on the structural stability of the euler voigt and navier stokes voigt models abstract we consider the euler voigt equations and the navier stokes voigt equations which are obtained by an inviscid regularization from the corresponding equations the main result we show is the structural stability of the system in terms of the variations of both viscosity and regularization parameters

a series test of the scaling limit of self avoiding walks it is widely believed that the scaling limit of self avoiding walks saws at the critical temperature is conformally invariant and consequently describable by schramm loewner evolution with parameter 8 3 we consider saws in a rectangle which originate at its centre and end at the boundary we assume that the boundary density transforms covariantly in a way that depends precisely on as conjectured by lawler schramm and werner 2004 fractal geometry and applications a jubilee of benoit mandelbrot part 2 pp 339 64 it has previously been shown by guttmann and kennedy 2013 j eng math at press that in the limit of an infinitely large rectangle the ratio of the fraction of saws hitting the side of the rectangle to the fraction that hit the end of the rectangle can be calculated by considering rectangles of fixed aspect ratio 2 and also rectangles of aspect ratio 10 we calculate this ratio exactly for larger and larger rectangles by extrapolating this data to infinite rectangle size and invoking the above conjectures we obtain the estimate 2 666 64 0 000 07 for rectangles of aspect ratio 2 and 2 666 75 0 000 15 for rectangles of aspect ratio 10 we also provide numerical evidence supporting the conjectured distribution of saws striking the boundary at various points in the case of rectangles with aspect ratio 2

on the construction of suitable solutions to the navier stokes equations and questions regarding the definition of large eddy simulation abstract this paper discusses the notion of large eddy simulation les as a mathematical concept and proposes to use the concept of suitable weak solutions as a building block for the development of a mathematical theory of les various techniques for constructing suitable weak solutions to the three dimensional navier stokes equations are reviewed and many features of these mathematical constructions are shown to comply with well accepted heuristic characterizations of les

shape functions and integration formulas for three dimensional finite element analysis shape functions and numerical integration formulas for three dimensional finite element analysis as found in most finite element reference books are incomplete for example shape functions and integration formulas for a pyramid with a quadrilateral base are missing it is also difficult to find symmetric higher order integration formulas for triangular and tetrahedral elements in general these shape functions and integration formulas cannot be satisfactorily derived as degenerate cases of shape functions and integration formulas for hexahedral brick elements in this paper we present c continuous quadrilateral pyramid elements and integration formulas for two and three dimensional elements

on the existence of global attractors of the approximate deconvolution models of turbulence abstract in this report we consider the problem of the existence of global attractors of the weak solution of the approximate deconvolution models of turbulence the l 2 global attractors are known to exist for large eddy simulation models such as leray alpha leray deconvolution or ns omega models in the case of the approximate deconvolution model as the l 2 norm of the weak solution of the adm is not controlled by the l 2 norm of the initial data and the approximate deconvolution model is an approximation of the filtered nse we are led to the question of existence of h 1 global attractors we also prove better regularity of the weak solution of adm

a deconvolution enhancement of the navier stokes αβ model a deconvolution enhancement of the email protected b model for turbulent flow is introduced the energy and energy dissipation rate for the enhanced model are derived it is also shown that the consistency error relative to the navier stokes equations and the microscale of the enhanced model are less than those of the email protected b model the proposed model is used to simulate the taylor green vortex problem and results show a qualitatively improved representation of the mean square vorticity when compared to the email protected b model numerical studies of the energy spectrum and the alignment between the vorticity and the eigenvectors of the stretching tensor for three dimensional turbulent flows with re 200 are used to explore the utility of the model a benchmark problem of a two dimensional channel flow over a step for re 600 also indicates that this model can be applied to more general flows than those involving periodic boundary conditions

an eulerian lagrangian approach to the navier stokes equations we present a formulation of the incompressible viscous navier stokes equation based on a generalization of the inviscid weber formula in terms of a diffusive back to labels map and a virtual velocity we derive a generalization of the inviscid cauchy formula and obtain certain bounds for the objects introduced

on the leray deconvolution model for the incompressible magnetohydrodynamics equations abstract we extend leray deconvolution modeling to the incompressible magnetohydrodynamics mhd the resulting model is shown to be well posed and have attractive limiting behavior both in its filtering radius and order of deconvolution additionally we present and study a numerical scheme for the model based on an extrapolated crank nicolson finite element method we show the numerical scheme is unconditionally stable preserves energy and cross helicity and optimally converges to the mhd solution numerical experiments are provided that verify convergence rates and test the scheme on benchmark problems of channel flow over a step and the orszag tang vortex problem

