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

a hybrid finite volume finite element method for bulk surface coupled problems abstract the paper develops a hybrid method for solving a system of advection diffusion equations in a bulk domain coupled to advection diffusion equations on an embedded surface a monotone nonlinear finite volume method for equations posed in the bulk is combined with a trace finite element method for equations posed on the surface in our approach the surface is not fitted by the mesh and is allowed to cut through the background mesh in an arbitrary way moreover a triangulation of the surface into regular shaped elements is not required the background mesh is an octree grid with cubic cells as an example of an application we consider the modeling of contaminant transport in fractured porous media one standard model leads to a coupled system of advection diffusion equations in a bulk matrix and along a surface fracture a series of numerical experiments with both steady and unsteady problems and different embedded geometries illustrate the numerical properties of the hybrid approach the method demonstrates great flexibility in handling curvilinear or branching lower dimensional embedded structures

convergence analysis of the high order mimetic finite difference method we prove second order convergence of the conservative variable and its flux in the high order mfd method the convergence results are proved for unstructured polyhedral meshes and full tensor diffusion coefficients for the case of non constant coefficients we also develop a new family of high order mfd methods theoretical result are confirmed through numerical experiments

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

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

the finite volume scheme preserving maximum principle for diffusion equations with discontinuous coefficient abstract we construct a new scheme whose primary unknowns include both cell centered unknowns and edge unknowns on discontinuous line to solve diffusion equations with discontinuous coefficient first two linear fluxes are given on both sides of the cell edge respectively in order to deal with the defect of the existing scheme preserving maximum principle for solving diffusion problem with discontinuous coefficient in addition to cell centered unknowns we also introduce cell edge unknowns on discontinuous line as basic unknowns second the conservative flux is constructed by using nonlinear weighted combination of these two linear fluxes for the cell edge unknowns on discontinuous line we add an equation by using the continuity of normal flux compared to the classical cell centered nonlinear finite volume scheme the introduction of cell edge unknowns on discontinuous edge is the key point for our scheme to solve diffusion equations with discontinuous coefficient then we prove that the scheme satisfies the discrete maximum principle based on this the existence of a solution for the scheme is also obtained numerical results are presented to show that our scheme obtains almost second order accuracy for solution on random meshes preserves discrete maximum principle and is superior to the existing scheme in sheng and yuan 2011 preserving the maximum principle on dealing with the problems with strong anisotropic discontinuous coefficient on distorted meshes

failure of the discrete maximum principle for an elliptic finite element problem there has been a long standing question of whether certain mesh restrictions are required for a maximum condition to hold for the discrete equations arising from a finite element approximation of an elliptic problem this is related to knowing whether the discrete green s function is positive for triangular meshes allowing sufficiently good approximation of h 1 functions we study this question for the poisson problem in two dimensions discretized via the galerkin method with continuous piecewise linears we give examples which show that in general the answer is negative and furthermore we extend the number of cases where it is known to be positive our techniques utilize some new results about discrete green s functions that are of independent interest

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

third order maximum principle satisfying direct discontinuous galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes we develop 3rd order maximum principle satisfying direct discontinuous galerkin methods 8 9 19 21 for convection diffusion equations on unstructured triangular mesh we carefully calculate the normal derivative numerical flux across element edges and prove that with proper choice of parameter pair 0 1 in the numerical flux formula the quadratic polynomial solution satisfies strict maximum principle the polynomial solution is bounded within the given range and third order accuracy is maintained there is no geometric restriction on the meshes and obtuse triangles are allowed in the partition a sequence of numerical examples are carried out to demonstrate the accuracy and capability of the maximum principle satisfying limiter

the dirichlet problem for monge ampere equations in non convex domains and spacelike hypersurfaces of constant gauss curvature in this paper we extend the well known results on the existence and regularity of solutions of the dirichlet problem for monge ampere equations in a strictly convex domain to an arbitrary smooth bounded domain in r as well as in a general riemannian manifold we prove for the nondegenerate case that a sufficient and necessary condition for the classical solvability is the existence of a subsolution for the totally degenerate case we show that the solution is in c11 q if the given boundary data extends to a locally strictly convex c2 function ori q as an application we prove some existence results for spacelike hypersurfaces of constant gauss kronecker curvature in minkowski space spanning a prescribed boundary

