two level method based on finite element and crank nicolson extrapolation for the time dependent navier stokes equations a fully discrete two level finite element method the two level method is presented for solving the two dimensional time dependent navier stokes problem the method requires a crank nicolson extrapolation solution u h tau 0 p h tau 0 on a spatial time coarse grid j h tau 0 and a backward euler solution u h tau p h tau on a space time fine grid j h tau the error estimates of optimal order of the discrete solution for the two level method are derived compared with the standard crank nicolson extrapolation method the one level method based on a space time fine grid j h tau the two level method is of the error estimates of the same order as the one level method in the h1 norm for velocity and the l2 norm for pressure however the two level method involves much less work than the one level method

discontinuous galerkin approximations of the stokes and navier stokes equations numerical schemes to compute approximate solutions of the evolutionary stokes and navier stokes equations are studied the schemes are discontinuous in time and conforming in space and of arbitrarily high order fully discrete error estimates are derived and dependence of the viscosity constant is carefully tracked it is shown that the errors are bounded by projection errors of the exact solution which exhibit optimal rates when the solutions are smooth

pairs of analytical eigenfunctions for the x2 λ x2 1 gx2 interaction a general method is discussed for deriving pairs of exact analytical eigenfunctions of the x2 x2 1 gx2 interaction recently discussed by blecher and leach and commented on by gallas it is shown that those pairs always consist of an odd and an even parity solution in each case and g are related by the conditions 0 and g as a result of their investigations the authors conjecture the possible existence of an infinite number of solution pairs having and g connected by several different relations

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

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 the classification of g spheres i equivariant transversality this paper is the first in a series of three stated in geometric terms the papers examine locally linear group actions on spheres for odd order groups g in essentially equivalent homotopy theoretic terms the papers study the homotopy types of the spaces plc v and tope v of equivariant pl homeomorphisms and homeomorphisms of a linear representation v in fact it is the homogeneous spaces f6 v plg v and top6 v pl6 v we study where fc v is the space of proper equivariant homotopy equivalences our results generalize theorems of haefliger kirby siebenmann sullivan and wall and others the link between classification of manifolds and classification of homotopy types is transversality and this is the subject of the present first paper the second paper generalizes wall s classification of fake lens spaces to the classification of g spheres which are equivariantly homotopy equivalent to a given linear action this involves the determination of the equivariant homotopy type of the g space f pl and a complete calculation of the pl equivariant surgery sequence for a linear g sphere as a result we show that the homotopy groups of pl v u plc v vanish in dimensions less than dim v g in the third paper we study g triangulation theory in particular we study the homotopy groups of top6 v u topc v in a range the homotopy groups of the stiefel spaces plc v u plc v and topc v u top6 v are in turn needed for equivariant transversality so the three papers are locked together in an inductive fashion the reader is referred to w 4 below and to the individual parts ii and iii for more information we now give a brief discussion of the content of the present paper let g denote an arbitrary finite group we consider the g transversality question in the locally linear

topological classification of systems of bilinear and sesquilinear forms abstract let a and b be two systems consisting of the same vector spaces c n 1 c n t and bilinear or sesquilinear forms a i b i c n k i c n l i c for i 1 s we prove that a is transformed to b by homeomorphisms within c n 1 c n t if and only if a is transformed to b by linear bijections within c n 1 c n t

