a semi analytic porosity evolution scheme for simulating wormhole propagation with the darcy brinkman forchheimer model abstract in this paper we consider numerical simulation of wormhole propagation with the darcy brinkman forchheimer model matrix acidization in carbonate reservoirs is a widely practiced technique in the product enhancement of the oil and gas reservoir a wormhole i e a flow channel with high porosity is generated during reactive dissolution of carbonates by the action of the injected acid in the wormhole forming process the porosity changes non uniformly in space and it even becomes close to unity in the central regions of a wormhole the darcy brinkman forchheimer model accounts for both the porous media and clear fluid area so it can be used to model wormhole propagation perfectly this model however strongly depends on porosity therefore the time schemes for solving the evolutionary equation of porosity have a significant effect on accuracy and stability of the numerical simulation for wormhole propagation we propose a semi analytic time scheme which solves the porosity equation analytically at each time step for given acid concentration the proposed numerical method can improve the accuracy and stability of numerical simulation significantly for theoretical analysis of the proposed time scheme for the wormhole simulation we first reconstruct the analytical functions of porosity to analyze the time error of the porosity and on the basis of error estimates of porosity we employ a coupled analysis approach to achieve the estimates of pressure velocity and solute concentration the time error estimates for velocity pressure concentration and porosity are obtained in different norms finally numerical results are provided to verify the effectiveness of the proposed scheme

fully mass conservative impes schemes for incompressible two phase flow in porous media abstract in this paper we consider fully mass conservative numerical schemes for the simulation of incompressible and immiscible two phase flow in porous media with capillary pressure compared with two kinds of conventional implicit pressure explicit saturation impes schemes the new schemes deserve a merit that the conservation of mass of both phases can be obtained the total conservation equation is obtained by the summation of the discretized conservation equation for each phase this approach is quite different from the conventional impes schemes we present two kinds of fully mass conservative impes schemes to solve the coupled systems for pressure auxiliary velocity and saturation of each phase the upwind mixed finite element methods are used to solve the pressure velocity systems which can be decoupled and the problems in the decoupled systems can be proved to be well posed moreover the new schemes are unbiased and the saturation of each phase can be proved to be bounds preserving if the time step size is smaller than a certain value the new schemes can also be applied to approximate the incompressible and immiscible two phase flow in heterogeneous porous media with different capillarity pressures several interesting examples of incompressible and immiscible two phase flow in porous media are presented to demonstrate the robustness of the new algorithms

comparison of fully implicit and impes formulations for simulation of water injection in fractured and unfractured media in this work we compare the fully implicit fi and implicit pressure explicit saturation impes formulations for the simulation of water injection in fractured media the system of partial differential equations is discretized within the discrete fracture framework using a control volume method a unique feature of the methodology is that there is no need for the computation of matrix fracture transfer terms the non linear system of equations resulting from the fi formulation is solved with state of the art newton and tensor methods direct and krylov iterative methods are employed to solve the system resulting from the newton linearization the performance of the fi and impes formulations is compared with numerical testing results show that the contrast between matrix and fracture properties affects the performance of both impes and fi formulations and that the tensor method outperforms all the newton solvers for the near singular jacobian matrix copyright 2006 john wiley sons ltd

energy norm a posteriori error estimates for mixed finite element methods this paper deals with the a posteriori error analysis of mixed finite element methods for second order elliptic equations it is shown that a reliable and efficient error estimator can be constructed using a postprocessed solution of the method the analysis is performed in two different ways under a saturation assumption and using a helmholtz decomposition for vector fields

iterative operator splitting methods with higher order time integration methods and applications for parabolic partial differential equations in this paper we design higher order time integrators for systems of stiff ordinary differential equations we combine implicit runge kutta and bdf methods with iterative operator splitting methods to obtain higher order methods the idea of decoupling each complicated operator in simpler operators with an adapted time scale allows to solve the problems more efficiently we compare our new methods with the higher order fractional stepping runge kutta methods developed for stiff ordinary differential equations the benefit is the individual handling of each operator with adapted standard higher order time integrators the methods are applied to equations for convection diffusion reactions and we obtain higher order results finally we discuss the applications of the iterative operator splitting methods to multi dimensional and multi physical problems

