodology for the characterization of subsurface properties

generalized product type operators from weighted bergman orlicz spaces to bloch orlicz spaces let d be the open unit disk in the complex plane an analytic self map of d and an analytic function on d let dn be the nth differentiation operator and w the weighted composition operator in this paper the boundedness and compactness of the generalized product type operators dnw and w dn from weighted bergman orlicz spaces to bloch orlicz spaces are characterized

adaptive eulerian lagrangian finite element method for advection dispersion a new adaptive finite element method is proposed for the advection dispersion equation using an eulerian lagrangian formulation the method is based on a decomposition of the concentration field into two parts one advective and one dispersive in a rigorous manner that does not leave room for ambiguity the advective component of steep concentration fronts is tracked forward with the aid of moving particles clustered around each front away from such fronts the advection problem is handled by an efficient modified method of characteristics called single step reverse particle tracking when a front dissipates with time its forward tracking stops automatically and the corresponding cloud of particles is eliminated the dispersion problem is solved by an unconventional lagrangian finite element formulation on a fixed grid which involves only symmetric and diagonal matrices preliminary tests against analytical solutions of one and two dimensional dispersion in a uniform steady state velocity field suggest that the proposed adaptive method can handle the entire range of peclet numbers from 0 to with courant numbers well in excess of 1

mass balance errors when solving the convective form of the transport equation in transient flow problems 1 the original divergence form of the solute transport equation expresses the mass balance of solutes in groundwater the alternative convective form can be obtained by subtracting the flow equation multiplied by concentration from the original form many investigators prefer the latter because it facilitates the use of lagrangian or mixed eulerian lagrangian numerical methods for solution however it performs very poorly in transient contaminant transport problems with large temporal variations in water content we propose an alternative to overcome this difficulty it consists of evaluating the water content at time level k 1 if concentration in the convective term is evaluated at time level k for this reason we have called it reverse time weighting whereas we call an evaluation of the water content at time level k straight time weighting we show that reverse time weighting improves significantly over straight time weighting not only in terms of mass balance errors but also in terms of concentration errors

a finite element formulation of the outlet gradient boundary condition for convective diffusive transport problems a simple finite element formulation of the outlet gradient boundary condition is presented in the general context of convective diffusive transport processes basically the method is based on an upstream evaluation of the dependent variable gradient along open boundaries boundary normal unit vectors and gradient operators are evaluated using covariant bases and metric tensors which allow handling finite elements of mixed dimensions even though the presented method has implications for many fields where diffusion processes are involved discussion and illustrative examples address more particularly the framework of contaminant transport in porous media in which the outlet gradient concentration is classically but wrongly assumed to be zero copyright 2004 john wiley sons ltd

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

a fully mass and volume conserving implementation of a characteristic method for transport problems the characteristics mixed method considers the transport not of a single point or fluid particle but rather the mass in an entire region of fluid this mass is transported along the characteristic curves of the hyperbolic part of the transport equation and the scheme thereby produces very little numerical dispersion conserves mass locally and can use long time steps however since the shape of a characteristic trace back region must be approximated in numerical implementation its volume may be incorrect resulting in inaccurate concentration densities and further inaccurate reaction dynamics we present a simple modification to the characteristics mixed method that conserves both mass and volume of the transported fluid regions our algorithm also handles boundary conditions through a space time change of variables in the trace back routines which allows the boundary to be treated as if it were interior to the domain nearly point sources such as wells present special difficulties since characteristic trace back curves converge in their vicinity we also present techniques that allow one to conservatively implement wells the techniques are illustrated in five numerical examples

