quadrilateral mesh revisited several quadrilateral shape regular mesh conditions commonly used in the finite element method are proven to be equivalent the effect of the bi section condition on the degenerate mesh conditions is also checked

a mixed nonoverlapping covolume method on quadrilateral grids for elliptic problems a covolume method is proposed for the mixed formulation of second order elliptic problems the solution domain is divided by a quadrilateral grid corresponding to which a nonoverlapping dual grid is constructed the velocity and pressure are approximated by the lowest order raviart thomas space on quadrilaterals we prove its first order optimal rate of convergence for the approximate velocities in the h div norm as well as for the approximate pressures in the l 2 norm a second order error estimate between a suitable projection of the exact velocity or pressure and the approximate velocity or approximate pressure is also presented numerical experiments are provided to illustrate the error behavior of the scheme

proof of existence of global solutions for m component reaction diffusion systems with mixed boundary conditions via the lyapunov functional method to prove global existence for solutions of m component reaction diffusion systems presents fundamental difficulties in the case in which some components of the system satisfy neumann boundary conditions while others satisfy nonhomogeneous dirichlet boundary conditions and nonhomogeneous robin boundary conditions the purpose of this paper is to prove the existence of a global solution using a single inequality for the polynomial growth condition of the reaction terms our technique is based on the construction of polynomial functionals this result generalizes those obtained recently by kouachi et al at press kouachi 2002 electron j diff eqns 2002 1 kouachi 2001 electron j diff eqns 2001 1 and independently by malham and xin 1998 commun math phys 193 287

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

mixed upwinding covolume methods on rectangular grids for convection diffusion problems we consider an upwinding covolume or control volume method for a system of first order pdes resulting from the mixed formulation of a convection diffusion equation with a variable anisotropic diffusion tensor the system can be used to model the steady state of the transport of a contaminant carried by a flow we use the lowest order raviart thomas space and show that the concentration and concentration flux both converge at one half order provided that the exact flux is in h 1 omega 2 and the exact concentration is in h 1 omega some numerical experiments illustrating the error behavior of the scheme are provided

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

the accuracy of cell vertex finite volume methods on quadrilateral meshes for linear first order hyperbolic equations in two dimensions we restate the cell vertex finite volume scheme as a finite element method on structured meshes consisting of distorted quadrilaterals the global error is shown to be of second order in various mesh dependent norms provided that the quadrilaterals are close to parallelograms in the sense that the distance between the midpoints of the diagonals is of the same order as the measure of the guadrilateral on tensor product nonuniform meshes the cell vertex scheme coincides with the familiar box scheme in this case second order accuracy is shown without any additional assumption on the regularity of the mesh which explains the insensitivity of the cell vertex scheme to mesh stretching in the coordinate directions observed in practice 17 refs

enriched multi point flux approximation for general grids it is well known that the two point flux approximation a numerical scheme used in most commercial reservoir simulators has o 1 error when grids are not k orthogonal in the last decade the multi point flux approximations have been developed as a remedy however non physical oscillations can appear when the anisotropy is really strong we found out the oscillations are closely related to the poor approximation of pressure gradient in the flux computation in this paper we propose the control volume enriched multi point flux approximation empfa for general diffusion problems on polygonal and polyhedral meshes non physical oscillations are not observed for realistic and strongly anisotropic heterogeneous material properties described by a full tensor exact linear solutions are recovered for grids with non planar interfaces and a first and second order convergence are achieved for the flux and scalar unknowns respectively

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

analysis and convergence of a covolume method for the generalized stokes problem we introduce a covolume or mac like method for approximating the generalized stokes problem two grids are needed in the discretization a triangular one for the contimlity equation and a quadrilateral one for the momentum equation the velocity is approximated using nonconforming piecewise linears and the pressure piecewise constants error in the l 2 norm for the pressure and error in a mesh dependent h 1 norm as well as in the l 2 norm for the velocity are shown to be of first order provided that the exact velocity is in h 2 and the true pressure in h i we also introduce the concept of a network model into the discretized linear system so that an efficient pressure recovering technique can be used to simplify a great deal the computational work involved in the augmented lagrangian method given is a very general decomposition condition under which this technique is applicable to other fluid problems that can be formulated as a saddle point problem

