a bounded approximation of weakly compact operators abstract we study bounded approximation properties of operators it is shown that for every weakly compact operator t on a banach space x if t is uniformly approximated on each compact subset of x by finite rank operators then t 2 can be approximated in the same topology by finite rank operators with the operator norm less than or equal to t 2

the weak metric approximation property we introduce and investigate the weak metric approximation property of banach spaces which is strictly stronger than the approximation property and at least formally weaker than the metric approximation property among others we show that if a banach space has the approximation property and is 1 complemented in its bidual then it has the weak metric approximation property we also study the lifting of the weak metric approximation property from banach spaces to their dual spaces this enables us in particular to show that the subspace of c0 constructed by johnson and schechtman does not have the weak metric approximation property

a general mass conservative numerical solution for the unsaturated flow equation numerical approximations based on different forms of the governing partial differential equation can lead to significantly different results for unsaturated flow problems numerical solution based on the standard h based form of richards equation generally yields poor results characterized by large mass balance errors and erroneous estimates of infiltration depth conversely numerical solutions based on the mixed form of richards equation can be shown to possess the conservative property so that mass is perfectly conserved this leads to significant improvement in numerical solution performance while requiring no additional computational effort however use of the mass conservative method does not guarantee good solutions accurate solution of the unsaturated flow equation also requires use of a diagonal time or mass matrix only when diagonal time matrices are used can the solution be shown to obey a maximum principle which guarantees smooth nonoscillatory infiltration profiles this highlights the fact that proper treatment of the time derivative is critical in the numerical solution of unsaturated flow

accurate multiscale finite element methods for two phase flow simulations in this paper we propose a modified multiscale finite element method for two phase flow simulations in heterogeneous porous media the main idea of the method is to use the global fine scale solution at initial time to determine the boundary conditions of the basis functions this method provides a significant improvement in two phase flow simulations in porous media where the long range effects are important this is typical for some recent benchmark tests such as the spe comparative solution project m christie m blunt tenth spe comparative solution project a comparison of upscaling techniques spe reser eval eng 4 2001 308 317 where porous media have a channelized structure the use of global information allows us to capture the long range effects more accurately compared to the multiscale finite element methods that use only local information to construct the basis functions we present some analysis of the proposed method to illustrate that the method can indeed capture the long range effect in channelized media

iterative multiscale finite volume method the multiscale finite volume msfv method for the solution of elliptic problems is extended to an efficient iterative algorithm that converges to the fine scale numerical solution the localization errors in the msfv method are systematically reduced by updating the local boundary conditions with global information this iterative multiscale finite volume i msfv method allows the conservative reconstruction of the velocity field after any iteration and the msfv method is recovered if the velocity field is reconstructed after the first iteration both the i msfv and the msfv methods lead to substantial computational savings where an approximate but locally conservative solution of an elliptic problem is required in contrast to the msfv method the i msfv method allows a systematic reduction of the error in the multiscale approximation line relaxation in each direction is used as an efficient smoother at each iteration this smoother is essential to obtain convergence in complex highly anisotropic heterogeneous domains numerical convergence of the method is verified for different test cases ranging from a standard poisson equation to highly heterogeneous anisotropic elliptic problems finally to demonstrate the efficiency of the method for multiphase transport in porous media it is shown that it is sufficient to apply the iterative smoothing procedure for the improvement of the localization assumptions only infrequently i e not every time step this result is crucial since it shows that the overall efficiency of the i msfv algorithm is comparable with the original msfv method at the same time the solutions are significantly improved especially for very challenging cases

analytical decomposition of the nonlinear unsaturated flow equation an analytical procedure for the solution of the nonlinear unsaturated flow equation is proposed the horizontal and vertical flow equations subject to constant boundary conditions and continuous wetting are approximated using analytical decomposition series the results are verified with the philip 1955 and the parlange 1971 classical solutions and with experimental data a convergence theorem with proof and discussion on the numerical accuracy of decomposition series are included applications to the modeling of infiltration subject to hysteresis in the soil water functional relationships soil heterogeneity and time variable point rainfall are also discussed other potential applications in water management models under more realistic conditions are possible the methodology offers advantages over the common small perturbation solutions the possibility to study nonstationarity in the random quantities statistical separability from the physics of the problem i e independence between the system parameters and the input quantities rather than forced through closure approximations and the construction of an analytical series solution that converges uniformly to the true nonlinear solution finally because of stability and computational economy with respect to linearized numerical solutions an analytical solution appears promising

multidimensional thermodynamic uncertainty relations we extend a class of recently derived thermodynamic uncertainty relations to vector valued observables in contrast to the scalar valued observables examined previously this multidimensional thermodynamic uncertainty relation provides a natural way to study currents in high dimensional systems and to obtain relations between different observables our proof is based on the generalized cramer rao inequality which we interpret as a relation between physical observables and the fisher information this allows us to develop high dimensional versions of both the original steady state uncertainty relation and the more recently obtained generalized uncertainty relation for time periodic systems we apply the multidimensional uncertainty relation to obtain a new constraint on the performance of steady state heat engines which is tighter than previous bounds and reveals the role of heat work correlations as a second application we show that the uncertainty relation is connected to a bound on the differential mobility as a result of this connection we find that a necessary condition for equality in the uncertainty relation is that the system obeys the equilibrium fluctuation dissipation relation

adaptive algebraic multiscale solver for compressible flow in heterogeneous porous media this paper presents the development of an adaptive algebraic multiscale solver for compressible flow c ams in heterogeneous porous media similar to the recently developed ams for incompressible linear flows wang et al 2014 19 c ams operates by defining primal and dual coarse blocks on top of the fine scale grid these coarse grids facilitate the construction of a conservative finite volume coarse scale system and the computation of local basis functions respectively however unlike the incompressible elliptic case the choice of equations to solve for basis functions in compressible problems is not trivial therefore several basis function formulations incompressible and compressible with and without accumulation are considered in order to construct an efficient multiscale prolongation operator as for the restriction operator c ams allows for both multiscale finite volume msfv and finite element msfe methods finally in order to resolve high frequency errors fine scale pre and post smoother stages are employed in order to reduce computational expense the c ams operators prolongation restriction and smoothers are updated adaptively in addition to this the linear system in the newton raphson loop is infrequently updated systematic numerical experiments are performed to determine the effect of the various options outlined above on the c ams convergence behaviour an efficient c ams strategy for heterogeneous 3d compressible problems is developed based on overall cpu times finally c ams is compared against an industrial grade algebraic multigrid amg solver results of this comparison illustrate that the c ams is quite efficient as a nonlinear solver even when iterated to machine accuracy

