spurious transients of projection methods in microflow simulations the temporal behavior of projection methods for viscous incompressible lowreynolds number ows is adressed the methods considered result from algebraically splitting the linear system corresponding to each time step in such a way that the computation of velocity is segregated from that of pressure each method is characterized by two possibly equal approximate inverses b1 and b2 of the momentum equation velocity matrix plus a parameter which renders the method non incremental if 0 or incremental if 1 the classical rst order projection method together with more sophisticated methods perot s second order method yosida method pseudoexact factorization method and their incremental variants are put into the same algebraic form and their accuracy numerically tested splitting errors of rst

mathematical and numerical modeling of solute dynamics in blood flow and arterial walls the numerical modeling of solutes absorption processes by the arterial wall is of paramount interest for the understanding of the relationships between the local features of blood flow the nourishing of the inner arterial wall by the blood solutes and the pathologies that can appear when this process is for some reason perturbed in the present work two models for the solutes dynamics are investigated in the first model which is essentially based on the one introduced by rappitsch and perktold j biomech engrg 118 1996 pp 511 519 and rappitsch perktold and pernkopf internat j numer methods fluids 25 1997 pp 847 857 the navier stokes equations for an incompressible fluid describing the blood velocity and pressure fields are coupled with an advection diffusion equation for the solute concentration the wellposedness of this model is discussed the second model considers also the solutes dynamics inside the arterial wall described by a pure diffusion equation actually this is a heterogeneous model coupling different equations in different parts of the domain at hand its wellposedness is proven moreover in view of the numerical study an iterative finite element method by subdomains is proposed and its convergence properties are analyzed finally several numerical results comparing the different models in situations of physiologic interest are 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

on some nonunitary representations of the poincare group and their use for the construction of free quantum fields in this paper a class of nonunitary infinite dimensional hilbert space representations of a semidirect product is investigated the equivalence of this category with the category of finite dimensional representations of the stability subgroups is shown this theory is applied to the poincare group and to the construction of free quantum fields in an appendix a method is introduced for building an infinite family of finite dimensional indecomposable representations of the noncompact euclidean group in two dimensions such representations are used for carrying out the analysis of the massless fields

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

high order methods for the approximation of the incompressible navier stokes equations in a moving domain in this paper we address the numerical approximation of the incompressible navier stokes equations in a moving domain by the spectral element method and high order time integrators we present the arbitrary lagrangian eulerian ale formulation of the incompressible navier stokes equations and propose a numerical method based on the following kernels a lagrange basis associated with fekete points in the spectral element method context bdf time integrators an ale map of high degree and an algebraic linear solver in particular the high degree ale map is appropriate to deal with a computational domain whose boundary is described with curved elements finally we apply the proposed strategy to a test case c 2011 elsevier b v all rights reserved

a taxonomy and comparison of parallel block multi level preconditioners for the incompressible navier stokes equations in recent years considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible navier stokes equations based on preconditioned krylov methods these include physics based methods such as simple and purely algebraic preconditioners based on the approximation of the schur complement all these techniques can be represented as approximate block factorization abf type preconditioners the goal is to decompose the application of the preconditioner into simplified sub systems in which scalable multi level type solvers can be applied in this paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the navier stokes system first presented in a quarteroni f saleri a veneziani factorization methods for the numerical approximation of navier stokes equations computational methods in applied mechanical engineering 188 2000 505 526 this taxonomy illuminates the similarities and differences among these preconditioners and the central role played by efficient approximation of certain schur complement operators we then present a parallel computational study that examines the performance of these methods and compares them to an additive schwarz domain decomposition dd algorithm results are presented for two and three dimensional steady state problems for enclosed domains and inflow outflow systems on both structured and unstructured meshes the numerical experiments are performed using mpsalsa a stabilized finite element code

a modified nodal scheme for the time dependent incompressible navier stokes equations based on poisson equation for pressure a nodal numerical scheme is developed for the time dependent incompressible navier stokes equations derivation is based on local transverse integrations over finite size brick like cells that transform each partial differential equation to a set of ordinary differential equations odes solutions of these odes for the transverse averaged dependent variables are then utilized to develop the difference scheme the discrete variables are scalar velocities and pressure averaged over the faces of brick like cells in the x y t space cell interior variation of transverse averaged pressure in each spatial direction is quadratic cell interior variation of transverse averaged velocity in each spatial direction is a sum of a constant a linear and an exponential term due to the introduction of delayed coefficients the exponential functions are to be evaluated only once at each time step the semi implicit scheme has inherent upwinding results of applications to several test problems show that the scheme is very robust and leads to a second order error as expected in such coarse mesh schemes even relatively large size cells lead to small errors extension to three dimensions is straightforward