an elementary proof of the polar factorization of vector valued functions we present an elementary proof of an important result of y brenier br1 br2 namely that vector fields in d satisfying a nondegeneracy condition admit the polar factorization u x s x where is a convex function and s is a measure preserving mapping brenier solves a minimization problem using monge kantorovich theory whereas we turn our attention to a dual problem whose euler lagrange equation turns out to be

existence and uniqueness of monotone measure preserving maps introduction given a pair of borel probability measures and v on rd it is natural to ask whether v can be obtained from by redistributing its mass in a canonical way in the case of the line d 1 the answer is clear as long as both measures are free from atoms x v x 0mthere is a map y x of the line to itself for which uniquely determined almost everywhere this map may be taken to be nondecreasing by a suitable choice of y x e rw o at the remaining points interpreting and v as the initial and final distribution of a one dimensional fluid the transformation y x gives a rearrangement of fluid particles yielding final state v from the initial state this rearrangement is characterized by the fact that it preserves particle ordering obviating any need for two particles to cross although the generalization of this construction to higher dimensions is the focus of this paper the one dimensional case will be pursued slightly further when the measures are absolutely continuous with respect to lebesgue d x f x dx and dr y g y dymthen formally at least neglecting regularity issues the fundamental theorem of calculus yields

imex extensions of linear multistep methods with general monotonicity and boundedness properties for solving hyperbolic systems with stiff sources or relaxation terms time stepping methods should combine favorable monotonicity properties for shocks and steep solution gradients with good stability properties for stiff terms in this paper we consider implicit explicit imex multistep methods suitable methods will be constructed based on explicit methods with general monotonicity and boundedness properties for hyperbolic equations numerical comparisons are made with several implicit explicit runge kutta methods

constrained variational problem with applications to the ising model we continue our study of the behavior of the two dimensional nearest neighbor ferromagnetic ising model under an external magnetic fieldh initiated in our earlier work we strengthen further a result previously proven by martirosyan at low enough temperature which roughly states that for finite systems with boundary conditions under a positive external field the boundary effect dominates in the system if the linear size of the system is of orderb h withb small enough while ifb is large enough then the external field dominates in the system in our earlier work this result was extended to every subcritical value of the temperature here for every subcritical value of the temperature we show the existence of a critical valueb0 t which separates the two regimes specified above we also find the asymptotic shape of the region occupied by the phase in the second regime which turns out to be a squeezed wulff shape the main step in our study is the solution of the variational problem of finding the curve minimizing the wulff functional which curve is constrained to the unit square other tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field extended to all temperatures below the critical one

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

active set reduced space methods with nonlinear elimination for two phase flow problems in porous media fully implicit methods are drawing more attention in scientific and engineering applications due to the allowance of large time steps in extreme scale simulations when using a fully implicit method to solve two phase flow problems in porous media one major challenge is the solution of the resultant nonlinear system at each time step to solve such nonlinear systems traditional nonlinear iterative methods such as the class of the newton methods often fail to achieve the desired convergent rate due to the high nonlinearity of the system and or the violation of the boundedness requirement of the saturation in the paper we reformulate the two phase model as a variational inequality that naturally ensures the physical feasibility of the saturation variable the variational inequality is then solved by an active set reduced space method with a nonlinear elimination preconditioner to remove the high nonlinear components that often causes the failure of the nonlinear iteration for convergence to validate

the regularity of mappings with a convex potential in this work we apply the techniques developed in cl to the problem of mappings with a convex potential between domains that is given two bounded domains q q2 of rn and two nonnegative real functions fi defined in qi that are bounded away from zero and infinity we want to study the map vf v w for a lipschitz convex v such that v maps ql onto q 2 in the a e sense and in some weak sense 1 f2 vyv det dij v f1 x in recent work y brenier showed existence and uniqueness of such a map provided that jaqil 0 under the obvious compatibility condition

