on the finite volume element method for general self adjoint elliptic problems the finite volume element method fve is a discretization technique for partial differential equations this paper develops discretization energy error estimates for general self adjoint elliptic boundary value problems with fve based on triangulations on which there exist linear finite element spaces and a very general type of control volumes covolumes the energy error estimates of this paper are also optimal but the restriction conditions for the covolumes given in r e bank and d j rose siam j numer anal 24 1987 pp 777 787 z q cai numer math 58 1991 pp 713 735 are removed the authors finally provide a counterexample to show that an expected l2 error estimate does not exist in the usual sense it is conjectured that the optimal order of u u h 0 omega should be o h for the general case

finite volume element methods for non definite problems the error estimates for finite volume element method applied to 2 and 3 d non definite problems are derived a simple upwind scheme is proven to be unconditionally stable and first order accurate

discontinuous finite volume methods for the stationary stokes darcy problem summary in this paper we analyze discontinuous finite volume methods for the stationary stokes darcy problem that models coupled fluid flow and porous media flow the discontinuous finite volume methods are combinations of finite volume method and discontinuous galerkin method with three interior penalty types incomplete symmetric nonsymmetric and symmetric briefly using discontinuous functions as trial functions in the finite volume method optimal error estimates in broken h1 norm are obtained for the three discontinuous finite volume methods optimal error estimates in the standard l2 norm are derived for the symmetric interior penalty discontinuous finite volume method numerical experiments are presented to confirm the theoretical results with non matching meshes across the common interface of stokes region and darcy region copyright 2015 john wiley sons ltd

an organizing center for wave bifurcation in multiphase flow models we consider a one parameter family of nonstrictly hyperbolic systems of conservation laws modeling three phase flow in a porous medium for a particular value of the parameter the model has a shock wave solution that undergoes several bifurcations upon perturbation of its left and right states and the parameter in this paper we use singularity theory and bifurcation theory of dynamical systems including melnikov s method to find all nearby shock waves that are admissible according to the viscous profile criterion we use these results to construct a unique solution of the riemann problem for each left and right state and parameter value in a neighborhood of the unperturbed shock wave solution together with previous numerical work this construction completes the solution of the three phase flow model in the bifurcation analysis the unperturbed shock wave acts as an organizing center for the waves appearing in riemann solutions

variational bounds on energy dissipation in incompressible flows ii channel flow a variational principle for lower bounds on the time averaged mass flux for newtonian fluids driven by a pressure gradient in a channel is derived from the incompressible navier stokes equations when supplied with appropriate test background flow fields the variational formulation produces explicit estimates for the friction coefficient these rigorous bounds are compared with the predictions of conventional turbulence theory

a residual type a posteriori error estimate of finite volume element method for a quasi linear elliptic problem in this paper we analyze a residual type a posteriori error estimator of the finite volume element method for a quasi linear elliptic problem of nonmonotone type and derive computable upper and lower bounds on the error in the h 1 norm numerical experiments are provided to illustrate the performance of the proposed estimator

the finite volume element methods for poisson equation based on adini s element in this paper we construct and analyze the finite volume element scheme based on adini s element for poisson equation choosing trial and test spaces as the adini s finite element space and the piecewise linear function space respectively optimal order error estimates in h 1 and l 2 norms are derived in addition the global and locally pointwise superconvergence estimates are also discussed lastly a numerical experiment is carried out which confirms some results of theoretical analysis

the riemann problem near a hyperbolic singularity ii this paper is interested in classifying the solutions of riemann problems for the 2 times 2 conservation laws that have homogeneous quadratic flux functions such flux functions approximate an arbitrary 2 times 2 system in a neighborhood of an isolated point where strict hyperbolicity fails here the solution for the symmetric systems in region iii of the four region classification of schaeffer and shearer is given the solution is based on the qualitative shape of the integral curves described by schaeffer and shearer and a numerical calculation of the hugoniot loci and their shock types

an algorithm using the finite volume element method and its splitting extrapolation this paper is to present a new efficient algorithm by using the finite volume element method and its splitting extrapolation this method combines the local conservation property of the finite volume element method and the advantages of splitting extrapolation such as a high order of accuracy a high degree of parallelism less computational complexity and more flexibility than a richardson extrapolation because the splitting extrapolation formulas only require us to solve a set of smaller discrete subproblems on different coarser grids in parallel instead of on the globally fine grid a large scale multidimensional problem is turned into a set of smaller discrete subproblems additionally this method is efficient for solving interface problems if we regard the interfaces of the problems as the interfaces of the initial domain decomposition

