on the large time galerkin approximation of the navier stokes equations in this article we are interested in the difficult problem of relating the large time behaviour of the navier stokes equations to the large time behaviour of their galerkin approximation we restrict ourselves to the simplest situation dealing with stationary solutions and the question is the following one if the computed approximation of the time dependent equations seems to converge to some limit as t to infty is the same true for the exact problem and are the limits related we give here some sufficient conditions reasonably easy to verify which guarantee a positive answer to this question

the stability in _ and _ of the projection onto finite element function spaces on etablit la stabilite de la l 2 projection sur quelques espaces a elements finis v h consideree comme une application dans l p et w p 1 1 p sous des hypotheses de regularite faibles

a semi implicit scheme for stationary statistical properties of the infinite prandtl number model we propose a semidiscrete in time semi implicit numerical scheme for the infinite prandtl model for convection besides the usual finite time convergence this scheme enjoys the additional highly desirable feature that the stationary statistical properties of the scheme converge to those of the infinite prandtl number model at vanishing time step one of the key characteristics of the scheme is that it preserves the dissipativity of the infinite prandtl number model uniformly in terms of the time step so far as we know this is the first rigorous result on convergence of stationary statistical properties of numerical schemes for infinite dimensional dissipative complex systems

attractors and error estimates for discretizations of incompressible navier stokes equations by embedding sets of nodal values to function spaces variational arguments are applied to the finite difference approximations to the two dimensional incompressible navier stokes equations the error estimates for trajectories and the it time free error estimates with tolerance for attractors are proved the upper semicontinuity of the attractors with respect to the finite difference discretizations are also proved arguments of applying the techniques to the finite element approximations are also given

upper semicontinuity of attractors for approximations of semigroups and partial differential equations abstract suppose a given evolutionary equation has a compact attractor and the evolutionary equation is approximated by a finite dimensional system conditions are given to ensure the approximate system has a compact attractor which converges to the original one as the approximation is refined applications are given to parabolic and hyperbolic partial differential equations

compatibility conditions for time dependent partial differential equations and the rate of convergence of chebyshev and fourier spectral methods abstract compatibility conditions for partial differential equations pdes are an infinite set of relations between the initial conditions the pde and the boundary conditions which are necessary and sufficient for the solution to be c that is infinitely differentiable everywhere on the computational domain including the boundaries since the performance of chebyshev spectral and spectral element methods is dramatically reduced when the solution is not c one would expect that the compatibility conditions would be a major theme in the spectral literature instead it has been completely ignored therefore we pursue three goals here first we present a proof of the compatibility conditions in a simplified form that does not require functional analysis second we analyze the connection between the compatibility conditions and the rate of convergence of chebyshev methods lastly we describe strategies for slightly adjusting initial conditions so that the compatibility conditions are satisfied

a spectral element method for the navier stokes equations with improved accuracy we present a procedure to enhance the accuracy of spectral element methods for the evolutionary navier stokes equations the approximations to the velocity and the pressure obtained using a spectral element method are postprocessed by solving a discrete stokes problem only once when the time evolution is completed the analysis shows that this procedure improves the order of convergence for the approximations to the velocity and the pressure the choice of a discrete pressure space endowed with an inf sup condition independent of the discretization parameter leads to optimal error bounds for the pressure

upper semicontinuity of attractors for random lattice systems perturbed by small white noises abstract in this paper we first present some conditions for the upper semicontinuity of perturbed random attractors to a limiting random attractor then we apply this result to establish the upper semicontinuity of random attractors for the first order stochastic lattice differential equations with random coupled coefficients and multiplicative additive white noises in the weighted space of infinite sequences as the coefficient of the white noise term tends to zero

long term dissipativity of time stepping algorithms for an abstract evolution equation with applications to the incompressible mhd and navier stokes equations abstract this paper examines the long term dissipativity and unconditional non linear stability of time integration algorithms for the incompressible mhd equations the results apply as a particular case to the uncoupled incompressible navier stokes equations the analysis is performed in the setting defined by an abstract evolution equation possessing the same long term dissipative structure as the above physical systems a general class of time stepping schemes is analyzed in this context showing the non linear unconditional stability and long term dissipativity of particular members of this family of algorithms in particular algorithms showing these strong properties which additionally involve the solution of a linear problem at each time step are identified a particular linear algorithm is shown to exhibit optimal long term dissipative properties and to possess an algorithmic global attractor these abstract results apply rigorously to galerkin mixed finite element discretizations of the problem to this end it is shown that such galerkin type projection inherits the dissipative properties of the continuum problem computational aspects involved in the numerical implementation of these methods are addressed and illustrated in representative numerical simulations