the multiscale restriction smoothed basis method for fractured porous media f msrsb a novel multiscale method for multiphase flow in heterogeneous fractured porous media is devised the discrete fine scale system is described using an embedded fracture modeling approach in which the heterogeneous rock matrix and highly conductive fractures are represented on independent grids given this fine scale discrete system the method first partitions the fine scale volumetric grid representing the matrix and the lower dimensional grids representing fractures into independent coarse grids then basis functions for matrix and fractures are constructed by restricted smoothing which gives a flexible and robust treatment of complex geometrical features and heterogeneous coefficients from the basis functions one constructs a prolongation operator that maps between the coarse and fine scale systems the resulting method allows for general coupling of matrix and fracture basis functions giving efficient treatment of a large variety of fracture conductivities in addition basis functions can be adaptively updated using efficient global smoothing strategies to account for multiphase flow effects the method is conservative and because it is described and implemented in algebraic form it is straightforward to employ it to both rectilinear and unstructured grids through a series of challenging test cases for single and multiphase flow in which synthetic and realistic fracture maps are combined with heterogeneous petrophysical matrix properties we validate the method and conclude that it is an efficient and accurate approach for simulating flow in complex large scale fractured media

a mixed multiscale finite element method for elliptic problems with oscillating coefficients the recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large scale structure of the solutions without resolving all the fine scale structures motivated by the numerical simulation of flow transport in highly heterogeneous porous media we propose a mixed multiscale finite element method with an over sampling technique for solving second order elliptic equations with rapidly oscillating coefficients the multiscale finite element bases are constructed by locally solving neumann boundary value problems we provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain the asymptotic structure of the solutions numerical experiments are carried out for flow transport in a porous medium with a random log normal relative permeability to demonstrate the efficiency and accuracy of the proposed method

new criterion of the approximation property this paper is concerned with the approximation property which is an important property in banach space theory we show that a banach space x has the approximation property if and only if for every banach space y the set of finite rank operators from x to y is dense in the corresponding space of compact operators in the usual topology of uniform convergence on compact sets

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

a multi resolution multiscale finite volume method for simulation of fluid flows in heterogeneous porous media abstract this paper presents an extension of the multiscale finite volume msfv method to multi resolution coarse grid solvers for single phase incompressible flows to achieve this a grid one level coarser than the coarse grids used in the msfv method is constructed and the local problems are redefined to compute the basis and correction functions associated with this new grid to construct the coarse scale pressure equations the coarse scale transmissibility coefficients are calculated using a new multi point flux approximation mpfa method the estimated coarse scale pressures are utilized to compute the multiscale pressure solution finally a reconstruction step is performed to produce a conservative velocity field which is used to solve the transport equations the computational cost of the proposed method is compared with that of the msfv method and the relevant time complexity formulas are given several two dimensional test cases with permeability fields ranging from two scale to multi scale problems are solved the performance of the proposed method in handling problems with shale layers or discrete fractures is assessed also a number of layers from the tenth spe comparative study problem are used to examine general abilities of the method when facing realistic reservoir problems the results are compared with fine scale reference solutions to assess the accuracy of the proposed method

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

a simplified newton iteration method with linear finite elements for transient unsaturated flow a simplified newton iteration method with galerkin linear finite elements is developed for the solution of the nonlinear transient unsaturated flow equation in mixed form the use of the galerkin linear finite element method in the spatial discretization produces a recurrence equation in which the interaode hydraulic conductivity is represented by an integrated average expression which generates the exact darcy flow flux through each element under the steady horizontal flow condition the numerical computation of the integrated average hydraulic conductivity using simpson s 1 3 rule leads to an accurate solution even in the presence of a sharp front in the time discretization using a backward euler scheme the standard newton iteration method is simplified by taking account of the integrated average expression of the hydraulic conductivity and the diffusion dominated nature of the flow process the resulting two level scheme is of the same efficiency on a per iteration basis as the modified picard scheme but converges at a faster rate

estimation of finite difference interblock conductivities for simulation of infiltration into initially dry soils simulations of transient unsaturated vertical flow in different initially dry homogeneous soils using richards equation suggest that for a relatively coarse numerical grid there is no universal weighting scheme to estimate interblock conductivities capable of giving an accurate resolution of the resulting steep wetting fronts except for the scheme associated with the kirchhoff transformation kt the principal drawback of kt is that it can hardly be applied to heterogeneous soils a new asymptotic weighting aw scheme based on the asymptotic behavior of the hydraulic conductivity function in the vicinity of residual saturation is proposed numerical tests demonstrate that the aw scheme describes wetting fronts in homogeneous soils nearly as accurately as a scheme based on kt applicability and robustness of the aw scheme for simulating water infiltration into heterogeneous soils having substantial vertical variation of hydraulic properties are also demonstrated extension of the proposed scheme to higher dimensions is straightforward

finite difference modeling of the electroseismic logging in a fluid saturated porous formation in a fluid saturated porous medium an electromagnetic em wavefield induces an acoustic wavefield due to the electrokinetic effect a potential geophysical application of this effect is electroseismic es logging in which the converted acoustic wavefield is received in a fluid filled borehole to evaluate the parameters of the porous formation around the borehole in this paper a finite difference scheme is proposed to model the es logging responses to a vertical low frequency electric dipole along the borehole axis the em field excited by the electric dipole is calculated separately by finite difference first and is considered as a distributed exciting source term in a set of extended biot s equations for the converted acoustic wavefield in the formation this set of equations is solved by a modified finite difference time domain fdtd algorithm that allows for the calculation of dynamic permeability so that it is not restricted to low frequency poroelastic wave problems the perfectly matched layer pml technique without splitting the fields is applied to truncate the computational region the simulated es logging waveforms approximately agree with those obtained by the analytical method the fdtd algorithm applies also to acoustic logging simulation in porous formations

a combined mixed and discontinuous galerkin method for compressible miscible displacement problem in porous media in this paper we present a numerical scheme for solving the coupled system of compressible miscible displacement problem in porous media the flow equation is solved by the mixed finite element method and the transport equation is approximated by a discontinuous galerkin method the scheme is continuous in time and a priori hp error estimates is presented