transforming fixed length self avoiding walks into radial sle8 3 we conjecture a relationship between the scaling limit of the fixed length ensemble of self avoiding walks in the upper half plane and radial sle8 3 in this half plane from 0 to i the relationship is that if we take a curve from the fixed length scaling limit of the saw weight it by a suitable power of the distance to the endpoint of the curve and apply the conformal map of the half plane that takes the endpoint to i then we get the same probability measure on curves as radial sle8 3 in addition to a non rigorous derivation of this conjecture we support it with monte carlo simulations of the saw using the conjectured relationship between the saw and radial sle8 3 our simulations give estimates for both the interior and boundary scaling exponents the values we obtain are within a few hundredths of a percent of the conjectured values

toward a conformal field theory for schramm loewner evolutions we discuss the partition function point of view for chordal schramm loewner evolutions and their relationship with correlation functions in conformal field theory both are closely related to crossing probabilities and interfaces in critical models in two dimensional statistical mechanics we gather and supplement previous results with different perspectives point out remaining difficulties and suggest directions for future studies

chebychev optimized approximate deconvolution models of turbulence abstract if the navier stokes equations are averaged with a local spacial convolution type filter g the resulting system is not closed due to the filtered nonlinear term uu an approximate deconvolution operator d is a bounded linear operator which satisfies u d u o where is the filter width and 2 using a deconvolution operator as an approximate filter inverse yields the closure uu d u d u o the residual stress of this model and related models depends directly on the deconvolution error u d u this report derives deconvolution operators yielding an effective turbulence model which minimize the deconvolution error for velocity fields with finite kinetic energy we also give a convergence theory of deconvolution as 0 an ergodic theorem as the deconvolution order n and estimate the increase in accuracy obtained by parameter optimization the report concludes with numerical illustrations

on the smoothness of the partition function for multiple schramm loewner evolutions we consider the measure on multiple chordal schramm loewner evolution textit sle kappa curves we establish a derivative estimate and use it to give a direct proof that the partition function is c 2 if kappa 4

grad div stablilization for stokes equations in this paper a stabilizing augmented lagrangian technique for the stokes equations is studied the method is consistent and hence does not change the continuous solution we show that this stabilization improves the well posedness of the continuous problem for small values of the viscosity coefficient we analyze the influence of this stabilization on the accuracy of the finite element solution and on the convergence properties of the inexact uzawa method

truncation of scales by time relaxation abstract we study a time relaxation regularization of flow problems proposed and tested extensively by stolz and adams the aim of the relaxation term is to drive the unresolved fluctuations in a computational simulation to zero exponentially fast by an appropriate and often problem dependent choice of its coefficient this relaxation term is thus intermediate between a tunable numerical stabilization and a continuum modeling term our aim herein is to understand how this term by itself acts to truncate solution scales and to use this understanding to give insight into parameter selection

high order optimal edge elements for pyramids prisms and hexahedra edge elements are a popular method to solve maxwell s equations especially in time harmonic domain however when non affine elements are considered elements of the nedelec s first family 19 are not providing an optimal rate of the convergence of the numerical solution toward the solution of the exact problem in h curl norm we propose new finite element spaces for pyramids prisms and hexahedra in order to recover the optimal convergence in the particular case of pyramids a comparison with other existing elements found in the literature is performed numerical results show the good behavior of these new finite elements

a sensitivity study of the navier stokes α model abstract we present a sensitivity study of the navier stokes model with respect to perturbations of the differential filter length the parameter sensitivity is evaluated using the sensitivity equations method once formulated the sensitivity equations are discretized and computed alongside the ns model using the same finite elements in space and crank nicolson in time we provide a complete stability analysis of the scheme along with the results of several benchmark problems in both 2d and 3d we further demonstrate a practical technique to utilize sensitivity calculations to determine the reliability of the ns model in problem specific settings lastly we investigate the sensitivity and reliability of important functionals of the velocity and pressure solutions

