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

critical behavior of the annealed ising model on random regular graphs in 17 the authors have defined an annealed ising model on random graphs and proved limit theorems for the magnetization of this model on some random graphs including random 2 regular graphs then in 11 we generalized their results to the class of all random regular graphs in this paper we study the critical behavior of this model in particular we determine the critical exponents and prove a non standard limit theorem that the magnetization scaled by n 3 4 converges to a specific random variable with n the number of vertices of random regular graphs

co volume methods for degenerate parabolic problems a complementary volume co volume technique is used to develop a physically appealing algorithm for the solution of degenerate parabolic problems such as the stefan problem it is shown that these algorithms give rise to a discrete semigroup theory that parallels the continuous problem in particular the discrete stefan problem gives rise to nonlinear semigroups in both the discretel 1 andh 1 spaces

convergence analysis of a numerical method for a mean field model of superconducting vortices in the mean field models of superconductivity the individual vortex like structures occurring in practical type ii superconductors are averaged and a vortex density is solved for them the numerical solution of the mean field models makes large scale simulations of vortex phenomena possible in this paper we present a simple convergence analysis for a numerical method based on hybrid finite element finite volume finite difference approximations in the two dimensional case by providing various different interpretations to the discretization scheme

the orthogonal decomposition theorems for mimetic finite difference methods accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus provide reliable finite difference methods for approximating the solutions to a wide class of partial differential equations these methods mimic many fundamental properties of the underlying physical problem including conservation laws symmetries in the solution and the nondivergence of particular vector fields i e they are divergence free and should satisfy a discrete version of the orthogonal decomposition theorem this theorem plays a fundamental role in the theory of generalized solutions and in the numerical solution of physical models including the navier stokes equations and in electrodynamics we are deriving mimetic finite difference approximations of the divergence gradient and curl that satisfy discrete analogs of the integral identities satisfied by the differential operators we first define the natural discrete divergence gradient and curl operators based on coordinate invariant definitions such as gauss s theorem for the divergence next we use the formal adjoints of these natural operators to derive compatible divergence gradient and curl operators with complementary domains and ranges of values in this paper we prove that these operators satisfy discrete analogs of the orthogonal decomposition theorem and demonstrate how a discrete vector can be decomposed into two orthogonal vectors in a unique way satisfying a discrete analog of the formula vec a ggrad varphi curl vec b we also present a numerical example to illustrate the numerical procedure and calculate the convergence rate of the method for a spiral vector field

convergence analysis of a finite volume method for maxwell s equations in nonhomogeneous media in this paper we analyze a recently developed finite volume method for the time dependent maxwell s equations in a three dimensional polyhedral domain composed of two dielectric materials with different parameter values for the electric permittivity and the magnetic permeability convergence and error estimates of the numerical scheme are established for general nonuniform tetrahedral triangulations of the physical domain in the case of nonuniform rectangular grids the scheme converges with second order accuracy in the discrete l2 norm despite the low regularity of the true solution over the entire domain in particular the finite volume method is shown to be superconvergent in the discrete h curl norm in addition the explicit dependence of the error estimates on the material parameters is given

connection probabilities and rsw type bounds for the two dimensional fk ising model we prove russo seymour welsh type uniform bounds on crossing probabilities for the fk ising fk percolation with cluster weight q 2 model at criticality independent of the boundary conditions our proof relies mainly on smirnov s fermionic observable for the fk ising model 24 which allows us to get precise estimates on boundary connection probabilities we stay in a discrete setting in particular we do not make use of any continuum limit and our result can be used to derive directly several noteworthy properties including some new ones among which are the fact that there is no infinite cluster at criticality tightness properties for the interfaces and the existence of several critical exponents in particular the half plane one arm exponent such crossing bounds are also instrumental for important applications such as constructing the scaling limit of the ising spin field 6 and deriving polynomial bounds for the mixing time of the glauber dynamics at criticality 17

the fourier spectrum of critical percolation consider the indicator function f of a 2 dimensional percolation crossing event in this paper the fourier transform of f is studied and sharp bounds are obtained for its lower tail in several situations various applications of these bounds are derived in particular we show that the set of exceptional times of dynamical critical site percolation on the triangular grid in which the origin percolates has dimension frac 31 36 almost surely and the corresponding dimension in the half plane is frac 5 9 it is also proved that critical bond percolation on the square grid has exceptional times almost surely also the asymptotics of the number of sites that need to be resampled in order to significantly perturb the global percolation configuration in a large square is determined

the mimetic finite difference method for the 3d magnetostatic field problems on polyhedral meshes we extend the mimetic finite difference mfd method to the numerical treatment of magnetostatic fields problems in mixed div curl form for the divergence free magnetic vector potential to accomplish this task we introduce three sets of degrees of freedom that are attached to the vertices the edges and the faces of the mesh and two discrete operators mimicking the curl and the gradient operator of the differential setting then we present the construction of two suitable quadrature rules for the numerical discretization of the domain integrals of the div curl variational formulation of the magnetostatic equations this construction is based on an algebraic consistency condition that generalizes the usual construction of the inner products of the mfd method we also discuss the linear algebraic form of the resulting mfd scheme its practical implementation and discuss existence and uniqueness of the numerical solution by generalizing the concept of logically rectangular or cubic meshes by hyman and shashkov to the case of unstructured polyhedral meshes the accuracy of the method is illustrated by solving numerically a set of academic problems and a realistic engineering problem

