fully implicit mixed hybrid finite element discretization for general purpose subsurface reservoir simulation abstract we present a new fully implicit mixed hybrid finite element mhfe discretization scheme for general purpose compositional reservoir simulation the locally conservative scheme solves the coupled momentum and mass balance equations simultaneously and the fluid system is modeled using a cubic equation of state we introduce a new conservative flux approach for the mass balance equations for this fully implicit approach we discuss the nonlinear solution procedure for the proposed approach and we present extensive numerical tests to demonstrate the convergence and accuracy of the mhfe method using tetrahedral elements we also compare the method to other advanced discretization schemes for unstructured meshes and tensor permeability finally we illustrate the applicability and robustness of the method for highly heterogeneous reservoirs with unstructured grids

stable cell centered finite volume discretization for biot equations in this paper we discuss a new discretization for the biot equations the discretization treats the coupled system of deformation and flow directly as opposed to combining discretizations for the two separate subproblems the coupled discretization has the following key properties the combination of which is novel 1 the variables for the pressure and displacement are co located and are as sparse as possible e g one displacement vector and one scalar pressure per cell center 2 with locally computable restrictions on grid types the discretization is stable with respect to the limits of incompressible fluid and small time steps 3 no artificial stabilization term has been introduced furthermore due to the finite volume structure embedded in the discretization explicit local expressions for both momentum balancing forces and mass conservative fluid fluxes are available we prove stability of the proposed method with respect to all relevant limits together with consistency this proves conve

cell centered finite difference method for parabolic equation abstract we develop two cell centered finite difference schemes for parabolic problems on quadrilateral grids one scheme is backward euler scheme with first order accuracy in time increment while the other is crank nicolson scheme with second order accuracy in time increment the method is based on the lowest order brezzi douglas marini bdm mixed finite element method a quadrature rule gives a block diagonal mass matrix and allows for local velocity elimination which leads to a symmetric and positive definite cell centered system for the pressures theoretical results indicate first order convergence in spacial meshsize both for pressures and for subedge flues numerical experiments using the schemes show that the convergence rates are in agreement with the theoretical analysis

a global jacobian method for mortar discretizations of nonlinear porous media flows we describe a nonoverlapping domain decomposition algorithm for nonlinear porous media flows discretized with the multiscale mortar mixed finite element method there are two main ideas 1 linearize the global system in both subdomain and interface variables simultaneously to yield a single newton iteration and 2 algebraically eliminate subdomain velocities and optionally subdomain pressures to solve linear systems for the 1st or the 2nd schur complements solving the 1st schur complement system gives the multiscale solution without the need to solve an interface iteration solving the 2nd schur complement system gives a linear interface problem for a nonlinear model the methods are less complex than a previously developed nonlinear mortar algorithm which requires two nested newton iterations and a forward difference approximation furthermore efficient linear preconditioners can be applied to speed up the iteration the methods are implemented in parallel and a numerical study is performed

pores resolving simulation of darcy flows a theoretical formulation and corresponding numerical solutions are presented for microscopic fluid flows in porous media with the domain sufficiently large to reproduce integral darcy scale effects pore space geometry and topology influence flow through media but the difficulty of observing the configurations of real pore spaces limits understanding of their effects a rigorous direct numerical simulation dns of percolating flows is a formidable task due to intricacies of internal boundaries of the pore space representing the grain size distribution by means of repelling body forces in the equations governing fluid motion greatly simplifies computational efforts an accurate representation of pore scale geometry requires that within the solid the repelling forces attenuate flow to stagnation in a short time compared to the characteristic time scale of the pore scale flow in the computational model this is achieved by adopting an implicit immersed boundary method with the attenuation time scale smaller than the time step of an explicit fluid model a series of numerical simulations of the flow through randomly generated media of different porosities show that computational experiments can be equivalent to physical experiments with the added advantage of nearly complete observability besides obtaining macroscopic measures of permeability and tortuosity numerical experiments can shed light on the effect of the pore space structure on bulk properties of darcy scale flows

residual based a posteriori error estimation for multipoint flux mixed finite element methods a novel residual type a posteriori error analysis technique is developed for multipoint flux mixed finite element methods for flow in porous media in two or three space dimensions the derived a posteriori error estimator for the velocity and pressure error in l 2 l2 norm consists of discretization and quadrature indicators and is shown to be reliable and efficient the main tools of analysis are a locally postprocessed approximation to the pressure solution of an auxiliary problem and a quadrature error estimate numerical experiments are presented to illustrate the competitive behavior of the estimator