on the accuracy of the rotation form in simulations of the navier stokes equations the rotation form of the navier stokes equations nonlinearity is commonly used in high reynolds number flow simulations it was pointed out by a few authors and not widely known apparently that it can also lead to a less accurate approximate solution than the usual email protected u form we give a different explanation of this effect related to i resolution of the bernoulli pressure and ii the scaling of the coupling between velocity and pressure error with respect to the reynolds number we show analytically that i the difference between the two nonlinearities is governed by the difference in the resolution of the bernoulli and kinematic pressures and ii a simple linear grad div stabilization ameliorates much of the bad scaling of the velocity error with respect to re the rotation form does have superior conservation properties to the alternatives and it appears to be amenable to more efficient preconditioners thus the rotational form with grad div stabilization is a promising method we also give experiments that show bad velocity approximation is tied to poor pressure resolution in either form

on a leray α model of turbulence in this paper we introduce and study a new model for three dimensional turbulence the leray model this model is inspired by the lagrangian averaged navier stokes model of turbulence also known navier stokes model or the viscous camassa holm equations as in the case of the lagrangian averaged navier stokes model the leray model compares successfully with empirical data from turbulent channel and pipe flows for a wide range of reynolds numbers we establish here an upper bound for the dimension of the global attractor the number of degrees of freedom of the leray model of the order of l l d 12 7 where l is the size of the domain and l d is the dissipation length scale this upper bound is much smaller than what one would expect for three dimensional models i e l l d 3 this remarkable result suggests that the leray model has a great potential to become a good sub grid scale large eddy simulation model of turbulence we support this observation by studying analytically and computationally the energy spectrum and show that in addition to the usual k 5 3 kolmogorov power law the inertial range has a steeper power law spectrum for wavenumbers larger than 1 finally we propose a prandtl like boundary layer model induced by the leray model and show a very good agreement of this model with empirical data for turbulent boundary layers

the self avoiding walk spanning a strip we review the existence of the infinite length self avoiding walk in the half plane and its relationship to bridges we prove that this probability measure is also given by the limit as c of the probability measure on all finite length walks with the probability of proportional to where is the number of steps in c is the reciprocal of the connective constant the self avoiding walk in a strip z 0 operatorname im z y is defined by considering all self avoiding walks in the strip which start at the origin and end somewhere on the top boundary with probability proportional to beta c omega we prove that this probability measure may be obtained by conditioning the saw in the half plane to have a bridge at height y this observation is the basis for simulations to test conjectures on the distribution of the endpoint of the saw in a strip and the relationship between the distribution of this strip saw and sle8 3

an energy and helicity conserving finite element scheme for the navier stokes equations we present a new finite element scheme for solving the navier stokes equations that exactly conserves both energy int omega u 2 and helicity int omega u cdot nabla times u in the absence of viscosity and external force we prove stability exact conservation and convergence for the scheme energy and helicity are exactly conserved by using a combination of the usual convective form with the rotational form of the nonlinearity and solving for both velocity and a projected vorticity in a trapezoidal time discretization numerical results are presented that compare the scheme to the usual trapezoidal schemes

a 3 d interface enriched generalized fem for electromagnetic problems with nonconformal discretizations an interface enriched generalized finite element method igfem is introduced for efficient 3 d electromagnetic analysis of heterogeneous materials without using meshes that conform to the material microstructures which greatly lessens the burden of mesh generation the method assigns generalized degrees of freedom dofs at material interfaces to capture the discontinuities of the field and its derivatives the generalized dofs are supported by enriched vector basis functions vbfs which are constructed through a linear combination of the vbfs from the subelements several verification examples are provided to show that the igfem is not sensitive to the quality of the subelements and maintains the same level of solution accuracy and computational complexity as the standard finite element method fem based on conformal meshes the potential of the proposed igfem is demonstrated by simulating some engineering problems with complex periodic internal structures including composite materials with randomly distributed spherical particles and ellipsoidal inclusions and microvascular channels

analysis of a large eddy simulation model based on anisotropic filtering abstract we consider a new large eddy simulation model derived with the introduction of suitable horizontal anisotropic differential filters one main advantage of this filtering is that for channel flows there is no need for artificial boundary conditions hence we can deal with some realistic problems equipped with dirichlet boundary conditions in special bounded domains at least those bounded only in one direction recent numerical results for a similar model based on a derivation with wave number asymptotics are also recalled after a detailed analysis of the properties of the differential filter we prove that the resulting initial boundary value problem is well posed in suitable anisotropic sobolev spaces giving a strong mathematical support to the model we propose some remarks on higher accuracy approximate deconvolution models are also given in the last section