a posteriori error estimate for the mixed finite element method a computable error bound for mixed finite element methods is established in the model case of the poisson problem to control the error in the h div x l 2 norm the reliable and efficient a posteriori error estimate applies e g to raviart thomas brezzi douglas marini and brezzi douglas fortin marini elements

a mass conservative generalized multiscale finite element method applied to two phase flow in heterogeneous porous media in this paper we propose a method for the construction of locally conservative flux fields through a variation of the generalized multiscale finite element method gmsfem the flux values are obtained through the use of a ritz formulation in which we augment the resulting linear system of the continuous galerkin cg formulation in the higher order gmsfem approximation space in particular we impose the finite volume based restrictions through incorporating a scalar lagrange multiplier for each mass conservation constraint this approach can be equivalently viewed as a constraint minimization problem where we minimize the energy functional of the equation restricted to the subspace of functions that satisfy the desired conservation properties to test the performance of the method we consider equations with heterogeneous permeability coefficients that have high variation and discontinuities and couple the resulting fluxes to a two phase flow model the increase in accuracy associated with the computation of the gmsfem pressure solutions is inherited by the flux fields and saturation solutions and is closely correlated to the size of the reduced order systems in particular the addition of more basis functions to the enriched multiscale space produces solutions that more accurately capture the behavior of the fine scale model a variety of numerical examples are offered to validate the performance of the method

multiphase thermomechanics with interfacial structure 2 evolution of an isothermal interface paper 1 1988 g 1 of this series began an investigation whose goal is a thermomechanics of two phase continua based on gibbs s notion of a sharp phase interface endowed with thermomechanical structure in that paper a balance law balance of capillary forces was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics the chief results are thermodynamic restrictions on constitutive equations exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems the simplest versions of these problems the mullins sekerka problems are essentially the classical stefan problem with the free boundary condition u 0 for the temperature replaced by the condition u h k where k is the curvature of the free boundary and h 0 is a material constant this dependence on curvature renders the problem difficult and apart from numerical studies involving linearization stability there are almost no supporting theoretical results

an a posteriori error estimate for the variable degree raviart thomas method we propose a new a posteriori error analysis of the variable degree hybridized version of the raviart thomas method for second order elliptic problems on conforming meshes made of simplexes we establish both the reliability and eciency of the estimator for the l2 norm of the error of the ux we also nd the explicit dependence of the estimator on the order of the local spaces k 0 the only constants that are not explicitly computed are those depending on the shape regularity of the simplexes in particular the constant of the local eciency in equality is proven to behave like k 2 3 2 however we present numer ical experiments suggesting that such constant is actually independent of k

equilibrium shapes for planar crystals in an external field the equilibrium shape of a two dimensional crystal in a convex background potential g x is analyzed for g 0 the shape of minimum energy may be deduced from surface tension via the wulff construction but if g is not constant little is known beyond the case of a crystal sitting in a uniform field only an unpublished result of okikiolu shows each connected component of the equilibrium crystal to be convex here it will be shown that any such component minimizes energy uniquely among convex sets of its area if the wulff shape and g x are symmetric under x x it follows that the equilibrium crystal is unique convex and connected this last result leads to a new proof that convex crystals away from equilibrium remain convex as they evolve by curvature driven flow subsequent work with felix otto shows without assuming symmetry that no equilibrium crystal has more than two convex components

the wulff theorem revisited the parametrized indicator measures and the brunn minkowski inequality are used to prove that the wulff set w is a minimizer for the surface energy where the density is the support function of w the support of the indicator measures associated to minimizing sequences is characterized it is shown that if w is polyhedral then minimizing sequences cannot oscillate

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

a uniqueness proof for the wulff theorem the wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of small elastic crystals it consists in minimising the generally anisotropic surface energy among sets of given volume a solution of this problem is given by a geometric construction due to wulff in the class of sets of finite perimeter this was first shown by j e taylor who using methods of geometric measure theory also proved uniqueness here a more analytic uniqueness proof is presented the main ingredient is a sharpened version of the brunn minkowski inequality

