discretization on non orthogonal quadrilateral grids for inhomogeneous anisotropic media two classes of discretization methods are proposed for control volume formulations on quadrilateral grids in two space dimensions curvilinear grids are considered as a special case the methods are applicable for any system of conservation laws where the flux is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy and or non orthogonality of the grid further properties of the methods will be discussed and examples of their use will be presented

the generalized buckley leverett system solvability we propose a new approach to the mathematical modeling of the buckley leverett system which describes two phase flows in porous media considering the initial boundary value problem for a deduced model we prove the solvability of the problem the solvability result relies mostly on the kinetic method

on existence and uniqueness for a coupled system modeling immiscible flow through a porous medium we consider a system of nonlinear coupled partial differential equations that models immiscible two phase flow through a porous medium a primary difficulty with this problem is its degenerate nature under reasonable assumptions on the data and for appropriate boundary and initial conditions we prove the existence of a weak solution to the problem in a certain sense using a compactness argument this is accomplished by regularizing the problem and proving that the regularized problem has a unique solution which is bounded independently of the regularization parameter we also establish a priori estimates for uniqueness of a solution

h convergence and numerical schemes for elliptic equations we study the convergence of two coupled numerical schemes which are a discretization of a so called elliptic hyperbolic system only weak convergence properties are proved on the discrete diffusion of the elliptic problem and an adaptation of the h convergence method gives a convergence property of the elliptic part of the scheme the limit of the approximate solution is then the solution of an elliptic problem the diffusion of which is not in the general case the h limit of the discrete diffusion in a particular case a kind of weak limit is then obtained for the hyperbolic equation

a nonlinear mixed finite eelement method for a degenerate parabolic equation arising in flow in porous media we study a model nonlinear degenerate advection diffusion equation having application in petroleum reservoir and groundwater aquifer simulation the main difficulty is that the true solution is typically lacking in regularity therefore we consider the problem from the point of view of optimal approximation through time integration we develop a mixed variational form that respects the known minimal regularity and then we develop and analyze two versions of a mixed finite element approximation a simpler semidiscrete time continuous version and a fully discrete version our error bounds are optimal in the sense that all but one of the bounding terms reduce to standard approximation error the exceptional term is a nonstandard approximation error term we also consider our new formulation for the nondegenerate problem showing the usual optimal l 2 error bounds moreover superconvergence is obtained under special circumstances

group analysis and exact solutions of the time fractional fokker planck equation abstract in this paper the lie symmetry analysis method is extended to deal with the nonlinear time fractional fokker planck fp equation with riemann liouville derivative the erdelyi kober fractional derivative which is depending on a parameter is used for the reduction of fp equation symmetry reduction is provided and some exact analytic solutions to the time fractional fp equation are investigated by virtue of the reduction method introduced by m c nucci

measurement selection for vibration based structural damage identification many researchers have intensively investigated the methods of using vibration properties of a structure to identify its damage in the past two decades it has been realized that the modal data included in damage identification analysis have a great influence on the accuracy of identification results in this paper a technique is presented to select a subset of measurement points and corresponding mode to be used in model updating for damage identification a new concept of damage measurability is introduced in terms of two factors namely the sensitivity of a residual vector to the structural damage and the sensitivity of the damage to the measurement noise based on the two factors the structural damage measurability is estimated quantitatively it yields more accurate and reliable damage identification results if the mode and measurement points corresponding to the largest damage measurability value are chosen in field test and numerical analysis the advantage of the technique is that it is based on the undamaged state of structures and thus independent of the damage configuration therefore it is applicable in practice to determine the measurement selection prior to field testing and damage identification analysis a frame structure is used to demonstrate the validity and versatility of the method it demonstrates a significant improvement on the accuracy and reliability of damage identification results when the measurement points and mode are determined according to the present technique

random walk percolation based modeling of two phase flow in porous media breakthrough time and net to gross ratio estimation abstract fluid flow modeling in porous media has many applications in waste treatment hydrology and petroleum engineering in any geological model flow behavior is controlled by multiple properties these properties must be known in advance of common flow simulations when uncertainties are present deterministic modeling often produces poor results percolation and random walk rw methods have recently been used in flow modeling their stochastic basis is useful in dealing with uncertainty problems they are also useful in finding the relationship between porous media descriptions and flow behavior this paper employs a simple methodology based on random walk and percolation techniques the method is applied to a well defined model reservoir in which the breakthrough time distributions are estimated the results of this method and the conventional simulation are then compared the effect of the net to gross ratio on the breakthrough time distribution is studied in terms of shannon entropy use of the entropy plot allows one to assign the appropriate net to gross ratio to any porous medium

mixed finite elements for second order elliptic problems in three variables two families of mixed finite elements one based on simplices and the other on cubes are introduced as alternatives to the usual raviart thomas nedelec spaces these spaces are analogues of those introduced by brezzi douglas and marini in two space variables error estimates inl2 andh s are derived

degenerate two phase incompressible flow iii sharp error estimates this is the third paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic parabolic partial differential system which describes the flow of two incompressible immiscible fluids in porous media in this paper we consider a finite element approximation for this system the elliptic equation for the pressure and velocity is approximated by a mixed finite element method while the degenerate parabolic equation for the saturation is approximated by a galerkin finite element method a fully discrete approximation is analyzed sharp error estimates in energy norms are obtained for this approximation the error analysis does not use any regularization of the saturation equation the error estimates are derived directly from the degenerate equation also the analysis does not impose any restriction on the nature of degeneracy finally it respects the minimal regularity on the solution of the differential system