a method for implementing dirichlet and third type boundary conditions in ptrw simulations we present an efficient and accurate numerical method for implementing dirichlet boundary conditions in particle tracking random walk ptrw simulations of advective dispersive transport this is a challenge because defining concentrations for dirichlet boundary conditions requires invoking control volumes of some kind which are not natural to the lagrangian based ptrw concept our method performs a galerkin projection of ptrw based particle densities onto control volumes that discretize the boundary thus we obtain concentration values at the boundary condition and can control the particle release rates such that the prescribed boundary values are met this allows for complex shaped internal and external boundaries where concentration values are fixed to prescribed values third type boundary conditions can be addressed as well we test and illustrate the properties and behavior of our method in a series of test cases the results are benchmarked against the conceptually related semianalytical method masst multiple analytical source superposition technique and to those of a finite element method fem while masst is restricted to uniform velocity fields due to the underlying analytical solutions fem is limited in heterogeneous velocity fields at large peclet numbers by numerical dispersion in the feasible discretization range the results demonstrate that our proposed method performs better than the other methods in both regimes

multiblock hybrid grid finite volume method to solve flow in irregular geometries in this work a finite volume based finite element method is suitably developed for solving incompressible flow and heat transfer on collocated hybrid grid topologies the method is generally applicable to arbitrarily shaped elements and orientations and thus challenges the potential to unify many of the different grid topologies into a single formulation the key point in this formulation is the correct estimation of the convective and diffusive fluxes at the cell faces using a novel physical influence scheme this scheme remarkably enhances the achieved solution accuracy it is shown that the extended formulation is robust enough to treat any combination of multiblock meshes with dual element shape employments in this regard the solution domain is broken up into a number of different multiblock arrangements in which each block is filled with only one type of finite element shape the combined grid can decrease the computational time and memory requirements and increase the numerical accuracy the results of the extended formulation are validated against different benchmark and other solutions the current results are in excellent agreement with the other solutions without exhibiting any disturbances around the block boundaries

function theory on cartan domains and the berezin toeplitz symbol calculus cn with dv z normalized lebesgue measure on 9 we let l l2 q dv be the usual space of lebesgue square integrable complex valued functions on 9 by h2 h2 q dv we denote the bergman subspace of l2 consisting of holomorphic functions the orthogonal projection operator from l2 onto h2 is denoted by p forf in l q the space of essentially bounded measurable functions on 9 we will consider the multiplication operator

on a new integral type operator from the bloch space to bloch type spaces on the unit ball abstract we introduce the following integral type operator on the space h b of all holomorphic functions on the unit ball b c n p g f z 0 1 f t z g t z d t t z b where g h b g 0 0 and is a holomorphic self map of b the boundedness and compactness of the operator from the bloch space b or the little bloch space b 0 to the bloch type space b or the little bloch type space b 0 are characterized in the main results we calculate the essential norm of the operators p g b or b 0 b or b 0 in an elegant way

a numerical inversion of the laplace transform solution to radial dispersion in a porous medium a special form of the numerical inversion of the laplace transform described by stehfest 1970 is applied to the transformed solution of dispersion in a radial flow system in a porous medium the inversion is extremely simple to use because the weighting coefficients depend only on the number of terms used in the computation and not upon the transform solution as required by most numerical inversion techniques the result obtained by use of this approximate inversion of the radial dispersion problem reproduced the curves obtained previously by hoopes and harleman 1967 with a finite difference scheme

solving the advection diffusion equation with the eulerian lagrangian localized adjoint method on unstructured meshes and non uniform time stepping eulerian lagrangian localized adjoint method ellam is used to solve the advection diffusion equation ade which is a very common mathematical model in physics in this work ellam is extended to triangular meshes standard integration schemes which perform well for rectangular grids are improved to reduce oscillations with unstructured triangulations numerical experiments for grid peclet numbers ranking from 1 to 100 show the efficiency of the developed scheme a new algorithm is also developed in order to avoid excessive numerical diffusion when using many time steps with the ellam the basic idea of this approach is to keep the same characteristics for all time steps and to interpolate only the concentration variations due to the dispersion process at the end of each time step although ellam requires a lot of integration points for unstructured meshes it remains a competitive method when using a single or many time steps compared to explicit discontinuous galerkin finite element method