scalar curvature and stability of toric varieties we define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a kahler metric of constant scalar curvature the main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces we prove that under the stability hypothesis the mabuchi functional is bounded below on invariant metrics and that minimising sequences have a certain convergence property in the reverse direction we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature the proofs use a general framework developed by guillemin and abreu in which invariant kahler metrics correspond to convex functions on the moment polytope of a toric variety this paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a kahler metric of constant scalar curvature the pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of stability in the sense of geometric invariant theory this expectation is probably now an item of folklore going back to suggestions put forward by yau in the case of kahlereinstein metrics and the many results of tian and others in this case reinforced by a detailed formal picture which makes clear the analogy with the well established relation between the stability of vector bundles and yang mills connections 5 here we begin the investigation of

global existence for reaction diffusion systems modelling ignition the pair of parabolic equations with a 0 and b 0 models the temperature and concentration for an exothermic chemical reaction for which just one species controls the reaction rate f of particular interest is the case where which appears in the frank kamenetskii approximation to arrhenius type reactions we show here that for a large choice of the nonlinearity f u v in 1 2 including the model case 3 the corresponding initial value problem for 1 2 in the whole space with bounded initial data has a solution which exists for all times finite time blow up might occur though for other choices of function f u v and we discuss here a linear example which strongly hints at such behaviour

mixed finite element methods for general quadrilateral grids abstract we study a new mixed finite element of lowest order for general quadrilateral grids which gives optimal order error in the h div norm this new element is designed so that the h div projection h satisfies h p h div a rigorous optimal order error estimate is carried out by proving a modified version of the bramble hilbert lemma for vector variables we show that a local h div projection reproducing certain polynomials suffices to yield an optimal l 2 error estimate for the velocity and hence our approach also provides an improved error estimate for original raviart thomas element of lowest order numerical experiments are presented to verify our theory

mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems we construct and analyze a mixed finite volume method on quadrilateral grids for elliptic problems written as a system of two first order pdes in the state variable e g pressure and its flux e g darcy velocity an important point is that no staggered grids or covolumes are used to stabilize the system only a single primary grid system is adopted and the degrees of freedom are imposed on the interfaces the approximate flux is sought in the lowest order raviart thomas space and the pressure field in the rotated q1 nonconforming space furthermore we demonstrate that the present finite volume method can be interpreted as a rotated q1 nonconforming finite element method for the pressure with a simple local recovery of flux numerical results are presented for a variety of problems which confirm the usefulness and effectiveness of the method

on mori cone of bott towers abstract a bott tower of height r is a sequence of projective bundles x r r x r 1 r 1 2 x 1 p 1 1 x 0 p t where x i p o x i 1 l i 1 for a line bundle l i 1 over x i 1 for all 1 i r and p denotes the projectivization these are smooth projective toric varieties and we refer to the top object x r also as a bott tower in this article we study the mori cone and numerically effective nef cone of bott towers and we classify fano weak fano and log fano bott towers we prove some vanishing theorems for the cohomology of tangent bundle of bott towers

an lq lp theory for the time fractional evolution equations with variable coefficients abstract we introduce an l q l p theory for the semilinear fractional equations of the type 0 1 t u t x a i j t x u x i x j t x f t x u t 0 x r d here 0 2 p q 1 and t is the caupto fractional derivative of order uniqueness existence and l q l p estimates of solutions are obtained the leading coefficients a i j t x are assumed to be piecewise continuous in t and uniformly continuous in x in particular a i j t x are allowed to be discontinuous with respect to the time variable our approach is based on classical tools in pde theories such as the marcinkiewicz interpolation theorem the calderon zygmund theorem and perturbation arguments

the convergence of mimetic discretization for rough grids we prove that the mimetic finite difference discretizations of laplace s equation converges on rough logically rectangular grids with convex cells mimetic discretizations for the invariant operators divergence gradient and curl satisfy exact discrete analogs of many of the important theorems of vector calculus the mimetic discretization of the laplacian is given by the composition of the discrete divergence and gradient we first construct a mimetic discretization on a single cell by geometrically constructing inner products for discrete scalar and vector fields then constructing a finite volume discrete divergence and then constructing a discrete gradient that is consistent with the discrete divergence theorem this construction is then extended to the global grid we demonstrate the convergence for the two dimensional laplace equation with dirichlet boundary conditions on grids with a lower bound on the angles in the cell corners and an upper bound on the cell aspect ratios the best convergence rate to be expected is first order which is what we prove the techniques developed apply to far more general initial boundary value problems