the long time l2 and h1 stability of linearly extrapolated second order time stepping schemes for the 2d incompressible navier stokes equations abstract herein we present a study on the long time stability of finite element discretizations of a generalized class of semi implicit second order time stepping schemes for the 2d incompressible navier stokes equations these remarkably efficient schemes require only a single linear solve per time step through the use of a linearly extrapolated advective term our result develops a class of sufficient conditions such that if external forcing is uniformly bounded in time velocity solutions are uniformly bounded in time in both the l2 and h1 norms we provide numerical verification of these results we also demonstrate that divergence free finite elements are critical for long time h1 stability

large time behavior of deterministic particle approximations to the navier stokes equations we prove that for a class of deterministic vortex methods for the navier stokes equations in two dimensions the numerical solution decays for large time with the same rate as the exact solution we substantiate our result with numerical experiments and with a remark concerning the problem of reinitialization of a distribution of particles

a new contribution to nonlinear stability of a discrete velocity model for a class of discrete velocity models of kinetic theory we prove exponential nonlinear conditional stability of the constant basic state in the slab 0 1

elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems we derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations the space discretization uses finite element spaces that are allowed to change in time our main tool is an appropriate adaptation of the elliptic reconstruction technique introduced by makridakis and nochetto we derive novel a posteriori estimates for the norms of l 0 t l2 and the higher order spaces l 0 t h1 and h1 0 t l2 with optimal orders of convergence

asymptotic behavior and time discretization analysis for the non stationary navier stokes problem the asymptotic behavior and the euler time discretization analysis are presented for the two dimensional non stationary navier stokes problem if the data and f t satisfy a uniqueness condition corresponding to the stationary navier stokes problem we then obtain the convergence of the non stationary navier stokes problem to the stationary navier stokes problem and the uniform boundedness stability and error estimates of the euler time discretization for the non stationary navier stokes problem

analysis of finite element approximation of evolution problems in mixed form this paper deals with the finite element approximation of evolution problems in mixed form following d boffi f brezzi and l gastaldi math comp 69 2000 pp 121 140 we handle separately two types of problems a model for the first case is the heat equation in mixed form while the time dependent stokes problem fits within the second one for either case we give sufficient conditions for a good approximation in the natural functional spaces the results are not obvious in the first situation in this case the well known conditions for the well posedness and convergence of the corresponding steady problem are not sufficient for the good approximation of the time dependent problem this issue is demonstrated with a numerical counter example and justified analytically

unconditional long time stability of a velocity vorticity method for the 2d navier stokes equations we prove unconditional long time stability for a particular velocity vorticity discretization of the 2d navier stokes equations the scheme begins with a formulation that uses the lamb vector to couple the usual velocity pressure system to the vorticity dynamics equation and then discretizes with the finite element method in space and implicit explicit bdf2 in time with the vorticity equation decoupling at each time step we prove the method s vorticity and velocity are both long time stable in the l 2 l2 and h 1 h1 norms without any timestep restriction moreover our analysis avoids the use of gronwall type estimates which leads us to stability bounds with only polynomial instead of exponential dependence on the reynolds number numerical experiments are given that demonstrate the effectiveness of the method

numerical analysis of parabolic p laplacian approximation of trajectories the long time numerical approximation of the parabolic p laplacian problem with a time independent forcing term and sufficiently smooth initial data is studied convergence and stability results which are uniform for t in 0 infty are established in the l2 w1 p norms for the backward euler and the crank nicholson schemes with the finite element method fem this result extends the existing uniform convergence results for exponentially contractive semigroups generated by some semilinear systems to nonexponentially contractive semigroups generated by some quasi linear systems

stability and convergence of semi implicit time stepping algorithm for stationary incompressible magnetohydrodynamics abstract for the stationary incompressible magnetohydrodynamics mhd equations we provide a new uniqueness assumption a0 and show the exponential stability of the solution then the semi implicit time stepping algorithm is used to solve the stationary mhd equations the algorithm is proved to be unconditionally stable the discrete velocity and magnetic field are bounded in l 0 l for any space and time step sizes the error estimates for the algorithm are deduced under the uniqueness conditions finally numerical experiments are performed to testify our theoretical analysis