solution of the div curl problem in generalized curvilinear coordinates we propose a method for solving the div curl problem on a structured nonorthogonal curvilinear grid the differential operators are discretized using a mac scheme for the unknowns in such a way that the discrete counterparts of the usual vector analysis relations are satisfied the derived discrete problem is then solved by performing a helmholtz type decomposition of the unknown vector field this allows us to obtain a vector field for which both divergence and curl are satisfied to within machine accuracy the method is validated for several configurations in two and three dimensions and its accuracy is numerically checked

an inhomogeneous transference principle and diophantine approximation in a landmark paper 29 d y kleinbock and g a margulis established the fundamental baker sprindzuk conjecture on homogeneous diophantine approximation on manifolds subsequently there has been dramatic progress in this area of research however the techniques developed to date do not seem to be applicable to inhomogeneous approximation consequently the theory of inhomogeneous diophantine approximation on manifolds remains essentially non existent in this paper we develop an approach that enables us to transfer homogeneous statements to inhomogeneous ones this is rather surprising as the inhomogeneous theory contains the homogeneous theory and so is more general as a consequence we establish the inhomogeneous analogue of the baker sprindzuk conjecture furthermore we prove a complete inhomogeneous version of the profound theorem of kleinbock lindenstrauss weiss 27 on the extremality of friendly measures the results obtained in this paper constitute the first step towards developing a coherent inhomogeneous theory for manifolds in line with the homogeneous theory epsrc advanced research fellow grant ep c54076x 1 research supported by epsrc grant ep e061613 1 and intas grant 03 51 5070

mixed covolume methods for elliptic problems on triangular grids we consider a covolume or finite volume 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 represent either the darcy law and the mass conservation law in anisotropic porous media flow or fourier law and energy conservation the velocity and pressure are approximated by the lowest order raviart thomas space on triangles we prove its first order optimal rate of convergence for the approximate velocities in the l 2 and h mbox div omega norms as well as for the approximate pressures in the l 2 norm numerical experiments are included

analysis and computation of compatible least squares methods for div curl equations we develop and analyze least squares finite element methods for two complementary div curl elliptic boundary value problems the first one prescribes the tangential component of the vector field on the boundary and is solved using curl conforming elements the second problem specifies the normal component of the vector field and is handled by div conforming elements we prove that both least squares formulations are norm equivalent with respect to suitable discrete norms yield optimal asymptotic error estimates and give rise to algebraic systems that can be solved by efficient algebraic multigrid methods numerical results that illustrate scalability of iterative solvers and optimal rates of convergence are also included

discrete calculus methods for diffusion a general methodology for the solution of partial differential equations is described in which the discretization of the calculus is exact and all approximation occurs as an interpolation problem on the material constitutive equations the fact that the calculus is exact gives these methods the ability to capture the physics of pde systems well the construction of both node and cell based methods of first and second order are described for the problem of unsteady heat conduction though the method is applicable to any pde system the performance of these new methods are compared to classic solution methods on unstructured 2d and 3d meshes for a variety of simple and complex test cases

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

control volume approximation of degenerate two phase porous flows implicit euler approximations of the equations governing the porous flow of two immiscible incompressible fluids are shown to be the euler lagrange equations of a convex function tools from conve

conformal invariance in random cluster models i holmorphic fermions in the ising model we construct discrete holomorphic observables in the ising model at criticality and show that they have conformally covariant scaling limits as mesh of the lattice tends to zero in the sequel those observables are used to construct conformally invariant scaling limits of interfaces though ising model is often cited as a classical example of conformal invariance it seems that ours is the first paper where it is actually established

high quality 2d mesh generation without obtuse and small angles abstract in this paper we present an efficient method to eliminate the obtuse triangles for high quality 2d mesh generation given an initialization e g from centroidal voronoi tessellation cvt a limited number of point insertions and removals are performed to eliminate obtuse or small angle triangles a mesh smoothing and optimization step is then applied these steps are repeated till a desired good quality mesh is reached we tested our algorithm on various 2d polygonal domains and verified that our algorithm always converges after inserting a few number of new points and generates high quality triangulation with no obtuse triangles

