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

finite analytic method for 2d steady fluid flows in heterogeneous porous media with unstructured grids the finite analytic method is developed to solve the two dimensional steady fluid flows in heterogeneous porous media with full tensor permeability on unstructured grids the proposed fam is constructed based upon the power law analytic nodal solution in the angular domain with arbitrary shape when approaching the grid node joining the subdomains three different flow patterns may exist power law flow linear flow or the stagnant flow based on the nodal analytic solution the triangle based fam is proposed numerical examples show that the proposed numerical scheme makes the convergences much quickly than the traditional methods typically the weighted harmonic mean method under the cell refinement in practical applications the grid refinement parameter n 2 or n 3 is recommended and the relative error of the calculated equivalent permeability will below 4 independent of the heterogeneity in contrast when using the traditional numerical scheme the refinement ratio for the grid cell needs to increase dramatically to get an accurate result especially for strong heterogeneous porous medium

a small stencil and extremum preserving scheme for anisotropic diffusion problems on arbitrary 2d and 3d meshes in this paper a nonlinear extremum preserving scheme for the heterogeneous and anisotropic diffusion problems is proposed on general 2d and 3d meshes through a certain linearity preserving approach the so called harmonic averaging points located at the interface of heterogeneity are employed to define the auxiliary unknowns this new scheme is locally conservative has only cell centered unknowns and possesses a small stencil which is five point on the structured quadrilateral meshes and seven point on the structured hexahedral meshes the stability result in h 1 norm is obtained under quite general assumptions numerical results show that our scheme is robust and extremum preserving and the optimal convergence rates are verified on general distorted meshes in case that the diffusion tensor is taken to be anisotropic at times discontinuous

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

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

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

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 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

a nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes a nine point scheme is presented for discretizing diffusion operators on distorted quadrilateral meshes the advantage of this method is that highly distorted meshes can be used without the numerical results being altered remarkably and it treats material discontinuities rigorously and offers an explicit expression for the face centered flux moreover it has only the cell centered unknowns we prove that the method is stable and has first order convergence on distorted meshes numerical experiments show that the method has second order or nearly second order accuracy on distorted meshes

an improvement of the two point flux approximation scheme on polygonal meshes abstract a new diffusion scheme satisfying the maximum principle on two dimensional 2d polygonal meshes is proposed being different from the existing schemes which are based on the nonlinear combination of the two one side fluxes the scheme in this paper is constructed by a modification of the conventional linear two point flux approximation tpfa scheme in which the two side flux is constructed to improve its accuracy on 2d skewed meshes the tpfa scheme is rewritten by adding a tangential part i e the tangential component of the flux multiplied with a nonlinear factor a new method for the discretization of the tangential part is presented so that both the conservation condition and the maximum principle are satisfied while the tpfa part remains unchanged moreover by virtue of the harmonic averaging points for the calculation of the tangential part the scheme has a minimal stencil similar to the tpfa scheme finally some numerical examples are presented to verify that it is second order accurate on 2d polygonal meshes for anisotropic diffusion problems

cell centered finite volume methods with flexible stencils for diffusion equations on general nonconforming meshes a cell centered finite volume method is presented for discretizing diffusion operator on general nonconforming meshes the node values are accurately approximated using a new weighted interpolation formula in which the calculation of the weight is adaptive to both geometric parameters and diffusion coefficients it follows that an explicit expression composed of cell centered unknowns only is obtained for the discretization of normal flux numerical results demonstrate that linear solutions are reproduced exactly on the nonconforming random grids and that the convergence rate is close to second order for non linear or discontinuous problems

the finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes we construct a new nonlinear finite volume scheme for diffusion equation on polygonal meshes and prove that the scheme satisfies the discrete extremum principle our scheme is locally conservative and has only cell centered unknowns numerical results are presented to show how our scheme works for preserving discrete extremum principle and positivity on various distorted meshes