conformal invariance and the operator content of the xxz model with arbitrary spin the authors are concerned with the critical properties of the anisotropic heisenberg chain or xxz model with arbitrary integer or half integer spin the eigenspectra of these hamiltonians with periodic boundaries are calculated for finite chains by solving numerically their associated bethe ansatz equations the resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified for lattices with an even and odd number of sites the results for spin 1 and spin 3 2 indicate that the conformal anomaly c for these models in the gapless regime has the value c 3s 1 s independent of the anisotropy and the exponents vary continuously with the anisotropy as in the eight vertex model the operator content of these models indicate that the underlying field theory governing these critical spin s models is described by composite fields formed by the product of gaussian and z n fields with n 2s finally some of the irrelevant operators which produce the leading finite size corrections of the eigenenergies are identified

finite element approximation of the nonstationary navier stokes problem part iii smoothing property and higher order error estimates for spatial discretization this paper continues our error analysis of finite element galerkin approximation of the nonstationary navier stokes equations optimal order error estimates both local and global are derived for higher order finite elements under appropriate assumptions about the smoothness and stability of the solution these assumptions take into account the loss of regularity at t 0 that one generally has to expect in the absence of higher order nonlocal compatibility conditions for the data of the problem

on approximate inertial manifolds to the navier stokes equations abstract recently the theory of inertial manifolds has shown that the long time behavior the dynamics of certain dissipative partial differential equations can be fully discribed by that of a finite ordinary differential system although we are still unable to prove existence of inertial manifolds to the navier stokes equations we present here a nonlinear finite dimensional analytic manifold that approximates closely the global attractor in the two dimensional case and certain bounded invariant sets in the three dimensional case this approximate manifold and others allow us to introduce modified galerkin approximations

the behavior of finite element solutions of semilinear parabolic problems near stationary points the qualitative behavior of spatially semidiscrete finite element solutions of a semilinear parabolic problem near an unstable hyperbolic equilibrium overline u is studied it is shown that any continuous trajectory is approximated by an appropriate discrete trajectory and vice versa as long as they remain in a sufficiently small neighborhood of overline u error bounds of optimal order in the l 2 and h 1 norms hold uniformly over arbitrarily long time intervals in particular the local stable and unstable manifolds of the discrete problem converge to their continuous counterparts therefore the discretized dynamical system has the same qualitative behavior near overline u as the continuous system

on the convergence of a time discretization scheme for the navier stokes equations version linearisee du schema d euler implicite utilisee pour l approximation des solutions des equations de navier stokes dans un domaine bidimensionnel

finite element approximation for the viscoelastic fluid motion problem in this article the variational formulation of the two dimensional viscoelastic fluid motion problem and its finite element approximation are considered an local error estimate for the velocity with h1 norm and the pressure with l2 norm is obtained and a uniform error estimate for the velocity and pressure with the above norms is provided if the given data satisfies the uniqueness condition

a scalable approach to computing representative lowest common ancestor in directed acyclic graphs lca computation for vertex pairs in trees can be achieved in constant time after linear time preprocessing however extension of these techniques to compute lca for vertex pairs in dags has been not possible due to the non tree edges in a dag in this paper we present an algorithm for computing the lca for vertex pairs in a dag which treats the dag s spanning tree and its non tree edges separately our approach enables us to tap the efficiency of existing lca algorithms for trees furthermore our algorithm decomposes the dag into a set of component trees called clusters which significantly reduces the preprocessing necessary to incorporate non tree edges in the lca computation our algorithm seamlessly interpolates the performance graph between the best reported algorithms for trees and the best reported algorithms for dags depending on the incidence of non tree edges in the dag using the proposed techniques it is possible to achieve near linear preprocessing and constant query time for sparse dags

the treatment of spurious pressure modes in spectral incompressible flow calculations algorithms for the transient and steady state simulation of incompressible newtonian and non newtonian flows are described for the primitive variable formulation of the governing equations spectral approximations are used for the spatial discretization attention is given to the satisfaction of the incompressibility constraint and the determination of the pressure spurious pressure modes are removed by means of a singular value decomposition the corresponding velocity field is divergence free at all the collocation points numerical results are presented for newtonian flow in a regularized driven square cavity and for non newtonian flow in a planar channel and in a journal bearing for a realistic range of material parameters

