adaptive unstructured volume remeshing ii application to two and three dimensional level set simulations of multiphase flow in part i adaptive unstructured volume remeshing i the method j comput phys in press we presented an adaptive remeshing algorithm that automatically adjusts the size of the elements of meshes of unstructured triangles 2d and unstructured tetrahedra 3d with time and position in the computational domain in order to efficiently resolve the relevant physical scales here we illustrate the performance of an implementation of the algorithm in finite element level set simulations of deformable droplet and fluid fluid interface interactions breakup and coalescence in multiphase flows the wide range of length scales characterizing the dynamics are accurately resolved as demonstrated by comparison to experiments and to theoretical and sharp interface boundary integral numerical results the computational cost is found to be competitive even with respect to boundary integral methods for the first time using an interface capturing level set method we successfully simulate the inertia driven impact and rebound of a liquid droplet from a liquid interface and find agreement with recent experimental results

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

subsets of some families of real functions and their algebrability abstract we consider the following families of real valued functions darboux d quasi continuous q and wi tkowski functions s the aim of this paper is to study the level of algebrability for various intersections of families d q and s and their complements

independent bernstein sets and algebraic constructions abstract we present the method of constructing algebras and linear spaces of 2 c generators using independent bernstein sets as an application we obtain large algebras of special functions i strongly everywhere surjective functions which are not perfectly everywhere surjective ii nowhere continuous darboux functions iii nowhere continuous compact to compact functions iv functions which are continuous precisely on a fixed closed proper subset of r most conclusions obtained in this paper are improvements of some already known results

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

subgrid stabilized projection method for 2d unsteady flows at high reynolds numbers a subgrid stabilization technique is developed for solving the two dimensional incompressible navier stokes equations at high reynolds numbers the time marching algorithm is based on a well established fractional step pressure correction projection method the advection diffusion step is enriched by an implicit subgrid stabilizing term and by an explicit dissipative shock capturing term the former is calculated by means of a hierarchical finite element setting the latter is included to avoid gibbs phenomenon in the boundary layer convergence tests on prototypical two dimensional examples are reported and the method is used to simulate the viscous incompressible flows around the airfoil naca0012 at zero incidence and reynolds numbers ranging from 105 to 106

nonseparable spaceability and strong algebrability of sets of continuous singular functions abstract let cbv denote the banach algebra of all continuous real valued functions of bounded variation defined in 0 1 we show that the set of strongly singular functions in cbv is nonseparably spaceable we also prove that certain families of singular functions constitute strongly c algebrable sets the argument is based on a criterion of strong c algebrability using compositions with exponential like functions

a segregated implicit pressure projection method for incompressible flows in this paper a segregated implicit projection method is presented and investigated although high reynolds turbulent flow has been a motivating factor the present study is mainly restricted to the incompressible navier stokes equations the proposed method is analyzed for properties like stability accuracy and consistency aided by some test examples in addition two different flow cases are simulated to evaluate the ability and efficiency of the method compared with other results

lineability spaceability and additivity cardinals for darboux like functions we introduce the concept of maximal lineability cardinal number ml m of a subset m of a topological vector space and study its relation to the cardinal numbers known as additivity a m homogeneous lineability hl m and lineability l m of m in particular we will describe in terms of l the lineability and spaceability of the families of the following darboux like functions on r n n 1 extendable jones and almost continuous functions

error estimates of finite element methods for nonstationary thermal convection problems with temperature dependent coefficients general error estimates are proved for a class of finite element schemes for nonstationary thermal convection problems with temperature dependent coefficients these variable coefficients turn the diffusion and the buoyancy terms to be nonlinear which increases the nonlinearity of the problems an argument based on the energy method leads to optimal error estimates for the velocity and the temperature without any stability conditions error estimates are also provided for schemes modified by approximate coefficients which are used conveniently in practical computations