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 vertex centered and positivity preserving scheme for anisotropic diffusion problems on arbitrary polygonal grids abstract we suggest a new positivity preserving finite volume scheme for anisotropic diffusion problems on arbitrary polygonal grids the scheme has vertex centered edge midpoint and cell centered unknowns the vertex centered unknowns are primary and have finite volume equations associated with them the edge midpoint and cell centered unknowns are treated as auxiliary ones and are interpolated by the primary unknowns which makes the final scheme a pure vertex centered one unlike most existing positivity preserving schemes the construction of the scheme is based on a special nonlinear two point flux approximation that has a fixed stencil and does not require the convex decomposition of the co normal in order to solve efficiently the nonlinear systems resulting from the nonlinear scheme picard method and its anderson acceleration are discussed numerical experiments demonstrate the second order accuracy and well positivity of the solution for heterogeneous and anisotropic problems on severely distorted grids the high efficiency of the anderson acceleration is also shown on reduction of the number of nonlinear iterations moreover the proposed scheme does not have the so called numerical heat barrier issue suffered by most existing cell centered and hybrid schemes however further improvements have to be made if the solution is very close to the machine precision and the mesh distortion is very severe

weakened acute type condition for tetrahedral triangulations and the discrete maximum principle we prove that a discrete maximum principle holds for continuous piecewise linear finite element approximations for the poisson equation with the dirichlet boundary condition also under a condition of the existence of some obtuse internal angles between faces of terahedra of triangulations of a given space domain this result represents a weakened form of the acute type condition for the three dimensional case

a numerical approach for determining weight functions in fracture mechanics the paper deals with the development of a numerical method for determining weight functions in two dimensional problems after a short review of some recent numerical techniques an original approach is presented the method is based on finite element calculations with coarse meshes and on the knowledge of some values of the stress intensity factor for one reference loading condition the validity of the method is demonstrated for a theoretical case and its accuracy and suitability are discussed with reference to practical applications

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

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

high order mimetic finite difference method for diffusion problems on polygonal meshes the mimetic finite difference mfd methods mimic important properties of physical and mathematical models as a result conservation laws solution symmetries and the fundamental identities of the vector and tensor calculus are held for discrete models the mfd methods retain these attractive properties for full tensor coefficients and arbitrary polygonal meshes which may include non convex and degenerate elements the existing mfd methods for solving diffusion type problems are second order accurate for the conservative variable temperature pressure energy etc and only first order accurate for its flux we developed new high order mfd methods which are second order accurate for both scalar and vector variables the second order convergence rates are demonstrated with a few numerical examples on randomly perturbed quadrilateral and polygonal meshes

a higher order formulation of the mimetic finite difference method a new mimetic finite difference method for the diffusion problem is developed by using a linear interpolation for the numerical fluxes this approach provides a higher order accurate approximation to the flux of the exact solution in analogy with the original formulation a family of local scalar products is constructed to satisfy the fundamental properties of local consistency and spectral stability the scalar solution field is approximated by a piecewise constant function a computationally efficient postprocessing technique is also proposed to get a piecewise quadratic polynomial approximation to the exact scalar variable finally optimal convergence rates and accuracy improvement with respect to the lower order formulation are shown by numerical examples

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 new discretization methodology for diffusion problems on generalized polyhedral meshes we develop a family of inexpensive discretization schemes for diffusion problems on generalized polyhedral meshes with elements having non planar faces the material properties are described by a full tensor we also prove superconvergence for the scalar pressure variable under very general assumptions the theoretical results are confirmed with numerical experiments in the practically important case of logically cubic meshes with randomly perturbed nodes the mixed finite element with the lowest order raviart thomas elements does not converge while the proposed mimetic method has the optimal convergence rate 2007 elsevier b v all rights reserved