a staggered space time discontinuous galerkin method for the three dimensional incompressible navier stokes equations on unstructured tetrahedral meshes in this paper we propose a novel arbitrary high order accurate semi implicit space time discontinuous galerkin method for the solution of the three dimensional incompressible navier stokes equations on staggered unstructured curved tetrahedral meshes as is typical for space time dg schemes the discrete solution is represented in terms of space time basis functions this allows to achieve very high order of accuracy also in time which is not easy to obtain for the incompressible navier stokes equations similarly to staggered finite difference schemes in our approach the discrete pressure is defined on the primary tetrahedral grid while the discrete velocity is defined on a face based staggered dual grid while staggered meshes are state of the art in classical finite difference schemes for the incompressible navier stokes equations their use in high order dg schemes is still quite rare a very simple and efficient picard iteration is used in order to derive a space time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five point block system for the scalar pressure which is conveniently solved with a matrix free gmres algorithm from numerical experiments we find that the linear system seems to be reasonably well conditioned since all simulations shown in this paper could be run without the use of any preconditioner even up to very high polynomial degrees for a piecewise constant polynomial approximation in time and if pressure boundary conditions are specified at least in one point the resulting system is in addition symmetric and positive definite this allows us to use even faster iterative solvers like the conjugate gradient method the flexibility and accuracy of high order space time dg methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both space and time the proposed method is verified for approximation polynomials of degree up to four in space and time by solving a series of typical 3d test problems and by comparing the obtained numerical results with available exact analytical solutions or with other numerical or experimental reference data to the knowledge of the authors this is the first time that a space time discontinuous galerkin finite element method is presented for the three dimensional incompressible navier stokes equations on staggered unstructured tetrahedral grids high order staggered space time dg scheme for the 3d incompressible navier stokes equations primal grid based on curved tetrahedra dual grid based on non standard five point hexahedra new discretization of the viscous terms on the main grid using lifting operators on the dual grid very simple space time pressure correction algorithm providing also high order in time all linear systems have maximum sparsity and use the smallest possible optimal five point stencil

long term dissipativity of time stepping algorithms for an abstract evolution equation with applications to the incompressible mhd and navier stokes equations abstract this paper examines the long term dissipativity and unconditional non linear stability of time integration algorithms for the incompressible mhd equations the results apply as a particular case to the uncoupled incompressible navier stokes equations the analysis is performed in the setting defined by an abstract evolution equation possessing the same long term dissipative structure as the above physical systems a general class of time stepping schemes is analyzed in this context showing the non linear unconditional stability and long term dissipativity of particular members of this family of algorithms in particular algorithms showing these strong properties which additionally involve the solution of a linear problem at each time step are identified a particular linear algorithm is shown to exhibit optimal long term dissipative properties and to possess an algorithmic global attractor these abstract results apply rigorously to galerkin mixed finite element discretizations of the problem to this end it is shown that such galerkin type projection inherits the dissipative properties of the continuum problem computational aspects involved in the numerical implementation of these methods are addressed and illustrated in representative numerical simulations

the schrodinger equation for the x2 λx2 1 gx2 interaction certain exact odd and even wavefunctions for the potential x2 x2 1 gx2 x in are obtained the coefficients of the wavefunctions and expressions for the energies are obtained using the symbolic manipulation code reduce the chief use of these results is to provide a check for the effectiveness of numerical algorithms the results supplement those of flessas 1982 for the same potential on xe 0

nonlinear similarity begins in dimension six the real representations pi and p2 of a finite group g are topologically similar written pi t p2 if there is a homeomorphism h v p1 v pa where v pi denotes the vector space of the representation pi such that h p1 g v p2 g h v for v e v p1 and g e g i e the representation spaces are equivariantly homeomorphic de rham dr conjectured that topological similarity implies the linear equivalence of the two representations the first two authors showed that de rham s conjecture is true for representations of dimension less than six cs6 but false for representations of dimension greater than or equal to nine cs2 here we give examples of nonlinear similarity i e topological similarity between linearly inequivalent representations in dimension six which is therefore the minimal dimension in which nonlinear similarity may occur since linear equivalence is detected by cyclic subgroups a minimal counterexample to de rham s conjecture will occur for g cyclic moreover by hp and mr nonlinear similarity may only occur for groups whose order is divisible by four thus we shall restrict attention to g z4q the cyclic group of order 4q we shall construct six dimensional nonlinear similarities of z4q for every q greater than two moreover we shall show in a later paper cssww2 joint with weinberger that the examples here represent all six dimensional nonlinear similarities of cyclic groups actually we shall give the complete topological classification of representations of the form p 6 c where p is free i e z4q acts freely away from the origin in the representation space of p and 6 and e are the nontrivial and trivial one dimensional representations respectively representations of the form p 6 were studied in cs2 suppose that pi 6 c 1t p2 6 e with p1 and p2 free let lp be the orbit space of the unit