robust streamline tracing for the simulation of porous media flow on general triangular and quadrilateral grids streamline methods for subsurface flow simulation have received renewed attention as fast alternatives to traditional finite volume or finite element methods key aspects of streamline simulation are the accurate tracing of streamlines and the computation of travel time along individual streamlines in this paper we propose a new streamline tracing framework that enables the extension of streamline methods to unstructured grids composed of triangular or quadrilateral elements and populated with heterogeneous full tensor coefficients the proposed method is based on the mathematical framework of mixed finite element methods which provides approximations of the velocity field that are especially suitable for streamline tracing we identify and implement two classes of velocity spaces the lowest order raviart thomas space low order tracing and the brezzi douglas marini space of order one high order tracing both on triangles and quadrilaterals we discuss the implementation of the streamline tracing method in detail and we investigate the performance of the proposed tracing strategy by means of carefully designed test cases we conclude that for the same computational cost high order tracing is more accurate smaller error in the time of flight and robust less sensitive to grid distortion than low order tracing

degenerate two phase incompressible flow ii regularity stability and stabilization abstract in this paper we analyze a coupled system of highly degenerate elliptic parabolic partial differential equations for two phase incompressible flow in porous media this system involves a saturation and a global pressure or a total flow velocity first we show that the saturation is holder continuous both in space and time and the total velocity is holder continuous in space uniformly in time applying this regularity result we then establish the stability of the saturation and pressure with respect to initial and boundary data from which uniqueness of the solution to the system follows finally we establish a stabilization result on the asymptotic behavior of the saturation and pressure we prove that the solution to the present system converges in appropriate norms to the solution of a stationary system as time goes to infinity an example is given to show typical regularity of the saturation

existence of a solution for two phase flow in porous media the case that the porosity depends on the pressure in this paper we prove the existence of a solution of a coupled system involving a two phase incompressible flow in the ground and the mechanical deformation of the porous medium where the porosity is a function of the global pressure the model is strongly coupled and involves a nonlinear degenerate parabolic equation in order to show the existence of a weak solution we consider a sequence of related uniformly parabolic problems and apply the schauder fixed point theorem to show that they possess a classical solution we then prove the relative compactness of sequences of solutions by means of the frechet kolmogorov theorem this yields the convergence of a subsequence to a weak solution of the parabolic system

degenerate two phase incompressible flow i existence uniqueness and regularity of a weak solution abstract this is the first paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic parabolic partial differential system which describes the flow of two incompressible immiscible fluids in porous media in this paper we first show that this system possesses a weak solution under physically reasonable hypotheses on the data then we prove that this weak solution is unique finally we establish regularity on the weak solution which is needed in the uniqueness proof in particular the holder continuity of the saturation in space and time and the lipschitz continuity of the pressure in space are obtained

numerical analysis of a finite volume scheme for two incompressible phase flow with dynamic capillary pressure abstract in this paper we propose and analyze the convergence of a tpfa two points flux approximation finite volume scheme to approximate the two incompressible phase flow with dynamic capillary pressure in porous media the fully implicit scheme is based on nonstandard approximation on mobilities and capillary pressure on the dual mesh we derive a discrete variational formulation and we present a new result of convergence in a two or three dimensional porous medium in comparison with static capillary pressure the non equilibrium capillary model requires more powerful techniques especially the discrete energy estimates are not standard

an a posteriori error estimate for vertex centered finite volume discretizations of immiscible incompressible two phase flow in this paper we derive an a posteriori error estimate for the numerical approximation of the solution of a system modeling the flow of two incompressible and immiscible fluids in a porous medium we take into account the capillary pressure which leads to a coupled system of two equations parabolic and elliptic the parabolic equation may become degenerate i e the nonlinear diffusion coefficient may vanish over regions that are not known a priori we first show that under appropriate assumptions the energy type norm differences between the exact and the approximate nonwetting phase saturations the global pressures and the kirchhoff transforms of the nonwetting phase saturations can be bounded by the dual norm of the residuals we then bound the dual norm of the residuals by fully computable a posteriori estimators our analysis covers a large class of conforming vertex centered finite volume type discretizations with fully implicit time stepping as an example we focus here on two approaches a mathematical scheme derived from the weak formulation and a phase by phase upstream weighting engineering scheme finally we show how the different error components namely the space discretization error the time discretization error the linearization error the algebraic solver error and the quadrature error can be distinguished and used for making the calculations efficient

improvement on the iterated irs method for structural eigensolutions iterated improved reduced system iirs technique is a model reduction or condensation method by repeatedly updating a transformation matrix an improvement on this dynamic condensation technique is proposed in this paper to modify the iterative transformation matrix and achieve faster convergence meanwhile connection between the present algorithm and the subspace iteration method sim is demonstrated a proof of the convergence property is also presented applications of the method to two numerical examples have demonstrated that the proposed method can obtain the lowest eigensolutions of structures more accurately and efficiently as compared with the current iirs

multidisciplinary interactions in energy and environmental modeling abstract mathematical models have been widely used to understand predict or optimize many complex physical processes from semiconductor or pharmaceutical design to large scale applications such as global weather models to astrophysics in particular simulation of environmental effects of air polution is extensive here we address the need for using similar models to understand the fate and transport of groundwater contaminants and to design in situ remediation strategies three basic problem areas must be addressed in the modeling and simulation of the flow of groundwater contamination one must first obtain an effective model to describe the complex fluid fluid and fluid rock interactions that control the transport of contaminants in groundwater this includes the problem of obtaining accurate reservoir descriptions at various length scales and modeling the effects of this heterogeneity in the reservoir simulators next one must develop accurate discretization techniques that retain the important physical properties of the continuous models finally one should develop efficient numerical solution algorithms that utilize the potential of the emerging computing architectures we will discuss recent advances in each of these three areas

