a finite volume scheme for two phase immiscible flow in porous media in this paper we prove the convergence of a numerical method for solving two phase immiscible incompressible flow in porous media the method combines an upwind time implicit finite volume scheme for the saturation equation hyperbolic parabolic type and a centered finite volume scheme for the chavent global pressure equation elliptic type the capillary pressure is not neglected and we study the case when the diffusion term in the saturation equation is weakly degenerated estimates on the approximate solution are proven then by using compactness theorems we obtain a limit when the size of the discretization goes to zero and we prove that this limit is the unique weak solution of the problem that we study

optimality conditions for henig and globally proper efficient solutions with ordering cone has empty interior in this paper we consider the set valued vector optimization problems with constraint in locally convex spaces we present the necessary and sufficient conditions for henig efficient solution pair globally proper efficient solution pair and super efficient solution pair without the ordering cones having the nonempty interior

the fermat rule for multifunctions on banach spaces using variational analysis we study vector optimization problems with objectives being closed multifunctions on banach spaces or in asplund spaces in particular in terms of the coderivatives we present fermat s rules as necessary conditions for an optimal solution of the above problems as applications we also provide some necessary conditions in terms of clarke s normal cones or the limiting normal cones for pareto efficient points

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

control volume approximation of degenerate two phase porous flows implicit euler approximations of the equations governing the porous flow of two immiscible incompressible fluids are shown to be the euler lagrange equations of a convex function tools from conve

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

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

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

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

finite volume methods for degenerate chemotaxis model a finite volume method for solving the degenerate chemotaxis model is presented along with numerical examples this model consists of a degenerate parabolic convection diffusion pde for the density of the cell population coupled to a parabolic pde for the chemoattractant concentration it is shown that discrete solutions exist and the scheme converges

relative pareto minimizers for multiobjective problems existence and optimality conditions in this paper we introduce and study enhanced notions of relative pareto minimizers for constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of pareto and weak pareto efficiency minimality using advanced tools of variational analysis and generalized differentiation we establish the existence of relative pareto minimizers for general multiobjective problems under a refined version of the subdifferential palais smale condition for set valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers as well as for conventional efficient and weak efficient counterparts that are new in both finite dimensional and infinite dimensional settings our proofs are based on variational and extremal principles of variational analysis in particular on new versions of the ekeland variational principle and the subdifferential variational principle for set valued and single valued mappings in infinite dimensional spaces

super efficiency of vector optimization in banach spaces using variational analysis in terms of the clarke normal cone we consider super efficiency of vector optimization in banach spaces we establish some characterizations for super efficiency in particular dropping the assumption that the ordering cone has a bounded base we extend a result in borwein and zhuang j m borwein d zhuang super efficiency in vector optimization trans amer math soc 338 1993 105 122 to the nonconvex setting 2006 elsevier inc all rights reserved

large scale averaging analysis of single phase flow in fractured reservoirs the method of large scale averaging is introduced to derive and analyze the most general form of the dual porosity model of single phase flow in naturally fractured reservoirs the dual porosity model contains the usual equations based on darcy s law and the coupling terms representing the fluid transfer between the matrix and the fractures a transient closure problem is developed to obtain and analyze the fracture and matrix permeability tensors and the fluid transfer terms the spatial averaging theorem is presented from a mathematical point of view and proved rigorously by the distribution theory the problem of wellposedness of the dual porosity model is also considered the techniques developed here are not restricted to either regular geometric fractures or spatially periodic reservoirs

sensitivity results for a general class of generalized vector quasi equilibrium problems with set valued maps abstract in this paper we give sufficient conditions for the local openness local closedness properties and the lower upper semicontinuity properties of the solution sets of a general model which includes as special cases many generalized vector quasi equilibrium problems with set valued maps the obtained results generalize and improve several known results an application is also given for a model which can be interpreted as a system of generalized vector quasi equilibrium problems

sensitivity in mixed generalized vector quasiequilibrium problems with moving cones in this paper we consider a parametric generalized vector quasiequilibrium problem which is mixed in the sense that several different relations can simultaneously appear in this problem the moving cones and other data of the problem are assumed to be set valued maps defined in topological spaces and taking values in topological spaces or topological vector spaces the main result of this paper gives general verifiable conditions for the solution mapping of this problem to be semicontinuous with respect to a parameter varying in a topological space the result is proven with the help of notions of cone semicontinuity of set valued maps weaker than the usual concepts of semicontinuity and an assumption imposed on the set valued map whose values are the dual cones of the corresponding values of the moving cones