asymptotic behavior and time discretization analysis for the non stationary navier stokes problem the asymptotic behavior and the euler time discretization analysis are presented for the two dimensional non stationary navier stokes problem if the data and f t satisfy a uniqueness condition corresponding to the stationary navier stokes problem we then obtain the convergence of the non stationary navier stokes problem to the stationary navier stokes problem and the uniform boundedness stability and error estimates of the euler time discretization for the non stationary navier stokes problem

the l theory of laurent extensions and genus 0 function fields the stable classification of quadratic forms over a field is given by the witt group topology via surgery theory has embedded the witt groups in a general theory of forms over any ring with involution in this paper we use geometrically inspired methods to make computations general results on the l theory of a laurent polynomial extension are used to study the witt groups of genus 0 function fields

unconditional stability and convergence of fully discrete schemes for 2d viscous fluids models with mass diffusion in this work we develop fully discrete in time and space numerical schemes for two dimensional incompressible fluids with mass diffusion also so called kazhikhov smagulov models we propose at most h 1 conformed finite elements only globally continuous functions to approximate all unknowns velocity pressure and density although the limit density solution of continuous problem will have h 2 regularity a backward euler in time scheme is considered decoupling the computation of the density from the velocity and pressure unconditional stability of the schemes and convergence towards the unique global in time weak solution of the models is proved since a discrete maximum principle cannot be ensured we must use a different interpolation inequality to obtain the strong estimates for the discrete density from the used one in the continuous case this inequality is a discrete version of the gagliardo nirenberg interpolation inequality in 2d domains moreover the discrete density is truncated in some adequate terms of the velocity pressure problem

error estimates of a linear decoupled euler fem scheme for a mass diffusion model we present error estimates of a linear fully discrete scheme for a three dimensional mass diffusion model for incompressible fluids also called kazhikhov smagulov model all unknowns of the model velocity pressure and density are approximated in space by c 0 finite elements and in time an euler type scheme is used decoupling the density from the velocity pressure pair if we assume that the velocity and pressure finite element spaces satisfy the inf sup condition and the density finite element space contains the products of any two discrete velocities we first obtain point wise stability estimates for the density under the constraint lim h k 0 h k 0 h and k being the space and time discrete parameters respectively and error estimates for the velocity and density in energy type norms at the same time afterwards error estimates for the density in stronger norms are deduced all these error estimates will be optimal of order mathcal o h k for regular enough solutions without imposing nonlocal compatibility conditions at the initial time finally we also study two convergent iterative methods for the two problems to solve at each time step which hold constant matrices independent of iterations

eigenvalues from power series expansions an alternative approach an appropriate rational approximation to the eigenfunction of the schrodinger equation for anharmonic oscillators enables one to obtain the eigenvalue accurately as the limit of a sequence of roots of hankel determinants the convergence rate of this approach is greater than that for a well established method based on power series expansions weighted by a gaussian factor with an adjustable parameter the so called hill determinant method

lower bounds for quartic anharmonic and double well potentials rigorous and remarkably accurate lower bounds to the lower eigenvalue spectrum of the schrodinger equation with quartic anharmonic and symmetric double well potentials of the form v a b ax2 2 bx4 b 0 are presented this procedure exploits some exactly soluble model potentials and appears to be of quite general utility