a multipoint flux mixed finite element method we develop a mixed finite element method for single phase flow in porous media that reduces to cell centered finite differences on quadrilateral and simplicial grids and performs well for discontinuous full tensor coefficients motivated by the multipoint flux approximation method where subedge fluxes are introduced we consider the lowest order brezzi douglas marini bdm mixed finite element method a special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell centered system for the pressures theoretical and numerical results indicate second order convergence for pressures at the cell centers and first order convergence for subedge fluxes second order convergence for edge fluxes is also observed computationally if the grids are sufficiently regular

first order convergence of multi point flux approximation on triangular grids and comparison with mixed finite element methods in this paper we show first order convergence of a multi point flux approximation control volume method mpfa on unstructured triangular grids in this approach the flux approximation is derived directly in the physical space in order to do this we introduce a perturbed mixed finite element method that is equivalent to the mpfa scheme and prove the first order convergence of this approach moreover we carefully compare the computational performance properties of the mpfa method with those of a lowest order raviart thomas and brezzi douglas marini mixed finite element approximation

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

the numerical solution of diffusion problems in strongly heterogeneous non isotropic materials a new second order finite difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non isotropic media is constructed on problems with rough coefficients or highly nonuniform grids the new algorithm is superior to all other algorithms we have compared it with for problems with smooth coefficients on smooth grids the method is comparable with other second order methods the new algorithm is formulated for logically rectangular grids and is derived using thesupport operators method a key idea in deriving the method was to replace the usual inner product of vector functions by an inner product weighted by the inverse of the material properties tensor and to use the flux operator defined as the material properties tensor times the gradient rather than the gradient as one of the basic first order operators in the support operators method the discrete analog of the flux operator must also be the negative adjoint of the discrete divergence in an inner product that is a discrete analog of the continuum inner product the resulting method is conservative and the discrete analog of the variable coefficient laplacian is symmetric and negative definite on nonuniform grids in addition on any grid the discrete divergence is zero on constant vectors the null space for the gradient is the constant functions and when the material properties are piecewise constant the discrete flux operator is exact for piecewise linear functions we compare the methods on some of the most difficult examples to be found in the literature

a family of mpfa finite volume schemes with full pressure support for the general tensor pressure equation on cell centered triangular grids a new family of cell centered finite volume schemes is presented for solving the general full tensor pressure equation of subsurface flow in porous media on arbitary unstructured triangulations the new schemes are flux continuous and have full pressure support fps over each subcell with continuous pressure imposed across each control volume sub interface in contrast to earlier formulations the earlier methods are point wise continuous in pressure and flux with triangle pressure support tps which leads to a more limited quadrature range an m matrix analysis identifies bounding limits for the schemes to posses a local discrete maximum principle conditions for the schemes to be positive definite are also derived a range of computational examples are presented for unstructured triangular grids including highly irregular grids and the new fps schemes are compared against the earlier pointwise continuous tps formulations the earlier pointwise tps methods can induce strong spurious oscillations for problems involving strong full tensor anisotropy where the m matrix conditions are violated and can lead to decoupled solutions in such cases unstructured cell centered decoupling is investigated in contrast to tps the new fps formulation leads to well resolved solutions that are essentially free of spurious oscillations a substantial degree of improved convergence behavior for both pressure and velocity is also observed in all convergence tests this is particularly important for problems involving high anisotropy ratios also the new formulation proves to be highly beneficial for an upscaling example where enhancement of convergence is highly significant for certain quadrature points clearly demonstrating further advantages of the new formulation

a block centered finite difference method for an unsteady asymptotic coupled model in fractured media aquifer system abstract a block centered finite difference method is proposed for solving an unsteady asymptotic coupled model in which the flow is governed by darcy s law both in the one dimensional fracture and two dimensional porous media the second order error estimates in discrete norms are derived on nonuniform rectangular grids for both pressure and velocity the numerical scheme can be extended to nonmatching spatial and temporal grids without loss of accuracy numerical experiments are performed to verify the efficiency and accuracy of the proposed method it is shown that the pressure and velocity are discontinuous across the fracture interface and the fracture indeed acts as the fast pathway or geological barrier in the aquifer system