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

α amenable hypergroups let k denote a locally compact commutative hypergroup l1 k the hypergroup algebra and a real valued hermitian character of k we show that k is amenable if and only if l1 k is left amenable we also consider the amenability of hypergroup joins and polynomial hypergroups in several variables as well as a single variable

bad behavior of godunov mixed methods for strongly anisotropic advection dispersion equations we study the performance of godunov mixed methods which combine a mixed hybrid finite element solver and a godunov like shock capturing solver for the numerical treatment of the advection dispersion equation with strong anisotropic tensor coefficients it turns out that a mesh locking phenomenon may cause ill conditioning and reduce the accuracy of the numerical approximation especially on coarse meshes this problem may be partially alleviated by substituting the mixed hybrid finite element solver used in the discretization of the dispersive diffusive term with a linear galerkin finite element solver which does not display such a strong ill conditioning to illustrate the different mechanisms that come into play we investigate the spectral properties of such numerical discretizations when applied to a strongly anisotropic diffusive term on a small regular mesh a thorough comparison of the stiffness matrix eigenvalues reveals that the accuracy loss of the godunov mixed method is a structural feature of the mixed hybrid method in fact the varied response of the two methods is due to the different way the smallest and largest eigenvalues of the dispersion diffusion tensor influence the diagonal and off diagonal terms of the final stiffness matrix one and two dimensional test cases support our findings

monotone finite volume schemes for diffusion equations on unstructured triangular and shape regular polygonal meshes we consider a non linear finite volume fv scheme for stationary diffusion equation we prove that the scheme is monotone i e it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients the scheme is extended to regular star shaped polygonal meshes and isotropic heterogeneous coefficients

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

monotone corrections for generic cell centered finite volume approximations of anisotropic diffusion equations we present a nonlinear technique to correct a general finite volume scheme for anisotropic diffusion problems which provides a discrete maximum principle we point out general properties satisfied by many finite volume schemes and prove the proposed corrections also preserve these properties we then study two specific corrections proving under numerical assumptions that the corresponding approximate solutions converge to the continuous one as the size of the mesh tends to zero finally we present numerical results showing that these corrections suppress local minima produced by the original finite volume scheme

spectral element approximation of fredholm integral eigenvalue problems we consider the numerical approximation of homogeneous fredholm integral equations of second kind with emphasis on computing truncated karhunen loeve expansions we employ the spectral element method with gauss lobatto legendre gll collocation points similar to the piecewise constant finite elements this approach is simple to implement and does not lead to generalized discrete eigenvalue problems numerical experiments confirm the expected convergence rates for some classical kernels and illustrate how this approach can improve the finite element solution of partial differential equations with random input data

enforcing positivity with conservation for nine point scheme of nonlinear diffusion equations abstract the nine point scheme nps is a finite volume scheme for solving nonlinear diffusion equations on quadrilateral grids it s well known that the solution of the scheme on distorted grids cannot preserve positivity when all given data are nonnegative here for the nps of nonlinear diffusion problems we propose a conservative enforcing positivity algorithm cepa which is posterior correction of the discrete solution and allows the new solution preserves positivity as well as local conservation at each step of nonlinear iteration numerical results are presented to demonstrate that the accuracy of our algorithm of enforcing positivity with conservation is higher than some common procedures of positivity correction

monotone finite volume schemes for diffusion equations on polygonal meshes we construct a nonlinear finite volume fv scheme for diffusion equation on star shaped polygonal meshes and prove that the scheme is monotone i e it preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficients our scheme has only cell centered unknowns and it treats material discontinuities rigorously and offers an explicit expression for the normal flux numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes for both smooth and non smooth highly anisotropic solutions and numerical results show that our scheme appears to be approximate second order accuracy for the solution and first order accuracy for the flux

an improved monotone finite volume scheme for diffusion equation on polygonal meshes we construct a new nonlinear monotone finite volume scheme for diffusion equation on polygonal meshes the new scheme uses the cell edge unknowns instead of cell vertex unknowns as the auxiliary unknowns in order to improve the accuracy of monotone scheme our scheme is locally conservative and has only cell centered unknowns numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes specially numerical results show that the new scheme is robust and more accurate than the existing monotone scheme on some kinds of meshes