a posteriori error estimates stopping criteria and adaptivity for multiphase compositional darcy flows in porous media in this paper we derive a posteriori error estimates for the compositional model of multiphase darcy flow in porous media consisting of a system of strongly coupled nonlinear unsteady partial differential and algebraic equations we show how to control the dual norm of the residual augmented by a nonconformity evaluation term by fully computable estimators we then decompose the estimators into the space time linearization and algebraic error components this allows to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver when the corresponding error components do not affect significantly the overall error moreover the spatial and temporal error components can be balanced by time step and space mesh adaptation our analysis applies to a broad class of standard numerical methods and is independent of the linearization and of the iterative algebraic solvers employed we exemplify it for the two point finite volume method with fully implicit euler time stepping the newton linearization and the gmres algebraic solver numerical results on two real life reservoir engineering examples confirm that significant computational gains can be achieved thanks to our adaptive stopping criteria already on fixed meshes without any noticeable loss of precision

solutions of the initial value problem for nonlinear fractional ordinary differential equations by the rach adomian meyers modified decomposition method abstract in this paper we present the generalized adomian rach theorem and the generalized rach adomian meyers modified decomposition method for solving multi order nonlinear fractional ordinary differential equations we consider different classes of initial value problems for nonlinear fractional ordinary differential equations including the case of real valued orders and another case of rational valued orders which are solved by the present method this method can treat any analytic nonlinearity the coefficients of the solution in the form of a generalized power series are determined by a convenient recurrence scheme which does not involve integration operations compared with the classic adomian decomposition method

adaptive moving grid methods for two phase flow in porous media a b s t r a c t in this paper we present an application of the moving mesh method for approximating numerical solutions of the two phase flow model in porous media the numerical schemes combine a mixed finite element method and a finite volume method which can handle the nonlinearities of the governing equations in an efficient way the adaptive moving grid method is then used to distribute more grid points near the sharp interfaces which enables us to obtain accurate numerical solutions with fewer computational resources the numerical experiments indicate that the proposed moving mesh strategy could be an effective way to approximate two phase flows in porous media

mathematical analysis of radionuclides displacement in porous media with nonlinear adsorption abstract this paper is devoted to the mathematical analysis of the miscible displacement of a set of radionuclides in a flow occurring in a heterogeneous porous medium the flow is governed by darcy s law and the motion of the chemical species is given by a nonclassical advection diffusion reaction equations system the novelty of the model lies in the adsorption phenomenon that leads to a time derivative of a nonlinear term in these equations a semi discretization method is used to establish the existence of weak solutions to this system uniform l estimates on the solutions are specified

a combined finite volume nonconforming finite element scheme for compressible two phase flow in porous media we propose and analyze a combined finite volume nonconforming finite element scheme on general meshes to simulate the two compressible phase flow in porous media the diffusion term which can be anisotropic and heterogeneous is discretized by piecewise linear nonconforming triangular finite elements the other terms are discretized by means of a cell centered finite volume scheme on a dual mesh where the dual volumes are constructed around the sides of the original mesh the relative permeability of each phase is decentred according the sign of the velocity at the dual interface this technique also ensures the validity of the discrete maximum principle for the saturation under a non restrictive shape regularity of the space mesh and the positiveness of all transmissibilities next a priori estimates on the pressures and a function of the saturation that denote capillary terms are established these stabilities results lead to some compactness arguments based on the use of the kolmogorov compactness theorem and allow us to derive the convergence of a subsequence of the sequence of approximate solutions to a weak solution of the continuous equations provided the mesh size tends to zero the proof is given for the complete system when the density of the each phase depends on its own pressure

numerical algorithm for the time fractional fokker planck equation anomalous diffusion is one of the most ubiquitous phenomena in nature and it is present in a wide variety of physical situations for instance transport of fluid in porous media diffusion of plasma diffusion at liquid surfaces etc the fractional approach proved to be highly effective in a rich variety of scenarios such as continuous time random walk models generalized langevin equations or the generalized master equation to investigate the subdiffusion of anomalous diffusion it would be useful to study a time fractional fokker planck equation in this paper firstly the time fractional the sense of riemann liouville derivative fokker planck equation is transformed into a time fractional ordinary differential equation fode in the sense of caputo derivative by discretizing the spatial derivatives and using the properties of riemann liouville derivative and caputo derivative then combining the predictor corrector approach with the method of lines the algorithm is designed for numerically solving fode with the numerical error o k m i n 1 2 a 2 o h 2 and the corresponding stability condition is got the effectiveness of this numerical algorithm is evaluated by comparing its numerical results for a 1 0 with the ones of directly discretizing classical fokker planck equation some numerical results for time fractional fokker planck equation with several different fractional orders are demonstrated and compared with each other moreover for a 0 8 the convergent order in space is confirmed and the numerical results with different time step sizes are shown

h convergence and numerical schemes for elliptic problems we study the convergence of two coupled numerical schemes which are a discretization of a so called elliptic hyperbolic system only weak convergence properties are proved on the discrete diffusion of the elliptic problem and an adaptation of the h convergence method gives a convergence property of the elliptic part of the scheme the limit of the approximate solution is then the solution of an elliptic problem the diffusion of which is not in the general case the h limit of the discrete diffusion in a particular case a kind of weak limit is then obtained for the hyperbolic equation