quantitative recurrence properties in conformal iterated function systems abstract we consider a quantitative version of poincare s recurrence theorem in a conformal iterated function system let i i be a conformal iterated function system on 0 1 d with a countable index set denote by j the attractor of let f 0 1 d r be a positive function s n f x be the sum f x f w 1 1 x f w 1 w n 1 1 x analogous to an ergodic sum and consider the set of points for which the inequality x w 1 w n 0 1 d x w 1 w n 1 x e s n f x is realized for infinitely many n n with w 1 w n n the set above contains the points x whose orbit returns very close to x infinitely many times with the quality of approximation depending on the time n and the point x this can be viewed as a quantitative version of poincare s recurrence theorem it is shown that its hausdorff dimension is the solution to some pressure function the setting considered in this paper includes the quantitative recurrence properties in the dynamical systems of b adic expansions continued fraction expansion as well as some dynamical systems defined on fractal sets applying the main result to diophantine approximation gives a partial answer to a question by k mahler

analysis and convergence of the mac scheme ii navier stokes equations the mac discretization scheme for the incompressible navier stokes equations is interpreted as a covolume approximation to the equations using some results from earlier papers dealing with covolume error estimates for div curl equation systems and under certain conditions on the data and the solutions of the navier stokes equations we obtain first order error estimates for both the vorticity and the pressure

a second order time accurate finite volume method for unsteady incompressible flow on hybrid unstructured grids a new second order time accurate fractional step method for solving unsteady incompressible navier stokes equations on hybrid unstructured grids is presented the nonstaggered grid method originally developed by rhie and chow 1983 aiaa j 21 1525 for steady flow and further extended by zang et al 1994 j comput phys 114 18 to unsteady flow on structured grids is employed in the present study to enforce mass conservation on hybrid unstructured grids the pressure and cartesian velocity components are defined at the center of each cell while the face normal velocities are defined at the mid points of the corresponding cell faces a second order fully implicit time advancement scheme is used for time integration and the resulting nonlinear equations are linearized without losing the overall time accuracy both the momentum and poisson equations are integrated with the finite volume method and the flow variables at the cell face are obtained using an interpolation scheme independent of cell shape the present numerical method is applied to four different benchmark problems and proves to be accurate and efficient

the discrete duality finite volume method for convection diffusion problems in this paper we extend the discrete duality finite volume ddfv formulation to the steady convection diffusion equation the discrete gradients defined in ddfv are used to define a cell based gradient for the control volumes of both the primal and dual meshes in order to achieve a higher order accurate numerical flux for the convection term a priori analysis is carried out to show convergence of the approximation and a global first order convergence rate is derived the theoretical results are confirmed by some numerical experiments

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

analysis and convergence of the mac scheme i the linear problem the mac marker and cell discretization of fluid flow is analysed for the stationary stokes equations it is proved that the discrete approximations do in fact converge to the exact solutions of the flow equations estimates using mesh dependent norms analogous to the standard bf h 1 and l 2 norms are given for the velocity and pressure respectively

discrete hodge operators many linear boundary value problems arising in computational physics can be formulated in the calculus of differential forms discrete differential forms provide a natural and canonical approach to their discretization however much freedom remains concerning the choice of discrete hodge operators that is discrete analogues of constitutive laws a generic discrete hodge operator is introduced and it turns out that most finite element and finite volume schemes emerge as its specializations we reap the possibility of a unified convergence analysis in the framework of discrete exterior calculus

a fast lattice green s function method for solving viscous incompressible flows on unbounded domains a computationally efficient method for solving three dimensional viscous incompressible flows on unbounded domains is presented the method formally discretizes the incompressible navier stokes equations on an unbounded staggered cartesian grid operations are limited to a finite computational domain through a lattice green s function technique this technique obtains solutions to inhomogeneous difference equations through the discrete convolution of source terms with the fundamental solutions of the discrete operators the differential algebraic equations describing the temporal evolution of the discrete momentum equation and incompressibility constraint are numerically solved by combining an integrating factor technique for the viscous term and a half explicit runge kutta scheme for the convective term a projection method that exploits the mimetic and commutativity properties of the discrete operators is used to efficiently solve the system of equations that arises in each stage of the time integration scheme linear complexity fast computation rates and parallel scalability are achieved using recently developed fast multipole methods for difference equations the accuracy and physical fidelity of solutions are verified through numerical simulations of vortex rings

direct discretization of planar div curl problems a control volume method is proposed for planar div curl systems the method is independent of potential and least squares formulations and works directly with the div curl system the novelty of the technique lies in its use of a single local vector field component and two control volumes rather than the other way round a discrete vector field theory comes quite naturally from this idea and is developed in the paper error estimates are proved for the method and other ramifications investigated

a discrete duality finite volume approach to hodge decomposition and div curl problems on almost arbitrary two dimensional meshes we define discrete differential operators such as gradient divergence and curl on general two dimensional nonorthogonal meshes these discrete operators verify discrete analogues of usual continuous theorems discrete green formulae discrete hodge decomposition of vector fields and vector curls have a vanishing divergence and gradients have a vanishing curl we apply these ideas to discretize div curl systems we give error estimates based on the reformulation of these systems into equivalent equations for the potentials numerical results illustrate the use of the method on several types of meshes some of which are degenerating triangular meshes and nonconforming locally refined meshes