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

spaceability of the set of continuous injections from blp into lp with nowhere continuous inverses abstract let p 1 we show that in the banach space of all bounded continuous mappings from b l p the open unit ball in l p into l p the subset consisting of all injections with nowhere continuous inverses is isometrically l p spaceable i e it contains an isometric copy of l p the proof uses some ideas of creswell s example

lineability and integrability in the sense of riemann lebesgue denjoy and khintchine abstract in this paper we continue the ongoing research on lineability related questions on this occasion we shall consider among others the classes of integrable functions in the sense of riemann lebesgue denjoy and khintchine improving some already known results and expanding the study of lineability to other famous integrable classes never considered before

an optimal galerkin scheme to solve the kinematic dynamo eigenvalue problem in a full sphere the kinematic dynamo approximation describes the generation of magnetic field in a prescribed flow of electrically conducting liquid one of its main uses is as a proof of concept tool to test hypotheses about self exciting dynamo action indeed it provided the very first quantitative evidence for the possibility of the geodynamo despite its utility due to the requirement of resolving fine structures historically numerical work has proven difficult and reported solutions were often plagued by poor convergence in this paper we demonstrate the numerical superiority of a galerkin scheme in solving the kinematic dynamo eigenvalue problem in a full sphere after adopting a poloidal toroidal decomposition and expanding in spherical harmonics we express the radial dependence in terms of a basis of exponentially convergent orthogonal polynomials each basis function is constructed from a terse sum of one sided jacobi polynomials that not only satisfies the boundary conditions of matching to an electrically insulating exterior but is everywhere infinitely differentiable including at the origin this galerkin method exhibits more rapid convergence for a given problem size than any other scheme hitherto reported as demonstrated by a benchmark of the magnetic diffusion problem and by comparison to numerous kinematic dynamos from the literature in the axisymmetric flows we consider in this paper at a magnetic reynolds number of o 100 a convergence of 9 significant figures in the most unstable eigenvalue requires only 40 radial basis functions alternatively 4 significant figures requires 20 radial functions the terse radial discretization becomes particularly advantageous when considering flows whose associated numerical solution requires a large number of coupled spherical harmonics we exploit this new method to confirm the tentatively proposed positive growth rate of the planar flow of bachtiar et al 4 thereby verifying a counter example to the zel dovich anti dynamo theorem in a spherical geometry

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

strong c algebrability of strong sierpinski zygmund smooth nowhere analytic and other sets of functions abstract we present a useful technique of proving strong c algebrability as an outcome we obtain the strong c algebrability of the following sets of functions strong sierpinski zygmund nowhere holder bruckner garg nowhere monotone differentiable a certain baire class smooth and nowhere analytic functions

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

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

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

structural aspects of the non uniformly continuous functions and the unbounded functions within c x abstract we prove in this paper that if a metric space supports a real continuous function which is not uniformly continuous then under appropriate mild assumptions there exists in fact a plethora of such functions in both topological and algebraical senses corresponding results are also obtained concerning unbounded continuous functions on a non compact metrizable space

a divergence free spectral expansions method for three dimensional flows in spherical gap geometries abstract a spectral method for the solution of the incompressible navier stokes equations in spherical gap geometries is presented the method uses divergence free vector expansions which inherently satisfy the boundary conditions 1 basis and test functions are constructed from chebyshev polynomials and vector spherical harmonics vsh yielding a petrov galerkin weighted residual method that produces spectral convergence no rotational nor equatorial symmetry of the flow field is implicitly imposed the approach makes extensive use of the convenient properties of the vsh which are presented in a computationally suitable form whenever possible the alias free implementation of the method rests upon a standard explicit implicit time integration technique a vsh chebyshev vector transform with two fast directions is also developed and briefly presented several test cases are used to validate the resulting initial boundary value code axisymmetric basic spherical couette flow computations are compared with available numerical results while a three dimensional spiral taylor gortler vortex flow simulation is tested against experimental measurements very good agreement is found in all cases

lineability criteria with applications lineability is a property enjoyed by some subsets within a vector space x a subset a of x is called lineable whenever a contains except for zero an infinite dimensional vector subspace if additionally x is endowed with richer structures then the more stringent notions of dense lineability maximal dense lineability and spaceability arise naturally in this paper several lineability criteria are provided and applied to specific topological vector spaces mainly function spaces sometimes such criteria furnish unified proofs of a number of scattered results in the related literature families of strict order integrable functions hypercyclic vectors non extendable holomorphic mappings riemann non lebesgue integrable functions sequences not satisfying the lebesgue dominated convergence theorem nowhere analytic functions bounded variation functions entire functions with fast growth and peano curves among others are analyzed from the point of view of lineability

on algebrability of nonabsolutely convergent series we prove that the set of all complex series which are nonabsolutely convergent is c algebrable we establish a similar result for the set of all divergent complex series with bounded partial sums