bounds on dissipation for navier strokes flow with kologorov forcing abstract in this paper bounds on average viscous dissipation are derived for kolmogorov flow in a spatially periodic domain with steady and unsteady forcing at arbitrarily large grashof number g for a force of the form f 0 sin mz i or f 0 sin mz cos t i we derive various bounds on the total dissipation in the flow d u as well as on the dissipation d m obtained from the x velocity averaged over the x y plane the mean velocity of the flow we derive upper bounds on d u and d v d u d m as well as lower bounds on d m and d m d u adopting constraints of the kind introduced by howard and busse and assuming a steady force the background flow method introduced by doering and constantin is used to obtain an improved lower bound on d m d u of o g 1 and a lower bound on d u of o g 1 2 where g f0l3 2 is the grashof number some of these results are then generalized to time periodic forcing direct numerical simulation of the flow indicates that these bounds leave substantial gaps at large grashof number g the calculated d m g and d u g being o g 1 2 and o 1 respectively as g our theoretical bounds on d m d u are shown to be attained by steady laminar type flows for neighboring forcing functions which seems to indicate that these bounds cannot be improved by adding further dynamical constraints however our elementary upper bound on d v can probably be improved by placing more constraints on the flows these results serve to emphasize the difference between boundary driven turbulence and body force driven turbulence where the appropriate dissipation bound is believed saturated at least up to logarithms

a global formalism for nonlinear waves in conservation laws we introduce a unifying framework for treating all of the fundamental waves occurring in general systems ofn conservation laws fundamental waves are represented as pairs of states statisfying the rankine hugoniot conditions after trivial solutions have been eliminated by means of a blow up procedure these pairs form an n 1 dimensional manifold the fundamental wave manifold there is a distinguishedn dimensional submanifold of containing a single one dimensonal foliation that represents the rarefaction curves for all families similarly there is a foliation of itself that represent shock curves we identify othern dimensional submanifolds of that are naturally interpreted as boundaries of regions of admissible shock waves these submanifolds also have one dimensional foliations which represent curves of composite waves this geometric framework promises to simplify greatly the study of the stability and bifurcation properties

structurally stable riemann solutions abstract we study the structure of solutions of riemann problems for systems of two conservation laws such a solution comprises a sequence of elementary waves viz rarefaction and shock waves of various types shock waves are required to have viscous profiles we construct a riemann solution by solving a system of equations characterizing its component waves a riemann solution is structurally stable if the number and types of its component waves are preserved when the initial data and the flux function are perturbed under the assumption that rarefaction waves and shock states lie in the stricly hyperbolic region we characterize riemann solutions for which the definition equations have maximal rank and we prove that such solutions are structurally stable structurally stable riemann solutions cannot contain overcompressive shock waves but they can contain transitional shock waves including doubly sonic transitional shock waves including doubly sonic transitional shock waves that have not been observed before

the classification of 2 2 systems of non strictly hyperbolic conservation laws with application to oil recovery on considere un systeme 2 2 a une variable d espace u f u x 0 0 ou u u x t r 2 et f r 2 r 2 on montre que pour des equations hyperboliques des points ombilicaux sont typiquement isoles et que en un point ombilical isole df est typiquement diagonalisable on donne une classification des equations hyperboliques a points ombilicaux isoles

the locally stabilized finite volume method for incompressible flow a locally stabilized finite volume method for the two dimensional incompressible flow governed by stationary stokes and navier stokes equations is investigated in this work a macroelement condition is introduced for constructing the local stabilized formulation for the problem the optimal l 2 error estimate of u h p h determined by stokes equations is also deduced some numerical examples are provided to check the efficiency of this method

two grid methods for finite volume element approximations of nonlinear parabolic equations two grid methods are studied for solving a two dimensional nonlinear parabolic equation using finite volume element method the methods are based on one coarse grid space and one fine grid space the nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine grid solution can be obtained in a single symmetric and linear step it is proved that the coarse grid can be much coarser than the fine grid the two grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h 3 lnh as a result solving such a large class of nonlinear parabolic equations will not be much more difficult than solving one single linearized equation

on the riemann problem for a prototype of a mixed type conservation law we study the system of conservation laws given by with initial value the system is elliptic when u2 v2 2 and hyperbolic when u2 v2 2 following liu s construction it is found that the system always has a weak solution which however is not necessarily unique