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

optimal biquadratic finite volume element methods on quadrilateral meshes in this paper an optimal biquadratic finite volume element scheme is established and analyzed for elliptic equations on quadrilateral meshes it is proved that the new scheme has optimal o h 2 convergence rate in h 1 norm and o h 3 convergence rate in l 2 norm with the assumption that each element is an h 2 parallelogram

kahler ricci solitons on toric manifolds with positive first chern class abstract in this paper we prove there exists a kahler ricci soliton unique up to holomorphic automorphisms on any toric kahler manifold with positive first chern class and the kahler ricci soliton is a kahler einstein metric if and only if the futaki invariant vanishes

hamiltonian 2 forms in kahler geometry iii extremal metrics and stability this paper concerns the existence and explicit construction of extremal kahler metrics on total spaces of projective bundles which have been studied in many places we present a unified approach motivated by the theory of hamiltonian 2 forms as introduced and studied in previous papers in the series but this paper is largely independent of that theory

extremal kahler metrics on projective bundles over a curve let m p e be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle e over a compact complex curve of genus 2 building on ideas of fujiki 1992 27 we prove that m admits a kahler metric of constant scalar curvature if and only if e is polystable we also address the more general existence problem of extremal kahler metrics on such bundles and prove that the splitting of e as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal kahler metrics in kahler classes sufficiently far from the boundary of the kahler cone the methods used to prove the above results apply to a wider class of manifolds called rigid toric bundles over a semisimple base which are fibrations associated to a principal torus bundle over a product of constant scalar curvature kahler manifolds with fibres isomorphic to a given toric kahler variety we discuss various ramifications of our approach to this class of manifolds

two families of functions related to the fractional powers of generators of strongly continuous contraction semigroups abstract two families of functions on 0 are related to the theory of fractional powers of generators of strongly continuous semigroups namely the family t t 0 of density functions of the one sided stable semigroup of order 0 1 and a family e 0 of mittag leffler type functions for the latter family we make this relation visible we collect some important properties of these functions and we improve a known result on the laplace transform of the functions e and their derivatives in the sense that we enlarge the domains of the laplace variable we furthermore find an expression for the laplace transform of the functions t x t x 0 in terms of the derivative e

the kahler geometry of bott manifolds abstract we study the kahler geometry of stage n bott manifolds which can be viewed as n dimensional generalizations of hirzebruch surfaces we show using a simple induction argument and the generalized calabi construction from 3 6 that any stage n bott manifold m n admits an extremal kahler metric we also give necessary conditions for m n to admit a constant scalar curvature kahler metric we obtain more precise results for stage 3 bott manifolds including in particular some interesting relations with c projective geometry and some explicit examples of almost kahler structures to place these results in context we review and develop the topology complex geometry and symplectic geometry of bott manifolds in particular we study the kahler cone the automorphism group and the fano condition we also relate the number of conjugacy classes of maximal tori in the symplectomorphism group to the number of biholomorphism classes compatible with the symplectic structure

topological classification of generalized bott towers if b is a toric manifold and e is a whitney sum of complex line bundles over b then the projectivization p e of e is again a toric manifold starting with b as a point and repeating this construction we obtain a sequence of complex projective bundles which we call a generalized bott tower we prove that if the top manifold in the tower has the same cohomology ring as a product of complex projective spaces then every fibration in the tower is trivial so that the top manifold is diffeomorphic to the product of complex projective spaces this gives supporting evidence to what we call the cohomological rigidity problem for toric manifolds are toric manifolds diffeomorphic or homeomorphic if their cohomology rings are isomorphic we provide two more results which support the cohomological rigidity problem