discontinuous galerkin finite element convergence for incompressible miscible displacement problems of low regularity in this article we analyze the numerical approximation of incompressible miscible displacement problems with a combined mixed finite element and discontinuous galerkin method under minimal regularity assumptions the main result is that sequences of discrete solutions weakly accumulate at weak solutions of the continuous problem in order to deal with the nonconformity of the method and to avoid overpenalization of jumps across interelement boundaries the careful construction of a reflexive subspace of the space of bounded variation which compactly embeds into l 2 omega and of a lifting operator which is compatible with the nonlinear diffusion coefficient are required an equivalent skew symmetric formulation of the convection and reaction terms of the nonlinear partial differential equation allows us to avoid flux limitation and nonetheless leads to an unconditionally stable and convergent numerical method numerical experiments underline the robustness of the proposed algorithm

a robust linearization scheme for finite volume based discretizations for simulation of two phase flow in porous media in this work we consider a mathematical model for two phase flow in porous media the fluids are assumed immiscible and incompressible and the solid matrix non deformable the mathematical model for the two phase flow is written in terms of the global pressure and a complementary pressure obtained by using the kirchhoff transformation as primary unknowns for the spatial discretization finite volumes have been used more precisely the multi point flux approximation method and in time the backward euler method has been employed we present here a new linearization scheme for the nonlinear system arising after the temporal and spatial discretization we show that the scheme is linearly convergent numerical experiments are presented that sustain the theoretical results

analysis of hp discontinuous galerkin methods for incompressible two phase flow in this paper we prove the convergence of a class of discontinuous galerkin methods for solving the fully coupled incompressible two phase flow problem in the non degenerate case estimates in both the mesh size and the polynomial degrees are obtained numerical convergence rates confirm the theoretical results

two phase flow equations with outflow boundary conditions in the hydrophobic hydrophilic case we introduce an approximation procedure and provide existence results for two phase flow equations in porous media the medium can have hydrophobic and hydrophilic components such that the capillary pressure function is degenerate for extreme saturations our main interest is the outflow boundary condition which models an interface with open space the approximate system introduces standard boundary conditions and can be used in numerical schemes it allows the derivation of maximum principles this is the basis for the derivation of the limiting system in the form of a variational inequality

a constructive method for deriving finite elements of nodal type in this paper we propose an algorithm to derive nodal methods corresponding to various two and three dimensional nonconforming and mixed finite elements we show that this algorithm can be used to obtain several classical schemes as well as some more recently developed schemes and that it leads to a simple proof of unisolvence for these methods finally we use our method to obtain a three dimensional nodal scheme of bdm type

improved methods of estimating critical indices via fractional calculus efficiencies of certain methods for the determination of critical indices from power series expansions are shown to be considerably improved by a suitable implementation of fractional differentiation in the context of the ratio method rm kinship of the modified strategy with the ad hoc shifted rm is established and the advantages are demonstrated further in the course of the estimation of critical points significant betterment of convergence properties of diagonal pade approximants is observed on several occasions by invoking this concept test calculations are performed on i various ising spin 1 2 lattice models for susceptibility series attended with a ferromagnetic phase transition ii complex model situations involving confluent and antiferromagnetic singularities and iii the chain generating functions for self avoiding walks on triangular square and simple cubic lattices

high order finite element methods for time fractional partial differential equations the aim of this paper is to develop high order methods for solving time fractional partial differential equations the proposed high order method is based on high order finite element method for space and finite difference method for time optimal convergence rate o dt 2 a n r is proved for the r 1 th order finite element method r 2

finite element method for the space and time fractional fokker planck equation we develop the finite element method for the numerical resolution of the space and time fractional fokker planck equation which is an effective tool for describing a process with both traps and flights the time fractional derivative of the equation is used to characterize the traps and the flights are depicted by the space fractional derivative the stability and error estimates are rigorously established and we prove that the convergent order is o k 2 alpha h mu where k is the time step size and h the space step size numerical computations are presented which demonstrate the effectiveness of the method and confirm the theoretical claims

modified riemann liouville derivative and fractional taylor series of nondifferentiable functions further results the paper gives some results and improves the derivation of the fractional taylor s series of nondifferentiable functions obtained recently in the form f g h e a h ad g a f g 0 a 1 where e a is the mittag leffier function here one defines fractional derivative as the limit of fractional difference and by this way one can circumvent the problem which arises with the definition of the fractional derivative of constant using riemann liouville definition as a result a modified riemann liouville definition is proposed which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one s in order to support this f taylor series one shows how its first term can be obtained directly in the form of a mean value formula the fractional derivative of the dirac delta function is obtained together with the fractional taylor s series of multivariate functions the relation with irreversibility of time and symmetry breaking is exhibited and to some extent this f taylor s series generalizes the fractional mean value formula obtained a few years ago by kolwantar

a linearization of a backward euler scheme for a class of degenerate nonlinear advection diffusion equations abstract in this paper we propose a linearization of a backward scheme for a class of degenerate nonlinear advection diffusion equations modeling a two phase flow through a porous medium this is done using a taylor expansion of the fractional flow function and of the inverse function of an antiderivative of the diffusion coefficient the coercivity of the matrix obtained in the linearization is shown for a sufficiently small time step

