projection method ii godunov ryabenki analysis this is the second of a series of papers on the subject of projection methods for viscous incompressible flow calculations the purpose of the present paper is to explain why the accuracy of the velocity approximation is not affected by 1 the numerical boundary layers in the approximation of pressure and the intermediate velocity field and 2 the noncommutativity of the projection operator and the laplacian this is done by using a godunov ryabenki type of analysis in a rigorous fashion by doing so we hope to be able to convey the message that normal mode analysis is basically sufficient for understanding the stability and accuracy of a finite difference method for the navier stokes equation even in the presence of boundaries as an example we analyze the second order projection method based on pressure increment formulations used by van kan and bell colella and glaz the leading order error term in this case is of o dt and behaves as high frequency oscillations over the whole domain compared with the o dt 1 2 numerical boundary layers found in the second order kim moin method

a time splitting finite element stable approximation for the ericksen leslie equations in this paper we propose an unconditional energy stable time splitting finite element scheme for approximating the ericksen leslie equations governing the flow of nematic liquid crystals these equations are to be solved for a velocity vector field and a scalar pressure as well as a director vector field representing the direction along which the molecules of the liquid crystal are oriented the algorithm is designed at two levels first at the variational level the velocity pressure and director are computed separately but the director field has to be computed together with an auxiliary variable associated to the equilibrium equation for the director in order to deduce a priori energy estimates second at the algebraic level one can avoid computing such an auxiliary variable if this is approximated by a piecewise constant finite element space therefore these two steps give rise to a numerical algorithm that computes separately only the primary variables velocity pressure and director vecto

a splitting method for incompressible flows with variable density based on a pressure poisson equation a new fractional time stepping technique for solving incompressible flows with variable density is proposed the main feature of this method is that as opposed to other known algorithms the pressure is determined by just solving one poisson equation per time step which greatly reduces the computational cost the stability of the method is proved and the performance of the method is numerically illustrated

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

error estimates for an operator splitting method for navier stokes equations second order schemes we present in this paper an error analysis of a fractional step method for the approximation of the unsteady incompressible navier stokes equations under mild regularity assumptions on the continuous solution we obtain second order error estimates in the time step size both for velocity and pressure numerical results in agreement with the error analysis are also presented

convergence analysis and error estimates for a parallel algorithm for solving the navier stokes equations this paper is concerned with the analysis of the convergence and the derivation of error estimates for a parallel algorithm which is used to solve the incompressible navier stokes equations as usual the main idea is to split the main differential operator this allows to consider independently the two main difficulties namely nonlinearity and incompressibility the results justify the observed accuracy of related numerical results

the accuracy of the fractional step method we analyze the accuracy of the fractional step method of kim and moin j comput phys 59 1985 pp 308 323 for the incompressible navier stokes equations we show that the boundary conditions cannot be exactly satisfied in the projection step and that this limits the accuracy of the method we also show that the pressure in any projection method can be at best first order accurate our analysis is simpler and more direct than the previous analyses of this method we also show that there is no numerical boundary layer for velocity or pressure but there is one for the auxiliary pressure variable

convergence analysis of a projection semi implicit coupling scheme for fluid structure interaction problems in this paper we provide a convergence analysis of a projection semi implicit scheme for the simulation of fluid structure systems involving an incompressible viscous fluid the error analysis is performed on a fully discretized linear coupled problem a finite element approximation and a semi implicit time stepping strategy are respectively used for space and time discretization the fluid is described by the stokes equations the structure by the classical linear elastodynamics equations linearized elasticity plate or shell models and all changes of geometry are neglected we derive an error estimate in finite time and we prove that the time discretization error for the coupling scheme is at least sqrt delta t finally some numerical experiments that confirm the theoretical analysis are presented

on error estimates of the projection methods for the navier stokes equations second order schemes we present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible navier stokes equations the error analysis is accomplished by interpreting the respective projection schemes as second order time discretizations of a perturbed system which approximates the navier stokes equations numerical results in agreement with the error analysis are also presented

on error estimates of some higher order projection and penalty projection methods for navier stokes equations this paper is a continuation of our previous work 10 on projection methods we study first existing higher order projection schemes in the semidiscretized form for the navier stokes equations one error analysis suggests that the precision of these schemes is most likely plagued by the inconsistent neumann boundary condition satisfied by the pressure approximations we then propose a penalty projection scheme for which we obtain improved error estimates

staggered grids hybrid dual spectral element method for second order elliptic problems application to high order time splitting methods for navier stokes equations abstract this paper is a presentation of a mixed spectral element method based on staggered grids strategy used for the discretization of second order elliptic problems set under hybrid dual formulation a numerical analysis is performed proving the expected optimality and some numerical tests are described to confirm the predicted convergence rates the second part deals with a high order time splitting spectral scheme for the simulation of unsteady incompressible navier stokes flows we show how to use such a solver as a filtering tool to obtain divergence free discrete velocity some computational examples are reported which quantify the improvements induced by such a filter we have observed a strengthening of the stability and of the accuracy

a bi projection method for bingham type flows we propose and study a new numerical scheme to compute the isothermal and unsteady flow of an incompressible viscoplastic bingham medium the main difficulty for both theoretical and numerical approaches is due to the non differentiability of the plastic part of the stress tensor in regions where the rate of strain tensor vanishes this is handled by reformulating the definition of the plastic stress tensor in terms of a projection a new time scheme based on the classical incremental projection method for the newtonian navier stokes equations is proposed the plastic tensor is treated implicitly in the first sub step of the projection scheme and is computed by using a fixed point procedure a pseudo time relaxation is added into the bingham projection whose effect is to ensure a geometric convergence of the fixed point algorithm this is a key feature of the bi projection scheme which provides a fast and accurate computation of the plastic tensor stability and error analyses of the numerical scheme are provided the error induced by the pseudo time relaxation term is controlled by a prescribed numerical parameter so that a first order estimate of the time error is derived for the velocity field a second order cell centred finite volume scheme on staggered grids is applied for the spatial discretisation the scheme is assessed against previously published benchmark results for both newtonian and bingham flows in a two dimensional lid driven cavity for reynolds number equals 1000 moreover the proposed numerical scheme is able to reproduce the fundamental property of cessation in finite time of a viscoplastic medium in the absence of any energy source term in the equations for a fixed value 100 of the bingham number various numerical simulations for a range of reynolds numbers up to 200000 were performed with the bi projection scheme on a grid with 1024 2 mesh points the effect of this physical parameter on the flow behaviour is discussed

pressure stability in fractional step finite element methods for incompressible flows abstract the objective of this paper is to analyze the pressure stability of fractional step finite element methods for incompressible flows that use a pressure poisson equation for the classical first order projection method it is shown that there is a pressure control which depends on the time step size and therefore there is a lower bound for this time step for stability reasons the situation is much worse for a second order scheme in which part of the pressure gradient is kept in the momentum equation the pressure stability in this case is extremely weak to overcome these shortcomings a stabilized fractional step finite element method is also considered and its stability is analyzed some simple numerical examples are presented to support the theoretical results

a separable space with no remote points in the model obtained by adding w02 side by side sacks reals to a model of ch there is a separable nonpseudocompact space with no remote points to prove this it is also shown that in this model the countable box product of cantor sets contains a subspace of size w02 such that every uncountable subset has density co furthermore assuming the existence of a measurable cardinal k with 2k k a space x is produced with no isolated points but with remote points in vx x it is also shown that a pseudocompact space does not have remote points

an incremental pressure correction finite element method for the time dependent oldroyd flows abstract in this paper we present an incremental pressure correction finite element method for the time dependent oldroyd flows this method is a fully discrete projection method as we all know most projection methods have been studied without space discretization then the ensuing analysis may not extend to this case we also give the stability analysis and the optimal error analysis the analysis is based on a time discrete error and a spatial discrete error in order to show the effectiveness of the method we also present some numerical results the numerical results confirm our analysis and show clearly the stability and optimal convergence of the incremental pressure correction finite element method for the time dependent oldroyd flows

an efficient method for the incompressible navier stokes equations on irregular domains with no slip boundary conditions high order up to the boundary common efficient schemes for the incompressible navier stokes equations such as projection or fractional step methods have limited temporal accuracy as a result of matrix splitting errors or introduce errors near the domain boundaries which destroy uniform convergence to the solution in this paper we recast the incompressible constant density navier stokes equations with the velocity prescribed at the boundary as an equivalent system for the primary variables velocity and pressure equation for the pressure the key difference from the usual approaches occurs at the boundaries where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time this avoids the common difficulty of an apparently over determined poisson problem since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity the recast equations are ideal for numerical marching methods the new system can be discretized using a variety of methods including semi implicit treatments of viscosity and in principle to any desired order of accuracy in this work we illustrate the approach with a 2 d second order finite difference scheme on a cartesian grid and devise an algorithm to solve the equations on domains with curved non conforming boundaries including a case with a non trivial topology a circular obstruction inside the domain this algorithm achieves second order accuracy in the l norm for both the velocity and the pressure the scheme has a natural extension to 3 d

domain decomposition and splitting methods for mortar mixed finite element approximations to parabolic equations we introduce in this article a new domain decomposition algorithm for parabolic problems that combines mortar mixed finite element methods for the space discretization with operator splitting schemes for the time discretization the main advantage of this method is to be fully parallel the algorithm is proven to be unconditionally stable and a convergence result in mathcal o t h 1 2 is presented

on error estimates of the projection method for the time dependent natural convection problem in this article a projection method or fractional step method is proposed and analyzed for the time dependent natural convection problem in two dimensions based on this method the considered problem is decoupled into two subproblems one is for the velocity and temperature and the other is for the pressure firstly a linear elliptic equation is treated for the temperature then a linear elliptic equation is solved for the velocity finally a poisson equation is considered for the pressure stability is performed and the convergence results of the projection method are derived furthermore a modified projection scheme which leads to some improved error estimates is also proposed finally some numerical examples are presented to verify the established theoretical findings

a finite element penalty projection method for incompressible flows the penalty projection method for the solution of navier stokes equations may be viewed as a projection scheme where an augmentation term is added in the first stage namely the solution of the momentum balance equation to constrain the divergence of the predicted velocity field after a presentation of the scheme in the time semi discrete formulation then in fully discrete form for a finite element discretization we assess its behaviour against a set of benchmark tests including in particular prescribed velocity and open boundary conditions the results demonstrate that the augmentation always produces beneficial effects as soon as the augmentation parameter takes a significant value the projection method splitting error is reduced pressure boundary layers are suppressed and the loss of spatial convergence of the incremental projection scheme in case of open boundary conditions does not occur anymore for high values of the augmentation parameter the results of coupled solvers are recovered consequently in contrast with standard penalty methods there is no need for a dependence of the augmentation parameter with the time step and this latter can be kept to reasonable values to avoid to degrade too severely the conditioning of the linear operator associated to the velocity prediction step

error analysis of projection methods for non inf sup stable mixed finite elements the transient stokes problem abstract a modified chorin teman euler non incremental projection method and a modified euler incremental projection method for non inf sup stable mixed finite elements are analyzed the analysis of the classical euler non incremental and euler incremental methods are obtained as a particular case we first prove that the modified euler non incremental scheme has an inherent stabilization that allows the use of non inf sup stable mixed finite elements without any kind of extra added stabilization we show that it is also true in the case of the classical chorin temam method for the second scheme we study a stabilization that allows the use of equal order pairs of finite elements the relation of the methods with the so called pressure stabilized petrov galerkin method pspg is established the influence of the chosen initial approximations in the computed approximations to the pressure is analyzed numerical tests confirm the theoretical results

on the error estimates for the rotational pressure correction projection methods in this paper we study the rotational form of the pressure correction method that was proposed by timmermans minev and van de vosse we show that the rotational form of the algorithm provides better accuracy in terms of the h1 norm of the velocity and of the l2 norm of the pressure than the standard form

an energy stable scheme for incompressible navier stokes equations with periodically updated coefficient matrix abstract we present an energy stable scheme for simulating the incompressible navier stokes equations based on the generalized positive auxiliary variable gpav framework in the gpav reformulated system the original nonlinear term is replaced by a linear term plus a correction term where the correction term is put under control by an auxiliary variable the proposed scheme incorporates a pressure correction type strategy into the gpav procedure and it satisfies a discrete energy stability property the scheme entails the computation of two copies of the velocity and pressure within a time step by solving an individual de coupled linear equation for each of these field variables upon discretization the pressure linear system involves a constant coefficient matrix that can be pre computed while the velocity linear system involves a coefficient matrix that is updated periodically once every k 0 time steps in the current work where k 0 is a user specified integer the auxiliary variable being a scalar valued number is computed by a well defined explicit formula which guarantees the positivity of its computed values it is observed that the current method can produce accurate simulation results at large or fairly large time step sizes for the incompressible navier stokes equations the impact of the periodic coefficient matrix update on the overall cost of the method is observed to be small in typical numerical simulations several flow problems have been simulated to demonstrate the accuracy and performance of the method developed herein

a stabilized implicit fractional step method for the time dependent navier stokes equations using equal order pairs abstract a stabilized implicit fractional step method for numerical solutions of the time dependent navier stokes equations is presented in this paper the time advancement is decomposed into a sequence of two steps the first step has the structure of the linear elliptic problem the second step can be seen as the generalized stokes problem the two problems satisfy the full homogeneous dirichlet boundary conditions on the velocity on the other hand a locally stabilized term is added in the second step of the schemes it allows one to enhance the numerical stability and efficiency by using the equal order pairs convergence analysis and error estimates for the velocity and pressure of the schemes are established via the energy method some numerical experiments are also used to demonstrate the efficiency of this new method

projection method i convergence and numerical boundary layers this is the first of a series of papers on the subject of projection methods for viscous incompressible flow calculations the purpose of these papers is to provide a thorough understanding of the

stability and convergence of the higher projection method for the time dependent viscoelastic flow problem abstract in this paper the time discrete higher order projection method is proposed and analyzed for the time dependent viscoelastic flow problem our numerical method is based on the time iterative discrete schemes by the projection method the considered problem is decoupled into two linear subproblems one is for the velocity and the other is for the pressure unconditional stability of the numerical schemes is established convergence results for the velocity and pressure are also derived our main results of this paper are that the convergence analysis for the velocity is weakly second order and for the pressure is weakly first order finally some numerical examples are provided to confirm the performances of the developed numerical algorithms

fourier spectral projection method and nonlinear convergence analysis for navier stokes equations in this paper we investigate the convergence rate of the fourier spectral projection methods for the periodic problem of n dimensional navier stokes equations based on some alternative formulations of the navier stokes equations and the related projection methods the error estimates are carried out by a global nonlinear error analysis it simplifies the analysis relaxes the restriction on the time step size weakens the regularity requirements on the genuine solution and leads to some improved convergence results a new correction technique is proposed for improving the accuracy of the numerical pressure 2003 elsevier inc all rights reserved

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

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

the gauge uzawa finite element method part i the navier stokes equations the gauge uzawa fem is a new first order fully discrete projection method which combines advantages of both the gauge and uzawa methods within a variational framework a time step consists of a sequence of d 1 poisson problems d being the space dimension thereby avoiding both the incompressibility constraint as well as dealing with boundary tangential derivatives as in the gauge method this allows for a simple finite element discretization in space of any order in both two and three dimensions this first part introduces the method for the navier stokes equations of incompressible fluids and shows unconditional stability and error estimates for both velocity and pressure via a variational approach under realistic regularity assumptions several numerical experiments document performance of the gauge uzawa fem and compare it with other projection methods

penalty finite element method for stokes problem with nonlinear slip boundary conditions abstract the penalty finite element method for stokes problem with nonlinear slip boundary conditions based on the finite element subspace v h m h which satisfies the discrete inf sup condition is investigated in this paper since this class of nonlinear slip boundary conditions include the subdifferential property the weak variational formulation associated with stokes problem is variational inequality under some regularity assumptions we obtain the optimal h 1 and l 2 error estimates between u and u h and between u and u h e where the error orders are e h for h 1 error and e h 2 for l 2 error

analysis and approximation of a fractional laplacian based closure model for turbulent flows and its connection to richardson pair dispersion abstract we study a turbulence closure model in which the fractional laplacian of the velocity field represents the turbulence diffusivity we investigate the energy spectrum of the model by applying pao s energy transfer theory for the case 1 3 the corresponding power law of the energy spectrum in the inertial range has a correction exponent on the regular kolmogorov 5 3 scaling exponent for this case this model represents richardson s particle pair distance superdiffusion of a fully developed homogeneous turbulent flow as well as levy jumps that lead to the superdiffusion for other values of the power law of the energy spectrum is consistent with the regular kolmogorov 5 3 scaling exponent we also propose and study a modular time stepping algorithm in semi discretized form the algorithm is minimally intrusive to a given legacy code for solving navier stokes equations by decoupling the local part and nonlocal part of the equations for the unknowns we prove the algorithm is first order accurate and unconditionally stable we also derive error estimates for full discretizations of the model which in addition to the time stepping algorithm involves a finite element spatial discretization and a domain truncation approximation to the range of the fractional laplacian

factorization methods for the numerical approximation of navier stokes equations we investigate a general approach for the numerical approximation of incompressible navier stokes equations based on splitting the original problem into successive subproblems which are cheaper to solve the splitting is obtained through an algebraic approximate factorization of the matrix arising from space and time discretization of the original equations several schemes based on approximate factorization are investigated for some of these methods a formal analogy with well known time advancing schemes such as the projection chorin temam s can be pointed out features and limits of this analogy that was earlier introduced in b perot j comp phys 108 1993 51 8 are addressed other new methods can also be formulated starting from this approach in particular we introduce the so called yosida method which can be investigated in the framework of quasi compressibility schemes numerical results illustrating the different performances of the different methods addressed are presented for a couple of test cases

numerical approximation of incompressible navier stokes equations based on an auxiliary energy variable abstract we present a numerical scheme for approximating the incompressible navier stokes equations based on an auxiliary variable associated with the total system energy by introducing a dynamic equation for the auxiliary variable and reformulating the navier stokes equations into an equivalent system the scheme satisfies a discrete energy stability property in terms of a modified energy and it allows for an efficient solution algorithm and implementation within each time step the algorithm involves the computations of two pressure fields and two velocity fields by solving several de coupled individual linear algebraic systems with constant coefficient matrices together with the solution of a nonlinear algebraic equation about a scalar number involving a negligible cost a number of numerical experiments are presented to demonstrate the accuracy and the performance of the presented algorithm

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

on error estimates of projection methods for navier stokes equations first order schemes in this paper projection methods or fractional step methods are studied in the semi discretized form for the navier stokes equations in a two or three dimensional bounded domain error estimates for the velocity and the pressure of the classical projection scheme are established via the energy method a modified projection scheme which leads to improved error estimates is also proposed

hopf bifurcation of the unsteady regularized driven cavity flow abstract a numerical simulation of the unsteady incompressible flow in the unit cavity is performed by using a chebyshev tau approximation for the space variables the high accuracy of the spectral methods and the condensed distribution of the chebyshev collocation points near the boundary enable us to obtain reliable results for high reynolds numbers with a moderate number of modes it is found that the flow converges to a stationary state for reynolds numbers re up to 10 000 for reynolds numbers larger than a critical value 10 000

error analysis of fully discrete velocity correction methods for incompressible flows a fully discrete version of the velocity correction method proposed by guermond and shen 2003 for the time dependent navier stokes equations is introduced and analyzed it is shown that when accounting for space discretization additional consistency terms which vanish when space is not discretized have to be added to establish stability and optimal convergence error estimates are derived for both the standard version and the rotational version of the method these error estimates are consistent with those by guermond and shen 2003 as far as time discretiztion is concerned and are optimal in space for finite elements satisfying the inf sup condition

a gauge uzawa finite element method for the time dependent viscoelastic oldroyd flows abstract in this paper we present the gauge uzawa finite element method for the time dependent viscoelastic oldroyd flow this method is a fully discrete projection method whereas most projection methods have been studied without space discretization and the ensuing analysis may not extend to this case we also give the stability analysis and the optimal error analysis the analysis is based on an iterated time discrete system with which the error function is split into a temporal error and a spatial error in order to show the effect of the method we present some numerical results the numerical results confirm our analysis and show clearly the stability and optimal convergence of the gauge uzawa finite element method for the viscoelastic oldroyd flow

an overview of projection methods for incompressible flows a series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed in this paper these methods are often referred to in the literature as projection methods and can be classified into three classes namely the pressure correction methods the velocity correction methods and the consistent splitting methods for each class of schemes theoretical and numerical convergence results available in the literature are reviewed and open questions are discussed the essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners published by elsevier b v msc 65n35 65n22 65f05 35j05

optimal error estimate of the penalty finite element method for the time dependent navier stokes equations a fully discrete penalty finite element method is presented for the two dimensional time dependent navier stokes equations the time discretization of the penalty navier stokes equations is based on the backward euler scheme the spatial discretization of the time discretized penalty navier stokes equations is based on a finite element space pair x h m h which satisfies some approximate assumption an optimal error estimate of the numerical velocity and pressure is provided for the fully discrete penalty finite element method when the parameters e at and h are sufficiently small

optimal first order error estimates of a fully segregated scheme for the navier stokes equations abstract a first order linear fully discrete scheme is studied for the incompressible time dependent navier stokes equations in three dimensional domains this scheme is based on an incremental pressure projection method and decouples each component of the velocity and the pressure solving in each time step a linear convection diffusion problem for each component of the velocity and a poisson neumann problem for the pressure using an inf sup stable and continuous finite elements approach of order o h in space unconditional optimal error estimates of order o k h are deduced for velocity and pressure without imposing constraints on the mesh size h and the time step k finally some numerical results are performed to validate the theoretical analysis and also to compare the studied scheme with other current first order segregated schemes

a sequential regularization method for time dependent incompressible navier stokes equations the objective of the paper is to present a method called the sequential regularization method srm for the nonstationary incompressible navier stokes equations from the viewpoint of regularization of differential algebraic equations daes and to provide a way to apply a dae method to partial differential algebraic equations pdaes the srm is a functional iterative procedure it is proved that its convergence rate is o epsilon m where m is the number of the srm iterations and epsilon is the regularization parameter the discretization and implementation issues of the method are considered in particular a simple explicit difference scheme is analyzed and its stability is proved under the usual step size condition of explicit schemes it appears that the srm formulation is new in the navier stokes context unlike other regularizations or pseudocompressibility methods in the navier stokes context the regularization parameter epsilon in the srm need not be very small and the regularized problem in the sequence may be essentially nonstiff in time direction for any epsilon hence the stability condition is independent of epsilon even for explicit time discretization numerical experiments are given to verify our theoretical results

lie brackets and singular type of real hypersurfaces this note is the sequel of barraud and mazzilli math z 248 757 772 2004 where the regular type of a real hypersurface h in mathbb c n eventually endowed with an almost complex structure was characterized in terms of lie brackets of complex tangent vector fields on h this note extends these results to the singular type

real hypersurfaces orders of contact and applications a fundamental problem in the modem theory of several complex variables concerns the boundary behavior of the cauchy riemann equations suppose that d is an open domain in cn and that its boundary m is a smooth real submanifold of cn how does the geometry of m influence the function theory on d one approach to this question is the a neumann problem 6 there one studies the inhomogeneous cauchy riemann equations au a where a is a 0 q form with distribution coefficients on the closed domain d u m satisfying the necessary compatibility condition a a 0 the a neumann problem constructs a particular solution u which is orthogonal to the null space of the complex laplacian kohn 10a has solved this problem and has shown that good local regularity properties for u in terms of a follow from so called subelliptic estimates it is natural to seek geometric conditions on m that are necessary and sufficient for these estimates in this paper we will be concerned with a certain geometric condition on m that we feel is the right one for subellipticity although the motivation for the questions discussed here comes from partial differential equations the techniques come from algebraic geometry to see why we state theorems of kohn greiner and catlin concerning subellipticity kohn 1oa has discovered a sufficient condition for a subelliptic estimate near a fixed point p in m in case d is pseudoconvex and is contained in c2 this condition is equivalent to the finiteness of order of tangency of every complex analytic one dimensional manifold with m at p greiner 13 established the necessity of this condition in

error estimates for semi discrete gauge methods for the navier stokes equations the gauge formulation of the navier stokes equations for incompressible fluids is a new projection method it splits the velocity u a in terms of auxiliary nonphysical variables a and and replaces the momentum equation by a heat like equation for a and the incompressibility constraint by a diffusion equation for this paper studies two time discrete algorithms based on this splitting and the backward euler method for a with explicit boundary conditions and shows their stability and rates of convergence for both velocity and pressure the analyses are variational and hinge on realistic regularity requirements on the exact solution and data both neumann and dirichlet boundary conditions are in principle admissible for but a compatibility restriction for the latter is uncovered which limits its applicability

role of the lbb condition in weak spectral projection methods 405 this paper investigates the relevance of the ladyshenskaya babuska brezzi con dition in spectral projection methods we consider the stability and convergence properties for a first order nonincremental projection method and a second order incremental projection method both based on a spectral galerkin legendre spatial discretization we show that the convergence of both projection methods is controlled by the ability of the spectral framework to approximate correctly the steady stokes problem c 2001 elsevier science

a penalty finite element method based on the euler implicit explicit scheme for the time dependent navier stokes equations a fully discrete penalty finite element method is presented for the two dimensional time dependent navier stokes equations where the time discretization is based on the euler implicit explicit scheme with some implicit linear terms and an explicit nonlinear term and the finite element spatial discretization is based on the p 1b p 1 element pair which satisfies the discrete inf sup condition this method allows us to separate the computation of the velocity from the computation of the pressure with a larger time step size dt so that the numerical velocity u e h n and the pressure p e h n are easily computed an optimal error estimate of the numerical velocity and the pressure is provided for the fully discrete penalty finite element method when the penalty parameter e the time step size dt and the mesh size h satisfy the following stability conditions ec 1 1 dt k 1 1 and h 2 b 1 dt respectively for some positive constants c 1 k 1 and b 1 finally some numerical tests to confirm the theoretical results of the penalty finite element method are provided

regular type of real hyper surfaces in almost complex manifolds the regular type of a real hyper surface m in an almost complex manifold at some point p is the maximal contact order at p of m with germs of non singular pseudo holomorphic disks the main purpose of this paper is to give two intrinsic characterizations the type one in terms of lie brackets of a complex tangent vector field on m the other in terms of some kind of derivatives of the levi form

a gpav based unconditionally energy stable scheme for incompressible flows with outflow open boundaries abstract we present an unconditionally energy stable scheme for approximating the incompressible navier stokes equations on domains with outflow open boundaries the scheme combines the generalized positive auxiliary variable gpav approach and a rotational velocity correction type strategy and the adoption of the auxiliary variable simplifies the numerical treatment for the open boundary conditions the discrete energy stability of the proposed scheme has been proven irrespective of the time step sizes within each time step the scheme entails the computation of two velocity fields and two pressure fields by solving an individual de coupled helmholtz including poisson type equation with a constant pre computable coefficient matrix for each of these field variables the auxiliary variable being a scalar number is given by a well defined explicit formula within a time step which ensures the positivity of its computed values extensive numerical experiments with several flows involving outflow open boundaries in regimes where the backflow instability becomes severe have been presented to test the performance of the proposed method and to demonstrate its stability at large time step sizes

corrigendum to error estimates for an operator splitting method for navier stokes equations second order schemes j comput appl math 231 2 2009 696 704