achieve higher efficiency at maximum power with finite time quantum otto cycle the optimization of heat engines was intensively explored to achieve higher efficiency while maintaining the output power however most investigations were limited to a few finite time cycles e g the carnot like cycle due to the complexity of the finite time thermodynamics in this paper we propose a class of finite time engine with quantum otto cycle and demonstrate a higher achievable efficiency at maximum power the current model can be widely utilized benefitting from the general c 2 scaling of extra work for a finite time adiabatic process with long control time we apply the adiabatic perturbation method to the quantum piston model and calculate the efficiency at maximum power which is validated with an exact solution

adaptive iterative multiscale finite volume method the multiscale finite volume msfv method is a computationally efficient numerical method for the solution of elliptic and parabolic problems with heterogeneous coefficients it has been shown for a wide range of test cases that the msfv results are in close agreement with those obtained with a classical computationally expensive technique the method however fails to give accurate results for highly anisotropic heterogeneous problems due to weak localization assumptions recently a convergent iterative msfv i msfv method was developed to enhance the quality of the multiscale results by improving the localization conditions although the i msfv method proved to be efficient for most practical problems it is still favorable to improve the localization condition adaptively i e only for a sub domain where the original msfv localization conditions are not acceptable e g near shale layers and long coherent structures with high permeability contrasts in this paper a space time adaptive i msfv ai msfv method is introduced it is shown how to improve the msfv results adaptively in space and simulation time the fine scale smoother which is necessary for convergence of the i msfv method is also applied locally finally for multiphase flow problems two criteria are investigated for adaptively updating the msfv interpolation functions 1 a criterion based on the total mobility change for the transient coefficients and 2 a criterion based on the pressure equation residual for the accuracy of the results for various challenging test cases it is demonstrated that iterations in order to obtain accurate results even for highly anisotropic heterogeneous problems are required only in small sub domains and not everywhere the findings show that the error introduced in the msfv framework can be controlled and improved very efficiently with very little additional computational cost compared to the original non iterative msfv method

techniques for making finite elements competitve in modeling flow in variably saturated porous media a galerkin finite element formulation is developed for the numerical simulation of water flow in variably saturated soil systems included in this formulation is a solution strategy based on picard and newton raphson algorithms both algorithms are designed especially to cope with severely nonlinear field problems the two algorithms are formulated for both rectangular and triangular elements the element matrices are evaluated in a simple and efficient manner using a technique referred to as the influence coefficient technique this technique avoids numerical integration and leads to a substantial saving of computational cost four examples are presented to demonstrate the effectiveness of the present finite element approach these examples show that the nonlinear solution schemes are capable of accomodating cases involving large variations in the saturated hydraulic conductivity as well as highly nonlinear soil moisture characteristics a comparative study of the picard and the newton raphson algorithms is also provided the study indicates that despite the higher cost per iteration of the newton raphson scheme it usually requires a substantially smaller number of iterations than the picard scheme in some instances where convergence difficulties are experienced with the latter scheme it is desirable to use the newton raphson scheme in order to obtain a cost effective solution to the problem

multiscale finite volume method for parabolic problems arising from compressible multiphase flow in porous media the multiscale finite volume msfv method was originally developed for the solution of heterogeneous elliptic problems with reduced computational cost recently some extensions of this method for parabolic problems have been proposed these extensions proved effective for many cases however they are neither general nor completely satisfactory for instance they are not suitable for correctly capturing the transient behavior described by the parabolic pressure equation in this paper we present a general multiscale finite volume method for parabolic problems arising for example from compressible multiphase flow in porous media opposed to previous methods here the basis and correction functions are solutions of full parabolic governing equations in localized domains at the same time to enhance the computational efficiency of the scheme the basis functions are kept pressure independent and do not have to be recalculated as pressure evolves this general approach requires no additional assumptions and its good efficiency and high accuracy is demonstrated for various challenging test cases finally to improve the quality of the results and also to extend the scheme for highly anisotropic heterogeneous problems it is combined with the iterative msfv i msfv method for parabolic problems as one iterates the i msfv solutions of compressible multiphase problems parabolic problems converge to the corresponding fine scale reference solutions in the same way as demonstrated recently for incompressible cases elliptic problems therefore the proposed msfv method can also be regarded as an efficient linear solver for parabolic problems and studies of its efficiency are presented for many test cases

convergence of a nonconforming multiscale finite element method the multiscale finite element method msfem t y hou x h wu and z cai math comp 1998 to appear t y hou and x h wu j comput phys 134 1997 pp 169 189 has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coefficients this is accomplished by constructing the multiscale base functions from the local solutions of the elliptic operator our previous study reveals that the leading order error in this approach is caused by the resonant sampling which leads to large error when the mesh size is close to the small scale of the continuous problem similar difficulty also arises in numerical upscaling methods an oversampling technique has been introduced to alleviate this difficulty t y hou and x h wu j comput phys 134 1997 pp 169 189 a consequence of the oversampling method is that the resulting finite element method is no longer conforming here we give a detailed analysis of the nonconforming error our analysis also reveals a new cell resonance error which is caused by the mismatch between the mesh size and the wavelength of the small scale we show that the cell resonance error is of lower order our numerical experiments demonstrate that the cell resonance error is generically small and is difficult to observe in practice

mass conservative numerical solutions of the head based richards equation numerical procedures for efficient mass conservative solutions of the head based form of the richards equation are presented mass conservative solutions are shown to result when the capacity coefficient c is formulated by equating the storage term and its chain rule expansion in their discretized forms equivalence in the storage term expansion is maintained in finite difference models when c is evaluated with a standard chord slope approximation this scheme is shown to produce excellent global mass balance accuracy in simulations of vertical moisture infiltration an analogous approach to the expansion of the storage term using finite elements results in element dependent expressions of c application of this approach produces mass balance accuracy with errors less than 1 but also exhibits slow convergence in the consistent form a nontraditional finite element procedure is presented which maintains equivalence in the storage term expansion when c is evaluated with the standard chord slope approximation this scheme exhibits excellent mass balance accuracy in either the consistent or lumped forms without significant loss in computational efficiency