boundary layer for the navier stokes alpha model of fluid turbulence we study boundary layer turbulence using the navier stokes alpha model obtaining an extension of the prandtl equations for the averaged flow in a turbulent boundary layer in the case of a zero pressure gradient flow along a flat plate we derive a nonlinear fifth order ordinary differential equation which is an extension of the blasius equation we study it analytically and prove the existence of a two parameter family of solutions satisfying physical boundary conditions matching these parameters with the skin friction coefficient and the reynolds number based on momentum thickness we get an agreement of the solutions with experimental data in the laminar and transitional boundary layers as well as in the turbulent boundary layer for moderately large reynolds numbers

h p finite element trace inequalities for the pyramid polynomial trace inequalities typically involve an unknown constant depending on the order n of the polynomial the dependence of these constants on n was made more explicit in warburton and hesthaven 2003 for the general d simplex and they were determined for quadrilaterals hexes and wedges in hillewaert 2013 in this note we derive explicit expressions for the constant in the trace inequality for two sets of basis functions on the pyramid

unconditional convergence and optimal l2 error estimates of the crank nicolson extrapolation fem for the nonstationary navier stokes equations abstract in this paper we study stability and convergence of fully discrete finite element method on large timestep which used crank nicolson extrapolation scheme for the nonstationary navier stokes equations this approach bases on a finite element approximation for the space discretization and the crank nicolson extrapolation scheme for the time discretization it reduces nonlinear equations to linear equations thus can greatly increase the computational efficiency we prove that this method is unconditionally stable and unconditionally convergent moreover taking the negative norm technique we derive the l 2 h 1 unconditionally optimal error estimates for the velocity and the l 2 unconditionally optimal error estimate for the pressure also numerical simulations on unconditional l 2 stability and convergent rates of this method are shown

the camassa holm equations and turbulence abstract in this paper we will survey our results on the camassa holm equations and their relation to turbulence as discussed in s chen c foias d d holm e olson e s titi s wynne the camassa holm equations as a closure model for turbulent channel and pipe flow phys rev lett 81 1998 5338 s chen c foias d d holm e olson e s titi s wynne a connection between the camassa holm equations and turbulent flows in channels and pipes phys fluids in press in particular we will provide a more detailed mathematical treatment of those equations for pipe flows which yield accurate predictions of turbulent flow profiles for very large reynolds numbers there are many facts connecting the camassa holm equations to turbulent fluid flows the dimension of the attractor agrees with the heuristic argument based on the kolmogorov statistical theory of turbulence the statistical properties of the energy spectrum agree in numerical simulation with the kolmogorov power law furthermore comparison of mean flow profiles for turbulent flow in channels and pipes given by experimental and numerical data show acceptable agreement with the profile of the corresponding solution of the camassa holm equations

an interpretation of the navier stokes alpha model as a frame indifferent leray regularization abstract we show in this note that the leray regularization of the navier stokes equations is almost frame indifferent but for one second order term neglecting this term we readily recover the navier stokes alpha equations introduced by holm et al adv math 137 1998 1 phys fluids 11 8 1999 2343 hence as an alternative to the exposition in adv math 137 1998 1 phys fluids 11 8 1999 2343 we propose to interpret the navier stokes alpha model as a frame indifferent perturbation of the leray regularization

a lowest order composite finite element exact sequence on pyramids abstract composite basis functions for pyramidal elements on the spaces h 1 h curl h div and l 2 are presented in particular we construct the lowest order composite pyramidal elements and show that they respect the de rham diagram i e we have an exact sequence and satisfy the commuting property moreover the finite elements are fully compatible with the standard finite elements for the lowest order raviart thomas nedelec sequence on tetrahedral and hexahedral elements that is to say the new elements have the same degrees of freedom on the shared interface with the neighbouring hexahedral or tetrahedra elements and the basis functions are conforming in the sense that they maintain the required level of continuity full tangential component normal component etc across the interface furthermore we study the approximation properties of the spaces as an initial partition consisting of tetrahedra hexahedra and pyramid elements are successively subdivided and show that the spaces result in the same optimal order of approximation in terms of the mesh size h as one would obtain using purely hexahedral or purely tetrahedral partitions

onsager s conjecture with physical boundaries and an application to the vanishing viscosity limit author s bardos claude titi edriss s wiedemann emil abstract we consider the incompressible euler equations in a bounded domain in three space dimensions recently the first two authors proved onsager s conjecture for bounded domains i e that the energy of a solution to these equations is conserved provided the solution is h quot older continuous with exponent greater than 1 3 uniformly up to the boundary in this contribution we relax this assumption requiring only interior h quot older regularity and continuity of the normal component of the energy flux near the boundary the significance of this improvement is given by the fact that our new condition is consistent with the possible formation of a prandtl type boundary layer in the vanishing viscosity limit