γ convergence of external approximations in boundary value problems involving the bi laplacian in addition to the various uses it was introduced for the theory of convergence offers a rather natural setting for discussing and developing nonorthodox approximation methods for variational problems for certain boundary value problems involving the bi laplacian sequences of discrete functionals are here defined and are shown to converge to the corresponding functionals of the continuous problems the minimizers of the discrete functionals provide converging approximations to the solution of the limit problem in question thus we obtain approximation schemes that are nonconforming but direct and that can be treated by current algorithms for symmetric and positive definite functionals the class of problems considered in this paper includes the stokes problem in fluid dynamics the loading problem of 2 d isotropic elastostatics and some boundary value problems of the kirchhoff love theory of plates also discussed is an extension of the discretization method that seems suitable for treating more general boundary value problems of elastic plates but whose convergence is conditional to a conjecture that remains to be proved a relevant application to the so called babuska paradox is presented

discrete exterior calculus discretization of incompressible navier stokes equations over surface simplicial meshes a conservative discretization of incompressible navier stokes equations is developed based on discrete exterior calculus dec a distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product the governing equations are first rewritten using the exterior calculus notation replacing vector calculus differential operators by the exterior derivative hodge star and wedge product operators the discretization is then carried out by substituting with the corresponding discrete operators based on the dec framework numerical experiments for flows over surfaces reveal a second order accuracy for the developed scheme when using structured triangular meshes and first order accuracy for otherwise unstructured meshes by construction the method is conservative in that both mass and vorticity are conserved up to machine precision the relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step

a framework for mimetic discretization of the rotating shallow water equations on arbitrary polygonal grids accurate simulation of atmospheric flow in weather and climate prediction models requires the discretization of the governing equations to have a number of desirable properties although these properties can be achieved relatively straightforwardly on a latitude longitude grid they are much more challenging on the various quasi uniform spherical grids that are now under consideration a recently developed scheme called trisk has these desirable properties on grids that have an orthogonal dual the present work extends the trisk scheme into a more general framework suitable for grids that have a nonorthogonal dual such as the equiangular cubed sphere we also show that this framework fits within the wider framework of mimetic discretizations and discrete exterior calculus one key ingredient is the definition of certain mapping operators that are discrete analogues of the hodge star operator enabling the definition of a compatible inner product discrete coriolis terms are also included within the mimet

conformal invariance in the fk representation of the quantum ising model and convergence of the interface to the mathrm sle _ 16 3 we study the interface in the fk representation of the 1d quantum ising model and show that in the limit it converges to the mathrm sle 16 3 curve

a finite volume method on general surfaces and its error estimates abstract in this paper we study a finite volume method and its error estimates for the numerical solution of some model second order elliptic partial differential equations defined on a smooth surface the discretization is defined via a surface mesh consisting of piecewise planar triangles and piecewise polygons the optimal error estimates of the approximate solution are proved in both the h 1 and l 2 norms which are of first order and second order respectively under mesh regularity assumptions some numerical tests are also carried out to experimentally verify our theoretical analysis

a unified approach to energy conservation and potential vorticity dynamics for arbitrarily structured c grids a numerical scheme applicable to arbitrarily structured c grids is presented for the nonlinear shallow water equations by discretizing the vector invariant form of the momentum equation the relationship between the nonlinear coriolis force and the potential vorticity flux can be used to guarantee that mass velocity and potential vorticity evolve in a consistent and compatible manner underpinning the consistency and compatibility of the discrete system is the construction of an auxiliary thickness equation that is staggered from the primary thickness equation and collocated with the vorticity field the numerical scheme also exhibits conservation of total energy to within time truncation error simulations of the standard shallow water test cases confirm the analysis and show convergence rates between 1st and 2nd order accuracy when discretizing the system with quasi uniform spherical voronoi diagrams the numerical method is applicable to a wide class of meshes including latitude longitude grids voronoi diagrams delaunay triangulations and conformally mapped cubed sphere meshes

covolume techniques for anisotropic media the covolume method a new approach applicable on general meshes is extended to discretize and numerically solve the div curl system in anisotropic media the covolume method gives simple schemes and good approximations to the solution of the div curl system it works directly with the system and utilizes dual pairs of meshes that are orthogonally related central to the approach is the introduction of field components tangent and normal to the edges of one of the meshes and the employment of dual discretization on the dual mesh pairs the discretization procedures schemes and error analysis are presented the convergence of the method is proved

modified block preconditioners for the discretized time harmonic maxwell equations in mixed form in this paper building on the previous work by greif and schotzau preconditioners for the discretized time harmonic maxwell equations in mixed form numer linear algebra appl 14 2007 281 297 and benzi and olshanskii an augmented lagrangian based approach to the oseen problem siam j sci comput 28 2006 2095 2113 we present the improved preconditioning techniques for the iterative solution of the saddle point linear systems which arise from the finite element discretization of the mixed formulation of the time harmonic maxwell equations the modified block diagonal and triangular preconditioners considered are based on augmentation with using the symmetric nonsingular weighted matrix we discuss the spectral properties of the preconditioned matrix in detail and generalize the results of the above mentioned paper by greif and schotzau numerical experiments are given to demonstrate the efficiency of the presented preconditioners