a transformed pressure head based approach to solve richards equation for variably saturated soils this paper presents a new approach to solve richards equation it introduces a nonlinear transformed pressure pt as the dependent variable with the modified picard method the new approach was compared to and contrasted with two efficient existing methods the based transformation method kirkland et al 1992 and the h based modified picard method celia et al 1990 a total of 12 different one dimensional cases were considered saturated unsaturated layered and uniform soil profiles with pressure and flux type boundary conditions the results show that the new method offers excellent cpu efficiency and unlike the h based method is numerically robust for all cases of variably saturated heterogeneous media and first or second type boundary conditions the method does not require difficult numerical coding and its cpu efficiency is not affected by complicated heterogeneous and hysteretic media the pt transformation is easy to incorporate into existing h based codes

multiscale finite element method for linear elastic geomechanics the demand for accurate and efficient simulation of geomechanical effects is widely increasing in the geoscience community high resolution characterizations of the mechanical properties of subsurface formations are essential for improving modeling predictions such detailed descriptions impose severe computational challenges and motivate the development of multiscale solution strategies we propose a multiscale solution framework for the geomechanical equilibrium problem of heterogeneous porous media based on the finite element method after imposing a coarse scale grid on the given fine scale problem the coarse scale basis functions are obtained by solving local equilibrium problems within coarse elements these basis functions form the restriction and prolongation operators used to obtain the coarse scale system for the displacement vector then a two stage preconditioner that couples the multiscale system with a smoother is derived for the iterative solution of the fine scale linear system various numerical experiments are presented to demonstrate accuracy and robustness of the method

parameter estimation in flow through partially saturated porous materials a class of numerical simulators were developed and critically evaluated to be incorporated as the solver of a forward problem in the framework of an inverse modeling strategy the strategy couples a mass lumped galerkin linear finite element solution of the mixed form richards equation with an experimental time space series and the osborne more revised form of the levenberg marquardt algorithm to retrieve hydraulic parameters of a partially saturated porous medium the numerical simulator shows excellent agreement with a reference solution obtained on a dense grid and infinitesimal time step in terms of fluid pressure head fluid content and fluid volumetric flux density and perfectly conserves the global mass an adaptive algorithm was implemented to estimate sensitivity matrix in the inverse algorithm a multi criterion stopping rule was developed and successfully implemented to end the inverse code at the solution the result of the optimization was compared with a large scale in situ soil moisture space time series measured during the course of a drainage experiment and excellent agreements were found analysis of the parameter response surfaces and hyper space plots closeness of the gradient of the penalty function at minimum to zero and positive definiteness of the approximation for the hessian at the solution eigs h 0 indicate that the obtained solution is a strong local minimum a state of the art sensitivity analysis carried out to quantify sensitivity of the state variable with respect to uncertainty and changes in different model parameters

algebraic multiscale solver for flow in heterogeneous porous media an algebraic multiscale solver ams for the pressure equations arising from incompressible flow in heterogeneous porous media is described in addition to the fine scale system of equations ams requires information about the superimposed multiscale dual and primal coarse grids ams employs a global solver only at the coarse scale and allows for several types of local preconditioners at the fine scale the convergence properties of ams are studied for various combinations of global and local stages these include multiscale finite element msfe and multiscale finite volume msfv methods as the global stage and correction functions cf block incomplete lower upper factorization bilu and ilu as local stages the performance of the different preconditioning options is analyzed for a wide range of challenging test cases the best overall performance is obtained by combining msfe and ilu as the global and local preconditioners respectively followed by msfv to ensure local mass conservation comparison between ams and a widely used algebraic multigrid amg solver 1 indicates that ams is quite efficient a very important advantage of ams is that a conservative fine scale velocity can be constructed after any msfv stage

an efficient multi scale poisson solver for the incompressible navier stokes equations with immersed boundaries the iterative multi scale finite volume imsfv procedure is applied as an efficient solver for the pressure poisson equation arising in numerical methods for the simulation of incompressible flows with the immersed interface method iim motivated by the requirements of the specific iim implementation a modified version of the imsfv algorithm is presented to allow the solution of problems in which the varying coefficient of the elliptic equation e g the permeability of the medium in the context of the simulation of flows in porous media varies over several orders of magnitude or even becomes zero within the integration domain furthermore a strategy is proposed to incorporate the iterative procedure needed by the iim to converge out constraints at immersed boundaries into the iterative imsfv cycle no significant deterioration of performance of the imsfv method is observed with respect to cases in which no iterative improvement of the boundary conditions is considered

multiscale direction splitting algorithms for parabolic equations with highly heterogeneous coefficients in this paper we discuss two methods for upscaling of highly heterogeneous data for parabolic problems in the context of a direction splitting time approximation the first method is a direct application of the idea of jenny et al 2003 in the context of the direction splitting approach the second method devises the approximation from the schur complement corresponding to the interface unknowns of the coarse grid by applying a proper l 2 projection operator to it the spatial discretization employed in this paper is based on a mac finite volume stencil but the same approach can be implemented within a proper finite element discretization a key feature of the present approach is that it can extend to 3d problems with very little computational overhead the properties of the resulting approximations are demonstrated numerically on some benchmark coefficient data available in the literature

a mass conservative switching algorithm for modeling fluid flow in variably saturated porous media numerical solutions of flow equation in fluid content based form or in fluid pressure head based form are often tradeoffs between speed accuracy and convenience the fluid content based form can be solved quite rapidly with low cpu time and perfect mass balance however it cannot be used in saturated regions as diffusivity function becomes infinite and strictly becomes invalid in composite layered and real heterogeneous porous materials due to singularity and discontinuity in fluid content profile this formulation also gives misleading impression that gradient in fluid content causes the flow of fluid in porous materials where in reality gravity and fluid pressure potential gradient produce the motion the pressure head based form on the other hand is more flexible but due to its highly nonlinear nature is much more time consuming and produces poor global mass balance for dry initial conditions very fine spatial and temporal discretizations are needed to maintain mass balance property for these scenarios the mixed form of the flow equation partially solves these issues as it maintains acceptable mass balance and is applicable to layered heterogeneous and composite fractured foundations however it is only applicable in unsaturated zones in this study a switching algorithm was proposed and implemented in which the mass conservative mixed form and the pressure head based form were respectively used in the unsaturated and saturated zones of an initial boundary value flow problem involving a variably saturated porous medium the algorithm showed excellent agreement with a reference solution obtained on a very fine spatiotemporal mesh the simulator was then calibrated with several real world large scale experimental datasets in all cases the proposed algorithm exhibited close agreements with the experimental time space series the algorithm poses excellent mass balance property and can easily be used in both saturated and unsaturated regions without special treatment of fluid content discontinuities in heterogeneous and layered porous media the proposed algorithm can also be extended to simulate multiphase and multidimensional flow problems