a numerical study of the navier stokes αβ model abstract we present a numerical study of the ns model which is a recently proposed multiscale variation of the ns model that attempts to recapture scales lost through over regularization by separately modeling dissipation range scales we develop a similarity theory for the new model which shows that it is better equipped than the ns model to capture smaller scale behavior next we propose and study an unconditionally stable optimally accurate and efficient finite element implementation for the ns model rigorous proofs for stability and convergence are provided finally we present results from two numerical experiments that demonstrate the advantages of the ns model over the ns model

the anisotropic lagrangian averaged euler and navier stokes equations the purpose of this paper is twofold first we give a derivation of the lagrangian averaged euler lae and navier stokes lans equations this theory involves a spatial scale and the equations are designed to accurately capture the dynamics of the euler and navier stokes equations at length scales larger than while averaging the motion at scales smaller than the derivation involves an averaging procedure that combines ideas from both the material lagrangian and spatial eulerian viewpoints this framework allows the use of a variant of g i taylor s frozen turbulence hypothesis as the foundation for the model equations more precisely the derivation is based on the strong physical assumption that fluctutations are frozen into the mean flow in this article we use this hypothesis to derive the averaged lagrangian for the theory and all the terms up to and including order 2 are accounted for

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

existence of smooth attractors for the navier stokes omega model of turbulence abstract we consider the navier stokes model given by u t u u p f u u 0 subject to periodic boundary conditions with zero mean the ns model is an outgrowth of ideas in approximate deconvolution models and in ns alpha models like the ns alpha model it is simple and conserves in the appropriate context kinetic energy and helicity 3d or energy and enstrophy 2d in first tests ns was found to be accurate robust and amenable to efficient numerical simulation in this note we prove existence and regularity of a global attractor for the model

stabilized finite element methods for the generalized oseen problem the numerical solution of the non stationary incompressible navierstokes model can be split into linearized auxiliary problems of oseen type we present in a unique way different stabilization techniques of finite element schemes on isotropic meshes first we describe the state of the art for the classical residual based supg pspg method then we discuss recent symmetric stabilization techniques which avoid some drawbacks of the classical method these methods are closely related to the concept of variational multiscale methods which seems to provide a new approach to large eddy simulation finally we give a critical comparison of these methods

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

an efficient solver for the incompressible navier stokes equations in rotation form we consider preconditioned iterative methods applied to discretizations of the linearized navier stokes equations in two and three dimensional bounded domains both unsteady and steady flows are considered the equations are linearized by picard iteration we make use of the rotation form of the momentum equations which has several advantages from the linear algebra point of view we focus on a preconditioning technique based on the hermitian skew hermitian splitting of the resulting nonsymmetric saddle point matrix we show that this technique can be implemented efficiently when the rotation form is used we study the performance of the solvers as a function of mesh size reynolds number time step and algorithm parameters our results indicate that fast convergence independent of problem parameters is achieved in many cases the preconditioner appears to be especially attractive in the case of low viscosity and for unsteady problems

a note on third order structure functions in turbulence starting from the navier stokes equation we rigorously prove that a modified third order structure function stilde3 r asymptotically equals 4 over 3 r is the dissipation rate in an inertial regime from this result we rigorously confirm the kolmogorov four fifths law without the kolmogorov assumption on isotropy our definition of the structure function involves a solid angle averaging over all possible orientations of the displacement vector y besides space time averaging direct numerical simulation for a highly symmetric flow for a taylor reynolds number of up to 155 shows that the flow remains significantly anisotropic and that without solid angle averaging the resulting structure functions approximately satisfy these scaling relations over some range of r l y l for some orientation of y but not for another

grad div stabilization and subgrid pressure models for the incompressible navier stokes equations in this paper the grad div stabilization for the incompressible navier stokes finite element approximations is considered from two different viewpoints i as a variational multiscale approach for the pressure subgrid modeling and ii as a stabilization procedure of least square type some new error estimates for the linearized problem with the grad div stabilization are proved with the help of norms induced by the pressure schur complement operator we discuss the stabilization parameter choice arising in the frameworks of least square and multiscale methods and consider assumptions which allow to relate both approaches