algebraic structures in the sets of surjective functions abstract in the paper we construct several algebraic structures vector spaces algebras and free algebras inside sets of different types of surjective functions among many results we prove that the set of everywhere but not strongly everywhere surjective complex functions is strongly c algebrable and that its 2 c algebrability is consistent with zfc under ch the set of everywhere surjective complex functions which are sierpinski zygmund in the sense of continuous but not borel functions is strongly c algebrable the set of jones complex functions is strongly 2 c algebrable

a conforming finite element method for the time dependent navier stokes equations we study a spatial discretization of the stokes problem in a domain of mathbb r 2 or mathbb r 3 by a finite element method the unknowns being the velocity and the pressure optimal error

infinite dimensional holomorphic non extendability and algebraic genericity abstract in this note the linear structure of the family h e g of holomorphic functions in a domain g of a complex banach space that are not holomorphically continuable beyond the boundary of g is analyzed more particularly we prove that h e g contains except for zero a closed and a dense vector space having maximal dimension as well as a maximally generated free algebra the results obtained complete a number of previous ones by several authors

a pressure correction scheme for convection dominated incompressible flows with discontinuous velocity and continuous pressure in this work we present a pressure correction scheme for the incompressible navier stokes equations combining a discontinuous galerkin approximation for the velocity and a standard continuous galerkin approximation for the pressure the main interest of pressure correction algorithms is the reduced computational cost compared to monolithic strategies in this work we show how a proper discretization of the decoupled momentum equation can render this method suitable to simulate high reynolds regimes the proposed spatial velocity pressure approximation is lbb stable for equal polynomial orders and it allows adaptive p refinement for velocity and global p refinement for pressure the method is validated against a large set of classical two and three dimensional test cases covering a wide range of reynolds numbers in which it proves effective both in terms of accuracy and computational cost

spectral radial basis functions for full sphere computations the singularity of cylindrical or spherical coordinate systems at the origin imposes certain regularity conditions on the spectral expansion of any infinitely differentiable function there are two efficient choices of a set of radial basis functions suitable for discretising the solution of a partial differential equation posed in either such geometry one choice is methods based on standard chebyshev polynomials although these may be efficiently computed using fast transforms differentiability to all orders of the obtained solution at the origin is not guaranteed the second is the so called one sided jacobi polynomials that explicitly satisfy the required behavioural conditions in this paper we compare these two approaches in their accuracy differentiability and computational speed we find that the most accurate and concise representation is in terms of one sided jacobi polynomials however due to the lack of a competitive fast transform chebyshev methods may be a better choice for some computationally intensive timestepping problems and indeed will yield sufficiently although not infinitely differentiable solutions provided they are adequately converged

on the approximation of the unsteady navier stokes equations by finite element projection methods this paper provides an analysis of a fractional step projection method to compute incompressible viscous flows by means of finite element approximations the analysis is based on the idea that the appropriate functional setting for projection methods must accommodate two different spaces for representing the velocity fields calculated respectively in the viscous and the incompressible half steps of the method such a theoretical distinction leads to a finite element projection method with a poisson equation for the incremental pressure unknown and to a very practical implementation of the method with only the intermediate velocity appearing in the numerical algorithm error estimates in finite time are given an extension of the method to a problem with unconventional boundary conditions is also considered to illustrate the flexibility of the proposed method

a mixed basis spectral projection method a new grid less no collocation spectral projection method is presented the unsteady navier stokes equations are approximated according to the variational framework of guermond and quartapelle which accommodates two vector spaces for the velocity fields obtained in the two half steps of the fractional step method but retains only one in the final solution algorithm two different bases built on legendre polynomials are used for the velocity and pressure to solve the corresponding helmholtz and poisson equations by direct spectral elliptic solvers interpolations pn and pn 2 are employed for velocity and pressure to satisfy the lbb stability requirement and a gauss legendre quadrature formula with 3 2n integration points is used to prevent aliasing error in the pseudospectral evaluation of the nonlinear terms a bdf second order time stepping is implemented to provide accurate numerical results about the stability of the singular driven cavity problem