stability and convergence of semi implicit time stepping algorithm for stationary incompressible magnetohydrodynamics abstract for the stationary incompressible magnetohydrodynamics mhd equations we provide a new uniqueness assumption a0 and show the exponential stability of the solution then the semi implicit time stepping algorithm is used to solve the stationary mhd equations the algorithm is proved to be unconditionally stable the discrete velocity and magnetic field are bounded in l 0 l for any space and time step sizes the error estimates for the algorithm are deduced under the uniqueness conditions finally numerical experiments are performed to testify our theoretical analysis

on an efficient second order backward difference newton scheme for mhd system abstract for the 2d 3d time dependent magnetohydrodynamics mhd system we propose a new second order backward difference formula newton scheme sbdfn which is a combination of second order backward difference approximation of the time derivative terms and newton treatment of the nonlinear terms meanwhile the finite element method is applied as spatial discretization firstly the optimal convergence of spatially semi discrete form is deduced secondly the stability and well posedness of sbdfn scheme are provided under c where c is independent of h based on these results optimal error estimates of the scheme in time are proved by negative norm technique finally numerical experiments are carried out to validate the theoretical analysis

high order direct stokes solvers with or without temporal splitting numerical investigations of their comparative properties a recently proposed direct stokes solver which decouples the velocity and pressure operators without calling for a temporal scheme is numerically analyzed in comparison first with the splitting scheme proposed by g karniadakis m israeli and s orszag in j comput phys 97 1991 pp 414 443 and with the unique grid bbb p n bbb p n 2 uzawa approach for the space accuracy and computational costs the chebyshev collocation approximation is used to analyze the spectra of the continuous temporal evolution operators and their discrete time versions from the first to the fourth order an explicit boundary condition is also involved in the proposed stokes solver and it is numerically shown that the trace of bf delta bf u on the boundary must be evaluated through its bf nabla times bf nabla times contribution only otherwise the ellipticity is lost before proceeding to the time discretization the explicit evaluation of the rotational boundary term does not prevent the first and second order in time schemes from being unconditionally stable while the schemes built at the next two higher orders are limited by the usual explicit cal o n 4 stability criterion on the time step the proposed bf u p decoupling gives this limitation a less restrictive coefficient and supplies the expected temporal orders on the whole explored range of time step sizes the effective accuracy obtained for the navier stokes 2d solutions has been measured for the re 1000 lid driven cavity problem and has been found equivalent to what is supplied by the much more expensive uzawa decoupling method

dynamic group approach to the x2 λx2 1 gx2 potential a modified operator method based upon the so 2 1 dynamic group is applied to the one dimensional anharmonic oscillator potential v x x2 x2 1 gx2 a tilting transformation is carried out in order to improve the rate of convergence of the algebraic perturbation series very accurate results are obtained for the energy eigenvalues and for the wave functions especially in the case of small g values

analysis of a sequential regularization method for the unsteady navier stokes equations the incompressibility constraint makes navier stokes equations difficult a reformulation to a better posed problem is needed before solving it numerically the sequential regularization method srm is a reformulation which combines the penalty method with a stabilization method in the context of constrained dynamical systems and has the benefit of both methods in the paper we study the existence and uniqueness for the solution of the srm and provide a simple proof of the convergence of the solution of the srm to the solution of the navier stokes equations we also give error estimates for the time discretized srm formulation

the k theoretic farrell jones conjecture for hyperbolic groups we prove the k theoretic farrell jones conjecture for hyperbolic groups with twisted coefficients in any associative ring with unit