a finite volume method for approximating 3d diffusion operators on general meshes a finite volume method is presented for discretizing 3d diffusion operators with variable full tensor coefficients this method handles anisotropic non symmetric or discontinuous variable tensor coefficients while distorted non matching or non convex n faced polyhedron meshes can be used for meshes of polyhedra whose faces have not more than four edges the associated matrix is positive definite and symmetric if the diffusion tensor is symmetric a second order resp first order accuracy is numerically observed for the solution resp gradient of the solution

finite volume method for 2d linear and nonlinear elliptic problems with discontinuities in this paper we study the approximation of solutions to linear and nonlinear elliptic problems with discontinuous coefficients in the discrete duality finite volume ddfv framework this family of schemes allows very general meshes and inherits the main properties of the continuous problem in order to take into account the discontinuities and to prevent consistency defect in the scheme we propose to modify the definition of the numerical fluxes on the edges of the mesh where the discontinuity occurs we first illustrate our approach by the study of the 1d situation then we show how to design our new scheme called m ddfv and we propose its analysis we also describe an iterative solver whose convergence is proved which can be used to solve the nonlinear discrete equations defining the finite volume scheme finally we provide numerical results which confirm that the m ddfv scheme significantly improves the convergence rate of the usual ddfv method for both linear and nonlinear problems

spectral refinement based on superconvergent nystrom and degenerate kernel methods abstract in this paper are studied two spectral refinement schemes elementary and double iteration for the approximate solution of eigenvalue problems for an integral operator with a smooth kernel we use new superconvergent nystrom and degenerate kernel approximating operators based on interpolatory projection at gauss points onto the space of discontinuous piecewise polynomials of degree r 1 we show that for the eigenvalue iterates the elementary iteration scheme exhibits order k 4 r while the double iteration scheme exhibits order 3 k 4 r where k 0 1 2 denotes the iteration step we show that these orders of convergence are preserved in the corresponding discrete methods obtained by replacing the integration by a numerical quadrature formula similar improvement is observed for eigenvector iterates our results are extended to the case of discontinuous kernels along the diagonal for which superconvergence results are also obtained finally numerical validation is given

a reconstruction algorithm with flexible stencils for anisotropic diffusion equations on 2d skewed meshes a new reconstruction algorithm is developed to obtain diffusion schemes with cell centered unknowns only the main characteristic of the new algorithm is the flexibility of stencils when the auxiliary unknowns are reconstructed with cell centered unknowns the stencils are selected depending on the mesh geometry and discontinuities of diffusion coefficients moreover an explicit expression is derived for interpolating the auxiliary unknowns in terms of cell centered unknowns and the auxiliary unknowns can be defined at any point on the edge the algorithm is applied to construct several new diffusion schemes whose effectiveness is illustrated by numerical experiments for anisotropic problems with or without discontinuities nearly second order accuracy is achieved on skewed meshes

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

a new superconvergent collocation method for eigenvalue problems here we propose a new method based on projections for the approximate solution of eigenvalue problems for an integral operator with a smooth kernel using an interpolatory projection at gauss points onto the space of discontinuous piecewise polynomials of degree r 1 we show that the proposed method exhibits an error of the order of 4r for eigenvalue approximation and of the order of 3r for spectral subspace approximation in the case of a simple eigenvalue we show that by using an iteration technique an eigenvector approximation of the order 4r can be obtained this improves upon the order 2r for eigenvalue approximation in the collocation iterated collocation method and the orders r and 2r for spectral subspace approximation in the collocation method and the iterated collocation method respectively we illustrate this improvement in the order of convergence by numerical examples

a monotone finite volume scheme for diffusion equations on general non conforming meshes a nonlinear monotone finite volume scheme on general non conforming meshes for diffusion equations is introduced which deals with discontinuous tensor coefficients rigorously since the expression of normal flux depends on auxiliary unknowns defined at cell vertex including hanging nodes we propose a new method to eliminate vertex unknown by using primary unknowns at the centers of the cells sharing the vertex especially the unknowns defined on hanging nodes are eliminated by flux continuous conditions the resulting scheme is monotone and preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficient problems numerical results show that the convergent order of the monotone scheme by different methods of eliminating vertex unknowns will vary remarkably and our new method can assure that it has almost second order accuracy and more accurate than some existing methods