the convergence of two numerical schemes for the navier stokes equations this paper deals with some convergence stability results concerning two numerical methods for solving the incompressible nonstationary navier stokes equations the algorithms are of a particular kind in what regards time discretization more precisely of the peaceman rachford and the strang type resp and have been obtained by modifying slightly the numerical treatment of the nonlinear terms in other schemes due to glowinski et al 1980 we first describe the full discretization of the homogeneous dirichlet problem using a general external approximation of the spatial functional spaces involved a particular and simple choice of such an approximation is the standardp 2 lagrange finite element for the velocity field when the fluid is bidimensional then we establish and prove convergence and stability and make some comments on the numerical treatment of other generally nonhomogeneous boundary conditions the theoretical results show that the schemes are at least conditionally stable and convergent which justifies the success of glowinski s methods

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

logarithmic corrections to finite size scaling in strips abstract the corrections to the finite size scaling behaviour of the eigenvalues of the transfer matrix of a critical theory defined on an infinitely long strip of finite width which occur when the hamiltonian contains a marginal operator are computed using conformal invariance they show a calculable universal logarthmic character for the four state potts model they agree with numerical data

analysis of the bethe ansatz equations for the anisotropic su 2 chain the bethe ansatz equations for the anisotropic spin s heisenberg chain are analysed numerically the position of the roots for the whole spectrum is calculated for the spin 1 model in a chain of width n 4 some states are presented which violate the normally accepted string hypothesis

convergence of three iterative methods based on the finite element discretization for the stationary navier stokes equations abstract this paper considers three iterative methods for solving the stationary navier stokes equations iterative method i consists in solving the stationary stokes equations iterative method ii consists in solving the stationary linearized navier stokes equations and iterative method iii consists in solving the stationary oseen equations under the finite element discretization respectively at each iterative step also we discuss the stability and convergence of three iterative methods the iterative methods i and ii are stability and convergence under the strong uniqueness conditions where the iterative method ii is the second order convergence furthermore the iterative method iii is uncondition stability and convergence under the uniqueness condition finally some numerical tests show that the efficiency of the theoretical analysis

perturbations of attractors of differential equations abstract in this paper we study small c 1 perturbations of a differential equation that has a hyperbolic attractor k we show that if k has a suitable lipschitz property and if the perturbation is small enough then there is a homeomorphism h k k y where k y is a hyperbolic attractor for the perturbed equation examples are included

numerical comparisons of time space iterative method and spatial iterative methods for the stationary navier stokes equations this paper makes some numerical comparisons of time space iterative method and spatial iterative methods for solving the stationary navier stokes equations the time space iterative method consists in solving the nonstationary stokes equations based on the time space discretization by the euler implicit explicit scheme under a weak uniqueness condition a2 the spatial iterative methods consist in solving the stationary stokes scheme newton scheme oseen scheme based on the spatial discretization under some strong uniqueness assumptions we compare the stability and convergence conditions of the time space iterative method and the spatial iterative methods moreover the numerical tests show that the time space iterative method is the more simple than the spatial iterative methods for solving the stationary navier stokes problem furthermore the time space iterative method can solve the stationary navier stokes equations with some small viscosity and the spatial iterative methods can only solve the stationary navier stokes equations with some large viscosities

comment on critical behaviour of spin s heisenberg antiferromagnetic chains analytic and numerical results makes some observations on a recent paper by affleck et al ibid vol 22 p 511 1989 about the singlet excitation in the spin s 1 integrable model and logarithmic corrections for integrable spin models

long time behavior of the difference solutions to the 2d gks equation we study the long time behavior of the finite difference solution to the generalized kuramoto sivashinsky equation in two space dimensions with periodic boundary conditions the unique solvability of numerical solution is shown it is proved that there exists a global attractor of the discrete dynamical system and the upper semicontinuity d a h t a 0 finally we obtain the long time stability and convergence of the difference scheme our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems

faster algorithms for finding lowest common ancestors in directed acyclic graphs we present two new methods for finding a lowest common ancestor lca for each pair of vertices of a directed acyclic graph dag on n vertices and m edges the first method is surprisingly natural and solves the all pairs lca problem for the input dag on n vertices and m edges in time o nm the second method relies on a novel reduction of the all pairs lca problem to the problem of finding maximum witnesses for boolean matrix product we solve the latter problem and hence also the all pairs lca problem in time o n 2 l where l satisfies the equation w 1 l 1 email protected and w 1 l 1 is the exponent of the multiplication of an nxn l matrix by an n lxn matrix by the currently best known bounds on w 1 l 1 the running time of our algorithm is o n 2 5 7 5 our algorithm improves the previously known o n 2 6 8 8 time bound for the general all pairs lca problem in dags by bender et al our additional contribution is a faster algorithm for solving the all pairs lowest common ancestor problem in dags of small depth where the depth of a dag is defined as the length of the longest path in the dag for all dags of depth at most email protected n a where a 0 294 our algorithm runs in a time that is asymptotically the same as that required for multiplying two nxn matrices that is o n w we also prove that this running time is optimal even for dags of depth 1 for dags with depth h n a the running time of our algorithm is at most o n email protected h 0 4 6 8 this algorithm is faster than our algorithm for arbitrary dags for all values of email protected n 0 4 2

euler implicit explicit iterative scheme for the stationary navier stokes equations in this paper a new uniqueness assumption a2 of the solution for the stationary navier stokes equations is presented under assumption a2 the exponential stability of the solution bar u bar p for the stationary navier stokes equations is proven moreover the euler implicit explicit scheme based on the mixed finite element is applied to solve the stationary navier stokes equations finally the almost unconditionally stability is proven and the optimal error estimates uniform in time are provided for the scheme

long term stability estimates and existence of a global attractor in a finite element approximation of the navier stokes equations with numerical subgrid scale modeling variational multiscale methods lead to stable finite element approximations of the navier stokes equations dealing with both the indefinite nature of the system pressure stability and the velocity stability loss for high reynolds numbers these methods enrich the galerkin formulation with a subgrid component that is modeled in fact the effect of the subgrid scale on the captured scales has been proved to dissipate the proper amount of energy needed to approximate the correct energy spectrum thus they also act as effective large eddy simulation turbulence models and allow one to compute flows without the need to capture all the scales in the system in this article we consider a dynamic subgrid model that enforces the subgrid component to be orthogonal to the finite element space in the l 2 sense we analyze the long term behavior of the algorithm proving the existence of appropriate absorbing sets and a compact global attractor the improvements with respect to a finite element galerkin approximation are the long term estimates for the subgrid component which are translated to effective pressure and velocity stability thus the stabilization introduced by the subgrid model into the finite element problem does not deteriorate for infinite time intervals of computation

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data on considere les approximations spatialement discretes continues par rapport au temps des problemes mixtes par la methode des elements finis de galerkin pour les equations paraboliques semi lineaires avec des donnees initiales irregulieres ou incompatibles on etablit des estimations d erreur

error estimates with smooth and nonsmooth data for a finite element method for the cahn hilliard equation a finite element method for the cahn hilliard equation a semilinear parabolic equation of fourth order is analyzed both in a spatially semidiscrete case and in a completely discrete case based on the backward euler method error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data a detailed study of the regularity of the exact solution is included the analysis is based on local lipschitz conditions for the nonlinearity with respect to sobolev norms and the existence of a ljapunov functional for the exact and the discretized equations is essential a result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup upper semicontinuity with respect to the discretization parameters is obtained as a simple application of the nonsmooth data error estimate

attractivity properties of α contractions abstract it is known that if t x x is completely continuous where x is a banach space then point dissipative and compact dissipative are equivalent and imply the existence of a maximal compact invariant set which is uniformly asymptotically stable and attracts bounded sets uniformly if t is an contraction it is not known whether point dissipative and compact dissipative are equivalent however it is known that if t is an contraction and compact dissipative then there exists a maximal compact invariant set which is uniformly asymptotically stable and attracts a neighborhood of any compact set uniformly in this paper we show that for most practical examples which give rise to contraction point dissipative and compact dissipative are equivalent for example we show this is true for stable neutral functional differential equations retarded functional differential equations of infinite delay and strongly damped nonlinear wave equations we conjecture that this should be true for almost any practical application which gives rise to an contraction

upper semicontinuity of attractors for lattice systems under singular perturbations abstract consider the following first order lattice system u m 2 u m u m 1 u m 1 m u m f m u m g m m z which is perturbed by the small two order term u m u m 2 u m u m 1 u m 1 m u m f m u m g m m z under certain conditions on f m m and g m the original systems and the small perturbed systems have global attractors a in l 2 and a in l 2 l 2 respectively and a can be naturally embedded into a compact set a 0 in l 2 l 2 we prove the upper semicontinuity of a 0 with respect to the attractors a at zero by showing that for any neighborhood o a 0 of a 0 a enters o a 0 if is small enough

