local projection stabilization for the oseen problem and its interpretation as a variational multiscale method we propose to apply the recently introduced local projection stabilization to the numerical computation of the oseen equation at high reynolds number the discretization is done by nested finite element spaces using a priori error estimation techniques we prove the convergence of the method the a priori estimates are independent of the local peclet number and give a sufficient condition for the size of the stabilization parameters in order to ensure optimality of the approximation when the exact solution is smooth moreover we show how this method may be cast in the framework of variational multiscale methods we indicate what modeling assumptions must be made to use the method for large eddy simulations

a sharp lower bound for the hausdorff dimension of the global attractors of the 2d navier stokes equations for a special class of the navier stokes equations on the two dimensional torus we give a lower bound in the formg2 3 whereg is the grashof number for the hausdorff dimension of its global attractor which is optimal up to a logarithmic term

strongly order preserving semiflows generated by functional differential equations the aim of this paper is to apply some recently developed improvements in the theory of monotone dynamical systems due to the authors 12 13 j to certain systems of functional differential equations fde s which do not enjoy the quasimonotone property considered by one of us in 9 the present work extends our previous results in this direction for scalar equations in 111 to systems in the fundamental paper s m w hirsch establishes that most orbits of a strongly monotone scmiflow on a strongly ordered space x tend to the set e of equilibria employing ideas of hirsch and of h matano 7 the authors have extended the theory to obtain some improvements in the results of hirsch and matano under matano s weaker assumption that the semiflow is strongly order preserving recently in 13 we gave sufficient conditions for most orbits of a strongly order preserving semiflow to converge to an equilibrium these conditions require additional smoothness of the semiflow and a certain strong monotonicity condition for the linearized flow determined by the variational equation about each equilibrium p policik s has also given sufficient conditions for most orbits to converge for semilinear parabolic evolution equations possessing certain strong monotonicity properties in this paper wc apply the convergence results developed in 131 to nonquasimonotone fde s in this introduction wc briefly review the application of monotone dynamical systems theory to fde s enjoying the quasimonotone property this research appeared in 9 133 then we state one of the main results of this paper which applies when the quasimonotone property fails but when another property m holds some simple examples are given which

reconstruction of turbulent fluctuations using a hybrid rans les approach a hybrid rans les approach is presented and assessed considering several flows which can be interpreted as the most general case of the nlde approach as defined by morris et al p j morris l n long a bangalore and a wang j comput phys 133 56 1997 a decomposition into three parts of the exact solution of the navier stokes equations is considered mean flow resolved fluctuations and unresolved subgrid fluctuations the mean flow is computed using a classical rans method while resolved fluctuations are derived from a les method this approach is succesfully assessed on the stationary and the pulsed plane channel flow configurations the case of the flow around a low pressure turbine blade is discussed in a second step the hybrid method is also demonstrated to be robust with respect to several sources of error and provides good results in the case of a coarsened grid simulation a reduced computational domain simulation and nonconsistent mean flow fields

galerkin and subspace decomposition methods in space and time for the navier stokes equations abstract the galerkin method and the subspace decomposition method in space and time for the two dimensional incompressible navier stokes equations with the h 2 initial data are considered the subspace decomposition method consists of splitting the approximate solution as the sum of a low frequency component discretized by the small time step t and a high frequency one discretized by the large time step p t with p 1 the h 2 stability and l 2 error analysis for the subspace decomposition method are obtained finally some numerical tests to confirm the theoretical results are provided

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

approximation of exponential order of the attractor of a turbulent flow abstract in this work we consider the two dimensional navier stokes equations with periodic boundary conditions describing the evolution of homogeneous turbulent flows we use the method of debussche and temam a debussche and r temam convergent families of approximate inertial manifolds j math pures appl to appear to construct approximate inertial manifolds aims whose order decreases exponentially fast with respect to the dimension of the manifold we recall that an aim is a smooth manifold of solutions m such that the attractor a is contained in a neighborhood of m of thinness is called the order of the manifold the dependence of all the constants with respect to physical parameters especially the grashof number is made explicit