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

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

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

error analysis of a fractional time stepping technique for incompressible flows with variable density in this paper we analyze the convergence properties of a new fractional time stepping technique for the solution of the variable density incompressible navier stokes equations the main feature of this method is that contrary to other existing algorithms the pressure is determined by just solving one poisson equation per time step first order error estimates are proved and stability of a formally second order variant of the method is established

locally recurrent functions density topologies and algebraic genericity we study the algebraic and topological genericity of certain subsets of locally recurrent functions obtaining among other results algebrability and spaceability within these classes

non lipschitz functions with bounded gradient and related problems let e be a topological vector space and let us consider a property p we say that the subset m of e formed by the vectors in e which satisfy p is lineable for certain cardinal finite or infinite if m 0 contains an infinite dimensional linear space of dimension in this note we prove that there exist uncountably infinite dimensional linear spaces of functions enjoying the following properties 1 being continuous on 0 1 a e differentiable with a e bounded derivative and not lipschitz 2 differentiable in r2 r and not enjoying the mean value theorem 3 real valued differentiable on an open connected and non convex set having bounded gradient non lipschitz and therefore not verifying the mean value theorem

linear subsets of nonlinear sets in topological vector spaces for the last decade there has been a generalized trend in mathematics on the search for large algebraic structures linear spaces closed subspaces or infinitely generated algebras composed of mathematical objects enjoying certain special properties this trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis operator theory summability theory polynomials in banach spaces hypercyclicity and chaos and general functional analysis this expository paper is devoted to providing an account on the advances and on the state of the art of this trend nowadays known as lineability and spaceability

on the penalty projection method for the navier stokes equations with the mac mesh we deal with the time dependent navier stokes equations nse with dirichlet boundary conditions on the whole domain or on a part of the domain and open boundary conditions on the other part it is shown numerically that combining the penalty projection method with spatial discretization by the marker and cell scheme mac yields reasonably good results for solving the above mentioned problem the scheme which has been introduced combines the backward difference formula of second order bdf2 namely gear s scheme for the temporal approximation the second order richardson extrapolation for the nonlinear term and the penalty projection to split the velocity and pressure unknowns similarly to the results obtained for other projection methods we estimate the errors for the velocity and pressure in adequate norms via the energy method

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

collective fixed points and maximal elements with applications to abstract economies abstract in this paper we first establish collective fixed points theorems for a family of multivalued maps with or without assuming that the product of these multivalued maps is condensing as an application of our collective fixed points theorem we derive the coincidence theorem for two families of multivalued maps defined on product spaces then we give some existence results for maximal elements for a family of ls majorized multivalued maps whose product is condensing we also prove some existence results for maximal elements for a family of multivalued maps which are not ls majorized but their product is condensing as applications of our results some existence results for equilibria of abstract economies are also derived the results of this paper are more general than those given in the literature

convergence analysis of the fem approximation of the first order projection method for incompressible flows with and without the inf sup condition in this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the incompressible navier stokes equations using a finite element approximation for the space discretization we consider two situations in the first one the analysis relies on the satisfaction of the inf sup condition for the velocity pressure finite element spaces after that we study a fully discrete fractional step method using a poisson equation for the pressure in this case the velocity pressure interpolations do not need to accomplish the inf sup condition and in fact we consider the case in which equal velocity pressure interpolation is used optimal convergence results in time and space have been obtained in both cases

navier stokes spectral solver in a sphere the paper presents the first implementation of a primitive variable spectral method for calculating viscous flows inside a sphere a variational formulation of the navier stokes equations is adopted using a fractional step time discretization with the classical second order backward difference scheme combined with explicit extrapolation of the nonlinear term the resulting scalar and vector elliptic equations are solved by means of the direct spectral solvers developed recently by the authors the spectral matrices for radial operators are characterized by a minimal sparsity diagonal stiffness and tridiagonal mass matrix closed form expressions of their nonzero elements are provided here for the first time showing that the condition number of the relevant matrices grows as the second power of the truncation order a new spectral elliptic solver for the velocity unknown in spherical coordinates is also described that includes implicitly the coriolis force in a rotating frame but requires a minimal coupling between the modal velocity components in the fourier space the numerical tests confirm that the proposed method achieves spectral accuracy and ensures infinite differentiability to all orders of the numerical solution by construction these results indicate that the new primitive variable spectral solver is an effective alternative to the spectral method recently proposed by kida and nakayama where the velocity field is represented in terms of poloidal and toroidal functions

peano curves on topological vector spaces the starting point of this paper is the existence of peano curves that is continuous surjections mapping the unit interval onto the unit square from this fact one can easily construct of a continuous surjection from the real line r to any euclidean space rn the algebraic structure of the set of these functions as well as extensions to spaces with higher dimensions is analyzed from the modern point of view of lineability and large algebras are found within the families studied we also investigate topological vector spaces that are continuous image of the real line providing an optimal lineability result