variational principles for lagrangian averaged fluid dynamics the lagrangian average la of the ideal fluid equations preserves their fundamental transport structure this transport structure is responsible for the kelvin circulation theorem of the la flow and hence for its potential vorticity convection and helicity conservation we show that lagrangian averaging also preserves the euler poincar e variational framework that implies the exact ideal fluid equations in the eulerian representation this is expressed in the lagrangian averaged euler poincar e laep theorem proved here we illustrate the laep theorem by applying it to incompressible ideal fluids to derive the lagrangian averaged euler equations and thereby recover the generalized lagrangian mean motion equation finally we discuss recent progress in applications of these equations as the basis for new la closure models of fluid turbulence

on degrees of freedom of certain conservative turbulence models for the navier stokes equations abstract we study the degrees of freedom of several conservative computational turbulence models that are derived via a non dissipative regularizations of the navier stokes equations for the navier stokes the leray and the navier stokes equations we prove that the longtime behavior of their respective solutions is completely determined by a finite set of grid values and by a finite set of fourier modes for each turbulence model the number of determining nodes and of determining modes is estimated in terms of flow parameters such as viscosity smoothing length forcing and domain size these estimates are global as they do not depend on an individual solution

random attractors for locally monotone stochastic partial differential equations abstract we prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive levy noise the main result is applicable to various types of spde such as stochastic burgers type equations stochastic 2d navier stokes equations the stochastic 3d leray model stochastic power law fluids the stochastic ladyzhenskaya model stochastic cahn hilliard type equations stochastic kuramoto sivashinsky type equations stochastic generalized porous media equations and stochastic generalized p laplace equations

discrete models of fluids spatial averaging closure and model reduction we consider semidiscrete ode models of single phase fluids and two fluid mixtures in the presence of multiple fine scale heterogeneities the size of these ode systems can be very large spatial averaging is then a useful tool for reducing computational complexity of the problem the averages satisfy exact balance equations of mass momentum and energy these equations do not form a satisfactory continuum model because evaluation of stress and heat flux requires solving the underlying odes to produce continuum equations that can be simulated without resolving microscale dynamics we recently proposed a closure method based on the use of regularized deconvolution here we continue the investigation of deconvolution closure with the long term objective of developing consistent computational upscaling for multiphase particle methods the structure of the fine scale particle solvers is reminiscent of molecular dynamics for this reason we use nonlinear averaging introduced for atomistic systems by noll ha

navier stokes equations in the whole space with an eddy viscosity abstract we study the navier stokes equations with an extra eddy viscosity term in the whole space i r 3 we introduce a suitable regularized system for which we prove the existence of a regular solution defined for all time we prove that when the regularizing parameter goes to zero the solution of the regularized system converges to a turbulent solution of the initial system

a low order galerkin finite element method for the navier stokes equations of steady incompressible flow a stabilization issue and iterative methods a galerkin finite element method is considered to approximate the incompressible navier stokes equations together with iterative methods to solve a resulting system of algebraic equations this system couples velocity and pressure unknowns thus requiring a special technique for handling we consider the navier stokes equations in velocity kinematic pressure variables as well as in velocity bernoulli pressure variables the latter leads to the rotation form of nonlinear terms this form of the equations plays an important role in our studies a consistent stabilization method is considered from a new view point theory and numerical results in the paper address both the accuracy of the discrete solutions and the effectiveness of solvers and a mutual interplay between these issues when particular stabilization techniques are applied 2002 published by elsevier science b v

bernstein bezier finite elements on tetrahedral hexahedral pyramidal partitions abstract a construction for high order continuous finite elements on partitions consisting of tetrahedra hexahedra and pyramids based on polynomial bernstein bezier shape functions is presented along with algorithms that allow the computation of the system matrices in optimal complexity o 1 per entry

analysis of numerical errors in large eddy simulation we consider the question of numerical errors in large eddy simulation it is often claimed that straightforward discretization and solution using centered methods of models for large eddy motion can simulate the motion of turbulent flows with complexity independent of the reynolds number and dependence only on the resolution delta of the eddies sought this report considers this question analytically is it possible to prove error estimates for discretizations of actually used large eddy models whose error constants depend only on delta but not re we consider the most common simplest and most mathematically tractable model and the most mathematically clear discretization in two cases we prove such an error estimate and try to explain why our technique of proof fails in the most general case our analysis aims to assume as little time regularity on the true solution as possible