explicit small scale velocity simulation for high re turbulent flows a new approach two level simulation tls is developed based on the explicit reconstruction of the small scale velocity by solving the small scale governing equations on the domain with reduced dimension representing a collection of one dimensional lines embedded in the three dimensional flow domain a coupled system of equations that is not based on an eddy viscosity hypothesis is derived based on the decomposition of flow variables into the large scale and the small scale components without introducing the concept of filtering simplified treatment of the small scale equations is proposed based on neglecting the small scale advective and dissipative terms in the line orthogonal directions fully coupled simulation of the forced isotropic turbulence at re l 65 114 demonstrates potential benefits and challenges of the tls approach for high re turbulent flows using coarse grids

robust stability and stabilizing controller design of fuzzy systems with discrete and distributed delays in this paper we consider delay dependent stability conditions of takagi sugeno fuzzy systems with discrete and distributed delays although many kinds of stability conditions for fuzzy systems with discrete delays have already been obtained almost no stability condition for fuzzy systems with distributed delays has appeared in the literature this is also true in case of the robust stability for uncertain fuzzy systems with distributed delays here we employ a generalized lyapunov functional to obtain delay dependent stability conditions of fuzzy systems with discrete and distributed delays we introduce some free weighting matrices to such a lyapunov functional in order to reduce the conservatism in stability conditions these techniques lead to generalized and less conservative stability conditions we also consider the robust stability of fuzzy time delay systems with uncertain parameters applying the same techniques made on the stability conditions we obtain delay dependent sufficient conditions for the robust stability of uncertain fuzzy systems with discrete and distributed delays moreover we consider the state feedback stabilization based on stability and robust stability conditions we obtain conditions for the state feedback controller to stabilize the fuzzy time delay systems finally we give two examples to illustrate our results delay dependent stability conditions obtained here are shown to guarantee a wide stability region

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

nonlinear galerkin methods the finite elements case with the increase in the computing power and the advent of supercomputers the approximation of evolution equations on large intervals of time is emerging as a new type of numerical problem in this article we consider the approximation of evolution equations on large intervals of time when the space discretization is accomplished by finite elements the algorithm that we propose called the nonlinear galerkin method stems from the theory of dynamical systems and amounts to some approximation of the attractor in the discrete finite elements space essential here is the utilization of incremental unknown which is accomplished in finite elements by using hierarchical bases beside a detailed description of the algorithm the article includes some technical results on finite elements spaces and a full study of the stability and convergence of the method

a multiscale v p discretization for flow problems this paper gives a comprehensive numerical analysis of a multiscale method for equilibrium navier stokes equations the method includes pressure regularization and eddy viscosity stabilizations both acting only on the finest scales this method allows for equal order velocity pressure spaces as well as the linear constant pair and the usual pk pk 1 pair we show the method is optimal in a natural energy norm for all of these pairs of spaces and provide guidance in choosing the regularization parameters

multilevel methods in space and time for the navier stokes equations we investigate the discretization in time of numerical schemes based on multilevel spatial splittings for the two dimensional periodic navier stokes equations the approximate solution is computed as the sum of a low frequency component and a high frequency one these two terms are advanced in time using different stepsizes we show improved stability conditions with respect to the classical galerkin method we derive error estimates that indicate that the high frequency term can be integrated less often we address implementation issues and show that the method should yield a significant gain in computing time

a multilevel algorithm for large eddy simulation of turbulent compressible flows abstract a multilevel method for large eddy simulation of turbulent compressible flows is proposed the method relies on the splitting of the turbulent flowfield into several frequency bands in space and time each band being associated to a specific computational grid in physical space this allows to take into account in a deterministic way the information contained on finer grids a subgrid model adapted to such a decomposition based on a generalization of the germano s identity to multilevel decomposition is also introduced the approach is validated by several multilevel simulations in a subsonic plane channel flow configuration for a low and a high value of the reynolds number while reductions of the cpu times up to 80 are obtained

the basic equations for the large eddy simulation of turbulent flows in complex geometry abstract the equations for large eddy simulation les of turbulent flows are derived by applying a filtering operation to the navier stokes equations les of inhomogeneous turbulent flows requires the use of filters with variable filter width the use of such filters invalidates the standard derivation of the basic equations for the filtered fields since the filtering operation in general does not commute with the operation of differentiation in this paper an alternate definition of the filtering operation based on the mapping function of the nonuniform grid is introduced it is shown that with this modified definition the filtering and differentiation operations commute up to an error which is second order in the filter width it is also shown that the commutation error can be expressed in terms of the filtered field and its derivatives as an asymptotic series in the square of the filter width these results are then applied to the navier stokes equations to derive the basic equations satisfied by the filtered fields