modified shrinking target problem in beta dynamical systems abstract in the paper we investigate a modified version of the shrinking target problem in beta dynamical systems which is induced by the transformation t x x mod 1 the hausdorff dimension of the lim sup set w f x 0 1 t n x f x n for infinitely many n n is determined where f 0 1 0 1 is a lipschitz function and is a positive function defined on n the set w f can be viewed as a dynamical analogue of the inhomogeneous diophantine approximation in which the inhomogeneous factor is allowed to vary

conservation properties of unstructured staggered mesh schemes classic cartesian staggered mesh schemes have a number of attractive properties they do not display spurious pressure modes and they have been shown to locally conserve mass momentum kinetic energy and circulation to machine precision recently a number of generalizations of the staggered mesh approach have been proposed for unstructured triangular or tetrahedral meshes these unstructured staggered mesh methods have been created to retain the attractive pressure aspects and mass conservation properties of the classic cartesian mesh method this work addresses the momentum kinetic energy and circulation conservation properties of unstructured staggered mesh methods it is shown that with certain choices of the velocity interpolation unstructured staggered mesh discretizations of the divergence form of the navier stokes equations can conserve kinetic energy and momentum both locally and globally in addition it is shown that unstructured staggered mesh discretizations of the rotational form of the navier stokes equations can conserve kinetic energy and circulation both locally and globally the analysis includes viscous terms and a generalization of the concept of conservation in the presence of viscosity to include a negative definite dissipation term in the kinetic energy equation these novel conserving unstructured staggered mesh schemes have not been previously analyzed it is shown that they are first order accurate on nonuniform two dimensional unstructured meshes and second order accurate on uniform unstructured meshes numerical confirmation of the conservation properties and the order of accuracy of these unstructured staggered mesh methods is presented

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

infinite conformal symmetry of critical fluctuations in two dimensions we study the massless quantum field theories describing the critical points in two dimensional statistical systems these theories are invariant with respect to the infinite dimensional group of conformal analytic transformations it is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the virasoro algebra exactly solvable theories associated with degenerate representations are analized in these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations

analysis of the cell vertex finite volume method for the cauchy riemann equations this paper initiates a study of finite volume methods for linear first order elliptic systems by performing a stability and convergence analysis of the cell vertex approximation of the cauchy riemann equations the approach is based on reformulating the scheme as a petrov galerkin finite element method with continuous bilinear trial functions and piecewise constant test functions optimal error bounds are derived in a mesh dependent norm and the counting problem which may occur due to geometry and boundary conditions is considered

approximating second order vector differential operators on distorted meshes in two space dimensions a new finite volume method is presented for approximating second order vector differential operators in two space dimensions this method allows distorted triangle or quadrilateral meshes to be used without the numerical results being too much altered the matrices that need to be inverted are symmetric positive definite therefore the most powerful linear solvers can be applied the method has been tested on a few second order vector partial differential equations coming from elasticity and fluids mechanics areas these numerical experiments show that it is second order accurate and locking free copyright 2008 john wiley sons ltd

influence and sharp threshold theorems for monotonic measures the influence theorem for product measures on the discrete space 0 1 n may be extended to probability measures with the property of monotonicity which is equivalent to strong positive association corresponding results are valid for probability measures on the cube 0 1 n that are absolutely continuous with respect to lebesgue measure these results lead to a sharp threshold theorem for measures of random cluster type and this may be applied to box crossings in the two dimensional random cluster model

natural discretizations for the divergence gradient and curl on logically rectangular grids abstract this is the first in series of papers creating a discrete analog of vector analysis on logically rectangular nonorthogonal nonsmooth grids we introduce notations for 2 d logically rectangular grids describe both cell valued and nodal discretizations for scalar functions and construct the natural discretizations of vector fields using the vector components normal and tangential to the cell boundaries we then define natural discrete analogs of the divergence gradient and curl operators based on coordinate invariant definitions and interpret these formulas in terms of curvilinear coordinates such as length of elements of coordinate lines areas of elements of coordinate surfaces and elementary volumes we introduce the discrete volume integral of scalar functions the discrete surface integral and a discrete analog of the line integral and prove discrete versions of the main theorems relating these objects these theorems include the following the discrete analog of relationship div a 0 if and only if a curl b curl a 0 if and only if a grad if a grad then the line integral does not depend on path and if the line integral of a vector function is equal to zero for any closed path then this vector is the gradient of a scalar function last we define the discrete operators div grad and curl in terms of primitive differencing operators based on forward and backward differences and primitive metric operators related to multiplications of discrete functions by length of edges areas of surfaces and volumes of 3 d cells these formulations elucidate the structure of the discrete operators and are useful when investigating the relationships between operators and their adjoints