local global splitting for spatiotemporal adaptive multiscale methods we present a novel spatiotemporal adaptive multiscale finite volume msfv method which is based on the natural idea that the global coarse scale problem has longer characteristic time than the local fine scale problems as a consequence the global problem can be solved with larger time steps than the local problems in contrast to the pressure transport splitting usually employed in the standard msfv approach we propose to start directly with a local global splitting that allows to locally retain the original degree of coupling this is crucial for highly non linear systems or in the presence of physical instabilities to obtain an accurate and efficient algorithm we devise new adaptive criteria for global update that are based on changes of coarse scale quantities rather than on fine scale quantities as it is routinely done before in the adaptive msfv method by means of a complexity analysis we show that the adaptive approach gives a noticeable speed up with respect to the standard msfv algorithm in particular it is efficient in case of large upscaling factors which is important for multiphysics problems based on the observation that local time stepping acts as a smoother we devise a self correcting algorithm which incorporates the information from previous times to improve the quality of the multiscale approximation we present results of multiphase flow simulations both for darcy scale and multiphysics hybrid problems in which a local pore scale description is combined with a global darcy like description the novel spatiotemporal adaptive multiscale method based on the local global splitting is not limited to porous media flow problems but it can be extended to any system described by a set of conservation equations local global splitting as a framework for multiscale multiphysics simulations spatiotemporal adaptivity for the multiscale finite volume method adaptive criteria based on the changes of coarse scale quantities self correcting scheme to improve the quality of multiscale approximation multiscale multiphysics simulations of two phase flow in porous media

on convergence of finite volume schemes for one dimensional two phase flow in porous media this paper deals with development and analysis of finite volume schemes for a one dimensional nonlinear degenerate convection diffusion equation having application in petroleum reservoir and groundwater aquifer simulation the main difficulty is that the solution typically lacks regularity due to the degenerate nonlinear diffusion term we analyze and compare three families of numerical schemes corresponding to explicit semi implicit and implicit discretization of the diffusion term and a godunov scheme for the advection term l stability under appropriate cfl conditions and bv estimates are obtained it is shown that the schemes satisfy a discrete maximum principle then we prove convergence of the approximate solution to the weak solution of the problem results of numerical experiments using the present approach are reported

the strong approximation property we introduce and investigate the strong approximation property of banach spaces which is strictly stronger than the approximation property and at least formally weaker than the weak bounded approximation property among others we show that the weak bounded approximation property is equivalent to a quantitative strengthening of the strong approximation property some recent results on the approximation property of banach spaces and their dual spaces are improved

cubic approximation of hydraulic properties for simulations of unsaturated flow an adequate description of soil hydraulic properties is critical to modeling unsaturated flow however flexible descriptions based on the functions normally used may increase simulation times this disadvantage can be overcome by using piecewise cubic approximation of such descriptions based on a table of function and derivative values a further advantage of this technique is that the kirchhofif transform variable can be approximated in the same way even when it cannot be obtained analytically results for simulated infiltration into an aggregated soil indicated that hydraulic properties over the entire matric potential range could be approximated using 8 12 points without causing significant errors in simulations whether the kirchhoff transform was used or not run times using the original van genuchten functions were 40 or 280 greater depending on whether an incomplete beta function was used the cubic approximation method thus allows more flexible use of the popular van genuchten hydraulic property functions since these involve incomplete beta functions unless unsatisfactory restrictions are placed on their parameters

diverging but negligible power at carnot efficiency theory and experiment we discuss the possibility of reaching the carnot efficiency by heat engines hes out of quasistatic conditions at nonzero power output we focus on several models widely used to describe the performance of actual hes these models comprise quantum thermoelectric devices linear irreversible hes minimally nonlinear irreversible hes hes working in the regime of low dissipation overdamped stochastic hes and an underdamped stochastic he although some of these hes can reach the carnot efficiency at nonzero and even diverging power the magnitude of this power is always negligible compared to the maximum power attainable in these systems we provide conditions for attaining the carnot efficiency in the individual models and explain practical aspects connected with reaching the carnot efficiency at large power output furthermore we show how our findings can be tested in practice using a standard brownian he realizable with available micromanipulation techniques

zonal multiscale finite volume framework in this work the zonal multiscale finite volume zmsfv framework is presented the framework generalizes previous extensions of the multiscale finite volume msfv method and allows to integrate additional models in a unified way the new approach is based on splitting the domain of interest into a classical msfv domain and additional zones within each zone the solution space is spanned by so called zonal functions coupling of the zonal solutions to the surrounding domain is achieved via extended zonal functions the construction of basis zonal and extended zonal functions can be performed such that the functions form a partition of unity the zmsfv framework allows to include the influence of the zonal functions directly in the coarse scale problem similar to previous msfv method formulations the zmsfv framework can ensure conservative coarse scale solutions and allows for the construction of conservative fine scale solutions the main features of the zmsfv framework are used to derive robust preconditioners for iterative solutions of high contrast problems as encountered in subsurface flow and transport simulations it is shown that the obtained convergence rates are independent of the contrast

a hybrid global optimization method for inverse estimation of hydraulic parameters annealing simplex method inverse estimation of unsaturated hydraulic parameters is often a highly nonlinear optimization problem with multiple parameters the objective functions involved are often topographically complex and may contain many local minima because of these reasons the inverse solutions are commonly very sensitive to the initial guess of the parameters when conventional optimizers are used this paper presents an annealing simplex method that incorporates simulated annealing strategies into a classical downhill simplex method an upward infiltration experiment was used as an example of inverse estimation to test the method numerical experiments of both minimizing an algebraic function and inversion of upward infiltration data showed that the new method successfully converged to the global minimum in all cases irrespective of the initial hydraulic parameter estimates while the classical downhill method often converged to unfavorable local minima the cpu times needed for the annealing simplex method to estimate 5 and 7 hydraulic parameters simultaneously are about a half hour and 1 hour on a pc respectively additionally no special requirements are imposed on the objective function and the method is independent of the details of the simulation submodel therefore the proposed method should be applicable to other optimization problems in water resources when it is important to have a robust global search capability

