a chebyshev spectral method for gas transients in pipelines a chebyshev pseudospectral method is proposed for slow transients simulation of gas transportation systems the scheme is theoretically investigated comparisons with the lax wendroff scheme prove that the chebyshev method is more efficient in terms of storage and computer time infinite order accuracy for smooth solutions is predicted by the theory and observed in the experiments

spectral gap for the zero range process with constant rate we solve an open problem concerning the relaxation time inverse spectral gap of the zero range process in z d lz d with constant rate proving a tight upper bound of o 1 2 l 2 where p is the density of particles

multigrid methods for combined finite difference and fourier problems abstract multigrid methods for combined finite difference and fourier pseudospectral problems are considered applications to two and three dimensional elliptic systems are given suitable relaxation schemes for isotropic and anisotropic problems are presented numerical results demonstrate the efficiency of these methods

the logarithmic sobolev inequality for discrete spin systems on a lattice for finite range lattice gases with a finite spin space it is shown that the dobrushin shlosman mixing condition is equivalent to the existence of a logarithmic sobolev inequality for the associated unique gibbs state in addition implications of these considerations for the ergodic properties of the corresponding glauber dynamics are examined

chebyshev pseudospectral solution of second order elliptic equations with finite element preconditioning abstract a new chebyshev pseudospectral algorithm for second order elliptic equations using finite element preconditioning is proposed and tested on various problems bilinear and biquadratic lagrange elements are considered as well as bicubic hermite elements the numerical results show that bilinear elements produce spectral accuracy with the minimum computational work l shaped regions are treated by a subdomain approach

domain decomposition methods for pseudo spectral approximations part i second order equations in one dimension multidomain pseudo spectral approximations of second order boundary value problems in one dimension are considered the equation is collocated at the chebyshev nodes inside each subinterval different patching conditions at the interfaces are analyzed results of stability and convergence are given

remarks on decay of correlations and witten laplacians brascamp lieb inequalities and semiclassical limit as it appears in recent articles by helffer or sjostrand and naddaf spencer the analysis in the context of the statistical mechanics of measures of the type exp x dxis connected with the analysis of suitable witten laplacians on 1 forms for illustrating this point of view we present here remarks about the brascamp lieb inequalities and its extensions and prove the decay of the correlation in some cases when is weakly non convex

gegenbauer approximation in certain hilbert spaces and its applications to singular differential equations gegenbauer approximation in certain hilbert spaces is investigated several weighted inverse inequalities and poincare inequalities are obtained some approximation results are given singular differential equations are approximated by using gegenbauer polynomials this method keeps the spectral accuracy the main idea and techniques used in this paper are also applicable to some singular problems in multiple dimensional spaces

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

spectral gap estimates for interacting particle systems via a bochner type identity abstract we develop a general technique based on a bochner type identity to estimate spectral gaps of a class of markov operator we apply this technique to various interacting particle systems in particular we give a simple and short proof of the diffusive scaling of the spectral gap of the kawasaki model at high temperature similar results are derived for kawasaki type dynamics in the lattice without exclusion and in the continuum new estimates for glauber type dynamics are also obtained

filtering in legendre spectral methods we discuss the impact of modal filtering in legendre spectral methods both on accuracy and stability for the former we derive sufficient conditions on the filter to recover high order accuracy away from points of discontinuity computational results confirm that less strict necessary conditions appear to be adequate we proceed to discuss a instability mechanism in polynomial spectral methods and prove that filtering suffices to ensure stability the results are illustrated by computational experiments

exact solutions for the coagulation fragmentation equation the authors present new exact equilibrium and time dependent solutions to the two main versions of the coagulation fragmentation equation equilibria are first found for the pure coagulation and pure fragmentation equations equilibrium solutions for the full coagulation fragmentation equation are then constructed when the kernels governing the reaction rates are chosen to be in a particular form time dependent solutions are also derived for the pure fragmentation equation

positive filtered p _n moment closures for linear kinetic equations we propose a positive preserving moment closure for linear kinetic transport equations based on a filtered spherical harmonic fp n expansion in the angular variable the recently proposed fp n moment equations are known to suffer from the occurrence of unphysical negative particle concentrations the origin of this problem is that the fp n approximation is not always positive at the kinetic level the new fp n closure is developed to address this issue a new spherical harmonic expansion is computed via the solution of an optimization problem with constraints that enforce positivity but only on a finite set of preselected points combined with an appropriate pde solver for the moment equations this ensures positivity of the particle concentration at each step in the time integration under an additional mild regularity assumption we prove that fp n has the same consistency as fp n that is the fp n approximation converges to a given target function in l 2 at the same rate a