smirnov s fermionic observable away from criticality in a recent and celebrated article smirnov ann of math 2 172 2010 1435 1467 defines an observable for the self dual random cluster model with cluster weight q 2 on the square lattice z2 and uses it to obtain conformal invariance in the scaling limit we study this observable away from the self dual point from this we obtain a new derivation of the fact that the self dual and critical points coincide which implies that the critical inverse temperature of the ising model equals 12log 1 2 moreover we relate the correlation length of the model to the large deviation behavior of a certain massive random walk thus confirming an observation by messikh the surface tension near criticality of the 2d ising model 2006 preprint which allows us to compute it explicitly

finite volume methods on spheres and spherical centroidal voronoi meshes we study in this paper a finite volume approximation of linear convection diffusion equations defined on a sphere using the spherical voronoi meshes in particular the spherical centroidal voronoi meshes the high quality of spherical centroidal voronoi meshes is illustrated through both theoretical analysis and computational experiments in particular we show that the l2 error of the approximate solution is of quadratic order when the underlying mesh is given by a spherical centroidal voronoi mesh we also demonstrate numerically the high accuracy and the superconvergence of the approximate solutions

accuracy and conservation properties of a three dimensional unstructured staggered mesh scheme for fluid dynamics a three dimensional unstructured mesh discretization of the rotational from of the incompressible navier stokes is presented the method uses novel and highly efficient algorithms for interpolating the velocity vector and constructing the convention term the resulting discretization is shown to conserve mass kinetic energy and vorticity to machine precision both locally and globally the spatial accuracy of the method is analyzed and found to be second order on regular meshes and first order on irregular meshes the numerical efficiency accuracy and conservation properties of the method are tested on three dimensional meshes and found to be in agreement with theory

the ergodic theory of shrinking targets to any dynamical system equipped with a metric we associate a class of well approximable sets in the case of an expanding rational map of the riemann sphere acting on its julia set we estimate and in some cases compute the hausdorff dimension of the associated well approximable sets the methods used show a clear link between distortion properties and the type of results obtained in this paper via ergodic theory and ubiquity

regular article finite difference schemes for u t x α δg δu that inherit energy conservation or dissipation property we propose a new procedure for designing by rote finite difference schemes that inherit energy conservation or dissipation property from nonlinear partial differential equations such as the korteweg de vries kdv equation and the cahn hilliard equation the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly since the inherited properties are kept even if the time mesh size changes in the time evolution process we can use some appropriate time mesh adaptive methods to obtain numerical solutions through the derived schemes because of these properties the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions and sufficiently flexible to treat the inheritance of the energy conservation and dissipation properties are verified numerically for the kdv equation and the cahn hilliard equation

covolume method for new nonconforming rectangle element for the stokes problem abstract we analyze a covolume method based on the new nonconforming element introduced by douglas et al 1 we show the h 1 optimal order convergence of the scheme for stokes problem and study the hybrid domain decomposition procedure for this covolume scheme the numerical experiment shows that the covolume scheme is somewhat better than finite element scheme in the computation of pressure

discrete gauge invariant approximations of a time dependent ginzburg landau model of superconductivity we present here a mathematical analysis of a nonstandard difference method for the numerical solution of the time dependent ginzburg landau models of superconductivity this type of method has been widely used in numerical simulations of the behavior of superconducting materials we also illustrate some of their nice properties such as the gauge invariance being retained in discrete approximations and the discrete order parameter having physically consistent pointwise bound

convergence analysis of a covolume scheme for maxwell s equations in three dimensions this paper contains error estimates for covolume discretizations of maxwell s equations in three space dimensions several estimates are proved first an estimate for a semi discrete scheme is given second the estimate is extended to cover the classical interlaced time marching technique third some of our unstructured mesh results are specialized to rectangular meshes both uniform and nonuniform by means of some additional analysis it is shown that the spatial convergence rate is one order higher than for the unstructured case

spectral partitioning works planar graphs and finite element meshes abstract spectral partitioning methods use the fiedler vector the eigenvector of the second smallest eigenvalue of the laplacian matrix to find a small separator of a graph these methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well in this paper we show that spectral partitioning methods work well on bounded degree planar graphs and finite element meshes the classes of graphs to which they are usually applied while naive spectral bisection does not necessarily work we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is o n for bounded degree planar graphs and two dimensional meshes and o n1 d for well shaped d dimensional meshes the heart of our analysis is an upper bound on the second smallest eigenvalues of the laplacian matrices of these graphs we prove a bound of o 1 n for bounded degree planar graphs and o 1 n2 d for well shaped d dimensional meshes

covolume solutions of three dimensional div curl equations the voronoi delaunay mesh systems provide a generalization of the classical rectangular staggered meshes to unstructured meshes in this work it is shown how such covolume discretizations may be applied to div curl systems in three dimensions error estimates are proved and confirmed by a numerical illustration