algorithms for solving richards equation for variably saturated soils previous work has shown that a one dimensional water content based finite difference algorithm can offer improved cpu efficiency in modeling infiltration into very dry heterogeneous soil when compared to similar pressure based algorithms the usefulness of the water content based method is limited however since it cannot be applied to saturated soils in this paper we develop two new methods which retain the advantages of water content based methods while still permitting fully saturated conditions in the first method we define a new variable for the transformed richards equation which has the characteristics of water content when soil is unsaturated and of pressure when soil is at or near saturation in addition to the transformed richards equation method an improved pressure based method which uses flux updating is presented both methods are implemented in algorithms using a preconditioned conjugate gradient method equation solver the performance of the algorithms is compared to that of an algorithm based on a mixed form of richards equation using modified picard iteration two test cases are examined the first test case examines two dimensional infiltration into very dry heterogeneous soil which remains unsaturated throughout the simulation period the second test case examines a two dimensional developing perched water table results indicate that the new methods retain the advantages of the water content based method for fully unsaturated heterogeneous problems while allowing for fully saturated conditions for the unsaturated problem the new algorithms were from 16 to 59 times faster than the algorithm based on the mixed form of richards equation using modified picard iteration for the saturated problem the new algorithms were from 6 to 25 times faster than the mixed base algorithm results also indicate that the performance of the mixed base algorithm could be significantly improved if a more effective adaptive time step were developed

a stabilized stochastic finite element second order projection method for modeling natural convection in random porous media we consider natural convection in flow saturated porous media with random porosity the porosity is treated as a random field and a stochastic finite element method is developed the stochastic projection method is considered for the solution of the high dimensional stochastic navier stokes equations since it leads to the uncoupling of the velocity and pressure degrees of freedom because of the porosity dependence of the pressure gradient term in the governing flow equations one cannot use the first order projection method a stabilized stochastic finite element second order projection method is presented based on a pressure gradient projection a two dimensional stochastic problem with moderate and large variation in the random porosity field is examined and the results are compared with monte carlo and sparse grid smolyak collocation approaches excellent agreement between these results indicates the effectiveness and accuracy of the proposed methodology

flux formulation of parabolic equations with highly heterogeneous coefficients abstract in this paper we study the flux formulation of unsteady diffusion equations with highly heterogeneous permeability coefficients and their discretization in the proposed approach first an equation governing the flux of the unknown scalar quantity is solved and then the scalar is recovered from its flux the problem for the flux is further discretized by splitting schemes that yield locally one dimensional problems and therefore the resulting linear systems are tridiagonal if the spatial discretization uses cartesian grids a first and a formally second order time discretization splitting scheme have been implemented in both two and three dimensions and we present results for a few model problems using a challenging benchmark dataset

multiscale finite volume method for compressible multiphase flow in porous media the multiscale finite volume msfv method has been recently developed and tested for multiphase flow problems with simplified physics i e incompressible flow without gravity and capillary effects and proved robust accurate and efficient however applications to practical problems necessitate extensions that enable the method to deal with more complex processes in this paper we present a modified version of the msfv algorithm that provides a suitable and natural framework to include additional physics the algorithm consists of four main steps computation of the local basis functions which are used to extract the coarse scale effective transmissibilities solution of the coarse scale pressure equation reconstruction of conservative fine scale fluxes and solution of the transport equations within this framework we develop a msfv method for compressible multiphase flow the basic idea is to compute the basis functions as in the case of incompressible flow such that they remain independent of the pressure the effects of compressibility are taken into account in the solution of the coarse scale pressure equation and if necessary in the reconstruction of the fine scale fluxes we consider three models with an increasing level of complexity in the flux reconstruction and test them for highly compressible flows tracer transport in gas flow imbibition and drainage of partially saturated reservoirs depletion of gas water reservoirs and flooding of oil gas reservoirs we demonstrate that the msfv method provides accurate solutions for compressible multiphase flow problems whereas slightly compressible flows can be treated with a very simple model a more sophisticate flux reconstruction is needed to obtain accurate fine scale saturation fields in highly compressible flows

three dimensional perfectly matched layer for the absorption of electromagnetic waves the perfectly matched layer is a technique of free space simulation developed for solving unbounded electromagnetic problems with the finite difference time domain method referred to as pml this technique has been described in a previous paper for two dimensional problems the present paper is devoted to the three dimensional case the theory of the perfectly matched layer is generalized to three dimensions and some numerical experiments are shown to illustrate the efficiency of this new method of free space simulation

analyses of infiltration events in relation to determining soil hydraulic properties by inverse problem methodology at the field scale because of the relatively large number of observation points required to estimate the spatial distributions of hydraulic properties application of inversion procedures must be based on a characteristic of the flow that for a given point in the field can be measured with relative ease and can be used as input for the inversion procedure suggesting the use of infiltration data for this purpose and recognizing the fact that a basic requirement for the design of a transient flow experiment is that the resulting inverse problem be sufficiently well posed to allow solution this paper addresses the question under which circumstances the resulting inverse problem is well posed utilizing the concepts of identifiability uniqueness and stability for this purpose three different models of the soil hydraulic properties were analyzed the main conclusion of this study is that when infiltration data measured at the darcy scale are used as input for the inversion procedure the inclusion of prior information on a single measurable parameter the saturated conductivity ks in the estimation criterion will enhance the likelihood of uniqueness and stability of the inverse solution provided that the structure of the hydraulic model is sufficiently simple however since for a particular situation it is impossible to determine a priori whether the resultant inverse problem is well posed or not this must be carried out only a posteriori by solving the problem several times with different initial parameter estimates accompanied by an analysis of the associated estimation errors

a finite element collocation method for variably saturated flow in two space dimensions this paper introduces a finite element collocation technique for solving the equation governing two dimensional flow in a variably saturated porous medium the scheme uses a mass conserving formulation of richards equation as the basis for the finite difference time stepping method collocation in tensor product spaces of hermite cubics yields a computationally efficient finite element approximation of the spatial derivatives a newton like iteration gives a temporally stable implicit scheme the paper examines two sample problems including an initial boundary value problem involving subsurface irrigation