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

analysis of operator splitting for advection diffusion reaction problems from air pollution modelling abstract operator or time splitting is often used in the numerical solution of initial boundary value problems for differential equations it is for example standard practice in computational air pollution modelling where we encounter systems of three dimensional time dependent partial differential equations of the advection diffusion reaction type for such systems little attention has been devoted to the analysis of splitting and to the question why splitting can work so well from the theoretical point of view the success of splitting is primarily determined by the splitting error this paper presents an analysis of operator splitting aimed at providing insight into the splitting error using the lie operator formalism a general expression is derived for a three term strang splitting in the pure initial value case for a class of advection diffusion reaction problems the splitting error is analyzed in greater detail a special case is discussed in which the splitting error can be reduced also some attention is paid to the use of operator splitting in initial boundary value problems

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

curvature driven flows a variational approach this paper introduces a new mathematical approach to the study of time evolutions of solids k t in n space whose boundaries move with velocity equal to the weighted mean curvature derived from the boundary surface energy phi partial k t these flat phi curvature flows are limits of sequences of solutions to variational problems in which a sum of surface and bulk energy is minimized the construction works equally well for smooth elliptic phi s for nondifferentiable crystalline phi s and for anything in between the flows agree with classical smooth flows when the data is smooth and elliptic in any dimension and coincide with motion by crystalline curvature for polyhedral curves in the plane

linearly decoupled energy stable numerical methods for multicomponent two phase compressible flow in this paper for the first time we propose two linear decoupled energy stable numerical schemes for multicomponent two phase compressible flow with a realistic equation of state e g peng ro

discontinuous galerkin approximation of two phase flows in heterogeneous porous media with discontinuous capillary pressures we design and investigate a sequential discontinuous galerkin method to approximate two phase immiscible incompressible flows in heterogeneous porous media with discontinuous capillary pressures the nonlinear interface conditions are enforced weakly through an adequate design of the penalties on interelement jumps of the pressure and the saturation an accurate reconstruction of the total velocity is considered in the raviart thomas nedelec finite element spaces together with diffusivity dependent weighted averages to cope with degeneracies in the saturation equation and with media heterogeneities the proposed method is assessed on one dimensional test cases exhibiting rough solutions degeneracies and capillary barriers stable and accurate solutions are obtained without limiters

iterative solution methods for modeling multiphase flow in porous media fully implicitly we discuss several fully implicit techniques for solving the nonlinear algebraic system arising in an expanded mixed finite element or cell centered finite difference discretization of two and three phase porous media flow every outer nonlinear newton iteration requires solution of a nonsymmetric jacobian linear system two major types of preconditioners supercoarsening multigrid scmg and two stage are developed for the gmres iteration applied to the solution of the jacobian system the scmg reduces the three dimensional system to two dimensions using a vertical aggregation followed by a two dimensional multigrid the two stage preconditioners are based on decoupling the system into a pressure and concentration equations several pressure preconditioners of different types are described extensive numerical results are presented using the integrated parallel reservoir simulator ipars and indicate that these methods have low arithmetical complexity per iteration and good convergence rates

convergence of discontinuous galerkin methods for incompressible two phase flow in heterogeneous media a class of discontinuous galerkin methods with interior penalties is presented for incompressible two phase flow in heterogeneous porous media with capillary pressures the semidiscrete approximate schemes for fully coupled system of two phase flow are formulated in highly heterogeneous permeable media the saturation is discontinuous due to different capillary pressures and therefore the proposed methods incorporate the capillary pressures in the pressure equation instead of saturation equation by introducing a coupling approach for stability and error estimates instead of the conventional separate analysis for pressure and saturation the stability of the schemes in space and time and a priori hp error estimates are presented in the l 2 h 1 for pressure and in the l infty l 2 and l 2 h 1 for saturation two time discretization schemes are introduced for effectively computing the discrete solutions