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

heat equation with dynamical boundary conditions of reactive diffusive type this paper deals with the heat equation posed in a bounded reg ular domain of r n n 2 coupled with a dynamical boundary condition of reactive diffusive type in particular we study the problem 8 utu 0 in 0 1 ut ku lu on 0 1 u 0 x u0 x on where u u t x t 0 x 2 x denotes the laplacian operator with respect to the space variable whiledenotes the laplace beltrami operator on is the outward normal to and k and l are given real constants l 0 well posedness is proved for data u0 2 h 1 such that u0 2 h 1 we also study higher regularity of the solution

algebraic fractional step schemes with spectral methods for the incompressible navier stokes equations the numerical investigation of a recent family of algebraic fractional step methods for the solution of the incompressible time dependent navier stokes equations is presented these methods are improved versions of the yosida method proposed in a quarteroni f saleri a veneziani factorization methods for the numerical approximation of navier stokes equations comput methods appl mech engrg 188 1 3 2000 505 526 a quarteroni f saleri a veneziani j math pures appl 9 78 5 1999 473 503 and one of them the yosida4 method is proposed in this paper for the first time they rely on an approximate lu block factorization of the matrix obtained after the discretization in time and space of the navier stokes system yielding a splitting in the velocity and pressure computation in this paper we analyze the numerical performances of these schemes when the space discretization is carried out with a spectral element method with the aim of investigating the impact of the splitting on the global accuracy of the computation

a note on the consistency and stability properties of yosida fractional step schemes for the unsteady stokes equations in this short note we give some improvements to the stability and consistency analysis of algebraic splitting methods for solving the incompressible fluid equations carried out in p gervasio siam j numer anal 46 2008 pp 1682 1703 in particular we generalize to any value of the approximation index p in mathbb n proposition 3 1 and lemma 3 2 of p gervasio siam j numer anal 46 2008 pp 1682 1703 on the consistency error associated with the algebraic splitting originally proved for p 0 1 2 moreover we remove an assumption of lemma 3 5 of p gervasio siam j numer anal 46 2008 pp 1682 1703

pressure correction algebraic splitting methods for the incompressible navier stokes equations in this paper we present a new family of methods for the effective numerical solution of the incompressible unsteady navier stokes equations these methods resort to an algebraic splitting of the discretized problem based on inexact lu block factorizations of the corresponding matrix following a quarteroni f saleri and a veneziani comput methods appl mech engrg 188 2000 pp 505 526 in particular we will start from inexact algebraic factorizations of algebraic chorin temam and yosida type and introduce a pressure correction step aimed at improving the time accuracy one of the schemes obtained in this way the algebraic chorin temam pressure correction method resembles a method previously introduced in the framework of differential projection schemes see l timmermans p minev and f v de vosse internat j numer methods fluids 22 1996 pp 673 688 a prohl projection and quasi compressibility methods for solving the incompressible navier stokes equations teubner stuttgart 1997 the stability and the dependence of splitting error on the time step of the new methods is investigated and tested on several numerical cases

representations of hermitian kernels by means of krein spaces ii invariant kernels in this paper we study hermitian kernels invariant under the action of a semigroup with involution we characterize those hermitian kernels that realize the given action by bounded operators on a krein space this is motivated by the gns representation of algebras associated to hermitian functionals the dilation theory of hermitian maps on c algebras as well as others we explain the key role played by the technique of induced krein spaces and a lifting property associated to them

on the shear thinning and viscoelastic effects of blood flow under various flow rates the aim of this paper is to describe and discuss the results of numerical comparative study performed in order to demonstrate and quantify some of the most relevant non newtonian characteristics of blood flow in medium sized blood vessels namely its shear thinning and viscoelastic behavior the models studied in this work are the classical newtonian and oldroyd b models as well as their generalized shear thinning modifications numerical tests are performed on three dimensional geometries namely an idealized axisymmetric stenosis and a realistic stenosed carotid bifurcation reconstructed from medical images the numerical solution of the system of governing equations is obtained by a finite volume method on a structured grid model sensitivity tests are achieved with respect to the characteristic flow rate to evaluate its impact on the observed non newtonian effects