the finite volume element method for diffusion equations on general triangulations this paper develops discretization error estimates for the finite volume element method on general triangulations of a polygonal domain in mathcal r 2 using a special type of control volume the theory applies to diffusion equations of the form begin gathered nabla a nabla u f quad text in omega hfill u 0 quad text on partial omega hfill end gathered under fairly general conditions the theory establishes o h estimates of the error in a discrete mathcal h 1 seminorm under an additional assumption concerning local uniformity of the triangulation the estimate is improved to o h 2

a stabilized multi level method for non singular finite volume solutions of the stationary 3d navier stokes equations this paper proposes and analyzes a stabilized multi level finite volume method fvm for solving the stationary 3d navier stokes equations by using the lowest equal order finite element pair without relying on any solution uniqueness condition this multi level stabilized fvm consists of solving the nonlinear problem on the coarsest mesh and then performing one newton correction step on each subsequent mesh thus only solving a large linear system an optimal convergence rate for the finite volume approximations of nonsingular solutions is first obtained with the same order as that for the usual finite element solution by using a relationship between the stabilized fvm and a stabilized finite element method then the multi level finite volume approximate solution is shown to have a convergence rate of the same order as that of the stabilized finite volume solution of the stationary navier stokes equations on a fine mesh with an appropriate choice of the mesh size h j h j 1 2 j 1 ldots j finally numerical results presented validate our theoretical findings

the finite volume method and application in combinations abstract in this paper the conventional finite volume method fvm is interpreted as a new kind of galerkin finite element method fem where the same piecewise linear functions are chosen as in both trial and test spaces and some specific integration rules are adopted error analysis is made for the regular delaunay triangulation involving obtuse triangles separated to prove optimal convergence rates of the approximate solutions obtained the new interpretation makes the fvm analysis much easier because we may bypass verification of the nontrivial ladyzhenskaya babuska brezzi lbb condition by the petrov galerkin fem in the existing analysis of fvm more importantly the new interpretation and the simple fvm analysis enable us to construct easily the combinations of fvm with other popular numerical methods such as the finite element method fem the finite difference method fdm the ritz galerkin method rgm etc for solving complicated problems of partial differential equations pde for example for solving singularity problems the combination of rgm fvm is superior to the combination of rgm fdm in flexibility of arbitrary solution domains and also superior to the combination of rgm fem in substantial saving of cpu time since the conservative law of flux may be maintained exactly in the numerical solutions and since obtuse triangles may be included in the delaunay triangulation the fvm and its combinations become very promising for solving elliptic boundary value problems in particular those where singularity solutions exist and those where the obeying of conservative law is crucial the numerical examples of combinations of rgm fvm are given for solving motz s problems to verify the optimal convergence rates the techniques and analysis of fvm and its combinations can be extended to the convection diffusion problems most importantly an important aspect in this paper is the possibility to include obtuse triangles in the error analysis

the finite volume method based on stabilized finite element for the stationary navier stokes problem a finite volume method based on stabilized finite element for the two dimensional stationary navier stokes equations is investigated in this work a macroelement condition is introduced for constructing the local stabilized formulation for the problem we obtain the well posedness of the fvm based on stabilized finite element for the stationary navier stokes equations moreover for quadrilateral and triangular partition the optimal h 1 error estimate of the finite volume solution u h and l 2 error estimate for p h are introduced finally we provide a numerical example to confirm the efficiency of the fvm

analysis of a finite volume element method for the stokes problem in this paper we propose a stabilized conforming finite volume element method for the stokes equations on stating the convergence of the method optimal a priori error estimates in different norms are obtained by establishing the adequate connection between the finite volume and stabilized finite element formulations a superconvergence result is also derived by using a postprocessing projection method in particular the stabilization of the continuous lowest equal order pair finite volume element discretization is achieved by enriching the velocity space with local functions that do not necessarily vanish on the element boundaries finally some numerical experiments that confirm the predicted behavior of the method are provided

some errors estimates for the box method we define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane our estimates show the error to be comparable to a standard galerkin finite element method using piecewise linear polynomials

the mortar finite volume method with the crouzeix raviart element for elliptic problems abstract we construct and analyse a mortar finite volume method for the discretization for selfadjoint elliptic boundary value problems in r 2 this method is based on the mortar crouzeix raviart non conforming finite element spaces we prove the optimal order h 1 norm and l 2 norm error estimates between the exact solution and the mortar finite volume approximation of the elliptic problems