solute transport in divergent radial flow with multistep pumping 1 an efficient approach is developed to analytically evaluate solute transport in a horizontal divergent radial flow field with a multistep injection flow rate and an arbitrary input concentration history by assuming a piecewise steady state flow and transforming the time domain to the cumulative injected flow domain the concentration distribution is found to be completely determined by the total volume of injected flow and independent of specific flow rates thus on the cumulative flow domain the transport problem with a temporally varying velocity field can be transformed into a steady state flow problem linear convolution can then be applied on the cumulative injected flow domain to evaluate the solution for an arbitrarily time dependent input concentration solutions on the regular time domain can be conveniently obtained by mapping the solution on the cumulative injected flow domain to the time domain furthermore we theoretically examine the conditions for the assumption of piecewise steady state flow to be valid on the basis of the critical time scale of the pseudosteady state condition defined as when velocity changes accomplish 99 of their steady state differences and the relative error in the mean travel time of plume front we obtain conditions for neglecting the transitional period between two pumping steps such conditions include the following 1 the duration of a pumping step tp must be longer than the critical time scale tc i e tp tc 25r2s t where r is the radial distance s is the storage coefficient and t is the transmissivity or similarly a maximum problem domain needs to be defined for a given pumping strategy 2 the maximum well pumping rate qmax should satisfy qmax t 25s where is the effective porosity when both conditions are satisfied transitional periods may be neglected

a finite volume eulerian lagrangian localized adjoint method for solution of the advection dispersion equation a new mass conservative method for solution of the one dimensional advection dispersion equation is derived and discussed test results demonstrate that the finite volume eulerian lagrangian localized adjoint method fvellam outperforms standard finite difference methods in terms of accuracy and efficiency for solute transport problems that are dominated by advection for dispersion dominated problems the performance of the method is similar to that of standard methods like previous ellam formulations fvellam systematically conserves mass globally with all types of boundary conditions fvellam differs from other ellam approaches in that integrated finite differences instead of finite elements are used to approximate the governing equation this approach in conjunction with a forward tracking scheme greatly facilitates mass conservation the mass storage integral is numerically evaluated at the current time level and quadrature points are then tracked forward in time to the next level forward tracking permits straightforward treatment of inflow boundaries thus avoiding the inherent problem in backtracking as used by most characteristic methods of characteristic lines intersecting inflow boundaries fvellam extends previous ellam results by obtaining mass conservation locally on lagrangian space time elements details of the integration tracking and boundary algorithms are presented test results are given for problems in cartesian and radial coordinates

weighted composition operators from weighted bergman spaces to weighted type spaces on the unit ball we calculate the operator norm of the weighted composition operator from a weighted bergman space to a weighted type space on the unit ball of c n we also characterize the compactness of the operator

the local structure of zero mode producing magnetic potentials we consider the class of continuous magnetic potentials on 3 which decay as o x 1 within this class it is shown that the set of potentials whose associated weyl dirac operator produces zero modes with multiplicity m forms a smooth submanifold of co dimension m2 when m 0 1 2 and is contained in a smooth submanifold of co dimension 2m 1 when m 3

an alternating direction galerkin technique for simulation of contaminant transport in complex groundwater systems a general contaminant transport simulation technique based on a galerkin finite element representation in space and an alternating direction time stepping scheme is developed the formulation which is in terms of natural coordinates following the principal directions of hydraulic conductivity of the medium decouples the spatial components of the equations leading to tridiagonal matrices the numerical accuracy can be easily controlled in each principal direction through the well established peclet and courant numbers results are found to be comparable to those from the classic finite element scheme but the cost of computation is generally much less because the computational effort varies only linearly with the grid size the technique is thus well suited for transport simulations with complex layered anisotropic groundwater systems requiring large and detailed grids

generalized weighted composition operators from zygmund spaces to bloch orlicz type spaces let d be the open unit disk in the complex plane c and h d the class of all analytic functions on d let be an analytic self map of d and h d in this paper the boundedness and compactness of the generalized weighted composition operator d n f f n from the zygmund space to the bloch orlicz space and the little bloch orlicz space are characterized