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

bounds on steady states for a non systemically autocatalysed reaction diffusion system bounds on parameter values that lead to different types of solutions for a non systemically autocatalysed reaction diffusion system are derived and show the existence of a critical concentration beyond which the stability properties undergo change additionally bifurcating solutions at the zero eigenvalue are obtained

diffusion terms in systems of reaction diffusion equations can lead to blow up we present a system of reaction diffusion equations posed in r in which the diffusion terms are responsible for the finite time blow up of its solutions

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

positively invariant regions for strongly coupled reaction diffusion systems with a balance law are nonnegative constants systems such as theseand their quasilinear generalizations have received much attention be cause they arise in several chemical and biological models the reader isreferred to the expository article by ni 6 and to further references thatmay be found in 3 and 4 2 positively invariant sets for weaklycoupled systemswe will first state a result from 5 on positively invariant regions butsince we will not need it in its full generality we will only state a specialcase of theorem 16 from that paper and under much simplified hypothe ses in particular we will avoid tangential derivatives in the boundaryconditions and we will impose simple regularity requirements on thecoefficients let be a bounded domain in

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

global solutions of reaction diffusion systems with a balance law and nonlinearities of exponential growth abstract we consider the initial boundary value problem of the form u t a u f u v v t c u d v f u v x r n n 1 t r where f u v 0 f 0 v 0 v r f u v k u e v k and are positive constants is any continuous nonnegative locally lipschitzian function on r such that 0 0 d a and c d a with bounded continuous nonegative initial data this system contains in particular the frank kamenetskii approximation to an n th order exothermic chemical reaction of arrhenius type and non systemically autocatalysed reaction diffusion systems we prove the existence of global classical solutions and study their large time behaviour our main tools are estimates of the neumann function for the heat equation and local l p a priori estimates independent of time

a posteriori error analysis for locally conservative mixed methods in this work we present a theoretical analysis for a residual type error estimator for locally conservative mixed methods this estimator was first introduced by braess and verfurth for the raviart thomas mixed finite element method working in mesh dependent norms we improve and extend their results to cover any locally conservative mixed method under minimal assumptions in particular avoiding the saturation assumption made by braess and verfurth our analysis also takes into account discontinuous coefficients with possibly large jumps across interelement boundaries the main results are applied to the p1 nonconforming finite element method and the interior penalty discontinuous galerkin method as well as the mixed finite element method

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

a co volume scheme for the rotating shallow water equations on conforming non orthogonal grids a co volume scheme is introduced for the rotating shallow water equations in which both velocity components are specified on cell edges and the thickness variables evolve on both the primary and the dual cell centers the scheme applies to generic conforming and non orthogonal staggered grids including the widely used lat lon quadrilateral grids and the delaunay voronoi tessellations it can be viewed either as coupled c grid schemes on the primary and dual meshes or as an generalization of the traditional e grid scheme on a new non overlapping grid linear dispersive wave analysis shows that the dispersive relations resolved by either the primary or the dual mesh of a uniform quadrilateral staggered grid is the same as those of the z grid scheme the total wavenumber space resolved by the staggered grid is twice as large on which the co volume behaves exactly like the e grid scheme on a uniform hexagon triangular staggered grid the co volume has two steady modes and two inertial gravity modes on the hexagonal mesh and one steady mode two inertial gravity modes and two spurious modes on the triangular mesh on the wavenumber space resolved by either the hexagonal or the triangular mesh the inertial gravity wave modes remain positive and largely monotone for the nonlinear shallow water equations the co volume scheme is shown to preserve the potential vorticity dynamics and the total energy exactly numerical results are presented to corroborate and supplement the analyses

conservative p 1 conforming and nonconforming galerkin fems effective flux evaluation via a nonmixed method approach given a p1 conforming or nonconforming galerkin finite element method gfem solution ph which approximates the exact solution p of the diffusion reaction equation nabla cdot mathcal k nabla p alpha p f with full tensor variable coefficient mathcal k we evaluate the approximate flux uh to the exact flux bf u mathcal k nabla p by a simple but physically intuitive formula over each finite element the flux is sought in the continuous in normal component or the discontinuous raviart thomas space a systematic way of deriving such a formula is introduced this direct method retains local conservation property at the element level typical of mixed methods finite element or finite volume type but avoids solving an indefinite linear system in short the present method retains the best of the gfem and the mixed method but without their shortcomings thus we view our method as a conservative gfem and demonstrate its equivalence to a certain mixed finite volume box method the equivalence theorems explain how the pressure can decouple basically cost free from the mixed formulation the accuracy in the flux is of first order in the h dive omega norm numerical results are provided to support the theory