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

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

generalized vector quasi equilibrium problems with applications in this paper we consider the generalized vector quasi equilibrium problem with or without involving condensing maps and prove the existence of its solution by using known fixed point and maximal element theorems as applications of our results we derive some existence results for a solution to the vector quasi optimization problem for nondifferentiable functions and vector quasi saddle point problem

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

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

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

towards a definition of the quantum ergodic hierarchy ergodicity and mixing abstract in a previous paper we have given a general framework for addressing the definition of quantum chaos by identifying the conditions that a quantum system must satisfy to lead to non integrability in its classical limit in this paper we will generalize those results with the purpose of defining the two lower levels of the quantum ergodic hierarchy ergodicity and mixing we will also argue for the physical relevance of this approach by considering a particular example where our formalism has been successfully applied

optimality conditions for vector equilibrium problems in this paper we present the necessary and sufficient conditions for weakly efficient solution henig efficient solution globally efficient solution and superefficient solution to the vector equilibrium problems with constraints as applications we give the necessary and sufficient conditions for corresponding solution to the vector variational inequalities and vector optimization problems

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

existence results for doubly nonlinear higher order parabolic equations on unbounded domains on demontre un resultat d existence pour le probleme aux valeurs limites et initiales u q 1 sign u t a 1 u f sur 0 t q u satisfait des conditions de dirichlet sur 0 t u u 0 sur pour t 0 a 1 u 1 m m d d u p 1 sign d u pour p q 1 m n 1 et r n un ensemble ouvert

semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems abstract in this paper we establish sufficient conditions for the solution set of parametric multivalued vector quasiequilibrium problems to be semicontinuous all kinds of semicontinuity are considered lower semicontinuity upper semicontinuity hausdorff upper semicontinuity and closedness moreover we investigate both the weak and strong solutions of quasiequilibrium problems

scalarization and optimality conditions for vector equilibrium problems abstract in this paper we investigated vector equilibrium problems and gave the scalarization results for weakly efficient solutions henig efficient solutions and globally efficient solutions to the vector equilibrium problems without the convexity assumption using nonsmooth analysis and the scalarization results we provided the necessary conditions for weakly efficient solutions henig efficient solutions globally efficient solutions and superefficient solutions to vector equilibrium problems by the assumption of convexity we gave sufficient conditions for those solutions as applications we gave the necessary and sufficient conditions for corresponding solutions to vector variational inequalities and vector optimization problems

on the efficacy of a control volume finite element method for the capture of patterns for a volume filling chemotaxis model in this paper a control volume finite element scheme for the capture of spatial patterns for a volume filling chemotaxis model is proposed and analyzed the diffusion term which generally involves an anisotropic and heterogeneous diffusion tensor is discretized by piecewise linear conforming triangular finite elements p1 fem the other terms are discretized by means of an upstream finite volume scheme on a dual mesh where the dual volumes are constructed around the vertices of each element of the original mesh the scheme ensures the validity of the discrete maximum principle under the assumption that the transmissibility coefficients are nonnegative the convergence analysis is based on the establishment of a priori estimates on the cell density these estimates lead to some compactness arguments in l 2 based on the use of the kolmogorov compactness theorem finally we show some numerical results to illustrate the effectiveness of the scheme to capture the pattern formation for the mathematical model

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

error estimates for a mixed finite element discretization of a two phase porous media flow model with dynamic capillarity abstract we analyze a fully discrete numerical scheme for the model describing two phase immiscible flow in porous media with dynamic effects in the capillary pressure we employ the euler implicit method for the time discretization the spatial discretization is based on the mixed finite element method mfem specifically the lowest order raviart thomas elements are applied in this paper the error estimates for the saturation fluxes and phase pressures in l 0 t l 2 are derived for the temporal and spatial discretization to show the convergence of the scheme finally we present some numerical results to support the theoretical findings

analysis of large scale averaged models for two phase flow in fractured reservoirs the method of large scale averaging has been recently applied to derive a general dual porosity model of two phase flow in naturally fractured reservoirs the present paper is aimed to develop mathematical theory behind the derivation of this model in particular two volume averaging theorems which are key mathematical tools in the model derivation are proved rigorously using distribution theory existence and uniqueness of a weak solution to a simplied dual porosity model are shown