on open boundaries in the finite element approximation of two dimensional advection diffusion flows summary a steady state and transient finite element model has been developed to approximate with simple triangular elements the two dimensional advection di usion equation for practical river surface flow simulations essentially the space time crank nicolson galerkin formulation scheme was used to solve for a given conservative flow field several kinds of point sources and boundary conditions namely cauchy and open were theoretically and numerically analysed steady state and transient numerical tests investigated the accuracy of boundary conditions on inflow noflow and outflow boundaries where di usion is important di usive boundaries with the proper choice of boundary conditions the steady state galerkin and the transient crank nicolson galerkin finite element schemes gave stable and precise results for advectiondominated transport problems comparisons indicated that the present approach can give equivalent or more precise results than other streamline upwind and high order time stepping schemes di usive boundaries can be treated with cauchy conditions when the flow enters the domain inflow and with open conditions when the flow leaves the domain outflow or when it is parallel to the boundary noflow although systems with mainly di usive noflow boundaries may still be solved precisely with open conditions they are more susceptible to be influenced by other numerical sources of error moreover the treatment of open boundaries greatly increases the possibilities of correctly modelling restricted domains of actual and numerical interest 1997 by john wiley sons ltd

extended cesaro operators from bergman spaces to besov spaces in the unit ball in this paper we characterize the holomorphic symbols g h b for which the induced extended cesaro operator t g is bounded or compact from bergman spaces to besov spaces in the unit ball in addition we obtain the properties of the p q carleson measure

finite element algorithms for simulating three dimensional groundwater flow and solute transport in multilayer systems a finite element formulation is developed for the simulation of groundwater flow and solute transport in multilayer systems of several aquifers and aquitards this formulation is general flexibile and capable of taking full advantage of the nature of flow in such multilayer systems a fully three dimensional spatial representation can be performed for certain aquifers or for the entire flow system if needed those parts of the system that require three dimensional spatial discretization are handled effectively by combining two dimensional basis functions in the x y space and one dimensional basis functions in the z space furthermore the formulation has a desirable option to perform one dimensional representation of flow and transport in aquitards and areal representation of flow and transport in certain aquifers that do not require the fully three dimensional discretization when the one dimensional representation of an aquitard is used coupling of adjacent aquifer and aquitard layers is handled using a convolution integral approach a general solution strategy is also developed to allow systematic time stepping and cost effective matrix handling schemes for those parts of the system that require three dimensional discretization an algorithm referred to as the alternate sublayer and line sweep procedure is presented for decomposing three dimensional matrix equations this algorithm can accomodate several thousand nodal unknowns without requiring excessive core storage and cpu time four examples of transient flow and transport problems are provided these examples show verification of numerical results and demonstrate that the present finite element models are far more cost effective than earlier three dimensional models that are based on the conventional galerkin approach and direct matrix solution algorithms

macrodispersion in a radially diverging flow field with finite peclet numbers 1 perturbation theory approach we study the transport behavior of a tracer in a radially diverging heterogeneous velocity field making use of homogenization theory we derive effective transport equations these effective transport equations are very similar to those defined on the local scale however the local transport parameters such as local dispersion coefficients are replaced by effective dispersion coefficients for smoothly varying heterogeneous media explicit results for effective radial dispersion coefficients are derived starting with the purely advective transport behavior infinite peclet numbers we extend our calculations to transport with finite peclet numbers we find that the impact of molecular diffusion on the effective dispersivity differs from the impact of local dispersion including local dispersion leads to effective dispersivities which are constant and equivalent to the effective dispersivities found in uniform flow configurations in contrast effective dispersivities including diffusion are not constant but depend on the radial distance we compare the results found by homogenization theory with those derived by neuweiler et al this issue by standard method of moments

a criterion for the existence of zero modes for the pauli operator with fastly decaying fields we consider the pauli operator in 3 for magnetic fields in l3 2 that decay at infinity as x 2 with 0 in this case we are able to prove that the existence of a zero mode for this operator is equivalent to a quantity b defined below being equal to zero complementing a result from balinsky et al j phys a math gen 34 l19 l23 2001 this implies that for the class of magnetic fields considered sobolev hardy and cwikel lieb rosenblum clr inequalities hold whenever the magnetic field has no zero mode