rapid and brief communications kernel direct discriminant analysis and its theoretical foundation in this paper the method of kernel direct discriminant analysis is analyzed from a new viewpoint and its theoretical foundation is revealed based on this result an efficient and robust method is proposed that is the qr decomposition on the small size matrix is adopted and then a small eigenvalue problem is solved finally experimental results on orl face database show that the proposed method is effective and feasible

a clustering algorithm for stream data with lda based unsupervised localized dimension reduction we present an algorithm for clustering high dimensional streaming data the algorithm incorporates dimension reduction into the stream clustering framework when a new datum arrives the algorithm performs dimension reduction to find a local projected subspace using unsupervised lda linear discriminant analysis based method the obtained local subspace would maximally separate the nearby micro clusters with respect to the incoming point then the incoming point is assigned to a micro cluster in the projected space rather than in the full dimensional space the experimental results show that the proposed algorithm outperforms its counterpart streaming clustering algorithms moreover when compared with traditional clustering algorithms which require the whole data set the proposed algorithms shows comparable clustering performances with much less computation time for large data sets

variational approach to closure of nonlinear dynamical systems autonomous case a general approach for the derivation of nonlinear parameterizations of neglected scales is presented for nonlinear systems subject to an autonomous forcing in that respect dynamically based formulas are derived subject to a free scalar parameter to be determined per mode to parameterize for each high mode this free parameter is obtained by minimizing a cost functional a parameterization defect depending on solutions from direct numerical simulation dns but over short training periods of length comparable to a characteristic recurrence or decorrelation time of the dynamics an important class of dynamically based formulas for our parameterizations to optimize are obtained as parametric variations of manifolds approximating the invariant ones to better appreciate the origins of the modified manifolds thus obtained the standard approximation theory of invariant manifolds is revisited in part i of this article a special emphasis is put on backward forward bf systems naturally associated with the original system whose asymptotic integration provides the leading order approximation of invariant manifolds part ii presents then i the modifications of these approximating manifolds based also on integration of the same bf systems but this time over a finite time tau and ii the variational approach aimed at making an efficient selection of tau per mode to parameterize the parametric class of leading interaction approximation lia of the high modes obtained this way is completed by another parametric class built from the quasi stationary approximation qsa close to the first criticality the qsa is an approximation to the lia but it differs as one moves away from criticality rigorous results are derived that show that given a cutoff dimension the best manifolds that can be obtained through our variational approach are manifolds which are in general no longer invariant the minimizers are objects called the optimal parameterizing manifolds pms that are intimately tied to the conditional expectation of the original system i e the best vector field of the reduced state space resulting from averaging of the unresolved variables with respect to a probability measure conditioned on the resolved variables applications to the closure of low order models of atmospheric primitive equations and rayleigh benard convection are then discussed the approach is finally illustrated in the context of the kuramoto sivashinsky turbulence as providing efficient closures without slaving for a cutoff scale k mathfrak c placed within the inertial range and the reduced state space is just spanned by the unstable modes without inclusion of any stable modes whatsoever the underlying optimal pms obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by the lia or the qsa parameterizations in their standard forms when they are used at this cutoff wavelength

monotone semiflows in scalar non quasi monotone functional differential equations abstract by introducing a stronger than pointwise ordering conditions are found under which scalar functional differential equations generate monotone semiflows even if they are not quasi monotone typically the maximum delay must be the smaller the more quasi monotonicity is violated the theory of monotone semiflows is used to show that most solutions converge to equilibrium and that stability of equilibria is essentially the same as for ordinary differential equations