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

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 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

monotonicity of control volume methods robustness of numerical methods for multiphase flow problems in porous media is important for development of methods to be used in a wide range of applications here we discuss monotonicity for a simplified problem of single phase flow but where the simulation grids and media are allowed to be general posing challenges to control volume methods we discuss discrete formulations of the maximum principle and derive sufficient criteria for discrete monotonicity for arbitrary nine point control volume discretizations for conforming quadrilateral grids in 2d these criteria are less restrictive than the m matrix property it is shown that it is impossible to construct nine point methods which unconditionally satisfy the monotonicity criteria when the discretization satisfies local conservation and exact reproduction of linear potential fields numerical examples are presented which show the validity of the criteria for monotonicity further the impact of nonmonotonicity is studied different behavior for different discretization methods is illuminated and simple ideas are presented for improvement in terms of monotonicity

interpolation based second order monotone finite volume schemes for anisotropic diffusion equations on general grids we propose two interpolation based monotone schemes for the anisotropic diffusion problems on unstructured polygonal meshes through the linearity preserving approach the new schemes are characterized by their nonlinear two point flux approximation which is different from the existing ones and has no constraint on the associated interpolation algorithm for auxiliary unknowns thanks to the new nonlinear two point flux formulation it is no longer required that the interpolation algorithm should be a positivity preserving one the first scheme employs vertex unknowns as the auxiliary ones and a second order but not positivity preserving interpolation algorithm is utilized the second scheme uses the so called harmonic averaging points located on cell edges to define the auxiliary unknowns and a second order positivity preserving interpolation method is employed both schemes have nearly the same convergence rates as compared with their related second order linear schemes numerical results demonstrate that the new schemes are monotone and have the second order accuracy for the solution and first order for its gradient on severely distorted meshes

collocation methods for solving multivariable integral equations of the second kind in a recent paper allouch in press 5 on one dimensional integral equations of the second kind we have introduced new collocation methods these methods are based on an interpolatory projection at gauss points onto a space of discontinuous piecewise polynomials of degree r which are inspired by kulkarni s methods kulkarni 2003 10 and have been shown to give a 4r 4 convergence for suitable smooth kernels in this paper these methods are extended to multi dimensional second kind equations and are shown to have a convergence of order 2r 4 the size of the systems of equations that must be solved in implementing these methods remains the same as for kulkarni s methods a two grid iteration convergent method for solving the system of equations based on these new methods is also defined

finite volume scheme for anisotropic diffusion in this paper we apply a special finite volume scheme limited to smooth temperature distributions and cartesian grids to test the importance of connectivity of the finite volumes the area of application is nuclear fusion plasma with field line aligned temperature gradients and extreme anisotropy we apply the scheme to the anisotropic heat conduction equation and compare its results with those of existing finite volume schemes for anisotropic diffusion also we introduce a general model adaptation of the steady diffusion equation for extremely anisotropic diffusion problems with closed field lines

monotonic solution of heterogeneous anisotropic diffusion problems anisotropic problems arise in various areas of science and engineering for example groundwater transport and petroleum reservoir simulations the pure diffusive anisotropic time dependent transport problem is solved on a finite number of nodes that are selected inside and on the boundary of the given domain along with possible internal boundaries connecting some of the nodes an unstructured triangular mesh that attains the generalized anisotropic delaunay condition for all the triangle sides is automatically generated by properly connecting all the nodes starting from an arbitrary initial one the control volume of each node is the closed polygon given by the union of the midpoint of each side with the anisotropic circumcentre of each final triangle a structure of the flux across the control volume sides similar to the standard galerkin finite element scheme is derived a special treatment of the flux computation mainly based on edge swaps of the initial mesh triangles is proposed in order to obtain a stiffness m matrix system that guarantees the monotonicity of the solution the proposed scheme is tested using several literature tests and the results are compared with analytical solutions as well as with the results of other algorithms in terms of convergence order computational costs are also investigated