a conservative characteristic finite volume element method for solution of the advection diffusion equation abstract we develop a modified characteristic finite volume element method for two dimensional advection diffusion problem in a convex spacial domain using quasi uniform triangular discretization and piecewise linear element this scheme conserves mass globally and locally on the lagrangian space time control volume we give the optimal order h1 norm and optimal l2 norm error estimates finally we present some numerical examples

weighted composition operators from the weighted bergman space to the weighted hardy space on the unit ball we investigate the weighted composition operator from the weighted bergman space into the weighted hardy space on the unit ball as a consequence of the investigation we also give a characterization for the boundedness and compactness of the operator whose the target space is the hardy space

analysis of zero modes for dirac operators with magnetic links abstract in this paper we provide a means to approximate dirac operators with magnetic fields supported on links in s 3 and r 3 by dirac operators with smooth magnetic fields then we proceed to prove that under certain assumptions the spectral flow of paths along these operators is the same in both the smooth and the singular case we recently characterized the spectral flow of such paths in the singular case this allows us to show the existence of new smooth compactly supported magnetic fields in r 3 for which the associated dirac operator has a non trivial kernel using clifford analysis we also obtain criteria on the magnetic link for the non existence of zero modes

a curvilinear finite element model for simulating two well tracer tests and transport in stratified aquifers the problem of solute transport in steady nonuniform flow created by a recharging and discharging well pair is investigated numerical difficulties encountered with the standard galerkin formulation in cartesian coordinates are illustrated an improved finite element solution strategy is presented to overcome the numerical problem the strategy is based on a formulation performed in three dimensional curvilinear coordinates this formulation is developed especially for the simulation of two well injection withdrawal tracer tests conducted in homogeneous or stratified aquifers the formulation can also be applied to other types of problems involving transport in homogeneous and stratified aquifers a three dimensional finite element approximation is derived by two successive applications of upstream weighted residual and subdomain collocation procedures first the weighted residual procedure is formulated in the horizontal flow plane using curvilinear coordinates along and normal to streamlines upstream weighting is included but automatically controlled and limited to one dimension along the main flow direction second the subdomain collocation procedure is formulated in the vertical dimension spatial discretization of the three dimensional flow region is performed using a curvilinear finite element mesh that is point centered in the horizontal plane and block centered in the vertical direction general algorithms are developed for mesh generation time stepping and matrix reduction the solution algorithm is designed to accommodate several thousand nodal unknowns with minimal core storage and cpu time requirement two simulation examples are presented to verify and validate the numerical model in two dimensions using an analytical solution and experimental laboratory and field data a comparative study shows that the proposed formulation is superior to the standard galerkin formulation in cartesian coordinates both in terms of accuracy and computational efficiency the field simulation example indicates the need to conduct a more elaborate three dimensional simulation study allowing for vertical variation in hydraulic conductivity such a study has been completed and is reported in the follow up paper of this series

new examples of zero modes we construct smooth vector potentials a 1 and a 2 on bbb r3 such that the associated pauli operators scenterdot d a i i 1 2 have zero energy eigenfunctions or zero modes the first example a 1 has compact support and the associated zero mode is contained in l2 the second example can be written in the form a 2 x2 x1 0 t tilde a 2 where tilde a 2 is bounded and supported on x3 1 furthermore the associated zero mode is schwartz class

on the iterated logarithmic bloch space on the unit ball abstract we introduce a new space of analytic functions on the unit ball the so called iterated logarithmic bloch space b log k and study some of its properties among other features we study the boundedness and compactness of weighted composition operators on the unit disk as well as of a recently introduced integral type operator on the unit ball with i domain the iterated logarithmic bloch space b log k or the little iterated logarithmic bloch space b log k 0 and ii range the bloch type space b or the little bloch type space b 0 or the weighted space h or the little weighted space h 0

on the zero modes of pauli operators two results are proved for nul pa the dimension of the kernel of the pauli operator formula in l2 r3 2 i for formula where formula is the magnetic field nul pta 0 except for a finite number of values of t in any compact subset of 0 ii formula contains an open dense subset of l3 2 r3 3