regularizing decompositions for matrix pencils and a topological classification of pairs of linear mappings abstract by kronecker s theorem each matrix pencil a b over a field f is strictly equivalent to its regularizing decomposition i e a direct sum i r d m 1 n 1 m t n t where d is an r r nonsingular matrix and each m i n i is of the form i k j k 0 j k 0 i k l k r k or l k t r k t in which l k and r k are obtained from i k by deleting its last or respectively first row and j k 0 is a singular jordan block we give a method for constructing a regularizing decomposition of an m n matrix pencil a b which is formulated in terms of the linear mappings a b f n f m two m n pencils a b and a b over f r or c are said to be topologically equivalent if the pairs of linear mappings a b f n f m and a b f n f m coincide up to homeomorphisms of the spaces f n and f m we prove that two pencils are topologically equivalent if and only if their regularizing decompositions coincide up to permutation of summands and replacement of d by a nonsingular matrix d such that the linear operators d d f r f r coincide up to a homeomorphism of f r

flow rate boundary problems for an incompressible fluid in deformable domains formulations and solution methods in this paper we consider the numerical solution of the interaction of an incompressible fluid and an elastic structure in a truncated computational domain as well known in this case there is the problem of prescribing realistic boundary data on the artificial sections when only partial data are available this problem has been investigated extensively for the rigid case in this work we start considering the compliant case by focusing on the flow rate conditions for the fluid we propose three formulations of this problem different algorithms for its numerical solution and carry out several 2d numerical simulations with the aim of comparing the performances of the different algorithms

a posteriori error estimations for mixed finite element approximations to the navier stokes equations a posteriori estimates for mixed finite element discretizations of the navier stokes equations are derived we show that the task of estimating the error in the evolutionary navier stokes equations can be reduced to the estimation of the error in a steady stokes problem as a consequence any available procedure to estimate the error in a stokes problem can be used to estimate the error in the nonlinear evolutionary problem a practical procedure to estimate the error based on the so called postprocessed approximation is also considered both the semidiscrete in space and the fully discrete cases are analyzed some numerical experiments are provided

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

on the simultaneous eigenproblem for the x2 λ x2 1 gx2 1 interaction extension of gallas results an ansatz for the eigenfunctions of the x2 x2 1 gx2 1 interaction has been recently introduced by blecher and leach using this ansatz gallas has obtained five solution pairs for 2g 3g 2 gallas results are generalised and extended here

euler implicit explicit iterative scheme for the stationary navier stokes equations in this paper a new uniqueness assumption a2 of the solution for the stationary navier stokes equations is presented under assumption a2 the exponential stability of the solution bar u bar p for the stationary navier stokes equations is proven moreover the euler implicit explicit scheme based on the mixed finite element is applied to solve the stationary navier stokes equations finally the almost unconditionally stability is proven and the optimal error estimates uniform in time are provided for the scheme

error estimates for the spectral approximation of the nonstationary stokes problem this paper analyzes a discretization of the nonstationary stokes problem with periodic nonperiodic boundary conditions fourier legendre spectral approximations are considered with respect to the space variables while the time discretization is based on the second order crank nicolson scheme optimal convergence estimates are proved for the velocity field

a defect correction method for unsteady conduction convection problems ii time discretization in this paper a fully discrete defect correction mixed finite element method mfem for solving the non stationary conduction convection problems in two dimension which is leaded by combining the back euler time discretization with the two step defect correction in space is presented in this method we solve the nonlinear equations with an added artificial viscosity term on a finite element grid and correct these solutions on the same grid using a linearized defect correction technique the stability and the error analysis are derived the theory analysis shows that our method is stable and has a good convergence property some numerical results are also given which show that this method is highly efficient for the unsteady conduction convection problems

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

topological classification of sesquilinear forms reduction to the nonsingular case abstract two sesquilinear forms c m c m c and c n c n c are called topologically equivalent if there exists a homeomorphism c m c n i e a continuous bijection whose inverse is also a continuous bijection such that x y x y for all x y c m r a horn and v v sergeichuk in 2006 constructed a regularizing decomposition of a square complex matrix a that is a direct sum s a s r j n 1 j n p in which s and r are nonsingular and each j n i is the n i by n i singular jordan block in this paper we prove that and are topologically equivalent if and only if the regularizing decompositions of their matrices coincide up to permutation of the singular summands j n i and replacement of r c r r by a nonsingular matrix r c r r such that r and r are the matrices of topologically equivalent forms c r c r c analogous results for bilinear forms over c and over r are also obtained

the stabilized extrapolated trapezoidal finite element method for the navier stokes equations abstract we consider a fully discrete stabilized finite element method for the navier stokes equations which is unconditionally stable and has second order temporal accuracy of o k 2 hk spatial error the method involves a simple artificial viscosity stabilization of the linear system for the approximation of the new time level connected to anti diffusion of its effects at the old time level lowering the cell reynolds number of the oseen problem to o 1 the method requires only the solution of one linear system arising from an oseen problem per time step the effectiveness of the method is illustrated in several numerical experiments

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

double exponential sinc collocation method for solving the energy eigenvalues of harmonic oscillators perturbed by a rational function we show that the double exponential sinc collocation method provides an efficient uniformly accurate solution to the one dimensional time independent schrodinger equation for a general class of rational potentials of the form v x p x q x the derived algorithm is based on the discretization of the hamiltonian of the schrodinger equation using sinc expansions this discretization results in a generalized eigenvalue problem the eigenvalues of which correspond to approximations of the energy values of the starting hamiltonian a systematic numerical study is conducted beginning with test potentials with known eigenvalues and moving to rational potentials of increasing degree

some observations on the nature of solutions for the interaction v x x2 λx2 1 gx2 the authors study the nature of exact solutions for the nonpolynomial interaction in particular it is shown that all the solutions are supersymmetric it is also shown that this problem admits a dynamical su 2 symmetry so that an arbitrarily large part of the spectrum but not the whole spectrum can be determined exactly

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

calculation of incompressible viscous flows by an unconditionally stable projection fem this paper investigates the numerical performance of a finite element implementation of a new incremental fractional step method to compute steady and unsteady incompressible viscous flows under general boundary conditions and using unstructured meshes a variational framework is adopted which accommodates two different spaces for representing and approximating the velocity fields calculated respectively in the viscous and inviscid phases of the method but which leads to a very simple numerical scheme in terms of only one discrete velocity field an unconditionally stable semi implicit approximation of the nonlinear term is used to eliminate any time step restriction as far as the numerical stability is concerned numerical results for five test problems in two dimensions are reported to illustrate the flexibility of the proposed method

decoupled schemes for unsteady mhd equations ii finite element spatial discretization and numerical implementation in this article a decoupled fully discrete scheme for solving 2d magnetohydrodynamics mhd equations is proposed the decoupled scheme is used for time discretization and the finite element method is used for spatial discretization firstly the almost unconditional stability t c of this scheme is established then optimal l2 and h1 error estimates of numerical solution are provided finally a numerical example is presented to confirm our theoretical results and show the high efficiency of the decoupled scheme

study of a class of non polynomial oscillator potentials we develop a variational method to obtain accurate bounds for the eigenenergies of h v in arbitrary dimensions n 1 where v r is the nonpolynomial oscillator potential the variational bounds are compared with results previously obtained in the literature an infinite set of exact solutions is also obtained and used as a source of comparison eigenvalues

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

exact analytical eigenfunctions for the x2 λ x2 1 gx2 interaction the author discusses an ansatz for the eigenfunctions of the x2 x2 1 gx2 interaction introduced by blecher and leach 1987 the author shows that the ansatz provides pairs of solutions in five cases having and g connected by 6g2 4g and conjectures the possible existence of an infinite number of such solution pairs

semi discrete stabilized finite element methods for navier stokes equations with nonlinear slip boundary conditions based on regularization procedure based on the pressure projection stabilized methods the semi discrete finite element approximation to the time dependent navier stokes equations with nonlinear slip boundary conditions is considered in this paper because this class of boundary condition includes the subdifferential property then the variational formulation is the navier stokes type variational inequality problem using the regularization procedure we obtain a regularized problem and give the error estimate between the solutions of the variational inequality problem and the regularized problem with respect to the regularized parameter varepsilon which means that the solution of the regularized problem converges to the solution of the navier stokes type variational inequality problem as the parameter varepsilon longrightarrow 0 moreover some regularized estimates about the solution of the regularized problem are also derived under some assumptions about the physical data the pressure projection stabilized finite element methods are used to the regularized problem and some optimal error estimates of the finite element approximation solutions are derived

first order decoupled finite element method of the three dimensional primitive equations of the ocean this paper is concerned with a first order fully discrete decoupled method for solving the three dimensional 3d primitive equations of the ocean with the dirichlet boundary conditions on the side where a decoupled semi implicit scheme is used for the time discretization and the p 1 p 1 p 1 p 1 p 1 finite element for velocity pressure and density is used for the spatial discretization of these equations the h 1 l 2 h 1 optimal error estimates for the numerical solution u h n p h n theta h n and the l 2 optimal error estimate for u n h theta h n are established under the restriction of 0 h le beta 1 and 0 tau le beta 2 for some positive constants beta 1 and beta 2 moreover numerical investigations are provided to show that the first order decoupled method is of almost unconditional convergence with accuracy mathcal o h tau in the h 1 norm and mathcal o h 2 tau in the l 2 norm for solving the 3d primitive equations of the ocean numerical results are gi

on convergent schemes for diffuse interface models for two phase flow of incompressible fluids with general mass densities we are concerned with convergence results for fully discrete finite element schemes suggested in grun and klingbeil j comput phys in press doi 10 1016 j jcp 2013 10 028 they were developed for the diffuse interface model in h abels h garcke and g grun math models methods appl sci 22 2012 1150013 which describes two phase flow of immiscible incompressible viscous fluids we formulate general conditions on discretization spaces and projection operators which allow us to prove compactness of discrete solutions with respect to both time and space and which hence permit us to establish convergence of the scheme to a generalized solution we identify a simple quantitative and physical criterion to decide whether this generalized solution is in fact a weak solution in this case our analysis provides another pathway to establish existence of weak solutions to the aforementioned model in two and in three space dimensions our argument is particularly based on higher regularity results

approximation by an iterative method for regular solutions for incompressible fluids with mass diffusion we study the approximation by means of an iterative method towards strong and more regular solutions for incompressible navier stokes equations with mass diffusion in addition some convergence rates for the error between the approximation and the exact solution will be given for weak strong and more regular norms

surgery with finite fundamental group the relationship between the topology of manifolds and homotopy theory has been analysed by surgery 1 18 28 31 however the effectiveness of this analysis depends on being able to compute surgery obstructions in this paper we introduce a new technique for evaluating the weakly simple surgery obstructions for degree 1 normal maps of closed manifolds and apply it to oriented manifolds with finite fundamental group our main result is theorem a below let x dx be a finite poincare pair of dimension n and a top bundle over x which is stably fibre homotopy equivalent to the spivak normal fibre space v let m dm be a compact n dimensional manifold and let

application of finite element method in aeroelasticity the subject of this paper is the numerical simulation of the interaction of two dimensional incompressible viscous flow and a vibrating airfoil which can rotate around the elastic axis and oscillate in the vertical direction the numerical simulation consists of the finite element approximation of the navier stokes equations coupled with the system of ordinary differential equations describing the airfoil motion the arbitrary lagrangian eulerian ale formulation of the navier stokes equations stabilization the finite element discretization and coupling of both models is discussed moreover the reynolds averaged navier stokes rans system of equations together with the spallart almaras turbulence model is also discussed the computational results of aeroelastic calculations are presented and compared with the nastran code solutions

topological equivalence of linear representations for cyclic groups i in the two parts of this paper we prove that the reidemeister torsion invariants determine topological equivalence of g representations for g a finite cyclic group