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

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

the maximum principle violations of the mixed hybrid finite element method applied to diffusion equations the affluent literature of finite element methods applied to linear parabolic problems generally produces numerical procedures with satisfactory properties from a numerical approximation point of view however some initial boundary conditions or abrupt sink source terms may cause large gradients at some points and consequently jumps in the solution that usually needs a certain period of time to become more and more smooth this intuitive fact of the diffusion process necessitates when applying numerical methods varying the mesh size in time and space according to the smoothness of the solution in this work the numerical behavior of the time dependent solutions for such problems during small time duration obtained by using the mixed hybrid finite element method mhfem is investigated numerical comparisons with the standard galerkin finite element fe as well as the finite difference fd methods are checked up on due to the fact that the mixed methods violate the discrete maximum principle some numerical experiments showed that the mhfem leads sometimes to non physical peaks in the solution a diffusivity criterion relating the mesh steps for an artificial initial boundary value problem will be presented one of the propositions given to avoid any non physical oscillations is to use the mass lumping techniques

a monotone finite volume method for advection diffusion equations on unstructured polygonal meshes we present a new second order accurate monotone finite volume fv method for the steady state advection diffusion equation the method uses a nonlinear approximation for both diffusive and advective fluxes and guarantees solution non negativity the interpolation free approximation of the diffusive flux uses the nonlinear two point stencil proposed in lipnikov 23 approximation of the advective flux is based on the second order upwind method with a specially designed minimal nonlinear correction the second order convergence rate and monotonicity are verified with numerical experiments

physical bound preserving finite volume methods for the nagumo equation on distorted meshes abstract in this paper we present a boundedness preserving finite volume scheme for the nagumo equation in this method we use the implicit euler method for the time discretization and construct a maximum principle preserving discrete normal flux for the diffusion term for the nonlinear reaction term we design a type of picard iteration to ensure that at each iterative step it keeps physical boundedness moreover we prove that the numerical solution of the resulting scheme can preserve the bound of the solution for the nagumo equation on distorted meshes some numerical results are presented to verify the theoretical analysis

convergence of nonlinear finite volume schemes for heterogeneous anisotropic diffusion on general meshes abstract in the present work we deal with the convergence of cell centered nonlinear finite volume schemes for anisotropic and heterogeneous diffusion operators a general framework for the convergence study of finite volume methods is provided and used to establish the convergence of the new methods thorough assessment on a set of anisotropic heterogeneous problems as well as a comparison with linear finite volume schemes is provided

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

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

monotone finite volume schemes of nonequilibrium radiation diffusion equations on distorted meshes we construct a monotone finite volume scheme on distorted meshes for multimaterial nonequilibrium radiation diffusion problems which are described by the coupled radiation diffusion and material conduction equations moreover we prove theoretically that the scheme is monotone numerical results are presented to show that our scheme preserves positivity of solution on various distorted meshes and the contours of numerical solution obtained by our scheme on distorted meshes accord with that on rectangular meshes moreover numerical tests indicate that our monotone scheme is more computationally efficient than the nine point scheme these results show that our nonlinear monotone finite volume scheme is a practical and attractive method for solving nonlinear diffusion equations on distorted meshes

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

a finite volume scheme for diffusion problems on general meshes applying monotony constraints in order to increase the accuracy and the stability of a scheme dedicated to the approximation of diffusion operators on any type of grids we propose a method which locally reduces the curvature of the discrete solution where the loss of monotony is observed the discrete solution is shown to fulfill a variational formulation thanks to the use of lagrange multipliers we can then show its convergence to the solution of the continuous problem and an error estimate is derived a numerical method based on uzawa s algorithm is shown to provide accurate and stable approximate solutions to various problems numerical results show the increase of precision due to the application of the method