a family of new high order ns α models arising from helicity correction in leray turbulence models abstract this report gives a reinterpretation of the ns model which leads to a family of high order ns deconvolution models with ns as the zeroth order case first we show that the navier stokes model arises by adding helicity correction to leray model higher order leray models have recently been proposed in w layton r lewandowski a high accuracy leray deconvolution model of turbulence and its limiting behavior anal appl 6 1 2008 w layton c manica m neda l rebholz numerical analysis and computational testing of a high accuracy leray deconvolution model of turbulence numer methods partial differential equations in press the so called leray deconvolution models that employ van cittert approximate deconvolution to decrease consistency error we use an analogous helicity correction idea to develop a family of higher order accurate ns type models the ns deconvolution models we prove several mathematical and physical properties for this new family of models and discuss the design of efficient algorithms for them

intermittency and regularity issues in 3d navier stokes turbulence two related open problems in the theory of 3d navier stokes turbulence are discussed in this paper the first is the phenomenon of intermittency in the dissipation field dissipation range intermittency was first discovered experimentally by batchelor and townsend over fifty years ago it is characterized by spatio temporal binary behaviour in which long quiescent periods in the velocity signal are interrupted by short active events during which there are violent fluctuations away from the average the second and related problem is whether solutions of the 3d navier stokes equations develop finite time singularities during these events this paper shows that leray s weak solutions of the three dimensional incompressible navier stokes equations can have a binary character in time the time axis is split into good and bad intervals on the good intervals solutions are bounded and regular whereas singularities are still possible within the bad intervals an estimate for the width of the latter is very small and decreases with increasing reynolds number it also decreases relative to the lengths of the good intervals as the reynolds number increases within these bad intervals lower bounds on the local energy dissipation rate and other quantities such as u t and u t are very large resulting in strong dynamics at sub kolmogorov scales intersections of bad intervals for n 1 are related to the potentially singular set in time it is also proved that the navier stokes equations are conditionally regular provided in a given bad interval the energy has a lower bound that is decaying exponentially in time

numerical analysis and computational comparisons of the ns alpha and ns omega regularizations we study stability accuracy and efficiency of algorithms for a new regularization of the nse the ns model in which the vorticity term u is averaged given by u t u q q v u f u 0 this is similar to the ns model in which the nonlinear term is x v x u but the small difference opens attractive algorithmic possibilities we give tests both confirming the predicted rates of convergence and exhibiting some shared limitations of both models the experiments also show the discrete ns simulation has greater accuracy tables 3 and 4 at less cost table 2 and requires significantly fewer degrees of freedom section 1 1 than a comparable ns simulation the experiments suggest consideration of adding grad div stabilization and higher accuracy ns deconvolution models as a next logical step in fact this combination produced accurate results see figs 10 and 11 on the coarsest mesh mesh 1 upon which all other methods models and variants tested failed

boundary effects on exact solutions of the lagrangian averaged navier stokes α equations in order to clarify the behavior of solutions of the lagrangian averaged navier stokes lans equations in the presence of solid walls we identify a variety of exact solutions of the full equations and their boundary layer approximations the solutions demonstrate that boundary conditions suggested for the lans equations in the literature 1 for a bounded domain do not apply in a semi infinite domain the convergence to solutions of the navier stokes equations as 0 is elucidated for infinite energy solutions in a semi infinite domain and non uniqueness of these solutions is discussed we also study the boundary layer approximation of lans equations denoted the prandtl equations and report solutions for turbulent jets and wakes our version of the prandtl equations includes an extra term necessary to conserve linear momentum and corrects an earlier result of cheskidov 2

on the convergence rate of grad div stabilized taylor hood to scott vogelius solutions for incompressible flow problems abstract it was recently proven in case et al 2010 2 that under mild restrictions grad div stabilized taylor hood solutions of navier stokes problems converge to the scott vogelius solution of that same problem however even though the analytical rate was only shown to be 1 2 where is the stabilization parameter the computational results suggest the rate may be improvable to 1 we prove herein the analytical rate is indeed 1 and extend the result to other incompressible flow problems including leray and mhd numerical results are given that verify the theory