the fast adaptive composite grid fac method for elliptic equation the fast adaptive composite grid fag method is a systematic process for solving differential boundary value problems fac uses global and local uniform grids both to define the composite grid problem and to interact for its fast solution it can with little added cost substantially improve accuracy of the coarse grid solution and is very suitable for vector and parallel computation this paper develops both the theoretical and practical aspects of fac as it applies to elliptic problems 1 introduction the need for local resolution in physical models occurs frequently in practice special local features of the forcing function operator coefficients boundary and boundary conditions can demand resolution in restricted regions of the domain that is much finer than the required global resolution it is important that the discretization and solution processes account for this locally that is that the local phenomena do not precipitate a dramatic increase in the overall computation unfortunately this objective of efficiently adapting to local features is often in conflict with the solution process equation solvers can degrade or even fail to apply in the presence of varying discretization scales data structures that account for irregular grids can be cumbersome the computer architecture may not be able to effectively account for grid irregularity e g vectorizability may be inhibited etc in fact even the discretization process itself may find difficulty with this objective for finite differences it is problematic to develop accurate difference formulae for irregular grids for finite elements this objective is reflected in the substantial overhead costs needed to automate the discretization the fast adaptive composite grid method fac 11 is a discretization and solution method designed to achieve efficient local resolution by constructing the

variational bounds on shear driven turbulence and turbulent boussinesq convection abstract based on earlier studies by hopf 1941 doering and constantin 1992 1994 1995 have recently formulated a new background technique for obtaining upper bounds on turbulent fluid flow quantities this method produces upper bounds on the limit supremum of long time averages making no statistical assumptions about the flow in contrast to the well known howard busse approach the full optimisation problems posed by this method for the momentum transport in turbulent couette flow and the heat transport with zero background flow in turbulent boussinesq convection are solved here for the first time at asymptotically large reynolds number and rayleigh number within busse s multiple boundary layer approximation to extract the best lowest possible upper bounds available intriguingly the original bounds isolated by busse 1969 1970 within the confines of statistical stationarity are recovered exactly using this new formalism the optimal background velocity profile for turbulent couette flow is found to be shearless in the interior thus differing from busse s 1 4 mean shear result in the convective case an interesting degeneracy in the formulation of the background variational problem leads to an indeterminacy in the optimal background temperature profile only for one special choice is the isothermal core feature of busse s mean profile recovered

analysis of a singular hyperbolic system of conservation laws abstract we solve the riemann and cauchy problems globally for a singular system of n hyperbolic conservation laws the system which arises in the study of oil reservoir simulation has only two wave speeds and these coincide on a surface of codimension one in state space the analysis uses the random choice method of glimm comm pure appl math 18 1965 697 715

convective heat transport in a rotating fluid layer of infinite prandtl number optimum fields and upper bounds on nusselt number by means of the howard busse method of the optimum theory of turbulence we investigate numerically upper bounds on convective heat transport for the case of infinite fluid layer with stress free vertical boundaries rotating about a vertical axis we discuss the case of infinite prandtl number 1 ensuremath ensuremath alpha solution of the obtained variational problem and optimum fields possessing internal intermediate and boundary layers we investigate regions of rayleigh and taylor numbers r and ta where no analytical bounds can be derived and compare the analytical and numerical bounds for these regions of r and ta where such comparison is possible the increasing rotation has a different influence on the rescaled optimum fields of velocity w 1 temperature ensuremath theta 1 and the vertical component of the vorticity f 1 the increasing ta for fixed r leads to vanishing of the boundary layers of w 1 and ensuremath theta 1 opposite to this the increasing ta leads first to a formation of boundary layers of the field f 1 but further increasing the rotation causes vanishing of these boundary layers we obtain optimum profiles of the horizontal averaged total temperature field which could be used as hints for construction of the background fields when applying doering constantin method to the problems of rotating convection the wave number ensuremath alpha 1 corresponding to the optimum fields follows the asymptotic relationship ensuremath alpha 1 r 5 1 4 for intermediate rayleigh numbers however when r becomes large with respect to ta after a transition region the power law for ensuremath alpha 1 becomes close to the power law for the case without rotation the nusselt number nu is close to the nonrotational bound 0 32r 1 3 for the case of large r and small ta nu decreases with increasing taylor number thus the upper bounds reflect the tendency of inhibiting thermal convection by increasing rotation for a fixed rayleigh number for the regions of rayleigh and taylor numbers where the numerical and asymptotic bounds on nu can be compared the numerical bounds are about 70 lower than the asymptotic bounds

on the finite volume element method the finite volume element method fve is a discretization technique for partial differential equations it uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations then restricts the admissible functions to a finite element space to discretize the solution this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with fve based on triangulations with linear finite element spaces and a general type of control volume we establisho h estimates of the error in a discreteh 1 semi norm under an additional assumption of local uniformity of the triangulation the estimate is improved too h 2 results on the effects of numerical integration are also included