slow decay of gibbs measures with heavy tails we consider glauber dynamics reversible with respect to gibbs measures with heavy tails in the case when spins are unbounded the interactions are bounded and of finite range the self potential enters into two classes of measures concave probability measures and sub exponential laws for which it is known that no exponential decay can occur using coercive inequalities we prove that for concave probability measures the associated infinite volume semi group decays to equilibrium polynomially and stretched exponentially for sub exponential laws this improves and extends previous results by bobkov and zegarlinski

combined finite element and spectral approximation of the navier stokes equations we present a method for the numerical approximation of navier stokes equations with one direction of periodicity in this direction a fourier pseudospectral method is used in the two others a standard f e m is applied we prove optimal rate of convergence where the two parameters of discretization intervene independently on presente une methode d approximation numerique des equations de navier stokes possedant une direction de periodicite dans cette direction une methode pseudospectrale basee sur des developpements en serie de fourier est utilisee dans les deux autres on applique une methode d elements finis standard on montre que la convergence est optimale et que les deux parametres de discretisation peuvent etre choisis de facon independante

spectral tau approximation of the two dimensional stokes problem we analyse the spectral tau method for the approximation of the stokes system on a square with dirichlet boundary conditions we provide an error estimate in the norm of the sobolev spaceh s for the approximation of a divergence free vector field with polynomial divergence free vector fields we apply this result to prove some convergence estimates for the solution of the discrete stokes problem

pseudospectral least squares method for the second order elliptic boundary value problem the least squares legendre and chebyshev pseudospectral methods are presented for a first order system equivalent to a second order elliptic partial differential equation continuous and discrete homogeneous least squares functionals using legendre and chebyshev weights are shown to be equivalent to the h1 omega norm and chebyshev weighted div curl norm over appropriate polynomial spaces respectively the spectral error estimates are derived the block diagonal finite element preconditioner is developed for the both cases several numerical tests are demonstrated on the spectral discretization errors and on performances of the finite element preconditioner

entropy dissipation estimates in a zero range dynamics we study the exponential decay of relative entropy functionals for zero range processes on the complete graph for the standard model with rates increasing at infinity we prove entropy dissipation estimates uniformly over the number of particles and the number of vertices

approximation of infinite delay and volterra type equations we consider a class of infinite delay equations of neutral type which includes volterrra type integral und integrodifferential equations using abstract approximation results cf trotter kato type for strongly continuous semigroups we develop an approximation scheme which is based on approximation of the system state by laguerre and legendre polynomials numerical examples demonstrate the feasibility of the scheme and show infinite order convergence for smooth data

crank nicolson and legendre spectral collocation methods for a partial integro differential equation with a singular kernel abstract in this paper we present an efficient numerical method for the solution of a partial integro differential equation with a singular kernel in the time direction a crank nicolson finite difference scheme is used to approximate the differential term and the product trapezoidal method is employed to treat the integral term also for space discretization we apply legendre spectral collocation method we discuss the stability and convergence of proposed method and show that the method is unconditionally stable and convergent with order o 3 2 n s where n and s are time step size number of collocation points and regularity of exact solution respectively we compare the numerical results of proposed method with the results of other schemes in the literature in terms of accuracy computational order and cpu time to show the efficiency and applicability of it

spectral gap and logarithmic sobolev inequality for kawasaki and glauber dynamics we prove that the spectral gap of the kawasaki dynamics shrink at the rate of 1 l2 for cubes of sizel provided that some mixing conditions are satisfied we also prove that the logarithmic sobolev inequality for the glauber dynamics in standard cubes holds uniformly in the size of the cube if the dobrushin shlosman mixing condition holds for standard cubes

entropy inequalities for unbounded spin systems we consider nonconservative reversible spin systems with unbounded discrete spins we show that for a class of these dynamics in a high temperature regime the relative entropy with respect to the equilibrium distribution decays exponentially in time although the logarithmic sobolev inequality fails to this end we prove a weaker modification of the logarithmic sobolev inequality