relative k stability and modified k energy on toric manifolds in this paper we give a sufficient condition for both the relative k stability and the properness of modified k energy associated to calabi s extremal metrics on toric manifolds in addition several examples of toric manifolds which satisfy the sufficient condition are presented

on the regularity and uniformness conditions on quadrilateral grids in the literature there are various regularity conditions as well as many different so called almost parallelogram conditions on the subdivisions of quadrilateral grids in this paper we investigate and clarify their relations

on systems of reaction diffusion equations with a balance law the sequel abstract this paper is a sequel of the review paper on reaction diffusion systems with a balance law by martin and pierre 1990 here we present the results obtained after martin and pierre 1990 not only for classical systems but also for systems with either time fractional derivatives or space fractional derivatives accounting for anomalous diffusions

diffusive instability near hopf bifurcation for exponentially autocatalyzed reaction diffusion system the analysis of an exponentially autocatalyzed reaction diffusion system near the hopf bifurcation point has been carried out using a reductive perturbation approach to obtain a description in terms of the ginzburg landau equation the conditions for the occurrence of instability in the presence and absence of diffusion leading to hopf bifurcation are also derived the nature of the governing equations leads to multi valued instability conditions and eventually results in more than one region in parameter space where instability of uniform oscillations due to diffusion is possible

global bounds and asymptotics for a system of reaction diffusion equations on etudie le probleme aux valeurs limites non lineaire u t a u u v dans r v t c u d v u v dans r avec les conditions u v 0 sur r et u 0 x u 0 x v 0 x v 0 x dans on etablit l existence de solutions globales fortes et on etudie leur comportement quand t

on the heterogeneous multiscale method with various macroscopic solvers abstract the heterogeneous multiscale method hmm is a general method for efficient numerical solution of problems with multiscales it consists of two components an overall macroscopic solver for macrovariables on a macrogrid and an estimation of the missing macroscopic data from the microscopic model in this paper we present a state of the art review of the hmm with various macroscopic solvers including finite differences finite elements discontinuous galerkin mixed finite elements control volume finite elements nonconforming finite elements and mixed covolumes the first four solvers have been studied in the hmm setting the others are not as example the hmm with the nonconforming finite element macroscopic solver for nonlinear and random homogenization problems is also studied here

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

global existence and asymptotic behavior for a time fractional reactiondiffusion system this paper is concerned with the existence of global in time solutions of a time fractional reactiondiffusion system with time fractional derivatives furthermore the large time behavior of bounded solutions is investigated our method of proof relies on a maximal regularity result for fractional linear reactiondiffusion equations that has been derived by bajlekova 2001

maximum principle for the generalized time fractional diffusion equation abstract in the paper a maximum principle for the generalized time fractional diffusion equation over an open bounded domain g 0 t g r n is formulated and proved the proof of the maximum principle is based on an extremum principle for the caputo dzherbashyan fractional derivative that is given in the paper too the maximum principle is then applied to show that the initial boundary value problem for the generalized time fractional diffusion equation possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions

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

on the existence and nonexistence of extremal metrics on toric kahler surfaces abstract in this paper we study the existence of extremal metrics on toric kahler surfaces we show that on every toric kahler surface there exists a kahler class in which the surface admits an extremal metric of calabi we found a toric kahler surface of 9 t c 2 fixed points which admits an unstable kahler class and there is no extremal metric of calabi in it moreover we prove a characterization of the k stability of toric surfaces by simple piecewise linear functions as an application we show that among all toric kahler surfaces with 5 or 6 t c 2 fixed points cp 2 3 cp 2 is the only one which allows vanishing futaki invariant and admits extremal metrics of constant scalar curvature

mixed covolume methods on rectangular grids for elliptic problems we consider a covolume method for a system of first order pdes resulting from the mixed formulation of the variable coefficient matrix poisson equation with the neumann boundary condition the system may be used to represent the darcy law and the mass conservation law in anisotropic porous media flow the velocity and pressure are approximated by the lowest order raviart thomas space on rectangles the method was introduced by russell rigorous block centered discretizations on irregular grids improved simulation of complex reservoir systems reservoir simulation research corporation denver co 1995 as a control volume mixed method and has been extensively tested by jones a mixed finite volume elementary method for accurate computation of fluid velocities in porous media university of colorado at denver 1995 and cai et al computational geosciences 1 1997 pp 289 345 we reformulate it as a covolume method and prove its first order optimal rate of convergence for the approximate velocities as well as for the approximate pressures