kpca plus lda a complete kernel fisher discriminant framework for feature extraction and recognition this paper examines the theory of kernel fisher discriminant analysis kfd in a hilbert space and develops a two phase kfd framework i e kernel principal component analysis kpca plus fisher linear discriminant analysis lda this framework provides novel insights into the nature of kfd based on this framework the authors propose a complete kernel fisher discriminant analysis ckfd algorithm ckfd can be used to carry out discriminant analysis in double discriminant subspaces the fact that it can make full use of two kinds of discriminant information regular and irregular makes ckfd a more powerful discriminator the proposed algorithm was tested and evaluated using the feret face database and the cenparmi handwritten numeral database the experimental results show that ckfd outperforms other kfd algorithms

rapid and brief communication laplacian linear discriminant analysis a novel linear discriminant criterion function is proved to be equal to fisher s criterion function the analysis of the function is linked to spectral decomposition of the laplacian of a graph moreover the function is maximized using two algorithms experimental results show the effectiveness and some specific characteristics of our algorithms

face recognition using laplacianfaces we propose an appearance based face recognition method called the laplacianface approach by using locality preserving projections lpp the face images are mapped into a face subspace for analysis different from principal component analysis pca and linear discriminant analysis lda which effectively see only the euclidean structure of face space lpp finds an embedding that preserves local information and obtains a face subspace that best detects the essential face manifold structure the laplacianfaces are the optimal linear approximations to the eigenfunctions of the laplace beltrami operator on the face manifold in this way the unwanted variations resulting from changes in lighting facial expression and pose may be eliminated or reduced theoretical analysis shows that pca lda and lpp can be obtained from different graph models we compare the proposed laplacianface approach with eigenface and fisherface methods on three different face data sets experimental results suggest that the proposed laplacianface approach provides a better representation and achieves lower error rates in face recognition

incremental unknowns multilevel methods and the numerical simulation of turbulence abstract the purpose of this monograph is to describe the dynamic multilevel dml methodology applied to the numerical simulation of turbulence the general setting of the navier stokes equations is recalled and also a number of basic notions on the statistical theory of turbulence the practical limitations of direct numerical simulation dns and the needs for modeling are emphasized the objective of this article is to discuss the modeling and the numerical simulation of turbulent flows by multilevel methods related to the concept of approximate inertial manifolds aim this mathematical concept stemming from the dynamical systems theory is briefly presented aim are based on a decomposition of the velocity field into small and large scale components they give a slaving law of the small scales as a function of the large scales the novel aspect of the work presented here is the time adaptative dynamical implementation of these multilevel methods indeed the dml methodology is based on the properties of this decomposition of the velocity field and on numerical arguments numerical analysis of multilevel methods for a simple system is proposed dml methods with some numerical and physical justifications are described for homogeneous turbulence the numerical results obtained are discussed

on the relevance of linear discriminative features linear discriminant analysis lda has been widely used to extract linear features for classification in real applications the usefulness of the extracted features usually needs to be confirmed using an error rate of classification embedded in a classifier little attention has been paid to whether and how discriminative features themselves can be interpreted as indicators of usefulness we refer to this as relevance i e the capability of discriminative features to characterize the contribution of the original variables to classification we approach the relevance by considering how it could be lost while extracting optimal discriminative features then the discrepancy between the relevance and optimality of discriminative features is shown to originate from the angle between the space spanned by eigenvectors of the within class scattering matrix and the primary space in which the original variables reside in particular for a given dataset the larger the angle the less evident is the relevance discovered from optimal discriminative features furthermore the relevance and optimality are regarded as two constraint conditions or a tradeoff in order to extract relevant discriminative features at last a simulated experiment is used to show how the relevance is lost when the angle is changed experimental results based on both usps handwritten digitals and pie face databases show that a maximum margin criterion is a reasonable compromise between the relevance and optimality since it approximates the averaged class margin using euclidean distance measured in the primary space

on an alternative formulation of the fisher criterion that overcomes the small sample problem in two very recently published rapid and brief communications both from the same authors an alternative formulation of the well known fisher criterion is presented in order to overcome the small sample problem a theorem in the first of the two communications provides the basis for the equivalence of both formulations by providing a simple counterexample we disprove the theorem subsequently based on an illustrative example we demonstrate that their criterion differs from the classical one and argue that the proposed criterion is not a suitable measure of discriminability