convergence analysis of high order algebraic fractional step schemes for time dependent stokes equations in this paper we analyze the family of yosida algebraic fractional step schemes proposed in a quarteroni f saleri and a veneziani comput methods appl mech engrg 188 2000 pp 505 526 f saleri and a veneziani siam j numer anal 43 2005 pp 174 194 and p gervasio f saleri and a veneziani j comput phys 214 2006 pp 347 365 when applied to time dependent stokes equations under suitable regularity assumptions on the data splitting error estimates both for velocity and pressure are established in particular we analyze the first three methods of this family providing respectively convergence of the fractional step solution towards the numerical solution achieved without any operator splitting of orders 3 2 5 2 7 2 for the velocity and 1 2 3 for the pressure moreover a general way to set up higher order schemes is proposed the present analysis is carried out when spectral element methods are employed for space discretization

a spectral multidomain method for the numerical simulation of turbulent flows the primitive variable formulation of the unsteady incompressible navier stokes equations in three space dimensions is discretized with a combined fourier legendre spectral method a semi implicit pressure correction scheme is applied to decouple the velocity from the pressure the arising elliptic scalar problems are first diagonalized in the periodic fourier direction and then solved by a multidomain legendre collocation method in the two remaining space coordinates in particular both an iterative and a direct version of the so called projection decomposition method pdm are introduced to separate the equations for the internal nodes from the ones governing the interface unknowns the pdm method first introduced by v agoshkov and e ovtchinnikov and later applied to spectral methods by p gervasio e ovtchinnikov and a quarteroni is a domain decomposition technique for elliptic boundary value problems which is based on a galerkin approximation of the steklov poincare equation for the unknown variables associated to the grid points lying on the interface between subdomains after having shown the exponential convergence of the proposed discretization technique some issues on the efficient implementation of the method are given finally as an illustration of the potentialities of the algorithm for the numerical simulation of turbulent flows the results of a direct numerical simulation dns of a fully turbulent plane channel flow are presented

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

approximate factorization as a high order splitting for the implicit incompressible flow equations abstract we apply the method of approximate factorization to derive a second order accurate splitting of the incompressible flow equations stokes or navier stokes equations this is novel because the method of approximate factorization was believed inapplicable to this type of equation system we demonstrate the resulting splitting on a second order crank nicolson discretization and point out its intimate relationship to some existing second order accurate splitting projection methods further we use the approximate factorization method to derive entirely new splittings we first develop a new generalized second order accurate splitting which may be specialized to a variety of applications we indicate its application to finite elements checker board free cell centered discretizations and ocean modeling we then generalized the original splitting to an arbitrarily high order scheme

index theory and second quantization of boundary value problems the second quantization functor associates to each skew symmetric operator in one particle space a derivation 6 of the algebra which is based on the given commutation relations in this paper we characterize the spatial theory of 6 in the fock representation by an index which generalizes the one studied earlier by powers and arveson in connection with the spatial cohomological obstruction for semigroups of endomorphisms of b x it is well known that such semigroups corresponding to one sided boundary conditions are generated by derivations but derivations associated to iwo sided boundary conditions do not generate semigroups we show that the known index theory for semigroups generalizes to the quantization of arbitrary boundary conditions in one particle space our quantized twosided abstract boundary conditions dictate representations in a certain indetinite inner product space a krein space and our index is an isomorphism invariant for representation theory in krein spaces the representations are not unitarizable i e are not equivalent to hermitian representations in hilbert space

some current cfd issues relevant to the incompressible navier stokes equations abstract the goals of this paper are to carefully define a particular class of well set incompressible navier stokes problems in the continuum partial differential equation pde setting and to discuss some relevant and sometimes poorly understood issues related to these well posed pde problems both in the continuum world and in its computer counterpart emphasized are three common formulations used to generate computer codes the primitive variable velocity pressure formulation and two derived variable formulations pressure poisson equation and vorticity transport equation