a comparison of picard and newton iteration in the numerical solution of multidimensional variably saturated flow problems picard iteration is a widely used procedure for solving the nonlinear equation governing flow in variably saturated porous media the method is simple to code and computationally cheap but has been known to fail or converge slowly the newton method is more complex and expensive on a per iteration basis than picard and as such has not received very much attention its robustness and higher rate of convergence however make it an attractive alternative to the picard method particularly for strongly nonlinear problems in this paper the picard and newton schemes are implemented and compared in one two and three dimensional finite element simulations involving both steady state and transient flow the eight test cases presented highlight different aspects of the performance of the two iterative methods and the different factors that can affect their convergence and efficiency including problem size spatial and temporal discretization initial solution estimates convergence error norm mass lumping time weighting conductivity and moisture content characteristics boundary conditions seepage faces and the extent of fully saturated zones in the soil previous strategies for enhancing the performance of the picard and newton schemes are revisited and new ones are suggested the strategies include chord slope approximations for the derivatives of the characteristic equations relaxing convergence requirements along seepage faces dynamic time step control nonlinear relaxation and a mixed picard newton approach the tests show that the picard or relaxed picard schemes are often adequate for solving richards equation but that in cases where these fail to converge or converge slowly the newton method should be used the mixed picard newton approach can effectively overcome the newton scheme s sensitivity to initial solution estimates while comparatively poor performance is reported for the various chord slope approximations finally given the reliability and efficiency of current conjugate gradient like methods for solving linear nonsymmetric systems the only real drawback of using newton rather than picard iteration is the algebraic complexity and computational cost of assembling the derivative terms of the jacobian matrix and it is suggested that both methods can be effectively implemented and used in numerical models of richards equation

efficient numerical methods for infiltration using richards equation two efficient finite difference methods for solving richards equation in one dimension are presented and their use in a range of soils and conditions is investigated large time steps are possible when the mass conserving mixed form of richards equation is combined with an implicit iterative scheme while a hyperbolic sine transform for the matric potential allows large spatial increments even in dry inhomogeneous soil infiltration in a range of soils can be simulated in a few seconds on a personal computer with errors of only a few percent in the amount and distribution of soil water one of the methods adds points to the space grid as an infiltration or redistribution front advances thus gaining considerably in efficiency over the other fixed grid method for infiltration problems in 17 s computing this advancing front method simulated infiltration redistribution and drainage for 50 days in an inhomogeneous soil with nonuniform initial conditions only 16 space and 21 time steps were needed for the simulation which included early ponding with the development and dissipation of a perched water table

hybrid multiscale finite volume method for two phase flow in porous media we present a novel hybrid or multiphysics algorithm which couples pore scale and darcy descriptions of two phase flow in porous media the flow at the pore scale is described by the navier stokes equations and the volume of fluid vof method is used to model the evolution of the fluid fluid interface an extension of the multiscale finite volume msfv method is employed to construct the darcy scale problem first a set of local interpolators for pressure and velocity is constructed by solving the navier stokes equations then a coarse mass conservation problem is constructed by averaging the pore scale velocity over the cells of a coarse grid which act as control volumes finally a conservative pore scale velocity field is reconstructed and used to advect the fluid fluid interface the method relies on the localization assumptions used to compute the interpolators which are quite straightforward extensions of the standard msfv and on the postulate that the coarse scale fluxes are proportional to the coarse pressure differences by numerical simulations of two phase problems we demonstrate that these assumptions provide hybrid solutions that are in good agreement with reference pore scale solutions and are able to model the transition from stable to unstable flow regimes our hybrid method can naturally take advantage of several adaptive strategies and allows considering pore scale fluxes only in some regions while darcy fluxes are used in the rest of the domain moreover since the method relies on the assumption that the relationship between coarse scale fluxes and pressure differences is local it can be used as a numerical tool to investigate the limits of validity of darcy s law and to understand the link between pore scale quantities and their corresponding darcy scale variables

modeling complex wells with the multi scale finite volume method in this paper an extension of the multi scale finite volume msfv method is devised which allows to simulate flow and transport in reservoirs with complex well configurations the new framework fits nicely into the data structure of the original msfv method and has the important property that large patches covering the whole well are not required for each well an additional degree of freedom is introduced while the treatment of pressure constraint wells is trivial the well bore reference pressure is explicitly specified additional equations have to be solved to obtain the unknown well bore pressure of rate constraint wells numerical simulations of test cases with multiple complex wells demonstrate the ability of the new algorithm to capture the interference between the various wells and the reservoir accurately

multi objective optimization in variably saturated fluid flow a general purpose multi objective inverse modeling strategy was developed and implemented to quantify fluid flow parameters in variably saturated porous materials the strategy combines a robust and mass conservative numerical simulator of the flow equation with an optimization algorithm and an experimental data set to estimate the parameters the numerical simulator of the direct problem shows excellent agreement with a reference solution and conserves global mass with near perfection an adaptive method was proposed in which the sensitivity matrix was calculated by one sided finite difference approximation at the early stages of the optimization and the more accurate two sided differentiation as the search approaches the minimum a combined termination criterium was developed to stop the inverse code at the solution the results of the multi objective optimization were compared with those of single objective minimization while single objective optimization generates reasonable results for either the fluid pressure head profile or the fluid content data the proposed multi objective optimization shows excellent agreement with both profiles simultaneously

a multiscale restriction smoothed basis method for high contrast porous media represented on unstructured grids a wide variety of multiscale methods have been proposed in the literature to reduce runtime and provide better scaling for the solution of poisson type equations modeling flow in porous media we present a new multiscale restricted smoothed basis msrsb method that is designed to be applicable to both rectilinear grids and unstructured grids like many other multiscale methods msrsb relies on a coarse partition of the underlying fine grid and a set of local prolongation operators multiscale basis functions that map unknowns associated with the fine grid cells to unknowns associated with blocks in the coarse partition these mappings are constructed by restricted smoothing starting from a constant a localized iterative scheme is applied directly to the fine scale discretization to compute prolongation operators that are consistent with the local properties of the differential operators the resulting method has three main advantages first of all both the coarse and the fine grid can have general polyhedral geometry and unstructured topology this means that partitions and good prolongation operators can easily be constructed for complex models involving high media contrasts and unstructured cell connections introduced by faults pinch outs erosion local grid refinement etc in particular the coarse partition can be adapted to geological or flow field properties represented on cells or faces to improve accuracy secondly the method is accurate and robust when compared to existing multiscale methods and does not need expensive recomputation of local basis functions to account for transient behavior dynamic mobility changes are incorporated by continuing to iterate a few extra steps on existing basis functions this way the cost of updating the prolongation operators becomes proportional to the amount of change in fluid mobility and one reduces the need for expensive tolerance based updates finally since the msrsb method is formulated on top of a cell centered conservative finite volume method it is applicable to any flow model in which one can isolate a pressure equation herein we only discuss single and two phase incompressible models compressible flow e g as modeled by the black oil equations is discussed in a separate paper