some new error estimates of a semidiscrete finite volume element method for a parabolic integro differential equation with nonsmooth initial data a semidiscrete finite volume element fve approximation to a parabolic integro differential equation pide is analyzed in a two dimensional convex polygonal domain an optimal order l 2 error estimate for smooth initial data and nearly the same optimal order l 2 error estimate for nonsmooth initial data are obtained more precisely for homogeneous equations an elementary energy technique and a duality argument are used to derive an error estimate of order o left t 1 h 2 ln h right in the l 2 norm for positive time when the given initial function is only in l 2

a finite volume method based on the crouzeix raviart element for elliptic pde s in two dimensions we introduce and analyse a finite volume method for the discretization of elliptic boundary value problems in mathbb r 2 the method is based on nonuniform triangulations with piecewise linear nonconforming spaces we prove optimal order error estimates in the l 2 norm and a mesh dependent h 1 norm

rigorous bound on the plane shear flow dissipation rate the doering constantin approach phys rev lett 69 1648 1992 to derive an upper bound on the dissipation rate in terms of a properly chosen stationary flow profile can be slightly improved by casting it in the form of a variational principle for the profile only schwarz s inequality is needed to prove this its solution can be evaluated analytically the mean dissipation bound is var epsilon le ital u sup 3 ital l sup minus 1 9 radical 2 provided the reynolds number re ital ul nu exceeds 12 radical 2 and the mean transverse momentum flux is bounded by ital j sub 31 le nu ital l sup minus 1 sup 2 re sup 2 9 radical 2

mixed volume element combined with characteristic mixed finite volume element method for oil water two phase displacement problem abstract as the basic of numerical simulation of energy science the displacement of three dimensional oil water two phase in porous media is discussed in this paper for incompressible miscible displacement the pressure is described by a flow equation in an elliptic mathematical formulation and the saturation is defined by a convection diffusion equation the pressure exists in the saturation equation by darcy velocity and controls the whole flow we develop and improve substantially the work of arbogast and wheeler then put forward a mixed volume element combined with characteristic mixed finite volume element method for two phase displacement problem the flow equation is discretized by the conservative mixed volume element method which could improve the accuracy of an order for computing darcy velocity the saturation equation is solved by a characteristic mixed volume element method where mixed volume element method is used to compute the diffusion term and the method of characteristics is adopted to discretize the convection term the method of characteristics can confirm high stability of numerical simulation at the fronts avoid numerical dispersion and nonphysical oscillation and can adopt large time steps obtain smaller time truncation error and improve the computation accuracy mixed volume element can solve the convection term confirm conservation of mass at each element and approximate the saturation and the adjoint vector meanwhile it is most important in numerical computation of seepage mechanics optimal second order estimates in l 2 norm are derived by theory and special techniques of priori estimates finally numerical experiments are shown to illustrate the efficiency and practicability and to solve the international problem successfully

voronoi based finite volume methods optimal voronoi meshes and pdes on the sphere we first develop and analyze a finite volume scheme for the discretization of partial differential equations pdes on the sphere the scheme uses voronoi tessellations of the sphere for a model convection diffusion problem the finite volume scheme is shown to produce first order accurate approximations with respect to a mesh dependent discrete firstderivative norm then we introduce the notion of constrained centroidal voronoi tessellations ccvts of the sphere these are special voronoi tessellation of the sphere for which the generators of the voronoi cells are also the constrained centers of mass with respect to a prescribed density function of the cells after discussing an algorithm for determining ccvt meshes on the sphere we discuss and illustrate several desirable properties possessed by these meshes in particular it is shown that ccvt meshes define very high quality uniform and non uniform meshes on the sphere finally we discuss through some computational experiments the performance of the ccvt meshes used in conjunction with the finite volume scheme for the solution of simple model pdes on the sphere the experiments show for example that the ccvt based finite volume approximations are second order accurate if errors are measured in

a discontinuous finite volume method for a coupled fracture model abstract in this paper we present a discontinuous finite volume method for a hybrid dimensional coupled fracture model on triangular meshes in matrix and fracture the pressure is approximated by the discontinuous and continuous piecewise linear elements and in order to retain local conservation property the velocity is sought in the discontinuous lowest order raviart thomas element space and piecewise linear element space the proposed numerical scheme is only associated with the pressure so we avoid solving an indefinite saddle point problem in addition the discontinuous method eliminates the continuity of approximate functions across the interelement boundary so it has the advantage of high localizability and easy handling of complicated geometries optimal order error estimates for the pressure and velocity are proved and numerical experiments are carried out to confirm our theoretical analysis