error estimates in l 2 h 1 and l in covolume methods for elliptic and parabolic p roblems a unified approach in this paper we consider covolume or finite volume element methods for variable coefficient elliptic and parabolic problems on convex smooth domains in the plane we introduce a general approach for connecting these methods with finite element method analysis this unified approach is used to prove known convergence results in the h 1 l 2 norms and new results in the max norm for the elliptic problems we demonstrate that the error u u h between the exact solution u and the approximate solution u h in the maximum norm is o h 2 ln h in the linear element case furthermore the maximur norm error in the gradient is shown to be of first order similar results hold for the parabolic problems

convergence analysis of fully discrete finite volume methods for maxwell s equations in nonhomogeneous media we will consider both explicit and implicit fully discrete finite volume schemes for solving three dimensional maxwell s equations with discontinuous physical coefficients on general polyhedral domains stability and convergence for both schemes are analyzed we prove that the schemes are second order accurate in time both schemes are proved to be first order accurate in space for the voronoi delaunay grids and second order accurate for nonuniform rectangular grids we also derive explicit expressions for the dependence on the physical parameters in all estimates

a stabilized covolume method for the stokes problem abstract we present a covolume method for the modified stokes problem using the simplest approximation spaces q 1 p 0 this scheme turns out the stabilized covolume method for the stokes problem we prove that the covolume method in this paper has a unique solution and o h convergence order in h 1 semi norm for the velocity and in l 2 norm the pressure approximants respectively we also present numerical results corresponding to our analysis

finite element methods with matching and nonmatching meshes for maxwell equations with discontinuous coefficients we investigate the finite element methods for solving time dependent maxwell equations with discontinuous coefficients in general three dimensional lipschitz polyhedral domains both matching and nonmatching finite element meshes on the interfaces are considered and optimal error estimates for both cases are obtained the analysis of the latter case is based on an abstract framework for nested saddle point problems along with a characterization of the trace space for h curl d a new extension theorem for h curl d functions in any lipschitz domain d and a novel compactness argument for deriving discrete inf sup conditions

universality in the 2d ising model and conformal invariance of fermionic observables it is widely believed that the celebrated 2d ising model at criticality has a universal and conformally invariant scaling limit which is used in deriving many of its properties however no mathematical proof has ever been given and even physics arguments support a priori weaker mobius invariance we introduce discrete holomorphic fermions for the 2d ising model at criticality on a large family of planar graphs we show that on bounded domains with appropriate boundary conditions those have universal and conformally invariant scaling limits thus proving the universality and conformal invariance conjectures

inhomogeneous theory of dual diophantine approximation on manifolds the theory of inhomogeneous diophantine approximation on manifolds is developed in particular the notion of nice manifolds is introduced and the divergence part of the groshev type theory is established for all such manifolds our results naturally incorporate and generalize the homogeneous measure and dimension theorems for non degenerate manifolds established to date the results have natural applications beyond the standard inhomogeneous theory such as diophantine approximation by algebraic integers

the self dual point of the two dimensional random cluster model is critical for q 1 we prove a long standing conjecture on random cluster models namely that the critical point for such models with parameter q 1 on the square lattice is equal to the self dual point p sd q sqrt q 1 sqrt q this gives a proof that the critical temperature of the q state potts model is equal to log 1 sqrt q for all q 2 we further prove that the transition is sharp meaning that there is exponential decay of correlations in the sub critical phase the techniques of this paper are rigorous and valid for all q 1 in contrast to earlier methods valid only for certain given q the proof extends to the triangular and the hexagonal lattices as well

well centered triangulation meshes composed of well centered simplices have nice orthogonal dual meshes the dual voronoi diagram this is useful for certain numerical algorithms that prefer such primal dual mesh pairs we prove that well centered meshes also have optimality properties and relationships to delaunay and minmax angle triangulations we present an iterative algorithm that seeks to transform a given triangulation in two or three dimensions into a well centered one by minimizing a cost function and moving the interior vertices while keeping the mesh connectivity and boundary vertices fixed the cost function is a direct result of a new characterization of well centeredness in arbitrary dimensions that we present ours is the first optimization based heuristic for well centeredness and the first one that applies in both two and three dimensions we show the results of applying our algorithm to small and large two dimensional meshes some with a complex boundary and obtain a well centered tetrahedralization of the cube we also show numerical evidence that our algorithm preserves gradation and that it improves the maximum and minimum angles of acute triangulations created by the best known previous method

discrete complex analysis on isoradial graphs abstract we study discrete complex analysis and potential theory on a large family of planar graphs the so called isoradial ones along with discrete analogues of several classical results we prove uniform convergence of discrete harmonic measures green s functions and poisson kernels to their continuous counterparts among other applications the results can be used to establish universality of the critical ising and other lattice models

on the radius edge condition in the control volume method in this paper we show that the control volume algorithm for the solution of poisson s equation in three dimensions will converge even if the mesh contains a class of very flat tetrahedra slivers these tetrahedra are characterized by the fact that they have modest ratios of diameter to shortest edge but large circumscribing to inscribed sphere radius ratios and therefore may have poor interpolation properties elimination of slivers is a notoriously difficult problem for automatic mesh generation algorithms we also show that a discrete poincare inequality will continue to hold in the presence of slivers