a general framework for constructing and analyzing mixed finite volume methods on quadrilateral grids the overlapping covolume case we present a general framework for constructing and analyzing finite volume methods applied to the mixed formulation of second order elliptic problems on quadrilateral grids the control volumes or covolumes in the grids overlap an overlapping finite volume method of this type was first introduced by russell in t f russell tech report 3 reservoir simulation research corp tulsa ok 1995 and was tested for a variety of problems on rectangular and quadrilateral grids in z cai et al comput geosci 1 1997 pp 289 315 later in s h chou and d y kwak siam j numer anal 37 2000 pp 758 771 chou and kwak reformulated it as their mixed covolume method and proved optimal order error estimates using the covolume methodology from s h chou math comp 66 1997 pp 85 104 and s h chou and d y kwak siam j numer anal 35 1998 pp 494 507 however their treatment was restricted to the case of diagonal coefficient tensor and rectangular grids since a different approach was needed for the quadrilateral distorted rectangular case in this paper we give a new framework which can handle not only the rectangular anisotropic case but also the anisotropic and irregular grid cases in which the locally supported test functions are images of the natural unit coordinate vectors under the piola transformation our theory sheds light on how to create new test functions using quadratures and now covers russell s quadrilateral case

a coupled multipoint stress multipoint flux mixed finite element method for the biot system of poroelasticity abstract we present a mixed finite element method for a five field formulation of the biot system of poroelasticity that reduces to a cell centered pressure displacement system on simplicial and quadrilateral grids a mixed stress displacement rotation formulation for elasticity with weak stress symmetry is coupled with a mixed velocity pressure darcy formulation the spatial discretization is based on combining the multipoint stress mixed finite element msmfe method for elasticity and the multipoint flux mixed finite element mfmfe method for darcy flow it uses the lowest order brezzi douglas marini mixed finite element spaces for the poroelastic stress and darcy velocity piecewise constant displacement and pressure and continuous piecewise linear or bilinear rotation a vertex quadrature rule is applied to the velocity stress and stress rotation bilinear forms which block diagonalizes the corresponding matrices and allows for local velocity stress and rotation elimination this leads to a cell centered positive definite system for pressure and displacement at each time step we perform error analysis for the semidiscrete and fully discrete formulations establishing first order convergence for all variables in their natural norms the numerical tests confirm the theoretical convergence rates and illustrate the locking free property of the method

superconvergence of the velocity in mimetic finite difference methods on quadrilaterals superconvergence of the velocity is established for mimetic finite difference approximations of second order elliptic problems over h 2 uniform quadrilateral meshes the superconvergence result holds for a full tensor coefficient the analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space the theoretical results are confirmed by numerical experiments

mixed finite element methods on nonmatching multiblock grids we consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids a mortar finite element space is introduced on the nonmatching interfaces we approximate in this mortar space the trace of the solution and we impose weakly a continuity of flux condition a standard mixed finite element method is used within the blocks optimal order convergence is shown for both the solution and its flux moreover at certain discrete points superconvergence is obtained for the solution and also for the flux in special cases computational results using an efficient parallel domain decomposition algorithm are presented in confirmation of the theory

finite volume methods for elasticity with weak symmetry we introduce a new cell centered finite volume discretization for elasticity with weakly enforced symmetry of the stress tensor the method is motivated by the need for robust discretization methods for deformation and flow in porous media and falls in the category of multi point stress approximations mpsa by enforcing symmetry weakly the resulting method has additional flexibility beyond previous mpsa methods this allows for a construction of a method which is applicable to all grid types and in particular the method amends a crucial shortcoming in previous mpsa methods for simplex grids by formulating the method as a discrete variational problem we prove convergence of the new method for a wide range of problems with conditions that can be verified at the time of discretization we present the first set of comprehensive numerical tests for the mpsa methods in three dimensions covering cartesian and simplex grids with both heterogeneous and nearly incompressible media the tests show that the new method consistently is second order convergent in displacement despite being lowest order with a rate that mostly is between 1 and 2 for stresses the results further show that the new method is more robust and computationally cheaper than previous mpsa methods

quadrilateral h div finite elements we consider the approximation properties of quadrilateral finite element spaces of vector fields defined by the piola transform extending results previously obtained for scalar approximation the finite element spaces are constructed starting with a given finite dimensional space of vector fields on a square reference element which is then transformed to a space of vector fields on each convex quadrilateral element via the piola transform associated to a bilinear isomorphism of the square onto the element for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l2 is that each component of the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms the situation is more complicated and we give a precise characterization of what is needed for optimal order l2 approximation of the function and of its divergence as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for some standard finite element approximations of h div we also derive new estimates for approximation by quadrilateral raviart thomas elements requiring less regularity and propose a new quadrilateral finite element space which provides optimal order approximation in h div finally we demonstrate the theory with numerical computations of mixed and least squares finite element approximations of the solution of poisson s equation

a posteriori error estimates for the mortar mixed finite element method several a posteriori error estimators for mortar mixed finite element discretizations of elliptic equations are derived a residual based estimator provides optimal upper and lower bounds for the pressure error an efficient and reliable estimator for the velocity and mortar pressure error is also derived which is based on solving local element problems in a higher order space the interface flux jump term that appears in the estimators can be used as an indicator for driving an adaptive process for the mortar grids only