a characteristics mixed covolume method for a convection dominated transport problem in this paper we propose a characteristics mixed covolume method for approximating the solution to a convection dominated transport problem the method is a combination of characteristic approximation to handle the convection term in time and mixed covolume method spatial approximation to deal with the diffusion term the velocity and press are approximated by the lowest order raviart thomas mixed finite element space on rectangles the projection of a mixed covolume element is introduced we prove its first order optimal rate of convergence for the approximate velocities in the l 2 norm as well as for the approximate pressures in the l 2 norm

on a new operator from the logarithmic bloch space to the bloch type space on the unit ball abstract we study the boundedness and compactness of a recently introduced operator denoted by p g which is a kind of the product of composition and integral operators on the unit ball b c n from the logarithmic bloch space b log and the little logarithmic bloch space b log 0 to the bloch type space b or the little bloch type space b 0

on boundary conditions and point sources in the finite element integration of the transport equation in the finite element integration of the convection dispersion equation over a groundwater domain boundary conditions of the third or cauchy type may or may not be incorporated with the variational statement while the former development formulation a requires the application of green s first identity to both the convective and the dispersive terms the latter formulation b does not although both approaches are equivalent at the theoretical level they behave differently from a computational standpoint the numerical solution proves more accurate and stable if green s lemma is not applied to the convective term since significant errors arising from the representation of the darcy velocity over the elements are avoided point sources or sinks representing injection or pumping wells can be treated as either sources sinks distributed over a small volume or external boundaries with prescribed contaminant flux it is shown that when the well bore is excluded from the flow field formulation a yields the same matrix addition for an injection abstraction well as formulation b does for an abstraction injection well reciprocity of the results the computational behaviors of these alternative modeling procedures are illustrated with a number of significant numerical examples

eulerian lagrangian localized adjoint methods for a nonlinear advection diffusion equation eulerian lagrangian localized adjoint methods ellam are developed for the non linear buckley leverett equation which is characterized by degenerate diffusion and sharpening near shock solutions the ellam methodology employs space time finite elements with edges oriented along flow paths and space time test functions that satisfy a local adjoint condition this combination extends eulerian lagrangian concepts in a systematic mass conservative fashion to problems with general boundary conditions various kinds of boundary conditions are considered and a local time stepping procedure is developed for a no flow outlet condition that leads to a boundary layer numerical experiments illustrate the potential of these methods

an efficient characteristic galerkin scheme for the advection equation in 3 d conventional galerkin based finite element algorithms perform poorly when modeling advective transport here we develop and test a characteristic galerkin scheme to solve the unsteady three dimensional advective transport equation the algorithm uses tracking of the gaussian quadrature points to project the information from the eulerian background grid to the lagrangian grid numerical experiments were carried out to investigate the performance of this scheme despite speculations to the contrary in the literature this scheme does not suffer from instability problems further it is conservative and shows good phase characteristics with slight numerical diffusion we conclude that the characteristic galerkin method is a viable and efficient scheme for solving advection problems in three dimensions

weighted composition operators between weighted bergman spaces and hardy spaces on the unit ball of cn abstract in this paper we study the weighted composition operators w f f between weighted bergman spaces and hardy spaces on the unit ball of c n we characterize the boundedness and the compactness of the weighted composition operators w a p a q 0 q p 1 and w h p b h q b 0 q p

spatial filter averaging approach of probabilistic method to linear second order partial differential equations of the parabolic type a backward in time probabilistic method with spatial filter averaging is presented to solve linear second order partial differential equations of the parabolic type an advantage of this methodology is that while forward methods are subject to region with loss of density of particles and hence loss of spatial resolution of the solution the solution given by backward methods is given on any desired grid however traditional backward time probabilistic method using monte carlo averaging are computationally expensive we prove a convergence result and present several examples the method leads to important improvement in computational efficiency and is expected to perform well to solve high dimensional problems where a solution is needed on a large grid

composition operators between bergman and hardy spaces we study composition operators between weighted bergman spaces certain growth conditions for generalized nevanlinna counting functions of the inducing map are shown to be necessary and sufficient for such operators to be bounded or compact particular choices for the weights yield results on composition operators between the classical unweighted bergman and hardy spaces

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100