a compatible and conservative spectral element method on unstructured grids we derive a formulation of the spectral element method which is compatible on very general unstructured three dimensional grids here compatible means that the method retains discrete analogs of several key properties of the divergence gradient and curl operators the divergence and gradient are anti adjoints the negative transpose of each other the curl is self adjoint and annihilates the gradient operator and the divergence annihilates the curl the adjoint relations hold globally and at the element level with the inclusion of a natural discrete element boundary flux term we then discretize the shallow water equations on the sphere using the cubed sphere grid and show that compatibility allows us to locally conserve mass energy and potential vorticity conservation is obtained without requiring the equations to be in conservation form the conservation is exact assuming exact time integration

quantitative recurrence properties for beta dynamical system abstract this note is concerned with the quantitative recurrence properties of beta dynamical system 0 1 t for general 1 the size of points with the prescribed recurrence rate is determined more precisely hausdorff dimensions of the sets x 0 1 t n x x n for infinitely many n n and x 0 1 t n x x e j 0 n 1 f t j x for infinitely many n n are obtained completely where is a positive function defined on n and f is a positive continuous function on 0 1 besides the pressure function p f t with a continuous potential f is proven to be continuous with respect to

a nonconforming covolume method for elliptic problems abstract we consider a control volume covolume method for second order elliptic pdes with the rotated q1 nonconforming finite element on rectangular grids the coefficient may a variable diagonal tensor or discontinuous we prove first order convergence in h1 norm and second order convergence in l2 norm when the partition is square our numerical experiments show that our covolume scheme has about 30 less error than fem even when is discontinuous tensor

the ising magnetization exponent on z2 is 1 15 we prove that for the ising model defined on the plane mathbb z 2 at beta beta c the average magnetization under an external magnetic field h 0 behaves exactly like begin aligned langle sigma 0 rangle beta c h asymp h frac 1 15 end aligned the proof which is surprisingly simple compared to an analogous result for percolation i e that theta p p p c 5 36 o 1 on the triangular lattice kesten in commun math phys 109 1 109 156 1987 smirnov and werner in math res lett 8 5 6 729 744 2001 relies on the ghs inequality as well as the rsw theorem for fk percolation from duminil copin et al commun pure appl math 64 1165 1198 2011 the use of ghs to obtain inequalities involving critical exponents is not new in this paper we show how it can be combined with rsw to obtain matching upper and lower bounds for the average magnetization

a new approximation technique for div curl systems in this paper we describe an approximation technique for div curl systems based in l where is a domain in 3 we formulate this problem as a general variational problem with different test and trial spaces the analysis requires the verification of an appropriate inf sup condition this results is a very weak formulation where the solution space is l and the data reside in various negative norm spaces subsequently we consider finite element approximations based on this weak formulation the main approach of this paper involves the development of stable pairs of discrete test and trial spaces with this approach we enlarge the test space so that the discrete inf sup condition holds and use a negative norm least squares formulation to reduce to a uniquely solvable linear system this leads to optimal order estimates for problems with minimal regularity which is important since it is possible to construct magnetostatic field problems whose solutions have low sobolev regularity e g h with 0 s 1 2 the resulting algebraic equations are symmetric positive definite and well conditioned a second approach using a smaller test space which adds terms to the form for stabilization will also be mentioned some numerical results are also presented

an algebraic multigrid approach based on a compatible gauge reformulation of maxwell s equations with the rise in popularity of compatible finite element finite difference and finite volume discretizations for the time domain eddy current equations there has been a corresponding need for fast solvers of the resulting linear algebraic systems however the traits that make compatible discretizations a preferred choice for the maxwell s equations also render these linear systems essentially intractable by truly black box techniques we propose an algebraic reformulation of the discrete eddy current equations along with a new algebraic multigrid amg technique for this reformulated problem the reformulation process takes advantage of a discrete hodge decomposition on cochains to replace the discrete eddy current equations by an equivalent 2 times2 block linear system whose diagonal blocks are discrete hodge laplace operators acting on 1 cochains and 0 cochains respectively while this new amg technique requires somewhat specialized treatment on the finest mesh the coarser meshes can be handled using standard methods for laplace type problems our new amg method is applicable to a wide range of compatible methods on structured and unstructured grids including edge finite elements mimetic finite differences covolume methods and yee like schemes we illustrate the new technique using edge elements in the context of smoothed aggregation amg and present computational results for problems in both two and three dimensions

analysis and convergence of a covolume approximation of the ginzburg landau model of superconductivity in this paper we present the mathematical analysis of a covolume method for the approximations of the ginzburg landau gl model for superconductivity a nice feature of this approach is that the gauge invariance properties are retained in discrete approximations based on triangular grids we also use properties of discrete vector fields to study issues such as the gauge choices and their enforcement

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments