multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

a penalty method for the vorticity velocity formulation we present a new vorticity velocity formulation and implementation for the unsteady three dimensional navier stokes equations based on a penalty method it relies on an equivalence theorem that employs exact boundary conditions and the vorticity definition on the domain boundary this approach is particularly attractive for high order methods for which the often used influence matrix method fails to converge for t 0 the accuracy and the robustness of the new method is demonstrated in the context of several spectral element simulations of unsteady two and three dimensional internal and external flows in particular the flow past a finite span cylinder attached to end plates is studied in some detail in order to evaluate the effects of the aspect ratio on the formation length

velocity vorticity helicity formulation and a solver for the navier stokes equations for the three dimensional incompressible navier stokes equations we present a formulation featuring velocity vorticity and helical density as independent variables we find the helical density can be observed as a lagrange multiplier corresponding to the divergence free constraint on the vorticity variable similar to the pressure in the case of the incompressibility condition for velocity as one possible practical application of this new formulation we consider a time splitting numerical scheme based on an alternating procedure between vorticity helical density and velocity bernoulli pressure systems of equations results of numerical experiments include a comparison with some well known schemes based on pressure velocity formulation and illustrate the competitiveness on the new scheme as well as the soundness of the new formulation

on the helicity in 3d periodic navier stokes equations i the non statistical case we study the asymptotic behavior of the statistical solutions to the navier stokes equations using the normalization map 9 it is then applied to the study of mean energy mean dissipation rate of energy and mean helicity of the spatial periodic flows driven by potential body forces the statistical distribution of the asymptotic beltrami flows are also investigated we connect our mathematical analysis with the empirical theory of decaying turbulence with appropriate mathematically defined ensemble averages the kolmogorov universal features are shown to be transient in time we provide an estimate for the time interval in which those features may still be present our collaborator and friend basil nicolaenko passed away in september of 2007 after this work was completed honoring his contribution and friendship we dedicate this article to him

vorticity vector potential method for 3d viscous incompressible flows in time dependent curvilinear coordinates e and liu j comput phys 138 1997 57 82 put forward a finite difference method for 3d viscous incompressible flows in the vorticity vector potential formulation on non staggered grids in this paper we will extend this method to the case of flows in the presence of a deformable surface by use of two kinds of surface differential operators the implementation of boundary conditions on a plane is generalized to a curved smooth surface with given velocity distribution whether this be an inflow outflow interface or a curved wall to deal with the irregular and varying physical domain time dependent curvilinear coordinates are constructed and the corresponding tensor analysis is adopted in deriving the component form of the governing equations therefore the equations can be discretized and solved in a regular and fixed parametric domain numerical results are presented for a 3d lid driven cavity with a deforming surface and a 3d duct flow with a deforming boundary a new way to validate numerical simulations is proposed based on an expression for the rate of strain tensor on a deformable surface

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

on the finite element solution of the pure neumann problem this paper considers the finite element approximation and algebraic solution of the pure neumann problem our goal is to present a concise variational framework for the finite element solution of the neumann problem that focuses on the interplay between the algebraic and variational problems while many of the results that stem from our analysis are known by some experts they are seldom derived in a rigorous fashion and remain part of numerical folklore as a result this knowledge is not accessible or appreciated by many practitioners both novices and experts in one source our paper contributes a simple yet insightful link between the continuous and algebraic variational forms that will prove useful

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

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

finite element solution of the 3d compressible navier stokes equations by a velocity vorticity method this paper presents an alternative formulation to the primitive variable form of the navier stokes equations the approach is based on the use of the velocity and vorticity variables as a logical extension of the stream function vorticity method which is quite effective for two dimensional flows second order equations are obtained for the variables and discretization is through a weak galerkin finite element method followed by newton linearization the velocity and vorticity components are solved simultaneously by a direct solver the scheme is demonstrated for two dimensional and three dimensional incompressible and subsonic internal enclosed flow problems

a compact difference scheme for the navier stokes equations in vorticity velocity formulation this paper presents a new numerical method for solving the incompressible unsteady navier stokes equations in vorticity velocity formulation the method is applicable to spatial simulations of transitional and turbulent boundary layer flows it is based on a compact difference discretization of the streamwise and wall normal derivatives in cartesian coordinates a fourier collocation approach is used for the spanwise derivatives important new features of the numerical method are the use of nonequidistant differences in the wall normal direction the use of split compact differences in the streamwise direction a new fast iteration for a semi implicit time integration of the wall normal diffusion terms and an improvement of the buffer domain technique to prevent reflections of waves at the outflow boundary results of test calculations are presented to verify the improvements obtained by the use of these new techniques

inclusion of no slip boundary conditions in the meevc scheme abstract this work presents three methods for enforcing tangential velocity boundary conditions for the meevc scheme which was shown to be mass enstrophy energy and vorticity conserving scheme in the case of inviscid flow 1 while the normal velocity component can be strongly imposed in a div conforming formulation for the velocity field inclusion of the tangential velocity needs to be set through an appropriate choice of vorticity boundary conditions three methods to impose the tangential velocity boundary condition will be discussed the kinematic dirichlet formulation the kinematic neumann formulation and the dynamic neumann formulation the conservation properties of each of the resulting schemes are analyzed and numerical results are shown for the taylor green vortex and for the dipole collision test cases these confirm that kinematic neumann vorticity boundary conditions perform best

floer homology and homotopy planes we compute the floer homology of a certain family of homology 3spheres occuring as the boundary of a compactification of complex homotopy planes the floer complex turns out to be concentrated in even degrees and 2 periodic this is proved by using suitable bordisms to seifert fibred spaces