stabilization of spectral methods by finite element bubble functions abstract we show that a standard spectral collocation scheme can be stabilized by adding extra trial test functions with local support bubbles applications are given to a scalar advection diffusion problem and to the stokes problem

on exact solutions for the coagulation fragmentation equation it is pointed out that the equilibrium solutions for pure coagulation and pure fragmentation which dubovskii et al ibid vol 25 p 4737 1992 purport to obtain are in error and that such solutions do not in fact exist

global properties of pseudospectral methods abstract we apply polynomial interpolation methods both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations the derivative matrix for a general sequence of collocation points is explicitly constructed we explore the effect of several factors on the performance of these methods we show that global methods cannot be interpreted in terms of local methods for example the accuracy of the approximation differs when large gradients of the function occur near the center of the region or in the vicinity of the boundary this difference does not depend on the density of the collocation points near the boundaries hence intuition based on finite difference methods can lead to false results

hypercontractivity and spectral gap of symmetric diffusions with applications to the stochastic ising models abstract we describe some results relating the spectral gap and logarithmic sobolev constants we work initially in the context of symmetric diffusions on a finite dimensional manifold and later apply our results to the analysis of certain infinite systems of mildly interacting diffusions

comment on singular solutions to the stationary coagulation equation we comment on our paper exact solutions for the coagulation fragmentation equation ibid vol 25 p 4737 1992

memory universal prediction of stationary random processes we consider the problem of one step ahead prediction of a real valued stationary strongly mixing random process xi sub i spl infin sup spl infin the best mean square predictor of x sub 0 is its conditional mean given the entire infinite past x sub i sub i spl infin sup 1 given a sequence of observations x sub 1 x sub 2 x sub n we propose estimators for the conditional mean based on sequences of parametric models of increasing memory and of increasing dimension for example neural networks and legendre polynomials the proposed estimators select both the model memory and the model dimension in a data driven fashion by minimizing certain complexity regularized least squares criteria when the underlying predictor function has a finite memory we establish that the proposed estimators are memory universal the proposed estimators which do not know the true memory deliver the same statistical performance rates of integrated mean squared error as that delivered by estimators that know the true memory furthermore when the underlying predictor function does not have a finite memory we establish that the estimator based on legendre polynomials is consistent

spectral approximation of pattern forming nonlinear evolution equations with double well potentials of quadratic growth this paper is concerned with the analysis of a numerical algorithm for the approximate solution of a class of nonlinear evolution problems that arise as l 2 gradient flow for the modica mortola regularization of the functional v bv t d 1 1 e v 2 td v 1 2 k d k v k 2 here is the interfacial energy per unit length or unit area t d is the flat torus in d and is a nonnegative fourier multiplier that is continuous on d symmetric in the sense that for all d and that decays to zero at infinity such functionals feature in mathematical models of pattern formation in micromagnetics and models of diblock copolymers the resulting evolution equation is discretized by a fourier spectral method with respect to the spatial variables and a modified crank nicolson scheme in time optimal order a priori bounds are derived on the global error in the l 0 t l 2 t d norm

the logarithmic sobolev inequality for continuous spin systems on a lattice abstract in this article we give cf theorem 1 6 a sufficient condition on a local specification for the associated gibbs state to satisfy a logarithmic sobolev inequality in the final section cf theorem 3 10 we relate our condition to the type of mixing condition introduced by dobrushin and shlosman in connection with their work on complete analyticity

computation for the canonical measures of a colored disordered lattice gas and spectral gap in this work we deal with spectral gap and canonical measures related to a model called colored disordered lattice gas we consider the approach used in the work of dermoune and heinrich a small step towards the hydrodynamic limit of a colored disordered lattice gas c r math acad sci 339 507 511 2004 we suggest a new computation for the canonical measures also we propose the explicit form of the spectral gap for colored disordered lattice gas of exclusion processes which plays an important role in the study of hydrodynamic limit

logarithmic sobolev inequality for lattice gases with mixing conditions let mu lambda l lambda gc denote the grand canonical gibbs measure of a lattice gas in a cube of sizel with the chemical potential and a fixed boundary condition let mu lambda l n c be the corresponding canonical measure defined by conditioning mu lambda l lambda gc on sigma x in lambda eta x n consider the lattice gas dynamics for which each particle performs random walk with rates depending on near by particles the rates are chosen such that for everyn andl fixed mu lambda l n c is a reversible measure suppose that the dobrushin shlosman mixing conditions holds for mu l lambda gc forall chemical potentials we prove that smallint f log fd mu lambda l n c leqq const l 2 d sqrt f for any probability densityf with respect to mu lambda l n c here the constant is independent ofn orl andd denotes the dirichlet form of the dynamics the dependence onl is optimal