robust streamline tracing using inter cell fluxes in locally refined and unstructured grids 1 we present a comprehensive study of velocity interpolation methods in polygons these methods are often used as postprocessing procedures for numerical schemes that do not directly calculate the velocity field but only provide cell boundary flux conditions such as the finite volume schemes these methods extend the widely used velocity interpolation algorithms such as the pollock s algorithm to more complex geometries such as perpendicular bisection pebi grids unstructured triangular grids and grids with local refinement once the velocity field is interpolated streamline trajectories and time of flight along the streamlines can be calculated for reservoir simulation model calibration and waterflood management for instance these velocity interpolation methods assume known lower order or higher order cell boundary fluxes which satisfy global mass conservation and normal flux continuity however they differ in the interpolation of velocities within the interior of the cells the interpolating velocity may be locally conservative or nonconservative continuous or discontinuous lower order or higher order results show that the interpolated velocity field has to be locally conservative in order to guarantee the correct volumetric transformation for the calculated streamlines and the time of flight velocity continuity is not as important as local conservation for the purpose of streamline applications compared to higher order interpolation for the streamline trajectories lower order interpolation has the advantage of an analytic solution and an efficient implementation based on our analysis we recommend a lower order locally conservative method for the most robust and numerically efficient calculation of streamline trajectories on unstructured grids

a q1 finite volume element scheme for anisotropic diffusion problems on general convex quadrilateral mesh abstract in this paper we study a q 1 finite volume element scheme for anisotropic diffusion problems on general convex quadrilateral mesh it is known that the coercivity is the basement for some other theoretical results stability h 1 and l 2 error estimates etc and the existing results were mainly obtained on h 1 parallelogram meshes with scalar diffusion coefficients for the cases of full diffusion tensors and arbitrary convex quadrilateral meshes we obtain a necessary and sufficient condition for the positive definiteness of the cell matrix related to the cell bilinear form based on this result a sufficient condition is suggested to guarantee the coercivity of the scheme more interesting is that this sufficient condition covers the traditional h 1 parallelogram mesh assumption and has an explicit expression by which one can easily judge on any diffusion tensor and any mesh with arbitrary mesh size h 0 moreover an h 1 error estimate is obtained without the h 1 parallelogram assumption and some numerical results are also provided to validate the theoretical results

residual flux based a posteriori error estimates for finite volume and related locally conservative methods we derive in this paper a posteriori error estimates for discretizations of convection diffusion reaction equations in two or three space dimensions our estimates are valid for any cell centered finite volume scheme and in a larger sense for any locally conservative method such as the mimetic finite difference covolume and other we consider meshes consisting of simplices or rectangular parallelepipeds and also provide extensions to nonconvex cells and nonmatching interfaces we allow for the cases of inhomogeneous and anisotropic diffusion dispersion tensors and of convection dominance the estimates are established in the energy semi norm for a locally postprocessed approximate solution preserving the conservative fluxes and are of residual type they are fully computable all occurring constants are evaluated explicitly and locally efficient give a local lower bound on the error times an efficiency constant so that they can serve both as indicators for adaptive refinement and for the actual control of the error they are semi robust in the sense that the local efficiency constant only depends on local variations in the coefficients and becomes optimal as the local peclet number gets sufficiently small numerical experiments confirm their accuracy

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

error estimates for a finite volume element method for elliptic pdes in nonconvex polygonal domains we consider standard finite volume piecewise linear approximations for second order elliptic boundary value problems on a nonconvex polygonal domain based on sharp shift estimates we derive error estimations in h1 l2 and l infty norms taking into consideration the regularity of the data numerical experiments and counterexamples illustrate the theoretical results

discretization on unstructured grids for inhomogeneous anisotropic media part ii discussion and numerical results a companion paper part i siam j sci comput 19 1998 pp 1700 1716 presents discretization methods for control volume formulations in two space dimensions properties of the developed methods are further discussed including rotational invariance and stability under the influence of anisotropy strong anisotropy may have to be handled by use of stretched grid cells numerical examples that demonstrate strengths and weaknesses of the methods are presented the examples focus on inhomogeneity anisotropy and rotational invariance