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

finite element solution of the unsteady navier stokes equations by a fractional step method abstract a finite element method is presented for the numerical simulation of time dependent incompressible viscous flows the method is based on a fractional step approach to the time integration of the navier stokes equations in which only the incompressibility condition is treated implicitly this leads to a computational scheme of extremely simple algorithmic structure that is particularly attractive for cost effective solutions of large scale problems numerical results indicate the versatility and effectiveness of the proposed method

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

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

a spectral numerical method for the navier stokes equations with applications to taylor couette flow a new spectral method for solving the incompressible navier stokes equations in a plane channel and between concentric cylinders is presented the method uses spectral expansions which inherently satisfy the boundary conditions and the continuity equation and yield banded matrices which are efficiently solved at each time step in addition the number of dependent variables is reduced resulting in a reduction in computer memory requirements several test problems have been computed for the channel flow and for flow between concentric cylinders including taylor couette flow with axisymmetric taylor vortices and wavy vortices in all cases agreement with available experimental and theoretical results is very good

convergence analysis of a finite element projection lagrange galerkin method for the incompressible navier stokes equations this paper provides a convergence analysis of a fractional step method to compute incompressible viscous flows by means of finite element approximations in the proposed algorithm the convection the diffusion and the incompressibility are treated in three different substeps the convection is treated first by means of a lagrange galerkin technique whereas the diffusion and the incompressibility are treated separately in two subsequent substeps by means of a projection method it is shown that provided the time step delta t is of cal o h d 4 where h is the meshsize and d is the space dimension 2 leq d leq 3 the proposed method yields for finite time t an error of cal o h l 1 delta t in the l2 norm for the velocity and an error of cal o h l delta t in the h1 norm or the l2 norm for the pressure where l is the polynomial degree of the approximate velocity

fixed point and maximal element theorems with applications to abstract economies and minimax inequalities in this paper we establish some fixed point theorems for a family of multivalued maps under mild conditions by using our fixed point theorems we derive some maximal element theorems for a particular family of multivalued maps namely the condensing multivalued maps as applications of our results we prove some general equilibrium existence theorems in the generalized abstract economies with preference correspondences further applications of our results are also given to minimax inequalities for a family of functions

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

efficient spectral galerkin methods iv spherical geometries fast spectral galerkin algorithms are developed for elliptic equations on the sphere the algorithms are based on a double fourier expansion and have quasi optimal optimal up to a logarithmic term computational complexity numerical experiments indicate that they are significantly more efficient and or accurate when compared with the algorithms based on spherical harmonics and on finite difference extensions to problems in spherical layers and to vector equations are also discussed

spaceability of strict order integrability abstract in this paper sharp conditions on a measure space are provided in order that the subset of functions in the corresponding lebesgue space l p which are in no other l q contains except for zero an infinite dimensional closed subspace this completes earlier work of several authors the main result is derived from a new general criterion for spaceability on function spaces a further application to spaceability of the class of non absolutely continuous functions is supplied

an undecidable case of lineability in rr abstract recently it has been proved that assuming that there is an almost disjoint family of cardinality 2 c in c which is assured for instance by either martin s axiom or the continuum hypothesis or even 2 c c one has that the set of sierpinski zygmund functions is 2 c strongly algebrable and thus 2 c lineable here we prove that these two statements are actually equivalent and moreover that they both are undecidable this would be the first time in which one encounters an undecidable proposition in the recently coined theory of lineability and spaceability

a pseudospectral method for solution of the three dimensional incompressible navier stokes equations abstract a new chebyshev pseudospectral technique based on the projection method that was previously applied by the authors to the solution of two dimensional incompressible navier stokes equations in primitive variables for nonperiodic boundary conditions is extended to solve the three dimensional navier stokes equations the crucial point of the method is the requirement that the continuity equation be satisfied everywhere in the domain on the boundaries as well as in the interior the key feature of the work presented in this paper is that the computer storage requirements of the full matrix inversion resulting from direct solution of the pressure poisson equation in three dimensions is greatly reduced by considering an eigenfunction decomposition the method was tested on a two dimensional driven cavity flow and the results were compared with those of the most accurate finite difference calculation the three dimensional driven cavity flow was then calculated at the same reynolds numbers as the two dimensional cases i e re 100 400 and 1000 in the calculated results three dimensional boundary effects were observed in all cases and became more apparent with increasing reynolds number