approximation of stationary statistical properties of dissipative dynamical systems time discretization we consider temporal approximation of stationary statistical properties of dissipative infinite dimensional dynamical systems we demonstrate that stationary statistical properties of the time discrete approximations i e numerical scheme converge to those of the underlying continuous dissipative infinite dimensional dynamical system under three very natural assumptions as the time step approaches zero the three conditions that are sufficient for the convergence of the stationary statistical properties are 1 uniform dissipativity of the scheme in the sense that the union of the global attractors for the numerical approximations is pre compact in the phase space 2 convergence of the solutions of the numerical scheme to the solution of the continuous system on the unit time interval 0 1 uniformly with respect to initial data from the union of the global attractors 3 uniform continuity of the solutions to the continuous dynamical system on the unit time interval 0 1 uniformly for initial data from the union of the global attractors the convergence of the global attractors is established under weaker assumptions an application to the infinite prandtl number model for convection is discussed

adaptive finite element methods for parabolic problems v long time integration we continue our previous work on adaptive finite element methods for parabolic problems now with particular emphasis on long time integration for semidefinite problems

long time error estimation for semi linear parabolic equations the long time error estimation approach of sun and ewing dyn contin discrete impuls systems ser b appl algorithms 9 2002 115 129 is applied here for the error analysis and estimation of linear and semilinear parabolic partial differential equations the analysis is carried out using the stability smoothing indicator the smoothing assumption the moving attractor the exact error propagation and the two level error propagation analysis introduced by sun and ewing dyn contin discrete impuls systems ser b appl algorithms 9 2002 115 129 moreover an inverse elliptic projection is employed here as a key technique in dealing with the spatial discretization error the error estimates obtained are uniform in time the results are substantiated by a complete mathematical analysis and numerical experiments

on the discretization in time of semilinear parabolic equations with nonsmooth initial data single step discretization methods are considered for equations of the form u au f t u where a is a linear positive definite operator in a hubert space h it is shown that if the method is consistent with the differential equation then the convergence is essentially of first order in the stepsize even if the initial data v are only in h but also that in contrast to the situation in the linear homogeneous case higher order convergence is not possible in general without further assumptions on v

determination of the solutions of the navier stokes equations by a set of nodal values we consider the navier stokes equations of a viscous incompresible fluid and we want to see to what extent these solutions can be determined by a discrete set of nodal values of these solutions the results presented here are exact results and not approximate ones we show in several cases that the solutions are entirely determined by their values on a discrete set provided this set contains enough points and these points are sufficiently densely distributed in a sense described in the article two typical results are the following ones two stationary solutions coincide if they coincide on a set sufficiently dense but finite similarly if the large time behavior of the solutions to the navier stokes equations is known on an appropriate discrete set then the large time behavior of the solution itself is totally determined

on the rate of convergence of the nonlinear galerkin methods in this paper we provide estimates to the rate of convergence of the nonlinear galerkin approximation method in particular and by means of an illustrative example we show that the nonlinear galerkin method converges faster than the usual galerkin method

calculation of incompressible viscous flows by an unconditionally stable projection fem this paper investigates the numerical performance of a finite element implementation of a new incremental fractional step method to compute steady and unsteady incompressible viscous flows under general boundary conditions and using unstructured meshes a variational framework is adopted which accommodates two different spaces for representing and approximating the velocity fields calculated respectively in the viscous and inviscid phases of the method but which leads to a very simple numerical scheme in terms of only one discrete velocity field an unconditionally stable semi implicit approximation of the nonlinear term is used to eliminate any time step restriction as far as the numerical stability is concerned numerical results for five test problems in two dimensions are reported to illustrate the flexibility of the proposed method

fast algorithms for finding nearest common ancestors we consider the following problem given a collection of rooted trees answer on line queries of the form what is the nearest common ancester of vertices x and y we show that any pointer machine that solves this problem requires omega log log n time per query in the worst case where n is the total number of vertices in the trees on the other hand we present an algorithm for a random access machine with uniform cost measure and a bound of omega log n on the number of bits per word that requires o 1 time per query and o n preprocessing time assuming that the collection of trees is static for a version of the problem in which the trees can change between queries we obtain an almost linear time and linear space algorithm