from face to element unknowns by local static condensation with application to nonconforming finite elements we derive in this paper a new local static condensation strategy which allows to reduce significantly the number of unknowns in algebraic systems arising in discretization of partial differential equations we apply it to the discretization of a model linear elliptic diffusion and a model nonlinear parabolic advection diffusion reaction problem by crouzeix raviart nonconforming finite elements herein the unknowns originally associated with the mesh faces can be reduced to new unknowns associated with the mesh elements the resulting matrices are sparse with possibly only four nonzero entries per row in two space dimensions positive definite in dependence on the mesh geometry and the diffusion dispersion tensor but in general nonsymmetric our approach consists in introducing new element unknowns the identification of suitable local vertex based subproblems and the inversion of the corresponding local matrices we give sufficient conditions for the well posedness of the local problems as well as for the resulting global one in addition we provide a geometrical interpretation which suggests how to influence the form of the local and global matrices depending on the local mesh and data we finally present an abstract generalization allowing for a further reduction of the number of unknowns typically to one unknown per a set of mesh elements we conclude by numerical experiments which show that the condition number of the resulting matrices is quite insensitive to the mesh anisotropies and the diffusion tensor inhomogeneities

discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients this paper introduces a new conservative cell centred finite volume scheme for the accurate solution of elliptic diffusion equations with strongly varying coefficients the discretisation is designed to model a wide range of heterogeneities and anisotropies and in particular the case where the diffusivity is represented by a nondiagonal matrix which may occur if the medium is anisotropic in a general direction such problems arise for example in oil reservoir simulation when renormalisation techniques are used to model the reservoir geology it is in this context that the method is described although it is applicable to a far wider class of problems in heat conduction electrostatics and potential theory the paper also describes the application of a mulitigrid scheme for efficient numerical solution of the algebraic equations derived using the new discretisation

maximum principle in linear finite element approximations of anisotropic diffusion convection reaction problems a mesh condition is developed for linear finite element approximations of anisotropic diffusion convection reaction problems to satisfy a discrete maximum principle loosely speaking the condition requires that the mesh be simplicial and mathcal o vert varvec b vert infty h vert c vert infty h 2 o b h c h 2 nonobtuse when the dihedral angles are measured in the metric specified by the inverse of the diffusion matrix where h h denotes the mesh size and varvec b b and c c are the coefficients of the convection and reaction terms in two dimensions the condition can be replaced by a weaker mesh condition an mathcal o vert varvec b vert infty h vert c vert infty h 2 o b h c h 2 perturbation of a generalized delaunay condition these results include many existing mesh conditions as special cases numerical results are presented to verify the theoretical findings

unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients multipoint flux approximation mpfa techniques are commonly applied for discretizing the porous media flow equations within the context of finite volume numerical procedures although these methods can be applied to heterogeneous anisotropic systems on generally unstructured grids the inverse of the resulting linear operator can suffer from a loss of monotonicity at high permeability anisotropy ratios resulting in spurious oscillations of the pressure solution the purpose of this paper is to develop a method for optimizing unstructured grids in two and three dimensions such that the monotonicity behavior of the mpfa technique is significantly improved the method employs anisotropic triangulation and can be readily combined with permeability upscaling procedures results are presented for a variety of examples and the technique is shown to perform well on problems involving complex grid point distributions and heterogeneous permeability fields with strong anisotropy ratios of o 100 oscillation free pressure solutions are achieved in all cases considered even for examples in which the original grid shows large oscillations

a cell functional minimization scheme for parabolic problem we are interested in a robust and accurate finite volume scheme for 2 d parabolic problems derived from the cell functional minimization approach the scheme has a local stencil is locally conservative treats discontinuity rigorously and leads to a symmetric positive definite linear system since the scheme has both cell centered unknowns and cell edge unknowns the computational cost is an issue and a parallel algorithm is then suggested based on nonoverlapping domain decomposition approach the interface condition is of the dirichlet robin type and has a parameter l by choosing this parameter properly the convergence of the iteration process could be sped up numerical results for linear and nonlinear problems demonstrate the good performance of the cell functional minimization scheme and its parallel version on distorted meshes