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

on weak residual error estimation a general framework for weak residual error estimators applying to various types of boundary value problems in connection with finite element and finite volume approximations is developed basic ideas commonly shared by various applications in error estimation and adaptive computation are presented and illustrated some numerical results are given to show the effectiveness and efficiency of the estimators

symmetric finite volume method for second order variable coefficient hyperbolic equations in this paper we consider one semi discrete and two full discrete symmetric finite volume schemes for a class of second order variable coefficient hyperbolic equations based on a linear finite element space the optimal order error estimates in l2 h1 norms are derived for the semi discrete and full discrete schemes finally some numerical experiments are presented to confirm the performance of the symmetric schemes

the riemann problem for 2 2 systems of hyperbolic conservation laws with case i quadratic nonlinearities on etudie les problemes de riemann pour des systemes 2 2 de lois de conservation non strictement hyperboliques avec des non linearites quadratiques

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

riemann problems for nonstrictly hyperbolic 2 2 systems of conservation laws on resout le probleme de riemann pour des systemes 2 2 de lois de conservation hyperboliques ayant des fonctions flux quadratiques

hyperbolic conservation laws with umbilic degeneracy i in this paper a compactness framework for approximate solutions to nonlinear hyperbolic systems with umbilic degeneracy is established by combining techniques of compensated compactness with some classical methods and by a detailed analysis of a highly singular equation of euler poisson darboux type then this framework is successfully applied to prove the convergence of the viscosity method and to prove the existence of global entropy solutions for the cauchy problem with large initial data for a canonical class of the systems with quadratic flux form

error estimation of a quadratic finite volume method on right quadrangular prism grids in this paper we develop a finite volume element method with affine quadratic bases on right quadrangular prism meshes for three dimensional elliptic boundary value problems the optimal h 1 norm error estimate of second order accuracy is proved under certain assumptions about the meshes numerical results are presented to illustrate the theoretical analysis

is 2k conjecture valid for finite volume methods this paper is concerned with superconvergence properties of a class of finite volume methods of arbitrary order over rectangular meshes our main result is to prove the 2k conjecture at each vertex of the underlying rectangular mesh the bi k degree finite volume solution approximates the exact solution with an order o h 2k where h is the mesh size as byproducts superconvergence properties for finite volume discretization errors at lobatto and gauss points are also obtained all theoretical findings are confirmed by numerical experiments

on the accuracy of the finite volume element method for diffusion equations on composite grids this paper develops the finite volume element fve method which is similar to the so called control volume finite element method but tailored to composite grid applications fve is described for the general two dimensional diffusion equation in divergence form o h 3 2 discretization error estimates are then developed for the case of triangular elements these results are improved to o h 2 using a special modification involving rectilinear elements at the grid interface

improved variational principle for bounds on energy dissipation in turbulent shear flow we extend the doering constantin approach to upper bounds on energy dissipation in turbulent flows by introducing a balance parameter into the variational principle this parameter governs the relative weight of different contributions to the dissipation rate its optimization leads to improved bounds without entailing additional technical difficulties for plane couette flow the high re bounds obtainable with one dimensional background flows are methodically lowered by a factor of 2732

the riemann problem near a hyperbolic singularity the classification of solutions of quadratic riemann problems 1 the purpose of this paper is to classify the solutions of riemann problems near a hyperbolic singularity in a nonlinear system of conservation laws hyperbolic singularities play the role in the theory of riemann problems that rest points play in the theory of ordinary differential equations indeed generically only a finite number of structures can appear in a neighborhood of such a singularity in this the first of three papers the program of classification is discussed in general and the simplest structure that occurs is characterized

positive nonlinear ddfv scheme for a degenerate parabolic system describing chemotaxis abstract in this work we prove the convergence of a discretized anisotropic degenerate parabolic system modeling chemotaxis the used method belongs to a positive nonlinear discrete duality finite volume ddfv family its aim is to respect the physical bounds on the computed solution independently of the used mesh and the involved anisotropy to achieve that two main terms referred to as convection and diffusion are carefully approximated the transport contribution is discretized using a product of centered and upstream schemes the discretization of the oscillatory diffusive fluxes is treated by means of an upwind scheme based on the established a priori estimates we demonstrate the convergence of the approximate solutions thanks to compactness arguments adapted to degenerate parabolic problems numerical experiments are performed in two dimensions the goal is to test the efficiency and the robustness of the proposed scheme on a series of examples including anisotropic tensors and non smooth data over various types of meshes