stability and convergence of the numerical solution for the species equation of a model for pemfcs abstract many mathematical models for fuel cells are presented in the literature however few studies of numerical analysis of these models are found the novelty of this work is the proof of stability and convergence of the equation of chemical species providing greater reliability to the mathematical model in this work a mathematical model is proposed that calculates the flow and concentration of species for proton exchange membrane fuel cells using the finite element method for spatial discretization and the crank nicolson method for temporal discretization the model takes into account the losses overpotentials at the anode and at the cathode the numerical results confirm the proof of convergence which is demonstrated in this work finally a numerical result is shown for the fuel cell power relative to current density and the result has good relation with the experimental data

variational multiscale residual based turbulence modeling for large eddy simulation of incompressible flows we present an les type variational multiscale theory of turbulence our approach derives completely from the incompressible navier stokes equations and does not employ any ad hoc devices such as eddy viscosities we tested the formulation on forced homogeneous isotropic turbulence and turbulent channel flows in the calculations we employed linear quadratic and cubic nurbs a dispersion analysis of simple model problems revealed nurbs elements to be superior to classical finite elements in approximating advective and diffusive processes which play a significant role in turbulence computations the numerical results are very good and confirm the viability of the theoretical framework

partition functions loop measure and versions of sle we discuss the partition function view of the schramm loewner evolution after reviewing a number of known results in the framework of brownian loop measures and scaling rules for partition functions we give some speculation about multiply connected domains

the fixed irreducible bridge ensemble for self avoiding walks we define a new ensemble for self avoiding walks in the upper half plane the fixed irredicible bridge ensemble by considering self avoiding walks in the upper half plane up to their n th bridge height y n and scaling the walk by 1 y n to obtain a curve in the unit strip and then taking n rightarrow infty we then conjecture a relationship between this ensemble to sle 8 3 in the unit strip from 0 to a fixed point along the upper boundary of the strip integrated over the conjectured exit density of self avoiding walk spanning a strip in the scaling limit we conjecture that there exists a positive constant sigma such that n sigma y n converges in distribution to that of a stable random variable as n rightarrow infty then the conjectured relationship between the fixed irreducible bridge scaling limit and sle 8 3 can be described as follows if one takes a saw considered up to y n and scales by 1 y n and then weights the walk by y n to an appropriate power then in the limit n rightarrow infty one should obtain a curve from the scaling limit of the self avoiding walk spanning the unit strip in addition to a heuristic derivation we provide numerical evidence to support the conjecture and give estimates for the boundary scaling exponent

the random pseudo metric on a graph defined via the zero set of the gaussian free field on its metric graph we further investigate properties of the gaussian free field gff on the metric graph associated to a discrete weighted graph where the edges of the latter are replaced by continuous line segments of appropriate length that has been introduced by the first author on such a metric graph the gff is a random continuous function that generalises one dimensional brownian bridges so that one dimensional techniques can be used in the present paper we define and study the pseudo metric defined on the metric graph and therefore also on the discrete graph itself where the length of a path on the metric graph is defined to be the local time at level zero accumulated by the gaussian free field along this path we first derive a pathwise transformation that relates the gff on the metric graph with the reflected gff on the metric graph via the pseudo distance defined by the latter this is a generalisation of paul levy s result relating the local time at zero of brownian motion to the supremum of another brownian motion we also compute explicitly the distribution of certain functionals of this pseudo metric and of the gff in particular we point out that when the boundary consists of just two points the law of the pseudo distance between them depends solely on the resistance of the network between them we then discuss questions related to the scaling limit of this pseudo metric in the two dimensional case which should be the conformally invariant way to measure distances between cle 4 loops introduced and studied by the second author with wu and by sheffield watson and wu our explicit laws on metric graphs also lead to new conjectures for related functionals of the continuum gff on fairly general riemann surfaces

navier stokes equations in rotation form a robust multigrid solver for the velocity problem the topic of this paper is motivated by the navier stokes equations in rotation form linearization and application of an implicit time stepping scheme results in a linear stationary problem of oseen type in well known solution techniques for this problem such as the uzawa or schur complement method a subproblem consisting of a coupled nonsymmetric system of linear equations of diffusion reaction type must be solved to update the velocity vector field in this paper we analyze a standard finite element method for the discretization of this coupled system and we introduce and analyze a multigrid solver for the discrete problem both for the discretization method and the multigrid solver the question of robustness with respect to the amount of diffusion and variation in the convection field is addressed we prove stability results and discretization error bounds for the galerkin finite element method we present a convergence analysis of the multigrid method which shows the robustness of the solver results of numerical experiments are presented which illustrate the stability of the discretization method and the robustness of the multigrid solver