incremental unknowns for solving partial differential equations incremental unknowns have been proposed in t as a method to approximate fractal attractors by using finite difference approximations of evolution equations in the case of linear elliptic problems the utilization of incremental unknown methods provides a new way for solving such problems using several levels of discretization the method is similar but different from the classical multigrid method in this article we describe the application of incremental unknowns for solving laplace equations in dimensions one and two we provide theoretical results concerning two level approximations and we report on numerical tests done with multi level approximations

stability analysis of the nonlinear galerkin method our object in this article is to describe some numerical schemes for the approximation of nonlinear evolution equations and to study the stability of the schemes spatial discretization can be performed by either spectral or pseudospectral methods finite elements or finite differences time discretization is done by two level schemes partly or fully explicit the algorithms that we present stem from the study of the evolution equations from the dynamical systems point of view they are based on a differentiated treatment of the small and large wave lengths and they are particularly adapted to the integration of such equations on large intervals of time introduction the long time integration of dissipative evolution equations has emerged as a new type of problem in numerical analysis and its appearance is due to the considerable increase in computing power that we have seen during the last years much effort has been devoted in the past to the approximation of evolution equations on finite intervals of time or in the dynamically simple case where the solution converges to a steady stationary state as time goes to infinity however in many physically relevant situations the solutions to a dissipative evolution equation do not converge to a stationary solution they rather remain time dependent and they converge to a compact attractor that encompasses the nonlinear dynamics hence and whether this is addressed implicitly or explicitly the numerical integration of the evolution equation is then closely related to the approximation of the attractor that may be a complicated set even a fractal inertial algorithms are new integration algorithms that are based on the approximation of the attractor by smooth simple manifolds namely the approximate inertial manifolds these algorithms will be recalled below but at this point let us point out that one of their main features is a differentiated treatment of small and large wave lengths an inertial manifold is an exact law of dependence of small wave lengths in terms of the large ones the small wave received april 3 1990 1980 mathematics subject classification 1985 revision primary 65f10 65f35 65n20 35a40 1991 american mathematical society 0025 5718 91 1 00 25 perpage

analysis of a model representing stage structured population growth with state dependent time delay a stage structured model of population growth is proposed where the time to ma turity is itself state dependent it is shown that under appropriate assumptions all solutions are positive and bounded criteria for the existence of positive equilibria and further conditions for the uniqueness of the equilibria are given the stability of the equilibria are also discussed in addition an attracting region is determined for solutions such that this region collapses to the unique positive equilibrium in the state independent case

residual based variational multiscale turbulence models for unstructured tetrahedral meshes abstract this paper presents three level residual based turbulence models for the incompressible navier stokes equations employing the variational multiscale vms framework the velocity and pressure fields are decomposed into two overlapping hierarchical scales thereby leading to a system of coupled mixed field problems the mixed problem at the fine scales is stabilized via a subsequent vms application that results in a further decomposition of the fine scale velocity field into level i and level ii scales the level ii scales are modeled using higher order bubble functions that are then variationally embedded in the level i formulation to stabilize it the level i problem is modeled via a second set of bubble functions that are linearly independent of the bubbles employed at level ii finally the resulting level i fine scales are variationally embedded in the coarse scale formulation this yields a residual based turbulence model for the larger or coarser scales a significant feature of the proposed method is that it results in a concurrent top down and bottom up two way nesting of the scales in addition the resulting turbulence model does not possess any embedded tunable parameters another attribute of the formulation is that the fine scales at every level are driven by the residuals of the euler lagrange equations of the coarser scales at the preceding levels thereby resulting in a method that is variationally consistent various algorithmic generalizations of the method are presented that lead to computationally economic residual based turbulence models the proposed telescopic depth in scales approach helps make these models accurate for low order tetrahedral and hexahedral elements a feature that is facilitated by the higher order bubble functions over element interiors and it results in an enhanced representation of the fine scale terms modeling the fine viscous effects from a computational perspective this method results in easy to implement equal order pressure velocity elements and possesses the desirable p refinement feature numerical performance of the method is assessed on turbulent channel flow problems at re 395 and re 590 also presented are the results for turbulent sd 7003 airfoil at re 60 000 and comparison is made with the published experimental data and numerical results

stability and convergence in strongly monotone dynamical systems soit x un systeme dynamique x est un espace topologique et le flot t t r est une famille d applications continues de sous ensembles ouverts de x dans x on decrit qualitativement le comportement asymptotique quand t des trajectoires t t x pour les etats initiaux x dans le cas de systemes ayant un principe de comparaison fort