a new nonlinear finite volume scheme preserving positivity for diffusion equations in this paper we present a new nonlinear finite volume scheme preserving positivity for diffusion equations the main feature of the scheme is the assumption that the values of auxiliary unknowns are nonnegative is avoided two nonnegative parameters are introduced to define a new nonlinear two point flux in which one point is the cell center and the other is the midpoint of cell edge the final flux on the edge is obtained by the continuity of normal flux numerical results show that the accuracy of both solution and flux for our new scheme is superior to that of some existing monotone schemes

a multidimensional unified gas kinetic scheme for radiative transfer equations on unstructured mesh abstract in order to extend the unified gas kinetic scheme ugks to solve radiative transfer equations in a complex geometry a multidimensional asymptotic preserving implicit method on unstructured mesh is constructed in this paper with an implicit formulation the cfl condition for the determination of the time step in ugks can be much relaxed and a large time step is used in simulations differently from previous direction by direction ugks on orthogonal structured mesh on unstructured mesh the interface flux transport takes into account multi dimensional effect where gradients of radiation intensity and material temperature in both normal and tangential directions of a cell interface are included in the flux evaluation the multiple scale nature makes the ugks be able to capture the solutions in both optically thin and thick regions seamlessly in the optically thick region the condition of cell size being less than photon s mean free path is fully removed and the ugks recovers a solver for diffusion equation in such a limit on unstructured mesh for a distorted quadrilateral mesh the ugks goes to a nine point scheme for the diffusion equation and it naturally reduces to the standard five point scheme for a orthogonal quadrilateral mesh numerical computations covering a wide range of transport regimes on unstructured and distorted quadrilateral meshes will be presented to validate the current approach

unified primal formulation based a priori and a posteriori error analysis of mixed finite element methods we derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second order elliptic problems it is based on the classical primal weak formulation the postprocessing of the potential proposed in t arbogast and z chen on the implementation of mixed methods as nonconforming methods for second order elliptic problems math comp 64 1995 943 972 and the discrete friedrichs inequality our analysis in particular avoids any explicit use of the uniform discrete inf sup condition and in a straightforward manner and under minimal necessary assumptions known convergence and superconvergence results are recovered the same framework then turns out to lead to optimal a posteriori energy error bounds in particular estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived they give a guaranteed and fully computable upper bound on the energy error represent error local lower bounds and are robust under some conditions on the diffusion dispersion tensor they are thus suitable for both overall error control and adaptive mesh refinement moreover the developed abstract framework and a posteriori error estimates are quite general and apply to any locally conservative method we finally prove that in parallel and simultaneously in converse to galerkin finite element methods under some circumstances the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the h 1 0 space and also establish several links between mixed finite element approximations and some generalized weak solutions

sign preserving of principal eigenfunctions in p1 finite element approximation of eigenvalue problems of second order elliptic operators this paper is concerned with the p1 finite element approximation of the eigenvalue problem of second order elliptic operators subject to the dirichlet boundary condition the focus is on the preservation of the basic properties of the principal eigenvalue and eigenfunctions of the continuous problem it is shown that when the stiffness matrix is an irreducible m matrix the discrete eigenvalue problem maintains almost all of the basic properties such as the smallest eigenvalue being real and simple and the corresponding eigenfunctions being either positive or negative inside the physical domain mesh conditions leading to such a stiffness matrix are also studied a sufficient condition is that the mesh is simplicial interiorly connected and acute when measured in the metric specified by the inverse of the diffusion matrix the acute requirement can be replaced by the delaunay condition in two dimensions numerical results show that when the stiffness matrix is not an m matrix a finite element approximation can be structurally different from the continuous eigenvalue problem the eigenfunctions corresponding to the smallest eigenvalue can change sign over the physical domain and the smallest eigenvalue in modulus can even be complex for the case with nonsymmetric operators