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

a general mixed covolume framework for constructing conservative schemes for elliptic problems we present a general framework for the finite volume or covolume schemes developed for second order elliptic problems in mixed form i e written as first order systems we connect these schemes to standard mixed finite element methods via a one to one transfer operator between trial and test spaces in the nonsymmetric case convection diffusion equation we show one half order convergence rate for the flux variable which is approximated either by the lowest order raviart thomas space or by its image in the space of discontinuous piecewise constants in the symmetric case diffusion equation a first order convergence rate is obtained for both the state variable e g concentration and its flux numerical experiments are included

existence of extremal metrics on compact almost homogeneous kahler manifolds with two ends in this note we prove the existence and the uniqueness of extremal metrics in every kahler class of any compact almost homogeneous kahler manifold with two ends by considering the scalar curvature equations those manifolds might not be projective we also prove that there are extremal metrics ini some kahler classes of a completion of the multicanonical line bundle of a kahler einstein manifold of positive ricci curvature

hamiltonian 2 forms in kahler geometry ii global classification we present a classification of compact kahler manifolds admitting a hamiltonian 2 form which were classified locally in part i of this work this involves two components of independent interest the first is the notion of a rigid hamiltonian torus action this natural condition for torus actions on a kahler manifold was introduced locally in part i but such actions turn out to be remarkably well behaved globally leading to a fairly explicit classification up to a blow up compact kahler manifolds with a rigid hamiltonian torus action are bundles of toric kahler manifolds the second idea is a special case of toric geometry which we call orthotoric we prove that orthotoric kahler manifolds are diffeomorphic to complex projective space but we extend our analysis to orthotoric orbifolds where the geometry is much richer we thus obtain new examples of kahler einstein 4 orbifolds combining these two themes we prove that compact kahler manifolds with hamiltonian 2 forms are covered by blow downs of projective bundles over kahler products and we describe explicitly how the kahler metrics with a hamiltonian 2 form are parameterized we explain how this provides a context for constructing new examples of extremal kahler metrics in particular a subclass of such metrics which we call weakly bochner flat we also provide a self contained treatment of the theory of compact toric kahler manifolds since we need it and find the existing literature incomplete

superconvergence of new mixed finite element spaces in this paper we prove some superconvergence of a new family of mixed finite element spaces of higher order which we introduced in etna vol 37 pp 189 201 2010 among all the mixed finite element spaces having an optimal order of convergence on quadrilateral grids this space has the smallest unknowns however the scalar variable is only suboptimal in general thus we have employed a post processing technique for the scalar variable as a byproduct we have obtained a superconvergence on a rectangular grid the superconvergence of a velocity variable naturally holds and can be shown by a minor modification of existing theory but that of a scalar variable requires a new technique especially for k 1 numerical experiments are provided to support the theory

a covolume method based on rotated bilinears for the generalized stokes problem we introduce a covolume or marker and cell mac method for approximating the generalized stokes problem on an axiparallel domain two grids are needed the primal grid made up of rectangles and the dual grid of quadrilaterals the velocity is approximated by nonconforming rotated bilinear elements with degrees of freedom at midpoints of rectangular elements and the pressure by piecewise constants the error in the velocity in the h 1 h norm and the pressure in the l2 norm are of first order provided that the exact velocity is in h2 and the exact pressure in h1

mixed covolume method for parabolic problems on triangular grids in this paper we present a mixed covolume method for parabolic equations on triangular grids this method use the lowest order raviart thomas r t mixed finite element space as the trial space we prove the optimal order of convergence for the approximate pressure and velocity in l 2 norm furthermore we obtain the quasi optimal error estimates for the approximate pressure in l norm