efficient spectral galerkin method ii direct solvers of second and fourth order equations using chebyshev polynomials efficient direct solvers based on the chebyshev galerkin methods for second and fourth order equations are presented they are based on appropriate base functions for the galerkin formulation that lead to discrete systems with special structured matrices that can be efficiently inverted numerical results indicate that the direct solvers presented in this paper are significantly more accurate and efficient than those based on the chebyshev tau method

stage structured models of intra and inter specific competition within age classes abstract in some species larvae and adults experience competition in completely different ways simple stage structured models without larval competition usually yield a single delay equation for the adults using an age structured system incorporating competition among both larvae and adults we derive a system of distributed delay equations for the numbers of larvae and adults the system is neither cooperative nor reducible to a single equation for either variable positivity boundedness and uniform strong persistence are established linear stability analysis of equilibria is difficult due to the strong coupling but results are proved for small delays using monotone systems theory and exponential ordering for small delay we prove a theorem on generic convergence to equilibria which does not directly follow from standard theory but can be proved indirectly using comparison arguments finally we consider an extension to two strain competition and prove theorems on the linear stability of the boundary equilibria

a stage structured mathematical model for fish stock with harvesting in this paper we propose a mathematical model for a single species fish stock with a three stage structure juveniles small adults and large adults with two harvesting strategies for mature classes maturity and size selectivities the purpose of the work is to investigate the dynamical behavior of the model and discuss the effect of harvesting we identify the adult reproduction number mathcal r a for the model obtain the local and global stability of the trivial equilibrium when mathcal r a 1 numerical simulations are provided to investigate the influence of harvesting functions discuss the optimal harvesting rates and explore the effect of periodic coefficients on the dynamical system

monotone semiflows generated by functional differential equations abstract this obtains sufficient condition for an autonomous functional differential equation to generate a strongly monotone semiflow on a suitable state space this allows the application to functional differential equations of very powerful recent results on strongly monotone semiflows due to hirsch and matano in addition a very striking relationship is established between such functional differential equations and corresponding ordinary differential equations an example involving a biochemical feedback loop is considered keywords stability steady states

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

convergence in smooth strongly monotone flows defined by semilinear parabolic equations on considere des equations paraboliques semi lineaires dy dt ay f y ou a est un operateur sectoriel sur un banach x ayant une resolvante compacte et f est une c 2 application de l espace des puissances fractionnaires x 0 1 dans x soit y t une solution qui est quasi convergente mais non convergente alors il existe un voisinage u de y 0 y 0 avec la propriete suivante si z u y 0 a une trajectoire bornee et soit z y 0 soit z y 0 alors la solution partant de z converge vers un equilibre

nonlinear galerkin methods this article presents a new method of integrating evolution differential equations the non linear galerkin method that is well adapted to the long term integration of such equations while the usual galerkin method can be interpreted as a projection of the considered equation on a linear space the methods considered here are related to the projection of the equation on a nonlinear manifold from the practical point of view some terms have been identified as small and sometimes but not always disregarded

a variational multiscale method for incompressible turbulent flows bubble functions and fine scale fields this paper presents a residual based turbulence model for the incompressible navier stokes equations the method is derived employing the variational multiscale vms framework a multiscale decomposition of the continuous solution and a priori unique decomposition of the admissible spaces of functions lead to two coupled nonlinear problems termed as the coarse scale and the fine scale sub problems the fine scale velocity field is assumed to be nonlinear and time dependent and is modeled via the bubble functions approach applied directly to the fine scale sub problem a significant contribution in this paper is a systematic and consistent derivation of the fine scale variational operator commonly termed as the stabilization tensor that possesses the right order in the advective and diffusive limits and variationally projects the fine scale solution onto the coarse scale space a direct treatment of the fine scale problem via bubble functions offers several fine scale approximation options with varying degrees of mathematical sophistication that are investigated via benchmark problems numerical accuracy of the proposed method is shown on a forced isotropic turbulence problem statistically stationary turbulent channel flow problems at ret 395 and 590 and non equilibrium turbulent flow around a cylinder at re 3 900