the approximation theory for the p version of the finite element method the purpose of this article is to present the approximation theory underlying the p version of the finite element method by exploiting the relationship between polynomial approximation and certain weighted sobolev spaces which are identified as the domains of positive real powers of the legendre operator this one dimensional result is generalized via a tensor product construction to yield a nonconforming piecewise polynomial approximation result in the usual unweighted sobolev spaces on triangulated domains of r n it is then shown that essentially the same result holds for approximation by conforming piecewise polynomials provided that the function being approximated possesses the same degree of conformality across the common boundaries of adjacent simplices and the same homogeneous boundary conditions inverse results are given for the special case of approximation in l 2

on the spectral gap of kawasaki dynamics under a mixing condition revisited we consider a conservative stochastic spin exchange dynamics which is reversible with respect to the canonical gibbs measure of a lattice gas model we assume that the corresponding grand canonical measure satisfies a suitable strong mixing condition we give an alternative and quite natural from the physical point of view proof of the famous lu yau result which states that the relaxation time in a box of side l scales like l2 we then show how to use such an estimate to prove a decay to equilibrium for local functions of the form 1 t e where e is positive and arbitrarily small and 12 for d 1 1 for d 2

generalized clenshaw curtis quadrature rule with application to a collocation least squares method abstract this paper deals with an extension of one dimensional clenshaw curtis quadrature rule to r d d 2 on a convex domain as one of its applications we apply this quadrature rule to a collocation least squares method using arbitrary abscissas for a first order system of an elliptic boundary value problem so that the convergence analysis on a numerical solution can be shown in a standard sobolev norm

a spectral difference method for two dimensional viscous flow abstract we propose a spectral difference method for the 2 dimensional vorticity equation with a periodic boundary condition in one direction the solution satisfies a semidiscrete conservation law and thus better numerical results are obtained we also prove stability and convergence

polynomial approximation of divergence free functions etude de la meilleure approximation d une fonction sans divergence par un polynome algebrique ou trigonometrique sans divergence etablissement d une estimation optimale

error estimates for spectral and pseudospectral approximations of hyperbolic equations the linear one dimensional advection equation with variable coefficients and nonperiodic boundary conditions is considered we investigate some spectral and pseudospectral collocation approximations of jacobi and legendre type in the latter case the boundary condition is implicitly treated stability and convergence results in weighted norms are given some precise estimates of the rate of convergence in terms of the regularity of the continuous solution are derived

spectral and pseudo spectral approximations of the navier stokes equations galerkin and collocation approximations by trigonometric polynomials to the stationary navier stokes equations with periodic boundary conditions are analyzed stability results and optimal rates of convergence for both velocity and pressure in sobolev spaces are proven essential tools to carry out this analysis are some abstract results concerning general polynomial approximations to nonsingular solutions of nonlinear problems brezzi rappag raviart numer math 36 1980 pp 1 25 maday quarteroni rairo numer anal 1980

convergence analysis for a second order elliptic equation by a collocation method using scattered points a collocation method using scattered points applied to a second order elliptic differential equation is analyzed by establishing a new quadrature formula for the space of the polynomials we show that a polynomial solution possesses stability and preserves a similar convergence property occurred in the classical high order collocation method

parameter estimation by spectral approximation in this paper chebyshev and legendre approximations are proposed for estimating parameters in differential equations which are easy to be performed the convergence and the spectral accuracy are proved even without some conditions as imposed in other papers the numerical results show the advantages of this new approach

spectral methods with sparse matrices spectral methods employ global polynomials for approximation hence they give very accurate approximations for smooth solutions unfortunately for dirichlet problems the matrices involved are dense and have condition numbers growing aso n 4 for polynomials of degree n in each variable we propose a new spectral method for the helmholtz equation with a symmetric and sparse matrix whose condition number grows only aso n 2 certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system numerical results are presented which show that we have probably found the most effective solver for spectral systems