velocity correction projection methods for incompressible flows we introduce and study a new class of projection methods namely the velocity correction methods in standard form and in rotational form for solving the unsteady incompressible navier stokes equations we show that the rotational form provides improved error estimates in terms of the h1 norm for the velocity and of the l2 norm for the pressure we also show that the class of fractional step methods introduced in s a orsag m israeli and m deville j sci comput 1 1986 pp 75 111 and k e karniadakis m israeli and s a orsag j comput phys 97 1991 pp 414 443 can be interpreted as the rotational form of our velocity correction methods thus to the best of our knowledge our results provide the first rigorous proof of stability and convergence of the methods in those papers we also emphasize that contrary to those of the above groups our formulations are set in the standard l2 setting and consequently they can be easily implemented by means of any variational approximation techniques in particular the finite element methods

stability of approximate projection methods on cell centered grids projection methods are a popular class of methods for solving the incompressible navier stokes equations if a cell centered grid is chosen in order to use high resolution methods for the advection terms performing the projection exactly is problematic an attractive alternative is to use an approximate projection in which the velocity is required to be only approximately discretely divergence free the stability of the cell centered approximate projection is highly sensitive to the method used to update the pressure and compute the pressure gradient this is demonstrated by analyzing a model problem and conducting numerical simulations of the navier stokes equations

shear rate projection schemes for non newtonian fluids abstract the operator splitting approach applied to the navier stokes equations gave rise to various numerical methods for the simulations of the dynamics of fluids the separate work of chorin and temam on this subject gave birth to the so called projection methods the most basic of those schemes the incremental and non incremental variant see guermond et al 2006 induce an artificial neumann boundary condition on the pressure the so called rotational incremental pressure correction scheme proposed by timmermans et al 1996 gives a consistent equation for the pressure in case of a newtonian fluids with homogeneous viscosity in this work we propose a family of projection methods for generalized newtonian fluids based on an extension of the rotational projection scheme called shear rate projections these methods produce consistent pressure when applied to generalized newtonian fluids accuracy of the methods will be illustrated using a manufactured solution numerical experiments for the flow past a cylinder with a carreau rheological model will also be presented

surface pressure poisson equation formulation of the primitive equations numerical schemes numerical methods for the primitive equations pes of oceanic flow are presented in this paper first a two dimensional poisson equation with a suitable boundary condition is derived to solve the surface pressure consequently we derive a new formulation of the pes in which the surface pressure poisson equation replaces the nonlocal incompressibility constraint which is known to be inconvenient to implement based on this new formulation backward euler and crank nicolson schemes are presented the marker and cell scheme which gives values of physical variables on staggered mesh grid points are chosen as spatial discretization the convergence analysis of the fully discretized scheme is established in detail the accuracy check for the scheme is also shown

time discretization of stochastic 2 d navier stokes equations with a penalty projection method a time discretization of the stochastic incompressible navier stokes problem by penalty method is analyzed some error estimates are derived combined and eventually arrive at a speed of convergence in probability of order 1 4 of the main algorithm for the pair of variables velocity and pressure also using the law of total probability we obtain the strong convergence of the scheme for both variables

stability and convergence of second order time discrete projection method for the linearized oldroyd model abstract in this paper we consider the second order time discrete projection method for the linearized oldroyd model based on the time iterative discrete scheme by the projection method the original problem is decoupled into two linear subproblems and each subproblem can be solved easily unconditional stability and the corresponding convergence results of the numerical solutions are derived our main results are that the convergence orders in time for the velocity in l 2 norm is second order and for the pressure in h 1 norm is first order finally some numerical examples are provided to verify the performances of the developed numerical method

projection method iii spatial discretization on the staggered grid in e liu siam j numer anal 1995 we studied convergence and the structure of the error for several projection methods when the spatial variable was kept continuous we call this the semi discrete case in this paper we address similar questions for the fully discrete case when the spatial variables are discretized using a staggered grid we prove that the numerical solution in velocity has full accuracy up to the boundary despite the fact that there are numerical boundary layers present in the semi discrete solutions