enhanced cell centered finite differences for elliptic equations on general geometry we present an expanded mixed finite element method for solving second order elliptic partial differential equations on geometrically general domains for the lowest order raviart thomas approximating spaces we use quadrature rules to reduce the method to cell centered finite differences possibly enhanced with some face centered pressures this substantially reduces the computational complexity of the problem to a symmetric positive definite system for essentially only as many unknowns as elements our new method handles general shape elements triangles quadrilaterals and hexahedra and full tensor coefficients while the standard mixed formulation reduces to finite differences only in special cases with rectangular elements as in other mixed methods we maintain the local approximation of the divergence i e local mass conservation in contrast galerkin finite element methods facilitate general element shapes at the cost of achieving only global mass conservation our method is shown to be as accurate as the standard mixed method for a large class of smooth meshes on nonsmooth meshes or with nonsmooth coefficients one can add lagrange multiplier pressure unknowns on certain element edges or faces this enhanced cell centered procedure recovers full accuracy with little additional cost if the coefficients or mesh geometry are piecewise smooth theoretical error estimates and numerical examples are given illustrating the accuracy and efficiency of the methods

mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences we present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient the mixed method is expanded in the sense that three variables are explicitly approximated namely the scalar unknown the negative of its gradient and its flux the tensor coefficient times the negative gradient the resulting linear system is a saddle point problem in the case of the lowest order raviart thomas elements on rectangular parallelepipeds we approximate this expanded mixed method by incorporating certain quadrature rules this enables us to write the system as a simple cell centered finite difference method requiring the solution of a sparse positive semidefinite linear system for the scalar unknown for a general tensor coefficient the sparsity pattern for the scalar unknown is a 9 point stencil in two dimensions and 19 points in three dimensions existing theory shows that the expanded mixed method gives optimal order approximations in the l 2 and h s norms and superconvergence is obtained between the l 2 projection of the scalar variable and its approximation we show that these rates of convergence are retained for the finite difference method if h denotes the maximal mesh spacing then the optimal rate is o h the superconvergence rate o h 2 is obtained for the scalar unknown and rate o h 3 2 for its gradient and flux in certain discrete norms moreover the full o h 2 is obtained in the strict interior of the domain computational results illustrate these theoretical results

a mass lumped mixed finite element method for acoustic wave propagation we consider the numerical approximation of acoustic wave propagation in time domain by a mixed finite element method based on the text bdm 1 text p 0 spaces a mass lumping strategy for the text bdm 1 element originally proposed by wheeler and yotov in the context of subsurface flow is utilized to enable an efficient integration in time by this mass lumping strategy the accuracy of the mixed method is formally reduced to first order we will show however that the numerical approximation still carries global second order information which is expressed in the super convergence of the numerical approximation to certain projections of the true solution based on this fact we propose post processing strategies for the pressure and the velocity leading to piecewise linear approximations with second order accuracy a complete convergence analysis is provided for the semi discrete and corresponding fully discrete approximations which result from time discretization by the leapfrog method the efficiency of the proposed strategy is illustrated also in numerical tests

postprocessing of non conservative flux for compatibility with transport in heterogeneous media a conservative flux postprocessing algorithm is presented for both steady state and dynamic flow models the postprocessed flux is shown to have the same convergence order as the original flux an

block minimum perturbation algorithm based on block arnoldi process for nonsymmetric linear systems with multiple right hand sides abstract in many applications in scientific computing and engineering one has to solve huge sparse linear systems of equations with several right hand sides customarily one uses the residual error as a stopping condition but small residue does not imply accurate approximate solutions therefore minimizing quantities other than the residual norm may be more adequate in this context based on the above consideration a block minimum perturbation krylov subspace method bminpet for the solution xm satisfying a a x m b b at each step of nonsymmetric linear systems a x b is shown such that the normwise backward error meets some optimality condition with minimizing the block joint backward perturbation norm a b f of the matrix a b and a stopping criterion is given for consistent system with multiple right hand sides to get a more robust way to stop the iterations this process is a generalization of minimum perturbation algorithm minpert by kasenally and simoncini 1997 analysis of a minimum perturbation algorithm for nonsymmetric linear systems siam j numer anal 34 48 66 to multiple right hand sides for the benefit of the amount of calculation we give lower and upper bounds about min a b f that are easier to compute as a by product we come up with a conception of function f g which is a generalization of function that is a symmetric gauge sg function the algorithm is used to analyze the performance of bgmres method by evaluating the proximity of their solutions to block joint backward perturbation optimality the relationships between bminpet method and relative methods are investigated numerical examples show us that bminpert shows its superiority compared with bfgmres s m pf gsgmres bl bicg rq bgmres and barnoldi in solving the large sparse ill conditioned problems

stability of multirate explicit coupling of geomechanics with flow in a poroelastic medium abstract we consider single rate and multirate explicit schemes for the biot system modeling coupled flow and geomechanics in a poro elastic medium these schemes are widely used in practice that follows a sequential procedure in which the flow and mechanics problems are fully decoupled in such a scheme the flow problem is solved first with time lagging the displacement term followed by the mechanics solve the multirate explicit coupling scheme exploits the different time scales for the mechanics and flow problems by taking multiple finer time steps for flow within one coarse mechanics time step we provide fully discrete schemes for both the single and multirate approaches that use backward euler time discretization and mixed spaces for flow and conformal galerkin for mechanics we perform a rigorous stability analysis and derive the conditions on reservoir parameters and the number of finer flow solves to ensure stability for both schemes furthermore we investigate the computational time savings for explicit coupling schemes against iterative coupling schemes

a multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra in this paper we develop a new mixed finite element method for elliptic problems on general quadrilateral and hexahedral grids that reduces to a cell centered finite difference scheme a special non symmetric quadrature rule is employed that yields a positive definite cell centered system for the pressure by eliminating local velocities the method is shown to be accurate on highly distorted rough quadrilateral and hexahedral grids including hexahedra with non planar faces theoretical and numerical results indicate first order convergence for the pressure and face fluxes

a locally conservative enriched galerkin approximation and efficient solver for elliptic and parabolic problems we present and analyze an enriched galerkin finite element method eg to solve elliptic and parabolic equations with jump coefficients eg is formulated by enriching the conforming continuous galerkin finite element method cg with piecewise constant functions which can be considered as a penalty stabilization the method is shown to be locally and globally conservative while keeping fewer degrees of freedom in comparison with discontinuous galerkin finite element methods dg moreover we present and analyze a fast effective eg solver whose cost is roughly that of cg and which can handle an arbitrary order of approximations a number of numerical tests in two and three dimensions are presented to confirm our theoretical results as well as to demonstrate the advantages of eg when coupled with transport

quasi positive families of continuous darcy flux finite volume schemes on structured and unstructured grids new families of flux continuous control volume distributed finite volume schemes are presented for the general full tensor pressure equation arising in porous media and formulated for structured and unstructured grids these schemes offer the practical advantage of being flux continuous while only depending on one degree of freedom per control volume unlike rival approximations such as the mixed finite element method m matrix bounds are presented quasi qm matrices are defined and an optimal quadrilateral scheme is identified anisotropy favoring triangulation is also shown to yield an optimal scheme the new schemes prove to be relatively robust for the cases tested including strongly anisotropic full tensor fields strong oscillations encountered with the earlier formulations are removed or minimized

a quasi positive family of continuous darcy flux finite volume schemes with full pressure support a new family of flux continuous locally conservative finite volume schemes is presented for solving the general tensor pressure equation of subsurface flow in porous media the new schemes have full pressure continuity imposed across control volume faces previous families of flux continuous schemes are point wise continuous in pressure and flux when applying the earlier point wise flux continuous schemes to strongly anisotropic full tensor fields their failure to satisfy a maximum principle as with other fem and finite volume methods can result in loss of local stability for high anisotropy ratios which can cause strong spurious oscillations in the numerical pressure solution an m matrix analysis reveals the upper limits for guaranteeing a maximum principle for general 9 point schemes and aids in the design of schemes that minimize the occurrence of spurious oscillations in the discrete pressure field the full pressure continuity schemes are shown to possess a larger range of flux continuous schemes than the previous point wise counter parts for strongly anisotropic full tensor cases it is shown that the full quadrature range possessed by the new schemes permits these schemes to exploit quadrature points previously out of range that are shown to minimize spurious oscillations in discrete pressure solutions the new formulation leads to a more robust quasi positive family of flux continuous schemes applicable to general discontinuous full tensor fields

a family of linearity preserving schemes for anisotropic diffusion problems on arbitrary polyhedral grids a family of cell centered finite volume schemes are proposed for anisotropic diffusion problems on arbitrary polyhedral grids with planar facets the derivation of the schemes is done under a general framework through a certain linearity preserving approach the key ingredient of our algorithm is to employ solely the so called harmonic averaging points located at the cell interfaces to define the auxiliary unknowns which not only makes the interpolation procedure for auxiliary unknowns simple and positivity preserving but also reduces the stencil of the schemes the final schemes are cell centered with a small stencil of 25 point on the structured hexahedral grids moreover the schemes satisfy the local conservation condition treat discontinuity exactly and allow for a simple stability analysis a second order accuracy in the l2 norm and a first order accuracy in the h1 norm are observed numerically on general distorted meshes in case that the diffusion tensor is anisotropic and discontinuous

a block centered finite difference method for reduced fracture model in karst aquifer system abstract a block centered finite difference scheme is given for the approximation of reduced coupled model in the fractured media aquifer system the fluid flow of aquifer system is governed by darcy s law in both fracture and surrounding porous media the degrees of freedom are given separately on both sides of fracture to capture the discontinuity of pressure and velocity optimal error estimates in discrete l 2 norms are obtained numerical experiments are provided to verify the second order accuracy of presented method it is demonstrated whether the fracture acts as a fast pathway or geological barrier is totally determined by the value of its permeability tensor

a multipoint flux mixed finite element method on hexahedra we develop a mixed finite element method for elliptic problems on hexahedral grids that reduces to cell centered finite differences the paper is an extension of our earlier paper for quadrilateral and simplicial grids m f wheeler and i yotov siam j numer anal 44 2006 pp 2082 2106 the construction is motivated by the multipoint flux approximation method and it is based on an enhancement of the lowest order brezzi douglas duran fortin bddf mixed finite element spaces on hexahedra in particular there are four fluxes per face one associated with each vertex a special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell centered system for the pressures theoretical and numerical results indicate first order convergence for pressures and subface fluxes on sufficiently regular grids as well as second order convergence for pressures at the cell centers second order convergence for face fluxes is also observed computationally

local flux mimetic finite difference methods we develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes to approximate the velocity vector variable the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n gonal mesh face in three dimensions to approximate the pressure scalar variable the method uses one degree of freedom per element a specially chosen quadrature rule for the l 2 product of vector functions allows for a local flux elimination and reduction of the method to a cell centered finite difference scheme for the pressure unknowns under certain assumptions first order convergence is proved for both variables and second order convergence is proved for the pressure the assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method for general polyhedral meshes non symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions numerical results confirm the theory

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

discretization on unstructured grids for inhomogeneous anisotropic media part i derivation of the methods discretization methods are proposed for control volume formulations on polygonal and triangular grid cells in two space dimensions the methods are applicable for any system of conservation laws where the flow density 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 this paper gives a derivation of the methods and the relation to previously published methods is also discussed a further discussion of the methods including numerical examples is given in the companion paper part ii siam j sci comput pp 1717 1736

a fully coupled multiphase flow and geomechanics solver for highly heterogeneous porous media abstract this paper introduces a fully coupled multiphase flow and geomechanics solver that can be applied to modeling highly heterogeneous porous media multiphase flow in deformable porous media is a multiphysics problem that considers the flow physics and rock physics simultaneously to model this problem the multiphase flow equations and geomechanical equilibrium equation must be tightly coupled conventional finite element modeling of coupled flow and geomechanics does not conserve mass locally since it uses continuous basis functions mixed finite element discretization that satisfies local mass conservation of the flow equation can be a good solution for this problem in addition the stabilized finite element method for discretizing the saturation equation minimizes numerical diffusion and provides better resolution of saturation solution in this work we developed a coupled multiphase flow and geomechanics solver that solves fully coupled governing equations namely pressure velocity saturation and geomechanical equilibrium equations the solver can deal with full tensor permeability and elastic stiffness for modeling a highly heterogeneous reservoir system the results of the numerical experiments are very encouraging we used the solver to simulate a reservoir system that has very heterogeneous permeability and elastic stiffness fields and found that the numerical solution captures the complex multiphysics of the system in addition we obtained higher resolution of saturation solution than with the conventional finite volume discretization this would help us make a better estimate of the numerical solution of complex multiphysics problems

a fully coupled upwind discontinuous galerkin method for incompressible porous media flows high order computations of viscous fingering instabilities in complex geometry we present a new approach to the simulation of viscous fingering instabilities in incompressible miscible displacement flows in porous media in the past high resolution computational simulations of viscous fingering instabilities have always been performed using high order finite difference or fourier spectral methods which do not posses the flexibility to compute very complex subsurface geometries our approach instead by means of a fully coupled nonlinear implementation of the discontinuous galerkin method possesses a fundamental differentiating feature in that it maintains high order accuracy on fully unstructured meshes in addition the proposed method shows very low sensitivity to mesh orientation in contrast with classical finite volume approximation used in porous media flow simulations the robustness and accuracy of the method are demonstrated in a number of challenging computational problems