an impes scheme for a two phase flow in heterogeneous porous media using a structured grid abstract we develop a numerical scheme for a two phase immiscible flow in heterogeneous porous media using a structured grid finite element method which has been successfully used for the computation of various physical applications involving elliptic interface equations li et al 2003 2004 chou et al 2010 kwak et al 2010 chang and kwak 2011 the proposed method is based on the implicit pressure explicit saturation procedure to solve the pressure equation we use an ifem based on the rannacher turek rannacher and turek 1992 nonconforming space which is a modification of the work in kwak et al 2010 where broken p 1 nonconforming element of crouzeix raviart 1973 1 was developed for the darcy velocity we apply the mixed finite volume method studied in chou et al 2003 and kwak et al 2010 on the basis of immersed finite element method ifem in this way the darcy velocity of the flow can be computed cheaply locally after we solve the pressure equation the computed darcy velocity is used to solve the saturation equation explicitly the whole procedure can be implemented on a structured grid which is independent of the underlying heterogeneous porous media in fact we apply a new version of multigrid algorithm to solve the pressure equation for which the cpu time grows like o n numerical tests for analytic problems show that our method is almost optimal in all variables problem with sources sinks is also computed and it seems to work well

numerical study of compositional compressible degenerate two phase flow in saturated unsaturated heterogeneous porous media we study the convergence of a combined finite volume nonconforming finite element scheme on general meshes for a partially miscible two phase flow model in anisotropic porous media this model includes capillary effects and exchange between the phases the diffusion term which can be anisotropic and heterogeneous is discretized by piecewise linear nonconforming triangular finite elements the other terms are discretized by means of a cell centered finite volume scheme on a dual mesh the relative permeability of each phase is decentered according to the sign of the velocity at the dual interface the convergence of the scheme is proved thanks to an estimate on the two pressures which allows to show estimates on the discrete time and compactness results in the case of degenerate relative permeabilities a key point in the scheme is to use particular averaging formula for the dissolution function arising in the diffusion term we show also a simulation of hydrogen production in nuclear waste management numerical results are obtained by in house numerical code

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

a combined finite volume nonconforming mixed hybrid finite element scheme for degenerate parabolic problems we propose and analyze a numerical scheme for nonlinear degenerate parabolic convection diffusion reaction equations in two or three space dimensions we discretize the diffusion term which generally involves an inhomogeneous and anisotropic diffusion tensor over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming crouzeix raviart finite element method or using the stiffness matrix of the hybridization of the lowest order raviart thomas mixed finite element method the other terms are discretized by means of a cell centered finite volume scheme on a dual mesh where the dual volumes are constructed around the sides of the original mesh checking the local peclet number we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection dominated case this technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor we prove the convergence of the scheme only supposing the shape regularity condition for the original mesh we use a priori estimates and the kolmogorov relative compactness theorem for this purpose the proposed scheme is robust only 5 point 7 point in space dimension three locally conservative efficient and stable which is confirmed by numerical experiments

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

generalization of the diffusion equation by using the maximum entropy principle by using the so called maximum entropy principle in information theory one derives a generalization of the fokker planck kolmogorov equation which applies when the n first transition moments of the process are proportional to t while the other ones can be neglected

fully discrete finite element analysis of multiphase flow in groundwater hydrology this paper deals with the development and analysis of a fully discrete finite element method for a nonlinear differential system for describing an air water system in groundwater hydrology the nonlinear system is written in a fractional flow formulation i e in terms of a saturation and a global pressure the saturation equation is approximated by a finite element method while the pressure equation is treated by a mixed finite element method the analysis is carried out first for the case where the capillary diffusion coefficient is assumed to be uniformly positive and is then extended to a degenerate case where the diffusion coefficient can be zero it is shown that error estimates of optimal order in the l 2 norm and almost optimal order in the l infty norm can be obtained in the nondegenerate case in the degenerate case we consider a regularization of the saturation equation by perturbing the diffusion coefficient the norm of error estimates depends on the severity of the degeneracy in diffusivity with almost optimal order convergence for nonsevere degeneracy implementation of the fractional flow formulation with various nonhomogeneous boundary conditions is also discussed results of numerical experiments using the present approach for modeling groundwater flow in porous media are reported

convergence of a mfe fv method for two phase flow with applications to heap leaching of copper ores in this paper we describe error estimates for a finite element approximation to partial differential systems describing two phase immiscible flows in porous media with applications to heap leaching of copper ores these approximations are based on mixed finite element mfe methods for the pressure and velocity and finite volume fv for the saturation the fluids are considered incompressible numerical results for heap leaching simulation are presented

error analysis for characteristics based methods for degenerate parabolic problems we consider characteristics based finite element methods for solving nonlinear degenerate advection diffusion equations these equations have applications in the simulation of petroleum reservoirs and groundwater aquifers and in the modeling of free boundary problems standard finite element galerkin methods have been studied for these equations in this paper we analyze the characteristics based finite element methods for them the main difficulty in the analysis is that the equations are degenerate and the solution lacks regularity here we develop a technique that respects the degeneracy and the known minimal regularity this technique is based on the green operator for standard elliptic equations and is developed directly for the degenerate advection diffusion equations we concentrate our analysis on the modified method of characteristics mmoc and one of its variants the modified method of characteristics with adjusted advection mmocaa which conserves mass we derive error estimates in various norms the extension to other variants is discussed the present technique is also applied to nondegenerate problems error estimates previously obtained for the mmoc are derived under much weaker regularity assumptions on the solution and the error estimates for the mmocaa appear new even in the nondegenerate case finally numerical results are presented to show the sharpness of the error estimates derived