numerical evaluation of iterative and noniterative methods for the solution of the nonlinear richards equation several noniterative procedures for solving the nonlinear richards equation are introduced and compared to the conventional newton and picard iteration methods noniterative strategies for the numerical solution of transient nonlinear equations arise from explicit or linear time discretizations or they can be obtained by linearizing an implicit differencing scheme we present two first order accurate linearization methods a second order accurate two level implicit factored scheme and a second order accurate three level lees method the accuracy and computational efficiency of these four schemes and of the newton and picard methods are evaluated for a series of test problems simulating one dimensional flow processes in unsaturated porous media the results indicate that first order accurate schemes are inefficient compared to second order accurate methods that second order accurate noniterative schemes can be quite competitive with the iterative newton and picard methods and that the newton scheme is no less efficient than the picard method and for strongly nonlinear problems can outperform the picard scheme the two second order accurate noniterative schemes appear to be attractive alternatives to the iterative methods although there are concerns regarding the stability behavior of the three level scheme which need to be resolved we conclude that of the four noniterative strategies presented the implicit factored scheme is the most promising and we suggest improved formulations of the method

the strong approximation property and the weak bounded approximation property abstract we show that the strong approximation property strong ap respectively strong cap and the weak bounded approximation property respectively weak bcap are equivalent for every banach space this gives a negative answer to oja s conjecture as a consequence we show that each of the spaces c 0 and l 1 has a subspace which has the ap but fails to have the strong ap

multi scale finite volume method for elliptic problems in subsurface flow simulation in this paper we present a multi scale finite volume msfv method to solve elliptic problems with many spatial scales arising from flow in porous media the method efficiently captures the effects of small scales on a coarse grid is conservative and treats tensor permeabilities correctly the underlying idea is to construct transmissibilities that capture the local properties of the differential operator this leads to a multi point discretization scheme for the finite volume solution algorithm transmissibilities for the msfv have to be constructed only once as a preprocessing step and can be computed locally therefore this step is perfectly suited for massively parallel computers furthermore a conservative fine scale velocity field can be constructed from the coarse scale pressure solution two sets of locally computed basis functions are employed the first set of basis functions captures the small scale heterogeneity of the underlying permeability field and it is computed in order to construct the effective coarse scale transmissibilities a second set of basis functions is required to construct a conservative fine scale velocity field the accuracy and efficiency of our method is demonstrated by various numerical experiments

enriched galerkin methods for two phase flow in porous media with capillary pressure abstract in this paper we propose an enriched galerkin eg approximation for a two phase pressure saturation system with capillary pressure in heterogeneous porous media the eg methods are locally conservative have fewer degrees of freedom compared to discontinuous galerkin dg and have an efficient pressure solver to avoid non physical oscillations an entropy viscosity stabilization method is employed for high order saturation approximations entropy residuals are applied for dynamic mesh adaptivity to reduce the computational cost for larger computational domains the iterative and sequential implicit pressure and explicit saturation impes algorithms are treated in time numerical examples with different relative permeabilities and capillary pressures are included to verify and to demonstrate the capabilities of eg

fast computation of multiphase flow in porous media by implicit discontinuous galerkin schemes with optimal ordering of elements we present a family of implicit discontinuous galerkin schemes for purely advective multiphase flow in porous media in the absence of gravity and capillary forces to advance the solution one time step one must solve a discrete system of nonlinear equations by reordering the grid cells the nonlinear system can be shown to have a lower triangular block structure where each block corresponds to the degrees of freedom in a single or a small number of cells to reorder the system we view the grid cells and the fluxes over cell interfaces as vertices and edges in a directed graph and use a standard topological sorting algorithm then the global system can be computed by processing the blocks sequentially using a standard newton raphson algorithm for the degrees of freedom in each block decoupling the system offers greater control over the nonlinear solution procedure and reduces the computational costs memory requirements and complexity of the scheme significantly in particular the first order version of the method may be at least as efficient as modern streamline methods when accuracy requirements or the dynamics of the flow allow for large implicit time steps

iterative galerkin enriched multiscale finite volume method the multiscale finite volume msfv method was designed to efficiently compute approximate numerical solutions of elliptic and parabolic problems with highly heterogeneous coefficients for a wide range of test cases it has been demonstrated that msfv solutions are in excellent agreement with reference data obtained with a standard fine scale finite volume method however for some problems involving large coherent structures with strong contrasts e g channels and shale layers the msfv method produces unsatisfactory results due to the dwindling validity of the localization assumption recently the iterative msfv i msfv method was introduced to iteratively improve the localization assumption and it was shown that the method converges to the fine scale reference solution in this work it is explained how the convergence rate of the i msfv method can be enhanced by consistent enrichment of the initial coarse space spanned by nodal basis functions a new set of enrichment basis functions associated with additional coarse scale degrees of freedom dof is introduced by construction the sum of all nodal and enrichment basis functions is one in the entire computational domain further a hybrid finite volume galerkin formulation for the coarse scale problem and a generic placement strategy for the additional dof are devised the resulting iterative galerkin enriched msfv i ge msfv method has all features of the i msfv method but is more robust and has improved convergence properties

parametrization of flow processes in porous media by multiobjective inverse modeling a multiobjective optimization algorithm was developed and applied to parameterize biofluid flow processes in partially saturated porous media the forward problem was formulated as a nonlinear partial differential equation and solved by an efficient galerkin finite element method the numerical simulator was validated with reference and analytical solutions the inverse problem was formulated in a nonlinear optimization framework in which model parameters were estimated by minimizing a complex penalty function representing the discrepancies between the observed and predicted attributes of the physical system a physical model was designed built and operated to collect the experimental datasets needed for model calibration the proposed strategy was then coupled with the fluid pressure head fluid content and fluid flux density time space series to estimate model parameters several optimization scenarios were investigated and it was concluded that while single objective optimization approach has limited accuracy and results in reasonable outputs for one or some attributes of the physical system the proposed multiobjective optimization shows excellent agreements with the experimental datasets for all state variables

corrigendum and addendum to the strong approximation property and the weak bounded approximation property j funct anal 266 2014 5439 5447 abstract we correct a mistake in the proof of 2 theorem 1 1 and a proof of the main result in 2 is presented without depending on 2 theorem 1 1