variational inequalities for a system of elliptic parabolic equations abstract we create a general framework for mathematical study of variational inequalities for a system of elliptic parabolic equations in this paper we establish a solvability theorem concerning the existence of solutions for the vector valued elliptic parabolic variational inequality with time dependent constraint moreover we give some applications of the system for example time dependent boundary obstacle problem and time dependent interior obstacle problem

a natural normalization for the eigenstates of a hamiltonian with continuous spectrum abstract a mathematical formalism that allows to deal with many problems on quantum systems with continuous evolution spectrum is presented the usual hilbert space is generalized to a prehilbert one t where singular states can be represented and an extended dirac s notation can be introduced the obtained formalism contains the van hove one but in a more natural way it allows to explain decoherence and other phenomena

perturbative method for generalized spectral decompositions abstract imposing analytic properties to states and observables we construct a perturbative method to obtain a generalized biorthogonal system of eigenvalues and eigenvectors for quantum unstable systems a decay process can be described using this generalized spectral decomposition and the final generalized state is obtained

the classical statistical limit of quantum mechanics abstract the classical statistical limit of quantum mechanics is studied it is proved that the limit 0 coincides with that limit for the operator algebra but it is not so for the state convex set in the last case decoherence must be invoked to obtain the classical statistical limit

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

the lagrange multiplier rule for super efficiency in vector optimization in this paper we study constrained multiobjective optimization problems with objectives being closed graph multifunctions in banach spaces in terms of the coderivatives and clarke s normal cones we establish lagrange multiplier rules for super efficiency as necessary or sufficient optimality conditions of the above problems

on optimality conditions for vector variational inequalities abstract optimality conditions for weak efficient global efficient and efficient solutions of vector variational inequalities with constraints defined by equality cone and set constraints are derived under various constraint qualifications necessary optimality conditions for weak efficient global efficient and efficient solutions in terms of the clarke and michel penot subdifferentials are established with assumptions on quasiconvexity of constraint functions sufficient optimality conditions are also given

new nonlinear scalarization functions and applications abstract in this paper new nonlinear scalarization functions which are different from the gerstewitz function are introduced some properties of these functions are discussed and are used to prove new results on the existence of solutions of generalized vector quasi equilibrium problems with moving cones and the lower semicontinuity of solution mappings of parametric vector quasi equilibrium problems detailed comparisons between our results and those obtained by using the gerstewitz function for existence theorems and by other approaches for the case of solution stability are given illustrating examples are provided

approximate duality for vector quasi equilibrium problems and applications abstract in this paper we introduce new versions of dual problems of a vector quasi equilibrium problem with set valued maps and we give an duality result between approximate solutions of the primal and dual problems as the first application of the main result we obtain an duality for a vector quasi equilibrium problem whose solutions are understood in the sense of proper efficiency the second application is devoted to an duality for a vector optimization problem with set valued maps

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

upper semicontinuity in a parametric general variational problem and application in this paper we give sufficient conditions for the upper semicontinuity property of the solution mapping of a general model which includes many generalized vector quasi equilibrium problems with set valued maps as special cases the main result generalizes and improves several recent results an example is given to illustrate such generalization and improvement the main result is also applied to a model which can be interpreted as a system of generalized vector quasi equilibrium problems with moving cones the main tools of the paper are some new notions of cone semicontinuity properties and openness closedness properties of families of set valued maps

convergence of a finite volume scheme for nonlinear degenerate parabolic equations one approximates the entropy weak solution u of a nonlinear parabolic degenerate equation u t rm div mathbf q f u delta phi u 0 by a piecewise constant function u mathcal d using a discretization mathcal d in space and time and a finite volume scheme the convergence of u mathcal d to u is shown as the size of the space and time steps tend to zero in a first step estimates on u mathcal d are used to prove the convergence up to a subsequence of u mathcal d to a measure valued entropy solution called here an entropy process solution a result of uniqueness of the entropy process solution is proved yielding the strong convergence of u mathcal d to it u some on a model equation are shown

an extended extremal principle with applications to multiobjective optimization we develop an extended version of the extremal principle in variational analysis that can be treated as a variational counterpart to the classical separation results in the case of nonconvex sets and which plays an important role in the generalized differentiation theory and its applications to optimization related problems the main difference between the conventional extremal principle and the extended version developed below is that instead of the translation of sets involved in the extremal systems we allow deformations the new version seems to be more flexible in various applications and covers in particular multiobjective optimization problems with general preference relations in this way we obtain new necessary optimality conditions for constrained problems of multiobjective optimization with nonsmooth data and also for multiplayer multiobjective games

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