unification of variational principles for turbulent shear flows the background method of doering constantin and the mean fluctuation formulation of howard busse abstract the recent discovery of a new variational formulation for bounding turbulent flow quantities has rekindled interest in the whole field of producing rigorous inequality results directly from the navier stokes equations initiated by howard and busse in the 1960s the new background method introduced by doering and constantin 1992 1994 1995 1996 for bounding turbulent viscous dissipation appears to differ so fundamentally from that of howard and busse that any connection between the two methods has been fascinatingly unclear recently highly suggestive results obtained asymptotically kerswell 1997 have indicated that the variational problems produced for both cases of shear flow and convection may be equivalent here we confirm this for the shear flow case by uncovering the common underlying functional which both approaches seek to make stationary busse s upper bound problem is found to be the optimal dual or complementary variational problem to that which emerges from the doering constantin approach this ensures that busse s best bound exactly coincides with that available within the doering constantin formalism for all reynolds numbers

uniform convergence of finite volume element method with crouzeix raviart element for non self adjoint and indefinite elliptic problems in this paper we consider the finite volume element method based on the crouzeix raviart element and prove the existence uniqueness and uniform convergence of the finite volume element approximations for the non self adjoint and indefinite elliptic problems under minimal elliptic regularity assumption

a finite volume method for stokes problems on quadrilateral meshes abstract we present a finite volume method for stokes problems using the isoparametric q 1 q 0 element pair on quadrilateral meshes to offset the lack of the i n f s u p condition a jump term of discrete pressure stabilizing term is added to the continuity approximation equation thus we establish a stabilized finite volume scheme on quadrilateral meshes then based on some superclose estimates we derive the optimal error estimates in the h 1 and l 2 norms for velocity and in the l 2 norm for pressure respectively numerical examples are provided to illustrate our theoretical analysis we emphasize that our work is the first time to propose and analyze a finite volume method for stoke problems using isoparametric elements on quadrilateral meshes

nonlinear and stochastic phenomena the grand challenge for partial differential equations the major problems for partial differential equations are either nonlinear or stochastic or both considerable progress has occurred recently in these areas much of which has resulted from a striking interplay among theory computation and applications the nonlinear structures emphasized here are the nonlinear hyperbolic waves which occur in the solutions of nonlinear hyperbolic conservation laws and associated dissipative equations these equations are those most commonly used to model physical and chemical processes in continuum systems in one space dimension these nonlinear waves form traveling waves for which the methods of dynamical systems and bifurcation theory are very useful established points of view toward mathematical theories have been challenged by recent work e g uniqueness of solutions entropy conditions and elliptic regions in hyperbolic equations a type of geometry is defined in the state space of the conservation law by the nonlinear waves this geometry is generically sing

on the accuracy of the mortar finite volume element method with lagrange multipliers abstract in this paper the mortar finite volume element mfve method with lagrange multipliers is considered for second order elliptic boundary value problems we prove l 2 error estimates which are affected by regularities in both the exact solution and the source term meanwhile the uniform convergence of the solution is obtained for the mfve method under minimal regularity assumption

additive average schwarz method for a crouzeix raviart finite volume element discretization of elliptic problems with heterogeneous coefficients in this paper we introduce an additive schwarz method for a crouzeix raviart finite volume element discretization of a second order elliptic problem with discontinuous coefficients where the discontinuities are both inside the subdomains and across and along the subdomain boundaries we show that depending on the distribution of the coefficient in the model problem the parameters describing the generalized minimal residual method gmres convergence rate of the proposed method depend linearly on the mesh parameters inlineequation not available see fulltext also under certain restrictions on the distribution of the coefficient the convergence of the gmres method is independent of jumps in the coefficient

a two grid method for finite volume element approximations of second order nonlinear hyperbolic equations the two grid method is studied for solving a two dimensional second order nonlinear hyperbolic equation using finite volume element method the method is based on two different finite element spaces defined on one coarse grid with grid size h and one fine grid with grid size h respectively the nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine grid solution can be obtained in a single symmetric and linear step it is proved that the coarse grid can be much coarser than the fine grid a prior error estimate in the h 1 norm is proved to be o h h 3 lnh for the two grid semidiscrete finite volume element method with these proposed techniques solving such a large class of second order nonlinear hyperbolic equations will not be much more difficult than solving one single linearized equation finally a numerical example is presented to validate the usefulness and efficiency of the method