the existence of weak solutions to single porosity and simple dual porosity models of two phase incompressible flow abstract it is shown that there exists a weak solution to a degenerate and singular elliptic parabolic partial integro differential system of equations these equations model two phase incompressible flow of immiscible fluids in either an ordinary porous medium or in a naturally fractured porous medium the full model is of dual porosity type though the single porosity case is covered by setting the matrixto fracture flow terms to zero this matrix to fracture flow is modeled simply in terms of fracture quantities that is no distinct matrix equations arise the equations are considered in a global pressure formulation that is justified by appealing to a physical relation between the degeneracy of the wetting fluid s mobility and the singularity of the capillary pressure function in this formulation the elliptic and parabolic parts of the system are separated hence it is natural to consider various boundary conditions including mixed nonhomogeneous saturation dependent ones of the first three types a weak solution is obtained as a limit of solutions to discrete time problems the proof makes no use of the corresponding regularized system the hypotheses required for various earlier results on single porosity systems are weakened so that only physically relevant assumptions are made in particular the results cover the cases of a singular capillary pressure function a pure neumann boundary condition and an arbitrary initial condition

a linear backward euler scheme for the saturation equation regularity results and consistency we consider a linearization of a numerical scheme for the saturation equation or porous medium equation email protected email protected email protected f s email protected email protected k s s 0 through first order expansions of the fractional function f and the inverse of the function k s email protected 0 sk t email protected after a regularization of the porous medium equation we establish a regularity result for the continuous galerkin method and a regularity result for the linearized scheme analogous to the corresponding nonlinear scheme we then show that the linearized scheme is consistent with the nonlinear scheme analyzed in a previous work

on a fully coupled nonlinear parabolic problem modelling miscible compressible displacement in porous media we study a model describing a compressible and miscible displacement in a porous medium it consists of a coupled system of nonlinear parabolic partial differential equations using nonclassical estimates and renormalization tools we prove the existence of relevant weak solutions for the problem this is the first existence result obtained for a transport model containing both the coupling due to the compressibility assumption and the coupling due to the concentration dependent viscosity

positivity preserving finite volume scheme for compressible two phase flows in anisotropic porous media the densities are depending on the physical pressures abstract we are concerned with the approximation of solutions to a compressible two phase flow model in porous media thanks to an enhanced control volume finite element discretization the originality of the methodology consists in treating the case where the densities are depending on their own pressures without any major restriction neither on the permeability tensor nor on the mesh contrary to the ideas of 23 the point of the current scheme relies on a phase by phase sub unpwinding approach so that we can recover the coercivity like property it allows on a second place for the preservation of the physical bounds on the discrete saturation the convergence of the numerical scheme is therefore performed using classical compactness arguments numerical experiments are presented to exhibit the efficiency and illustrate the qualitative behavior of the implemented method

two phase porous media flows with dynamic capillary effects and hysteresis in this paper we obtain the uniqueness of weak solutions for a two phase flow model in a porous medium a particularity of the model is that the dynamic effects and hysteresis are included in the capillary pressure to prove the uniqueness we define an auxiliary elliptic system

space time domain decomposition for two phase flow between different rock types in previous work e ahmed s ali hassan c japhet m kern and m vohralik a posteriori error estimates and stopping criteria for space time domain decomposition for two phase flow between different rock types the smai journal of computational mathematics 5 2019 pp 195 227 https doi org 10 5802 smai jcm 47 a space time domain decomposition method was proposed for two phase ow in a porous medium composed of two different rock types so that the capillary pressure field is discontinuous at the interface between the rocks for this nonlinear and degenerate parabolic problem with nonlinear and discontinuous transmission conditions on the interface the optimized schwarz waveform relaxation method oswr with robin or ventcell transmission conditions was considered a guaranteed and fully computable a posteriori error estimate was derived which in particular took into account the domain decomposition error in this paper we provide a mathematical and numerical analysis of this space time domain decomposition method in the robin case complete numerical approximation is achieved by a finite volume scheme in space and the lowest order discontinuous galerkin method in time we prove the existence of a weak solution of the two phase ow subdomain problem with robin boundary conditions by analyzing the convergence of the finite volume scheme the domain decomposition algorithm is based on the solution of space time nonlinear subdomain problems over the whole time interval allowing for different time steps in dfferent parts of the domain adapted to the physical properties of each subdomain and we show that such an algorithm is well defined numerical experiments on three dimensional problems with different rock types illustrate the performance of the domain decomposition method

holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations abstract we prove interior and boundary holder continuity for bounded weak soluttions of a class of quasilinear parabolic equations whose prototype is u t div u m 1 du p 2 du

analysis of an interior penalty discontinuous galerkin scheme for two phase flow in porous media with dynamic capillary effects we present an interior penalty discontinuous galerkin scheme for a two phase porous media flow model that incorporates dynamic effects in the capillary pressure the approximation of the mass conservation laws is performed in their original formulation without introducing a global pressure we prove the existence of a solution to the emerging fully discrete systems and the convergence of the scheme error estimates are obtained for sufficiently smooth data

convergence of finite volume schemes for triangular systems of conservation laws we consider non strictly hyperbolic systems of conservation laws in triangular form which arise in applications like three phase flows in porous media we device simple and efficient finite volume schemes of godunov type for these systems that exploit the triangular structure we prove that the finite volume schemes converge to weak solutions as the discretization parameters tend to zero some numerical examples are presented one of which is related to flows in porous media

structural damage detection using modal test data references uang j n optimal design of a passive vibration absorber for a truss beam journal of guidance control and dynamics vol 7 no 6 1984 pp 733 739 balas m j active control of flexible systems journal of optimization theory and applications vol 25 no 3 1978 pp 415 436 aubrun j n theory of control structures by low authority controllers journal of guidance control and dynamics vol 3 no 5 1980 pp 444 451 meirovitch l and baruch h control of self adjoint distributed parameter systems journal of guidance control and dynamics vol 5 no 1 1982 pp 60 66 sun c t and wang r t enhancement of frequency and damping in large space structures with extendible members proceedings of the aiaa asme asce ahs 29th structures structural dynamics and materials conference aiaa washington dc 1988 pp 1774 1780 hughes t j r and liu w k implicit explicit finite elements in transient analysis stability theory journal of applied mechanics vol 45 no 2 1978 pp 371 374