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

global well posedness of the velocity vorticity voigt model of the 3d navier stokes equations abstract the velocity vorticity formulation of the 3d navier stokes equations was recently found to give excellent numerical results for flows with strong rotation in this work we propose a new regularization of the 3d navier stokes equations which we call the 3d velocity vorticity voigt vvv model with a voigt regularization term added to momentum equation in velocity vorticity form but with no regularizing term in the vorticity equation we prove global well posedness and regularity of this model under periodic boundary conditions we prove convergence of the model s velocity and vorticity to their counterparts in the 3d navier stokes equations as the voigt modeling parameter tends to zero we prove that the curl of the model s velocity converges to the model vorticity which is solved for directly as the voigt modeling parameter tends to zero finally we provide a criterion for finite time blow up of the 3d navier stokes equations based on this inviscid regularization

a connection between scott vogelius and grad div stabilized taylor hood fe approximations of the navier stokes equations this article studies two methods for obtaining excellent mass conservation in finite element computations of the navier stokes equations using continuous velocity fields with a particular mesh construction the scott vogelius element pair has recently been shown to be inf sup stable and have optimal approximation properties while also providing pointwise mass conservation we present herein the first numerical tests of this element pair for the time dependent navier stokes equations we also prove that the limit of the grad div stabilized taylor hood solutions to the navier stokes problem converges to the scott vogelius solution as the stabilization parameter tends to infinity that is we provide theoretical justification that choosing the grad div parameter large does not destroy the solution numerical tests are provided which verify the theory and show how both scott vogelius and grad div stabilized taylor hood with large stabilization parameter elements can provide accurate results with excellent mass conservation for navier stokes approximations

a novel velocity vorticity formulation of the navier stokes equations with applications to boundary layer disturbance evolution a novel velocity vorticity formulation of the unsteady three dimensional navier stokes equations is presented the formulation is particularly suitable for simulating the evolution of three dimensional disturbances in boundary layers a key advantage is that there are only three governing equations for three primary dependent variables another advantage is that no wall boundary conditions are needed for the vorticity instead the conditions placed on the velocity are linked to the vorticity field through integral constraints based on the definition of vorticity numerical methods are presented in the context of application to the three dimensional boundary layer over a rotating disc the discretization scheme uses spectral expansions in the wall normal and azimuthal or spanwise directions and compact finite differences in the radial or streamwise direction the scheme is implemented so that the advantages of spectral convergence can be combined with the use of an efficient line iteration solution procedure the linearized form of the new velocity vorticity method is validated for the case of convective instabilities evolving over both rigid and compliant discs

special lagrangian submanifolds with isolated conical singularities v survey and applications this is the last in a series of five papers studying compact special lagrangian submanifolds sl m folds x in almost calabi yau m folds m with singularities x1 x n locally modelled on special lagrangian cones c1 c n in c m with isolated singularities at 0 readers are advised to begin with this paper we survey the major results of the previous four papers giving brief explanations of the proofs we apply the results to describe the boundary of a moduli space of compact nonsingular sl m folds n in m we prove the existence of special lagrangian connected sums n1 nk of sl mfolds n1 n k in m we also study sl 3 folds with t 2 cone singularities proving results related to ideas of the author on invariants of calabi yau 3 folds and the syz conjecture let x be a compact sl m fold with isolated conical singularities xi and cones ci for i 1 n the first paper studied the regularity of x near its singular points and the the second the moduli space of deformations of x the third and fourth papers construct desingularizations of x realizing x as a limit of a family of compact nonsingular sl m folds n t in m for small t 0 let li be an asymptotically conical sl m fold in cm asymptotic to ci at infinity we make n t by gluing tli into x at xi for i 1 n

natural vorticity boundary conditions on solid walls abstract we derive a new kind of boundary conditions for the vorticity equation with solid wall boundaries for fluid flow problems the formulation uses a dirichlet condition for the normal component of vorticity and neumann type conditions for the tangential components in a galerkin integral formulation the tangential condition is natural i e it is enforced by a right hand side functional and does not impose a boundary constraint on trial and test spaces the functional involves the pressure variable and we discuss several velocity vorticity formulations where the proposed condition is appropriate several numerical experiments are given that illustrate the validity of the new approach

effective vorticity velocity formulations for three dimensional incompressible viscous flows the vorticity velocity formulation for incompressible viscous flows is studied the main concern is effective numerical solution of the kinematic cauchy riemann equations for velocity especially when the divergence free condition of vorticity is violated due to numerical error the mathematical formulation of a differential and an integral approaches are revisited a novel projection of vorticity onto the divergence free space and its application to the two approaches are studied as an illustration of the differential approach the 3d lid driven cavity problem is solved the projection scheme is justified through numerical tests by comparing it to a new fully divergence free scheme numerical tests indicate that keeping vorticity divergence free is important for obtaining correct and accurate solutions the theory and methods apply to other divergence free fields of physical interest as well such as that in electro magnetic dynamics

navier stokes equations green s matrices vorticity direction and regularity up to the boundary abstract we consider the initial boundary value problem for the 3d navier stokes equations the physical domain is a bounded open set with a smooth boundary on which we assume a condition of free boundary type we show that if a suitable hypothesis on the vorticity direction is assumed then weak solutions are regular the main tool we use in the proof is an explicit representation of the velocity in terms of the vorticity by means of green s matrices