variational bounds on energy dissipation in incompressible flows iii convection building on a method of analysis for the navier stokes equations introduced by hopf math ann 117 764 1941 a variational principle for upper bounds on the largest possible time averaged convective heat flux is derived from the boussinesq equations of motion when supplied with appropriate test background fields satisfying a spectral constraint reminiscent of an energy stability condition the variational formulation produces rigorous upper bounds on the nusselt number nu as a function of the rayleigh number ra for the case of vertical heat convection between parallel plates in the absence of sidewalls a simplified but rigorous formulation of the optimization problem yields the large rayleigh number bound nu ensuremath le 0 167 mathrm ra 1 2 1 nonlinear euler lagrange equations for the optimal background fields are also derived which allow us to make contact with the upper bound theory of howard j fluid mech 17 405 1963 for statistically stationary flows the structure of solutions of the euler lagrange equations are elucidated from the geometry of the variational constraints which sheds light on busse s j fluid mech 37 457 1969 asymptotic analysis of general solutions to howard s euler lagrange equations the results of our analysis are discussed in the context of theory recent experiments and direct numerical simulations textcopyright 1996 the american physical society

two grid finite volume element method for linear and nonlinear elliptic problems two grid finite volume element discretization techniques based on two linear conforming finite element spaces on one coarse and one fine grid are presented for the two dimensional second order non selfadjoint and indefinite linear elliptic problems and the two dimensional second order nonlinear elliptic problems with the proposed techniques solving the non selfadjoint and indefinite elliptic problem on the fine space is reduced into solving a symmetric and positive definite elliptic problem on the fine space and solving the non selfadjoint and indefinite elliptic problem on a much smaller space solving a nonlinear elliptic problem on the fine space is reduced into solving a linear problem on the fine space and solving the nonlinear elliptic problem on a much smaller space convergence estimates are derived to justify the efficiency of the proposed two grid algorithms a set of numerical examples are presented to confirm the estimates

mortar upwind finite volume element method for convection diffusion problems abstract in this paper we consider the mortar upwind finite volume element method with local linear conforming finite element space for the convection diffusion problems and show that the mortar upwind finite volume element approximations derived are convergent with optimal order in the h 1 norm

a symmetric finite volume scheme for selfadjoint elliptic problems based on a linear finite element space a symmetric finite volume scheme for a self adjoint elliptic boundary value problem is proposed error estimates in l2 norm h1 norm and l norm are derived some post processing techniques are also provided

on the construction and analysis of high order locally conservative finite volume type methods for one dimensional elliptic problems locally conservative finite volume type methods based on continuous piecewise polynomial functions of degree r ge 2 are introduced and analyzed in the context of indefinite elliptic problems in one space dimension the new methods extend and generalize the classical finite volume method based on piecewise linear functions we derive a priori error estimates in the l2 h1 and l infty norm and discuss superconvergence effects for the error and its derivative explicit residual based a posteriori error bounds in the l2 and energy norm are also derived we compute the experimental order of convergence and show the results of an adaptive algorithm based on the a posteriori error estimates

uniqueness of the riemann solution for three phase flow in a porous medium we solve injection problems for immiscible three phase flow described by a system of two conservation laws with fluxes originating from corey s model with quadratic permeabilities a mixture of water gas and oil is injected into a porous medium containing oil which is partially displaced we prove that the resulting riemann problems have solutions which are unique under technical hypotheses that can be verified numerically the wave curve method constructs the solutions straightforwardly despite loss of strict hyperbolicity at an isolated point in state space this umbilic point induces the separation of the solutions into two types according to the water gas proportion in the injection mixture

simplified weak galerkin and new finite difference schemes for the stokes equation abstract this article presents a simplified formulation for the weak galerkin finite element method for the stokes equations without using the degrees of freedom associated with the unknowns in the interior of each element as formulated in the original weak galerkin algorithm the simplified formulation preserves the important mass conservation property locally on each element and allows the use of general polygonal partitions a particular application of the simplified weak galerkin on rectangular partitions yields a new class of 5 and 7 point finite difference schemes for the stokes equations an explicit formula is presented for the computation of the element stiffness matrices on arbitrary polygonal elements error estimates of optimal order are established for the simplified weak galerkin finite element method in the h 1 and l 2 norms furthermore a superconvergence of order o h 3 2 is established on rectangular partitions for the velocity approximation in the h 1 norm at cell centers and a similar superconvergence is derived for the pressure approximation in the l 2 norm at cell centers some numerical results are reported to confirm the convergence and superconvergence theory