superconvergence of mixed finite element approximations over quadrilaterals a superconvergence result is established in this article for approximate solutions of second order elliptic equations by mixed finite element methods over quadrilaterals the superconvergence indicates an accuracy of cal o h k 2 for the mixed finite element approximation if the raviart thomas or brezzi douglas fortin marini elements of order k are employed with optimal error estimate of cal o h k 1 numerical experiments are presented to illustrate the theoretical result

the lowest order weak galerkin finite element method for the darcy equation on quadrilateral and hybrid meshes abstract this paper investigates the lowest order weak galerkin finite element method for solving the darcy equation on quadrilateral and hybrid meshes consisting of quadrilaterals and triangles in this approach the pressure is approximated by constants in element interiors and on edges the discrete weak gradients of these constant basis functions are specified in local raviart thomas spaces specifically r t 0 for triangles and unmapped r t 0 for quadrilaterals these discrete weak gradients are used to approximate the classical gradient when solving the darcy equation the method produces continuous normal fluxes and is locally mass conservative regardless of mesh quality and has optimal order convergence in pressure velocity and normal flux when the quadrilaterals are asymptotically parallelograms implementation is straightforward and results in symmetric positive definite discrete linear systems we present numerical experiments and comparisons with other existing methods

superconvergence for control volume mixed finite element methods on rectangular grids we consider control volume mixed finite element methods for the approximation of second order elliptic problems on rectangular grids these methods associate control volumes covolumes with the vector variable as well as the scalar obtaining local algebraic representation of the vector equation e g darcy s law as well as the scalar equation e g conservation of mass we establish o h 2 superconvergence for both the scalar variable in a discrete l 2 norm and the vector variable in a discrete h rm div norm the analysis exploits a relationship between control volume mixed finite element methods and the lowest order raviart thomas mixed finite element methods

a mortar mimetic finite difference method on non matching grids we consider mimetic finite difference approximations to second order elliptic problems on non matching multiblock grids mortar finite elements are employed on the non matching interfaces to impose weak flux continuity optimal convergence and in certain cases superconvergence is established for both the scalar variable and its flux the theory is confirmed by computational results

double families of quasi positive darcy flux approximations with highly anisotropic tensors on structured and unstructured grids this paper focuses on flux continuous pressure equation approximation for strongly anisotropic media previous work on families of flux continuous schemes for solving the general geometry permeability tensor pressure equation has focused on single parameter families these schemes have been shown to remove the o 1 errors introduced by standard two point flux reservoir simulation schemes when applied to full tensor flow approximation improved convergence of the schemes has also been established for specific quadrature points however these schemes have conditional m matrices depending on the strength of the off diagonal tensor coefficients when applied to cases involving full tensors arising from strongly anisotropic media the point wise continuous schemes can fail to satisfy the maximum principle and induce severe spurious oscillations in the numerical pressure solution new double family flux continuous locally conservative schemes are presented for the general geometry permeability tensor pressure equation the new double family formulation is shown to expand on the current single parameter range of existing conditional m matrix schemes the conditional m matrix bounds on a double family formulation are identified for both quadrilateral and triangle cell grids a quasi positive qm matrix analysis is presented that classifies the behaviour of the new schemes with respect to double family quadrature in regions beyond the m matrix bounds the extension to double family quadrature is shown to be beneficial resulting in novel optimal anisotropic quadrature schemes the new methods are applied to strongly anisotropic full tensor field problems and yield results with sharp resolution with only minor or practically zero spurious oscillations

equivalence groupoid of a class of variable coefficient korteweg de vries equations we classify the admissible transformations in a class of variable coefficient korteweg de vries equations as a result a full description of the structure of the equivalence groupoid of the class is given the class under study is partitioned into six disjoint normalized subclasses the widest possible equivalence group for each subclass is found which appears to be generalized extended in five cases ways for improvement of transformational properties of the subclasses are proposed using gaugings of arbitrary elements and mapping between classes the group classification of one of the subclasses is carried out as an illustrative example

goal oriented a posteriori error estimation for the travel time functional in porous media flows in this article we consider the a posteriori error estimation and adaptive mesh refinement for the numerical approximation of the travel time functional arising in porous media flows the key application of this work is in the safety assessment of radioactive waste facilities in this setting the travel time functional measures the time taken for a nonsorbing radioactive solute transported by groundwater to travel from a potential site deep underground to the biosphere to ensure the computability of the travel time functional we employ a mixed formulation of darcy s law and conservation of mass together with raviart thomas h mathrm div omega conforming finite elements the proposed a posteriori error bound is derived based on a variant of the standard dual weighted residual approximation which takes into account the lack of smoothness of the underlying functional of interest the proposed adaptive refinement strategy is tested on both a simple academic test case and a problem based on the