discretization on quadrilateral grids with improved monotonicity properties multi point flux approximation mpfa discretization methods have been applied in the oil industry since the mid 1990s the discretizations are based on a control volume formulation and the finite difference structure makes general skew grids and unstructured grids feasible in a fully implicit formulation mpfa methods are therefore suitable for flow problems in realistic reservoirs monotonicity issues are known to arise for high aspect ratios combined with skewness of computational grids for mpfa methods in this paper we improve the mpfa discretization techniques for general quadrilateral grids such that the above difficulties are handled to give a more robust discretization of the governing equations for fluid flow in porous media comparisons to the mpfa o method are made and the suggested discretization is shown to be an improvement in regards to monotonicity for smooth solutions the method performs equally well as the o method when the convergence is examined

superconvergent nystrom and degenerate kernel methods for eigenvalue problems the nystrom and degenerate kernel methods based on projections at gauss points onto the space of discontinuous piecewise polynomials of degree r 1 for the approximate solution of eigenvalue problems for an integral operator with a smooth kernel exhibit order 2r we propose new superconvergent nystrom and degenerate kernel methods that improve this convergence order to 4r for eigenvalue approximation and to 3r for spectral subspace approximation in the case where the kernel is sufficiently smooth moreover for a simple eigenvalue we show that by using an iteration technique an eigenvector approximation of order 4r can be obtained the methods introduced here are similar to that studied by kulkarni in 10 and exhibit the same convergence orders so a comparison with these methods is worked out in detail also the error terms are analyzed and the obtained methods are numerically tested finally these methods are extended to the case of discontinuous kernel along the diagonal and superconvergence results are also obtained

an anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics petroleum engineering and image processing standard numerical methods can produce spurious oscillations when they are used to solve those problems a common approach to avoid this difficulty is to design a proper numerical scheme and or a proper mesh so that the numerical solution validates the discrete counterpart dmp of the maximum principle satisfied by the continuous solution a well known mesh condition for the dmp satisfaction by the linear finite element solution of isotropic diffusion problems is the non obtuse angle condition that requires the dihedral angles of mesh elements to be non obtuse in this paper a generalization of the condition the so called anisotropic non obtuse angle condition is developed for the finite element solution of heterogeneous anisotropic diffusion problems the new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem several variants of the new condition are obtained based on one of them two metric tensors for use in anisotropic mesh generation are developed to account for dmp satisfaction and the combination of dmp satisfaction and mesh adaptivity numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors

the projection kantorovich method for eigenvalue problems abstract a modified projection method for eigenvalues and eigenvectors of a compact operator t on a banach space is defined and analyzed the method is derived from the kantorovich regularization for second kind equations involving the operator t it is shown that when t is a positive self adjoint operator on a hilbert space and the projections are orthogonal the modified method always gives eigenvalue approximations which are at least as accurate as those obtained from the projection method for self adjoint operators the required computation is essentially the same for both methods numerical computations for two integral operators are presented one has t positive self adjoint while in the other t is not self adjoint in both cases the eigenvalue approximations from the modified method are more accurate than those from the projection method

study of discrete duality finite volume schemes for the peaceman model in this paper we are interested in the finite volume approximation of a system describing miscible displacement in porous media this system is made of two coupled equations an anisotropic diffusion equation on the pressure and a convection diffusion dispersion equation on the concentration of invading fluid the anisotropic diffusion operators in both equations require a special care while discretizing by a finite volume method we focus here on the numerical approximation by some discrete duality finite volume methods after the presentation of the scheme we establish some a priori estimates satisfied by the numerical solution and prove existence and uniqueness of the solution to the scheme we show the efficiency of the schemes through numerical experiments

a triangle based mixed finite element finite volume technique for modeling two phase flow through porous media triangle based discretization techniques offer great advantages relative to standard finite difference methods for the modeling of flow through geometrically complex geological features the purpose of this paper is to develop and apply a triangle based method for the modeling of two phase flow through porous formations the formulation includes the effects of gravity compressibility and capillary pressure the technique entails a triangle based mixed finite element method for solution of the variable coefficient parabolic pressure equation and a second order tvd type total variation diminishing finite volume scheme for solution of the essentially hyperbolic saturation equation the method is applied to a variety of example problems and is shown to perform very well on problems involving geometric complexity coupled with heterogeneous generally anisotropic permeability descriptions