dynamic condensation in a damped system through rational selection of primary degrees of freedom abstract various system reduction methods have been proposed to calculate eigenvalues and eigenvectors with computational efficiency in the case of system condensation the method of selecting proper primary degrees of freedom pdofs is crucial because they are closely related to the accuracy and reliability of eigensystem prediction in a reduced system although reasonable selection methods for pdofs have been proposed for undamped systems these methods cannot be directly extended to damped systems we propose a pdof selection method for a damped system the proposed method is based on the ratio of dof wise energy distributions in order to estimate the energy distribution of the structure ritz vectors are obtained using a two sided lanczos algorithm energy distribution matrices are calculated from the obtained ritz vectors then degrees of freedom dofs corresponding to the lowest rayleigh quotients are selected as the pdofs the efficiency and reliability of the proposed method are demonstrated using numerical examples

two compressible immiscible fluids in porous media abstract we consider a model of flow of two compressible and immiscible phases in a three dimensional porous media the equations are obtained by the conservation of the mass of each phase this model is treated in its general form with the whole nonlinear terms the only assumption concerns the dependence of densities on a global pressure we obtain the existence of weak solutions under different kinds of degeneracies of the capillary terms

solution of the multivariate fokker planck equation by using a maximum path entropy principle this paper proposes an approach via the maximum entropy principle in order to determine the nonstationary solutions of the fokker planck equation with time varying coefficients the constraints are not the state moments as usual but their dynamic equations the maximum entropy principle herein utilized is a slight extension of jaynes principle which involves the path entropy of the stochastic process

multicomponent fluid flow by discontinuous galerkin and mixed methods in unfractured and fractured media 1 a discrete fracture model for the flow of compressible multicomponent fluids in homogeneous heterogeneous and fractured media is presented in single phase in the numerical model we combine the mixed finite element mfe and the discontinuous galerkin dg methods we use the cross flow equilibrium concept to approximate the fractured matrix mass transfer the discrete fracture model is numerically superior to the single porosity model and overcomes limitations of the dual porosity models including the use of a shape factor the mfe method provides a direct and accurate approximation for the velocity field which is crucial for the convective terms in the flow equations the dg method associated with a slope limiter is used to approximate the species balance equations this method can capture the sharp moving fronts the calculation of the fracture fracture flux across three and higher intersecting fracture branches is a challenge in this work we provide an accurate approximation of these fluxes by using the mfe formulation numerical examples in unfractured and fractured media illustrate the efficiency and robustness of the proposed numerical model

degenerate two phase porous media flow model with dynamic capillarity abstract in this paper we investigate a degenerate elliptic parabolic system which describes the flow of two incompressible immiscible fluids in porous media and including dynamic effects in the phase pressure difference first for the regularized case the existence and uniqueness of the weak solution are obtained then we let the regularization parameter go to zero to show the existence of weak solutions under degenerate case

a finite volume scheme for nonlinear parabolic equations derived from one dimensional local dirichlet problems in this paper we propose a new method to compute the numerical flux of a finite volume scheme used for the approximation of the solution of the nonlinear partial differential equation u t div q f u u 0 in a 1d 2d or 3d domain the function is supposed to be strictly increasing but some values s such that s 0 can exist the method is based on the solution at each interface between two control volumes of the nonlinear elliptic two point boundary value problem q f 0 with dirichlet boundary conditions given by the values of the discrete approximation in both control volumes we prove the existence of a solution to this two point boundary value problem we show that the expression for the numerical flux can be yielded without referring to this solution furthermore we prove that the so designed finite volume scheme has the expected stability properties and that its solution converges to the weak solution of the continuous problem numerical results show the increase of accuracy due to the use of this scheme compared to some other schemes

an existence result for a coupled system modeling a fully equivalent global pressure formulation for immiscible compressible two phase flow in porous media abstract a new model describing immiscible compressible two phase flow such as water gas through heterogeneous porous media is considered the main feature of this model is the introduction of a new global pressure and the full equivalence to the original equations the resulting equations are written in a fractional flow formulation and lead to a coupled system which consists of a nonlinear parabolic equation the global pressure equation and a nonlinear diffusion convection one the saturation equation under some realistic assumptions on the data we show an existence result with the help of appropriate regularizations and a time discretization we use suitable test functions to get a priori estimates in order to pass to the limit in nonlinear terms we also obtain compactness results which are nontrivial due to the degeneracy of the system