a multipoint flux mixed finite element method for the compressible darcy forchheimer models abstract we develop a multipoint flux mixed finite element method for the compressible darcy forchheimer models this method is motivated by the multipoint flux approximation method based on the lowest order brezzi douglas marini mixed finite element method and combined with a special quadrature rule that allows for local velocity elimination and leads to a cell centered system for the pressures we develop two approximation methods for the problems the semi discrete method and fully discrete method where the fully discrete method includes the backward euler scheme and the crank nicolson scheme theoretical results show the convergences for both pressure and velocity the numerical experiments show that the convergence rates of the method are in agreement with the theoretical analysis

solving global problem by considering multitude of local problems application to fluid flow in anisotropic porous media using the multipoint flux approximation abstract in this work we apply the experimenting pressure field approach to the numerical solution of the single phase flow problem in anisotropic porous media using the multipoint flux approximation we apply this method to the problem of flow in saturated anisotropic porous media in anisotropic media the component flux representation requires generally multiple pressure values in neighboring cells e g six pressure values of the neighboring cells is required in two dimensional rectangular meshes this apparently results in the need for a nine points stencil for the discretized pressure equation 27 points stencil in three dimensional rectangular mesh the coefficients associated with the discretized pressure equation are complex and require longer expressions which make their implementation prone to errors in the experimenting pressure field technique the matrix of coefficients is generated automatically within the solver a set of predefined pressure fields is operated on the domain through which the velocity field is obtained apparently such velocity fields do not satisfy the mass conservation equations entailed by the source sink term and boundary conditions from which the residual is calculated in this method the experimenting pressure fields are designed such that the residual reduces to the coefficients of the pressure equation matrix

convergence analysis of multirate fixed stress split iterative schemes for coupling flow with geomechanics abstract we consider multirate iterative schemes for the biot system modeling coupled flow and geomechanics in a poro elastic medium the multirate iterative coupling scheme exploits the different time scales for the mechanics and flow problems by taking multiple finer time steps for flow within one coarse mechanics time step we adapt the fixed stress split algorithm that decouples the flow and mechanics equations for the multirate case and perform an iteration between the two problems until convergence we provide a fully discrete scheme that uses backward euler time discretization and mixed spaces for flow and conformal galerkin for mechanics our analysis is based on studying the equations satisfied by the difference of iterates and using banach contraction argument to prove that the corresponding scheme is a fixed point contraction the analysis provides the value of an adjustable coefficient used in the proposed iterative coupling algorithms furthermore we show that the converged quantities satisfy the variational weak form for the coupled discrete system

robust convergence of multi point flux approximation on rough grids this paper establishes the convergence of a multi point flux approximation control volume method on rough quadrilateral grids by rough grids we refer to a family of refined quadrilateral grids where the cells are not required to approach parallelograms in the asymptotic limit in contrast to previous convergence results for these methods we consider here a version where the flux approximation is derived directly in the physical space and not on a reference cell as a consequence less regular grids are allowed however the extra cost is that the symmetry of the method is lost

error analysis of multipoint flux domain decomposition methods for evolutionary diffusion problems abstract we study space and time discretizations for mixed formulations of parabolic problems the spatial approximation is based on the multipoint flux mixed finite element method which reduces to an efficient cell centered pressure system on general grids including triangles quadrilaterals tetrahedra and hexahedra the time integration is performed by using a domain decomposition time splitting technique combined with multiterm fractional step diagonally implicit runge kutta methods the resulting scheme is unconditionally stable and computationally efficient as it reduces the global system to a collection of uncoupled subdomain problems that can be solved in parallel without the need for schwarz type iteration convergence analysis for both the semidiscrete and fully discrete schemes is presented

parametric representation of cell boundary flux distribution in well equations material transport is anticipated between adjacent porous media in capillary contact for which we have independent neumann function solutions to either poisson s equation or the heat equation these solutions can be extended by opening the boundary using green s theorem resulting in analytic solutions coupled through a boundary integral previously a parametric representation of the boundary flux was proposed as a linear combination of uniform flux and uniform pressure constituents which has advantages of analytic evaluation of contributing terms this boundary flux structure is shown to be exact for cells of identical size and permeability we highlight the extension to systems of either differing domain size or permeability using prolongation prolonged problems allow identification with symmetry of equal cell size problems and an exact solution for flux distribution correcting for prolongation requires additional uniform flux and circulation elements that are related to the degree of mismatch in cells in the originally posed problem since lengths are scaled with respect to transport properties we can claim the new method also allows significant bandwidth reduction in solving the heat equation for heterogeneous systems using parametric representation of flux in boundary integrals