continuous interior penalty finite element method for the time dependent navier stokes equations space discretization and convergence this paper focuses on the numerical analysis of a finite element method with stabilization for the unsteady incompressible navier stokes equations incompressibility and convective effects are both stabilized adding an interior penalty term giving l 2 control of the jump of the gradient of the approximate solution over the internal faces using continuous equal order finite elements for both velocities and pressures in a space semi discretized formulation we prove convergence of the approximate solution the error estimates hold irrespective of the reynolds number and hence also for the incompressible euler equations provided the exact solution is smooth

an efficient second order in time scheme for approximating long time statistical properties of the two dimensional navier stokes equations we investigate the long time behavior of the following efficient second order in time scheme for the 2d navier stokes equations in a periodic box begin array ll frac 3 omega n 1 4 omega n omega n 1 2k nabla perp 2 psi n psi n 1 cdot nabla 2 omega n omega n 1 nu delta omega n 1 f n 1 quad delta psi n omega n end array the scheme is a combination of a 2nd order in time backward differentiation and a particular explicit adams bashforth treatment of the advection term therefore only a linear constant coefficient poisson solver is needed at each time step we prove uniform in time bounds on this scheme in dot l 2 dot h 1 per and dot h 2 per provided that the time step is sufficiently small these time uniform estimates further lead to the convergence of long time statistics stationary statistical properties of the scheme to that of the nse itself at vanishing time step

attractors and inertial manifolds for finite difference approximations of the complex ginzburg landau equation a semidiscrete spatial finite difference approximation and two fully discrete finite difference approximations to the complex ginzburg landau equation are considered in this paper the existence of an inertial manifold is proved inside a discrete h1 absorbing ball for the semi discrete approximation by showing that certain spectral properties hold for the linear operator and certain lipschitz properties hold for the nonlinear term convergence of the true and discrete inertial manifolds is then shown by considering the contraction mappings used to construct the inertial manifolds a fully implicit scheme is shown to form a continuous semigroup in a discrete l2 space and absorbing balls of radii independent of the spatial and temporal mesh sizes in discrete l2 and h1 spaces are constructed existence of a global attractor follows furthermore the existence of an inertial manifold is proved by showing that certain spectral and lipschitz properties hold for a semi implicit scheme existence of absorbing balls in the discrete space l2 of radii independent of the spatial and temporal mesh is proved existence of a global attractor follows

numerical solution of the incompressible navier stokes equations with coriolis forces based on the discretization of the total time derivative in this paper we present a numerical formulation to solve the incompressible navier stokes equations written in a rotating frame of reference the method is based on a finite difference discretization in time and a finite element discretization in space when the viscosity is very small numerical oscillations may appear due both to the high reynolds number and to the presence of the coriolis forces to overcome these oscillations a special discretization in time is proposed the idea is to discretize the total time derivative in an inertial basis rather than only the partial time derivative in the rotating reference system after this is done a further high order approximation is introduced leading to a problem posed in the rotating frame of reference and in spatial coordinates in contrast with the straightforward discretization of the original equations some additional terms appear that enhance the stability of the numerical scheme in the absence of coriolis forces the method is a generalization of the characteristic galerkin technique for convection dominated flows

on a criterion for locating stable stationary solutions to the navier stokes equations on donne une estimation superieure uniforme explicite a la borne de constantin foias ternam et on donne une forme applicable explicite aux conditions suffisantes on considere seulement le cas a 2 dimensions

conformal invariance and critical exponents of the takhtajan babujian models the authors are concerned with the critical properties of antiferromagnetic takhtajan babujian models with spin s 1 3 2 and 2 the leading eigenenergies of this hamiltonian in a finite chain are calculated by investigating numerically and analytically the bethe ansatz equations for the finite system the critical exponents and the conformal anomaly are obtained from their relations with the eigenspectrum of the finite hamiltonian the appearance of logarithmic corrections produces poor estimates however a combination of analytical and numerical methods produces very good estimates their results strongly support the conjecture that the wess zumino witten novikov non linear sigma models with topological charge k 2s are the underlying field theories for these spin s statistical mechanics models

on dispersive difference schemes i etude du comportement oscillatoire et de la convergence faible des solutions des approximations aux differences dispersives de u t 1 2 u 2 x 0

persistence of invariant sets for dissipative evolution equations we show that results concerning the persistence of invariant sets of ordinary differential equations under perturbation may be applied directly to a certain class of partial differential equations our framework is particularly well suited to encompass numerical approximations of these partial differential equations specifically we show that for a class of pdes with ac 1 inertial form certain natural numerical approximations possess an inertial form close to that of the underlying pde in thec 1 norm