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

a fokker planck equation of fractional order with respect to time by combining the maximum entropy principle with some considerations related to derivatives of fractional order one is led to suggest a fokker planck of fractional order with respect to time which could be related to dynamical systems subject to fractional brownian motion the relation with the process associated with the equation p t 1 n 1 2np x2n is exhibited

vertex approximate gradient discretization preserving positivity for two phase darcy flows in heterogeneous porous media abstract in this article a new nodal discretization is proposed for two phase darcy flows in heterogeneous porous media the scheme combines the vertex approximate gradient vag scheme for the approximation of the gradient fluxes with an hybrid upwind hu approximation of the mobility terms in the saturation equation the discretization in space incorporates naturally nodal interface degrees of freedom d o f allowing to capture the transmission conditions at the interface between different rock types for general capillary pressure curves it is shown to guarantee the physical bounds for the saturation unknowns as well as a nonnegative lower bound on the capillary energy flux term numerical experiments on several test cases exhibit that the scheme is more robust compared with previous approaches allowing the simulation of 3d large discrete fracture matrix dfm models

flow of oil and water in a porous medium abstract the flow of two immiscible and incompressible fluids in a porous medium is described by a system of quasilinear degenerate partial differential equations in this paper the existence of a weak solution by regularization is shown

approximation of degenerate parabolic problems using numerical integration a class of multidimensional degenerate parabolic equations is considered the two phase stefan problem and the porous medium equation are analyzed as models of singular parabolic equations nonstat

optimal tracking of parameter drift in a chaotic system experiment and theory we present a method of optimal tracking for chaotic dynamical systems with a slowly drifting parameter the net drift in the parameter is assumed to be small this makes detecting and tracking the drift more difficult the method relies on the existence of underlying deterministic behavior in the dynamical system yet neither requires a system model nor develops one we begin by describing an experimental study where a heuristic optimality criterion gave good tracking performance the tracking method there was based on maximizing smoothness and overall variation in the drift observer which was found by solving an igenvalue problem we then develop a theory based on simplifying assumptions about the chaotic dynamics to explain the success of the tracking method for chaotic systems for signals from deterministic systems that are sufficiently complex in a sense that we make precise typical drift observers provide poor tracking performance and require the drift to be particularly slow in contrast our theory shows that the optimality criterion seeks out a special drift observer that both provides better tracking performance and allows the drift to be appreciably faster for periodic or quasiperiodic systems no chaos good tracking is easily achievable and the present method is irrelevant for stochastic systems no determinism the optimal tracking method does not asymptotically improve tracking performance exhaustive numerical simulations of a simple drifting chaotic map first without and then with stochastic forcing show agreement with theoretical predictions of tracking performance and validate the theory

adaptive mixed hybrid and penalty discontinuous galerkin method for two phase flow in heterogeneous media in this paper we present a hybrid method which consists of a mixed hybrid finite element method and a penalty discontinuous galerkin method for the approximation of a fractional flow formulation of a two phase flow problem in heterogeneous media with discontinuous capillary pressure the fractional flow formulation is comprised of a wetting phase pressure equation and a wetting phase saturation equation which are coupled through a total velocity and the saturation affected coefficients for the wetting phase pressure equation the continuous mixed hybrid finite element method space can be utilized due to a fundamental property that the wetting phase pressure is continuous while it can reduce the computational cost by using less degrees of freedom and avoiding the post processing of velocity reconstruction this method can also keep several good properties of the discontinuous galerkin method which are important to the fractional flow formulation such as the local mass balance continuous normal flux and capability of handling the discontinuous capillary pressure for the wetting phase saturation equation the penalty discontinuous galerkin method is utilized due to its capability of handling the discontinuous jump of the wetting phase saturation furthermore an adaptive algorithm for the hybrid method together with the centroidal voronoi delaunay triangulation technique is proposed five numerical examples are presented to illustrate the features of proposed numerical method such as the optimal convergence order the accurate and efficient velocity approximation and the applicability to the simulation of water flooding in oil field and the oil trapping or barrier effect phenomena

regularization of outflow problems in unsaturated porous media with dry regions we study a porous medium with saturated unsaturated and dry regions described by richards equation for the saturation s and the pressure p due to a degenerate permeability coefficient k x s and a degenerate capillary pressure function pc x s the equations may be of elliptic parabolic or of ode type we construct a parabolic regularization of the equations and find conditions that guarantee the convergence of the parabolic solutions to a solution of the degenerate system an example shows that the convergence fails in general our approach provides an existence result for the outflow problem in the case of x dependent coefficients and a method for a numerical approximation

an input output based procedure for fully evaluating and monitoring dynamic properties of structural systems via a subspace identification method abstract dynamic behaviour of complex structural systems may be modelled by a system of second order linear ordinary differential equations i e m t d t sw t f t by means of either structural analysis for finite degree of freedom systems or discretization procedures e g fe methods for continuous systems here w t and f t are the displacement vector and the force vector owing to erosion friction and internal damage and cracks etc a working process of a system always accompanies gradual degradation of the performance of this system the stiffness of the system weakens whereas the damping of the system strengthens to evaluate such degradation the usual way is to model the evolution of property of a system obtain system property parameters trace the history of motion and loading carry out complicated analysis and computation under prescribed initial and boundary value conditions and finally derive the degraded property and responses of the system this traditional way however might be cumbersome and unsatisfactory in some cases due to the lack of adequate experimental data and well founded theoretical basis etc another way is to apply inverse methods such as modal analysis methods with fft and a subspace identification method etc developed in the theory of system identification which extracts information about system properties directly from experimental input output measurement data and hence do not involve the foregoing traditional analysis the latter method however could not supply full information about system properties due to the assumption of the black box viewpoint in this work with suitable experimental input output measurement data a simple effective procedure is described by which the stiffness matrix s and the damping matrix d may be determined in a complete unique manner using a subspace identification method the possibility of such a procedure arises from the observation of the self evident fact the conservation of mass of any part of a structural system implies that the mass matrix m of this system is constant and hence is given by its initial value the stiffness and damping matrices s and d determined by the proposed procedure may be used to evaluate and monitor in a full sense the degradation of dynamic properties of structural systems further with the information about the stiffness distribution of constituent elements of a structural system it is shown that it may be possible to estimate the locations of the damaged or faulty elements in this system an example is given to illustrate the application of the proposed procedure