a finite element formulation for the stokes problem allowing equal velocity pressure interpolation abstract in this paper we study a variational formulation of the stokes problem that accommodates the use of equal velocity pressure finite element interpolations the motivation of this method relies on the analysis of a class of fractional step methods for the navier stokes equations for which it is known that equal interpolations yield good numerical results the reason for this turns out to be the difference between two discrete laplacian operators computed in a different manner the formulation of the stokes problem considered here aims to reproduce this effect from the analysis of the finite element approximation of the problem we obtain stability and optimal error estimates using velocity pressure interpolations satisfying a compatibility condition much weaker than the inf sup condition of the standard formulation in particular this condition is fulfilled by the most common equal order interpolations

indefinite harmonic forms and gauge theory indecomposable representations have been extensively used in the construction of conformal and de sitter gauge theories it is thus noteworthy that certain unitary highest weight representations have been given a geometric realization as the unitary quotient of an indecomposable representation using indefinite harmonic forms rsw we apply this construction tosu 2 2 and the de sitter group the relation is established between these representations and the massless positive energy representations ofsu 2 2 obtained in the physics literature we investigate the extent to which this construction allows twistors to be viewed as a gauge theory ofsu 2 2 for the de sitter group on which the gauge theory of singletons is based we find that this construction is not directly applicable

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

on gns representations on inner product spaces a generalization of the gns construction to hermitian linear functionals w defined on a unital algebra is considered along these lines a continuity condition h upon w is introduced such that h proves to be necessary and sufficient for the existence of a j representation on a krein space the property whether or not the gram operator j leaves the common and invariant domain of the representation invariant is characterized as well by properties of the functional w as by those of furthermore the interesting class of positively dominated functionals is introduced and investigated some applications to tensor algebras are finally discussed

unitarizable massless fields of higher order and higher spin explicit fock spaces for massless multispinors and tensors satisfying n 0 are constructed it is found that they carry indecomposable representations which allow gupta bleuler type quantization for symmetric tensors and odd n there is a unitary space of physical states

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

approximation of the incompressible navier stokes equations using orthogonal subscale stabilization and pressure segregation on anisotropic finite element meshes abstract this paper describes a finite element model to solve the incompressible navier stokes equations based on the stabilization with orthogonal subscales and a pressure segregation the former consists of adding a least square form of the component orthogonal to the finite element space of the convective and pressure gradient terms this allows to deal with convection dominated flows and to use equal velocity pressure interpolation the pressure segregation is inspired in fractional step schemes although the converged solution corresponds to that of a monolithic time integration likewise we put special emphasis on the use of anisotropic grids in particular we describe some possible choices for the calculation of the element length that appears in the stabilization parameters and check their behavior in two classical numerical examples the preconditioning strategy used to solve the resulting algebraic system for very anisotropic meshes is also briefly described

some remarks on the gupta bleuler triplet indecomposable representations of the poincare group associated to infrared singular field theory models are discussed in the framework of the general theory of the gupta bleuler triplet formulated by araki it is shown that the definition of maximal hilbert space structures related to the infrared properties of the states of the models can be exploited to construct representation spaces for the gupta bleuler triplet the examples of the two dimensional massless scalar field and of the electromagnetic field in the landau gauge are discussed in particular in the first example the relation between the gupta bleuler triplet and the algebraic treatment of the massless scalar field is investigated in the case of electromagnetism the structure of the representation of the poincare group in the landau gauge is clarified the explicit form of the corresponding gupta bleuler triplet for the one particle space of the electromagnetic field is exhibited

aladins an algebraic splitting time adaptive solver for the incompressible navier stokes equations we address a time adaptive solver specifically devised for the incompressible navier stokes ins equations one of the challenging issues in this context is the identification of a reliable a posteriori error estimator typical strategies are based on the combination of the solutions computed either with two different time steps or two schemes with different accuracy in this paper we move from the pressure correction algebraic factorizations formerly proposed by saleri veneziani 2005 these schemes feature an intrinsic hierarchical nature such that an accurate solution for the pressure is obtained by computing intermediate low order guesses the difference between the two estimates provide a natural a posteriori estimator after introducing the incremental formulation of the pressure correction schemes we address the properties of this approach including extensive implementation details numerical results presented refer to 2d and 3d unstructured problems with a particular emphasis on cardiovascular problems which are expected to largely benefit from time adaptive solvers in memory of f saleri 1965 2007

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

stability of solutions of quasilinear parabolic equations abstract we bound the difference between solutions u and v of u t a u div x f h and v t b v div x g k with initial data and respectively by u t v t l p e a e t l r n 2 p b t a b x f x g f u g u h k p e p here all functions a f and h are smooth and bounded and may depend on u x r n and t the functions a and h may in addition depend on u identical assumptions hold for the functions that determine the solutions v furthermore e r n is assumed to be a bounded set and p and p are fractions that depend on n and p the diffusion coefficients a and b are assumed to be strictly positive and the initial data are smooth

stability of parabolic problems with nonlinear wentzell boundary conditions abstract of concern is the nonlinear uniformly parabolic problem u t div a u u 0 x f x u t a u x u q lb u 0 for x r n and t 0 the last equation holds on the boundary here a a i j x i j is a real hermitian uniformly positive definite n n matrix c with 0 r r q 0 lb is the laplace beltrami operator on the boundary and a u is the conormal derivative of u with respect to a and everything is sufficiently regular the solution of this wellposed problem depends continuously on the ingredients of the problem namely a q f this is shown using semigroup methods in g m coclite a favini g r goldstein j a goldstein s romanelli continuous dependence on the boundary parameters for the wentzell laplacian semigroup forum 77 1 2008 101 108 here we prove explicit stability estimates of the solution u with respect to the coefficients a q and the initial condition f moreover we cover the singular case of a problem with q 0 which is approximated by problems with positive q

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

on some pressure segregation methods of fractional step type for the finite element approximation of incompressible flow problems abstract in this paper we treat several aspects related to time integration methods for the incompressible navier stokes equations that allow to uncouple the calculation of the velocities and the pressure the first family of schemes consists of classical fractional step methods of which we discuss several possibilities for the pressure extrapolation and the time integration of first and second order the second family consists of schemes based on an explicit treatment of the pressure in the momentum equation followed by a poisson equation for the pressure it turns out that this staggered treatment of the velocity and the pressure is stable finally we present predictor corrector methods based on the above schemes that aim to converge to the solution of the monolithic time integration method apart from presenting these schemes and check its numerical performance we also present a complete set of stability results for the fractional step methods that are independent of the space stability of the velocity pressure interpolation that is of the classical inf sup condition

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 massively parallel fractional step solver for incompressible flows this paper presents a parallel implementation of fractional solvers for the incompressible navier stokes equations using an algebraic approach under this framework predictor corrector and incremental projection schemes are seen as sub classes of the same class making apparent its differences and similarities an additional advantage of this approach is to set a common basis for a parallelization strategy which can be extended to other split techniques or to compressible flows the predictor corrector scheme consists in solving the momentum equation and a modified continuity equation namely a simple iteration for the pressure schur complement consecutively in order to converge to the monolithic solution thus avoiding fractional errors on the other hand the incremental projection scheme solves only one iteration of the predictor corrector per time step and adds a correction equation to fulfill the mass conservation as shown in the paper these two schemes are very well suited for massively parallel implementation in fact when compared with monolithic schemes simpler solvers and preconditioners can be used to solve the non symmetric momentum equations gmres bi cgstab and to solve the symmetric continuity equation cg deflated cg this gives good speedup properties of the algorithm the implementation of the mesh partitioning technique is presented as well as the parallel performances and speedups for thousands of processors

an analysis of the fractional step method the fractional step method for solving the incompressible navier stokes equations in primitive variables is analyzed as a block lu decomposition in this formulation the issues involving boundary conditions for the intermediate velocity variables and the pressure are clearly resolved in addition it is shown that poor temporal accuracy first order is not due to boundary conditions but due to the method itself a generalized block lu decomposition that overcomes this difficulty is presented allowing arbitrarily high temporal order of accuracy the generalized decomposition is shown to be useful for a wide range of problems including steady problems technical issues such as stability and the appropriate pressure update scheme are also addressed numerical simulations of the steady incompressible navier stokes equations in a square domain confirm the theoretical results 18 refs 4 figs

modified homotopy perturbation method for solving the stokes equations a new approach is proposed for the stationary stokes equations based on the homotopy perturbation method some iterative algorithms are constructed and four kinds of perturbation cases are considered respectively numerical experiments show that these algorithms are simple and effective

time step restrictions using semi explicit methods for the incompressible navier stokes equations time step restrictions using semi explicit methods for the incompressible navier stokes equations

singular unitary representations and indefinite harmonic theory abstract square integrable harmonic spaces are defined and studied in a homogeneous indefinite metric setting in the process dolbeault cohomologies are unitarized and singlar unitary representations are obtained and studied

analysis of algebraic chorin temam splitting for incompressible nse and comparison to yosida methods abstract algebraic splitting methods are a common approach to solving the saddle point linear systems that arise at each time step of an incompressible flow simulation there are two main classes of these methods those of yosida type and those of algebraic chorin temam act type with the yosida methods being predominantly used in practice we show herein through new analysis and extensive numerical testing that act methods that include the viscous term stiffness matrix in the modified a block are unconditionally stable and can be superior to yosida methods in a range of problems particular situations where the act type solvers are advantageous include problems where numerical stability is a concern as well as problems where strong enforcement of the divergence constraint is important

proper general decomposition pgd for the resolution of navier stokes equations in this work the pgd method will be considered for solving some problems of fluid mechanics by looking for the solution as a sum of tensor product functions in the first stage the equations of stokes and burgers will be solved then we will solve the navier stokes problem in the case of the lid driven cavity for different reynolds numbers re 100 1000 and 10 000 finally the pgd method will be compared to the standard resolution technique both in terms of cpu time and accuracy

splitting methods based on algebraic factorization for fluid structure interaction we discuss in this paper the numerical approximation of fluid structure interaction fsi problems dealing with strong added mass effect we propose new semi implicit algorithms based on inexact block lu factorization of the linear system obtained after the space time discretization and linearization of the fsi problem as a result the fluid velocity is computed separately from the coupled pressure structure velocity system at each iteration reducing the computational cost we investigate explicit implicit decomposition through algebraic splitting techniques originally designed for the fsi problem this approach leads to two different families of methods which extend to fsi the algebraic pressure correction method and the yosida method two schemes that were previously adopted for pure fluid problems furthermore we have considered the inexact factorization of the fluid structure system as a preconditioner the numerical properties of these methods have been tested on a model problem representing a blood vessel system

non unitary representations of nilpotent groups i cohomologies extensions and neutral cocycles let be a finite dimensional representation of a connected nilpotent group g and u be a unitary representation of g we investigate the structure of the extensions of by u and correspondingly the group h1 u of 1 cohomologies a spectral criterion of triviality of h1 u is proved and systematically used in the study of various types of decomposition of the extensions we consider a special type of u cocycles neutral cocycles which play a crucial role in the theory of j unitary representations of groups on pontryagin k spaces

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

a fast schur complement method for the spectral element discretization of the incompressible navier stokes equations the weak formulation of the incompressible navier stokes equations in three space dimensions is discretized with spectral element approximations and gauss lobatto legendre quadratures the uzawa algorithm is applied to decouple the velocities from the pressure the equation that results for the pressure is solved by an iterative method within each pressure iteration a helmholtz operator has to be inverted this can efficiently be done by separating the equations for the interior nodes from the equations at the interfaces according to the schur method fast diagonalization techniques are applied to the interior variables of the spectral elements several ways to deal with the resulting interface problem are discussed finally a comparison is made with a more classical method c 1995 academic press inc

nonautonomous semilinear second order evolution equations with generalized wentzell boundary conditions abstract of concern are the initial boundary value problems for nonautonomous semilinear second order evolution equations with generalized wentzell boundary conditions we succeed in establishing a global wellposedness theorem in a classical sense for these problems via a specifically designed operator theoretic approach moreover we obtain sharp estimations for the evolution of the solution along the characteristic curves which enable us to derive the uniform exponential decay of the associated energies the results obtained in this paper are still new even for the autonomous case

on discrete projection methods for the incompressible navier stokes equations an algorithmical approach abstract we derive a general class of iteration schemes for the incompressible navier stokes equations which contains fully coupled solution techniques as well as operator splitting projection methods we combine the advantages of both namely accuracy stability and efficiency and obtain a special form of discrete projection schemes in combination with a nonlinear iteration of quasi newton type one may use these schemes analogously to the well known pure projection schemes e g of chorin and van kan or apply them as preconditioners in a defect correction approach to obtain the fully coupled galerkin solution the corresponding complexity analysis shows that in combination with certain nonconforming finite element discretizations a huge gain in efficiency may be obtained particularly in the highly nonstationary case our theoretical results are confirmed by comparative numerical tests for both types of schemes it turns out that the appropriate time steps for the pure projection approach are only moderately smaller than those for the fully implicitly coupled schemes but that the work to obtain comparative results with the discrete projection methods as solvers is much lower an interesting observation is that in the case of higher reynolds numbers no significant pressure boundary layers occur even for the pure projection schemes these considerations and first numerical tests in 3d give hope to obtain a powerful cfd tool

error analysis of pressure correction schemes for the time dependent stokes equations with open boundary conditions the incompressible stokes equations with prescribed normal stress open boundary conditions on part of the boundary are considered it is shown that the standard pressure correction method is not suitable for approximating the stokes equations with open boundary conditions whereas the rotational pressure correction method yields reasonably good error estimates these results appear to be the first ever published for splitting schemes with open boundary conditions numerical results in agreement with the error estimates are presented

a parallel block multi level preconditioner for the 3d incompressible navier stokes equations the development of robust and efficient algorithms for both steady state simulations and fully implicit time integration of the navier stokes equations is an active research topic to be effective the linear subproblems generated by these methods require solution techniques that exhibit robust and rapid convergence in particular they should be insensitive to parameters in the problem such as mesh size time step and reynolds number in this context we explore a parallel preconditioner based on a block factorization of the coefficient matrix generated in an oseen nonlinear iteration for the primitive variable formulation of the system the key to this preconditioner is the approximation of a certain schur complement operator by a technique first proposed by kay loghin and wathen siam j sci comput 2002 and silvester elman kay and wathen j comput appl math 128 2001 261 the resulting operator entails subsidiary computations solutions of pressure poisson and convection diffusion subproblems that are similar to those required for decoupled solution methods however in this case these solutions are applied as preconditioners to the coupled oseen system one important aspect of this approach is that the convection diffusion and poisson subproblems are significantly easier to solve than the entire coupled system and a solver can be built using tools developed for the subproblems in this paper we apply smoothed aggregation algebraic multigrid to both subproblems previous work has focused on demonstrating the optimality of these preconditioners with respect to mesh size on serial two dimensional steady state computations employing geometric multi grid methods we focus on extending these methods to large scale parallel three dimensional transient and steady state simulations employing algebraic multigrid amg methods our results display nearly optimal convergence rates for steady state solutions as well as for transient solutions over a wide range of cfl numbers on the two dimensional and three dimensional lid driven cavity problem

proper generalized decomposition method for incompressible navier stokes equations with a spectral discretization proper generalized decomposition pgd is a method which consists in looking for the solution to a problem in a separate form this approach has been increasingly used over the last few years to solve mathematical problems the originality of this work consists in the association of pgd with a spectral collocation method to solve transfer equations as well as navier stokes equations in the first stage the pgd method and its association with spectral discretization is detailed this approach was tested for several problems the poisson equation the darcy problem navier stokes equations the taylor green problem and the lid driven cavity in the navier stokes problems the coupling between velocity and pressure was performed using a fractional step scheme and a p n p n 2 discretization for all problems considered the results from pgd simulations were compared with those obtained by a standard solver and or with the results found in the literature the simulations performed showed that pgd is as accurate as standard solvers pgd preserves the spectral behavior of the errors in velocity and pressure when the time step or the space step decreases moreover for a given number of discretization nodes pgd is faster than the standard solvers

a discrete projection method for incompressible viscous flow with coriolis force the paper presents a new discrete projection method for the numerical solution of the navier stokes equations with coriolis force term on an algebraic level we interpret one time step of the projection method as an incomplete factorization of the linearized navier stokes system and as the iteration of an uzawa type algorithm with special preconditioning for the pressure block this enables us to modify the well known projection method in a way to account for possibly dominating coriolis terms we consider a special multigrid method for solving the velocity subproblems and a modified projection pressure correction step results of numerical tests are presented for a model problem as well as for 3d flow simulations in stirred tank reactors

unconditional stability and long term behavior of transient algorithms for the incompressible navier stokes and euler equations abstract this paper examines the long term behavior dissipativity and unconditional nonlinear stability properties of time integration algorithms for the incompressible navier stokes equations including both direct schemes and fractional step projection methods the algorithms are termed nonlinear unconditionally stable if in the absence of a forcing term but for arbitrary initial conditions the computed kinetic energy decreases for arbitrary step sizes i e l 2 stability such an a priori stability estimate is a characteristic feature of the navier stokes system similarly the algorithms are said to exhibit asymptotic t independent long term dissipative behavior if as the continuum flow generated by the navier stokes equations the algorithmic flow also exhibits an absorbing set and a maximal attractor both direct and fractional step methods are described which inherit these fundamental properties of the navier stokes system for any time step size moreover a subclass of algorithms which retain these strong notions of nonlinear stability and long term dissipative behavior is identified which in addition has the remarkable property of being linear within the time step the key implication is that no iterations are required to compute the solution within a time step a specific member of this class is shown to possess the same absorbing set as the continuum navier stokes system a second order accurate unconditionally stable method is also identified which retains the property of linearity within a time step numerical analysis and computational aspects involved in the implementation of these methods are addressed in detail and illustrated in representative numerical simulations

schur complements and matrix inequalities in the lowner ordering abstract the purpose of this paper is to present a matrix inequality on the kronecker product that unifies the proofs of many existing matrix inequalities in the lowner partial ordering on the sum ordinary product and hadamard schur product schur complements serve as the basic tool

a fractional step method for steady state flow abstract a fractional step method for solving the steady state incompressible navier stokes equations is presented the proposed iterative method uses an estimate for the optimal under relaxation coefficient at each iteration this estimate uses prior information that is already available in the iterative method so the calculation has negligible cost numerical tests show that the convergence rate of the fractional step method using this optimal relaxation parameter is nearly independent of the mesh size resulting in significant cost savings for problems with fine meshes the ideas demonstrated in this paper are general and can be used to improve many existing iterative methods for steady flows as well as methods for the unsteady navier stokes equations and methods for other saddle point problems

on the use of higher order projection methods for incompressible turbulent flow an important issue in the development of higher order methods for incompressible flow is how they perform when the flow is turbulent a useful diagnostic of a method for turbulent flow is the minimum resolution that is required to adequately resolve the turbulent energy cascade at a given reynolds number in this paper we present careful numerical experiments to assess the utility of higher order numerical methods based on this metric we first introduce a numerical method for the incompressible navier stokes equations based on fourth order discretizations in both space and time the method is based on an auxiliary variable formulation and combines fourth order finite volume differencing with a semi implicit spectral deferred correction temporal integration scheme we also introduce for comparison purposes versions based on second order spatial and or temporal discretizations we demonstrate that for smooth problems each of the methods exhibits the expected order of convergence in time and space we

stability estimates for parabolic problems with wentzell boundary conditions abstract of concern is the uniformly parabolic problem u t div a u u 0 x f x u t a u u q lb u 0 for x r n and t 0 here a a i j x i j is a real hermitian uniformly positive definite n n matrix c with 0 q 0 and a u is the conormal derivative of u with respect to a and everything is sufficiently regular the solution of this well posed problem depends continuously on the ingredients of the problem namely a q f this is shown using semigroup methods in g m coclite a favini g r goldstein j a goldstein s romanelli continuous dependence on the boundary parameters for the wentzell laplacian semigroup forum in press more precisely if we have a sequence of such problems with solutions u n and if a n a n etc in a suitable sense then u n u the solution of the limiting problem the abstract analysis associated with operator semigroup theory gives this conclusion but no rate of convergence determining how fast the convergence of the solutions is requires detailed estimates such estimates are provided in this paper

robin robin preconditioned krylov methods for fluid structure interaction problems in this work we propose a robin robin preconditioner combined with krylov iterations for the solution of the interface system arising in fluid structure interaction fsi problems it can be seen as a partitioned fsi procedure and in this respect it generalizes the ideas introduced in s badia f nobile c vergara j comput phys 227 2008 7027 7051 we analyze the convergence of gmres iterations with the robin robin preconditioner on a model problem and compare its efficiency with some existing algorithms the method is shown to be very efficient for many challenging fluid structure interaction problems such as those characterized by a large added mass effect or by enclosed fluids in particular the possibility to solve balloon type problems without any special treatment makes this algorithm very appealing compared to the computationally intensive existing approaches c 2009 elsevier b v all rights reserved