hopf bifurcation in the driven cavity abstract incompressible two dimensional calculations are reported for the impulsively started lid driven cavity with aspect ratio two the algorithm is based on the time dependent stream function equation with a crank nicolson differencing scheme for the diffusion terms and with an adams bashforth scheme for the convection terms a multigrid method is used to solve the linear implicit equations at each time step periodic asymptotic solutions have been found for re 10000 and for re 5000 the re 5000 results are validated by grid refinement calculations the solutions are shown to be precisely periodic and care is taken to demonstrate that asymptotic states have been reached a discussion is included about the indicators that are used to show that an asymptotic state has been reached and to show that the asymptotic state is indeed periodic

new pod error expressions error bounds and asymptotic results for reduced order models of parabolic pdes the derivations of existing error bounds for reduced order models of time varying partial differential equations pdes constructed using proper orthogonal decomposition pod have relied on bounding the error between the pod data and various pod projections of that data furthermore the asymptotic behavior of the model reduction error bounds depends on the asymptotic behavior of the pod data approximation error bounds we consider time varying data taking values in two different hilbert spaces h and v with v subset h and prove exact expressions for the pod data approximation errors considering four different pod projections and the two different hilbert space error norms furthermore the exact error expressions can be computed using only the pod eigenvalues and modes and we prove the errors converge to zero as the number of pod modes increases we consider the pod error estimation approaches of kunisch and volkwein siam j numer anal 40 2002 pp 492 515 and chapelle gariah an

wall driven incompressible viscous flow in a two dimensional semi circular cavity the main goal of this article is to investigate the capability of an operator splitting finite elements based methodology at handling accurately incompressible viscous flow at large reynolds number re in regions with corners and curved boundaries to achieve this goal the authors have selected a wall driven flow in a semi circular cavity on the basis of the numerical experiments reported in this article it seems that the method under investigation has no difficulty at capturing the formation of primary secondary and tertiary vortices as re increases it has also the capability of identifying a hopf bifurcation phenomenon taking place around re 6600

numerics and hydrodynamic stability toward error control in computational fluid dynamics we critically review the available error analysis in computational fluid dynamics cfd and come to the conclusion that the existing error estimates are meaningless in most cases of interest we propose a new approach to error analysis in cfd aiming at reliable and efficient adaptive quantitative error control this is based on a precise analysis of hydrodynamic stability coupled with galerkin orthogonality we prove a priori and a posteriori type error estimates in a model case for pipe flow formulate corresponding adaptive algorithms and discuss the potential of this approach for adaptive error control in cfd

estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems we show that the long time behavior of the projection of the exact solutions to the navier stokes equations and other dissipative evolution equations on the finite dimensional space of interpolant polynomials determines the long time behavior of the solution itself provided that the spatial mesh is fine enough we also provide an explicit estimate on the size of the mesh moreover we show that if the evolution equation has an inertial manifold then the dynamics of the evolution equation is equivalent to the dynamics of the projection of the solutions on the finite dimensional space spanned by the approximating polynomials our results suggest that certain numerical schemes may capture the essential dynamics of the underlying evolution equation

isogeometric divergence conforming b splines for the unsteady navier stokes equations divergence conforming b splines are developed for application to the incompressible navier stokes equations on geometrically mapped domains these enable smooth pointwise divergence free solutions and thus satisfy mass conservation in the strongest possible sense semi discrete methods based on divergence conforming b splines are shown to conserve linear and angular momentum and satisfy balance laws for energy vorticity enstrophy and helicity these are geometric structure preserving quantities and numerical simulations that are sensitive to them are shown to be qualitatively correct and quantitatively accurate the methods developed are anticipated to open new doors to the practical calculation of complex flows and to studies of their physical behavior

dynamics of numerical approximations assuming that a differential equation has a hyperbolic invariant set we discuss the existence of a similar invariant set for the map defined by a numerical approximation as well as the closeness of these sets in the c 1 topology particular emphasis is given to equilibrium points periodic orbits and invariant tori for the differential equation we present also a result on the preservation of structural stability for gradient systems

finite element approximation of the nonstationary navier stokes problem part ii stability of solutions and error estimates uniform in time in this paper assumptions about the stability of a solution are introduced in the numerical analysis of the nonstationary navier stokes problem for the purpose of extending local a priori error estimates and local a posteriori error estimates globally in time