approach to equilibrium of glauber dynamics in the one phase region ii the general case we develop a new method based on renormalization group ideas block decimation procedure to prove under an assumption of strong mixing in a finite cube 0 a logarithmic sobolev inequality for the gibbs state of a discrete spin system as a consequence we derive the hypercontractivity of the markov semigroup of the associated glauber dynamics and the exponential convergence to equilibrium in the uniform norm in all volumes multiples of the cube 0

two completely explicit and unconditionally convergent fourier pseudo spectral methods for solving the nonlinear schrodinger equation abstract this paper aims to construct and analyze two new fourier pseudo spectral fps methods for the general nonlinear schrodinger nls equation the two fps methods have two merits unconditional convergence and complete explicitness in the practical computation further more by introducing a modified mass functional and a modified energy functional the two fps methods are proved to preserve the total mass and energy in the discrete sense besides the standard energy method the key techniques used in our numerical analysis are a mathematical induction argument and a lifting technique without any restriction on the grid ratio and initial value we establish the optimal error estimate of the two fps methods for solving the general nls equation while previous work just is valid for the cubic nls equation and requires small initial value for the focusing case these two fps methods are proved to be spectrally accurate in space and second order accurate in time respectively the analysis framework can be used to prove the unconditional convergence of many other fourier pseudo spectral methods for solving the nls type equations we investigate the effect of the nonlinear term on the progression simulation of the plane wave the conservation of the invariants and the effect of initial data on the blow up solution via different parameters numerical results are reported to show the accuracy and efficiency of the proposed methods

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

boundary conditions in chebyshev and legendre methods we discuss two different ways of treating non dirichlet boundary conditions in chebyshev and legendre collocation methods for second order differential problems an error analysis is provided the effect of preconditioning the corresponding spectral operators by finite difference matrices is also investigated

optimal equivalent preconditioners preconditioning strategies for elliptic partial differential operators and corresponding difference operators are compared by way of a model problem the technique considered here makes use of a selfadjoint positive definite operator b as the preconditioner for the nonselfadjoint convection diffusion operator a preconditioning by a s leading term plus a positive zeroth order term sigma i and optimizing over sigma is proposed the analysis of a model problem indicates that when a has large first order terms both the spectral and norm condition of b 1 a are minimized by sigma gg 0

the fourier chebyshev spectral method for solving two dimensional unsteady vorticity equations abstract we propose a mixed method for solving the two dimensional unsteady vorticity equations by using the fourier spectral approximation in the periodic direction and chebyshev spectral approximation in the non periodic direction some numerical results are given which are compared with those of other methods stability of the scheme and the optimal rate of convergence are proved

legendre and chebyshev spectral approximations of burgers equation we give error estimates for both legendre and chebyshev spectral approximations of the steady state burgers problem u xx lambda uu x f 0 in i 1 1 u 1 u 1 0 to do that we prove some abstract approximation properties of orthogonal projection operators in some weighted sobolev spacesh s i for both legendre and chebyshev weights

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

konvergenzaussagen fur kollokationsverfahren bei elliptischen randwertaufgaben we examine the convergence of collocation with polynomials for elliptic problems the general theory is imbedded in thel p theory from which known results of existence and regularity are adopted convergence results are deduced in the cases of periodic dirichlet and mixed boundary conditions it becomes obvious that the order of convergence essentially depends on the smoothness of the solution

preconditioning chebyshev spectral collocation by finite difference operators in 1979 orszag proposed a finite difference preconditioning of the chebyshev collocation discretization of the poisson equation in 1984 haldenwang labrosse abboudi and deville gave analytic formulae for the eigenvalues of this preconditioned operator in the one dimensional case experimental results over many years have shown the effectiveness of this procedure and appropriate bounds on the eigenvalues in two dimensions however there have been no mathematical proofs describing the behavior of the eigenvalues in two or more dimensions in this work we consider the generalized field of values u hat a nu u l nu where hat a n is the matrix of the chebyshev collocation scheme and l n is the matrix of the finite difference operator for the case of the chebyshev collocation of the helmoltz operator au delta u au a ge 0 preconditioned by the finite difference operator associated with the helmholtz operator bu delta u bu b ge 0 we prove that there are two constants 0 0 and u hat a nu u l nu le lambda 1 these results extend to higher dimensions and to bounds on l n 1 hat a n 1 w and hat a n 1 l n 1 w in the general case where au delta u a 1u x a 2u y au