a fast and accurate numerical scheme for the primitive equations of the atmosphere we present a fast and accurate numerical scheme for the approximation of the primitive equations of the atmosphere the temporal variable is discretized by using a special semi implicit scheme which requires only to solve a helmholtz equation and a nonlocal stokes problem at each time step the spatial variables are discretized by a spectral galerkin procedure with the horizontal components of vectorial spherical harmonics for the horizontal variables and legendre or chebyshev polynomials for the vertical variable the new scheme has two distinct features i it is unconditionally stable given fixed physical parameters and ii the helmholtz equation and the nonlocal stokes problem which need to be solved at each time step can be decomposed into a sequence of one dimensional equations in the vertical variable which can be solved by a spectral galerkin method with optimal computational complexity

existence of a mixed equilibrium for a compact generalized strategic game abstract in this paper we first introduce new concepts of generalized mixed strategic game and mixed equilibrium which generalize the mixed equilibrium for a compact strategic game due to nash next by using an equilibrium existence theorem for the compact convex strategic game g x i t i f i we prove a mixed equilibrium existence theorem for the generalized mixed strategic game g x i t i f i without assuming the convexity of strategy space x i finally two examples of the nash equilibrium and mixed equilibrium for generalized strategic games g x i t i f i are given

stabilized finite element method for the transient navier stokes equations based on a pressure gradient projection abstract in this paper we present a stabilized finite element formulation for the transient incompressible navier stokes equations the main idea is to introduce as a new unknown of the problem the projection of the pressure gradient onto the velocity space and to add to the incompresibility equation the difference between the laplacian of the pressure and the divergence of this new vector field this leads to a pressure stabilization effect that allows the use of equal interpolation for both velocities and pressures in the case of the transient equations we consider the possibility of treating the pressure gradient projection either implicitly or explicity in the first case the number of unknowns of the problem is substantially increased with respect to the standard galerkin formulation nevertheless iterative techniques may be used in order to uncouple the calculation of the pressure gradient projection from the rest of unknowns velocity and pressure when this vector field is treated explicitly the increment of computational cost of the stabilized formulation with respect to the galerkin method is very low we provide a stability estimate for the case of the simple backward euler time integration scheme for both the implicit and the explicit treatment of the pressure gradient projection

spectral solvers for spherical elliptic problems a new family of direct spectral solvers for the 3d helmholtz equation in a spherical gap and inside a sphere for nonaxisymmetric problems is presented a variational formulation no collocation is adopted based on the fourier expansion and the associated legendre functions to represent the angular dependence over the sphere and using basis functions generated by legendre or jacobi polynomials to represent the radial structure of the solution in the present method boundary conditions on the polar axis and at the sphere center are not required and never mentioned by construction the spectral solution of the vector dirichlet problem is also considered by employing a transformation that uncouples the spherical components of the fourier modes and that is implemented here for the first time the condition numbers of the matrices involved in the scalar solvers are computed and the spectral convergence of all the proposed solution algorithms is verified by numerical tests

an adaptive pressure correction method without spurious velocities for diffuse interface models of incompressible flows in this article we propose to study two issues associated with the use of the incremental projection method for solving the incompressible navier stokes equation the first one is the combination of this time splitting algorithm with an adaptive local refinement method the second one is the reduction of spurious velocities due to the right hand side of the momentum balance we propose a new variant of the incremental projection method for solving the navier stokes equations with variable density and illustrate its behavior with the example of two phase flows simulations using a cahn hilliard navier stokes model

holomorphic functions on non runge domains and related problems let u be a domain in cn that is not a runge domain we study the topological and algebraic properties of the family of holomorphic functions on u which cannot be approximated by polynomials

on convergence of chorin s projection method to a leray hopf weak solution the projection method to solve the incompressible navier stokes equations was first studied by chorin math comput 1969 in the framework of a finite difference method and temam arch ration mech anal 1969 in the framework of a finite element method chorin showed convergence of approximation and its error estimates in problems with periodic boundary conditions assuming existence of a c 5 solution while temam demonstrated an abstract argument to obtain a leray hopf weak solution in problems on a bounded domain with the no slip boundary condition in the present paper the authors extend chorin s result with full details to obtain convergent finite difference approximation of a leray hopf weak solution to the incompressible navier stokes equations on an arbitrary bounded lipschitz domain of mathbb r 3 with the no slip boundary condition and an external force we prove unconditional solvability of our implicit scheme and strong l 2 convergence up to a subsequence under the scaling condition h 3 alpha le tau no upper bound is necessary where h tau are space time discretization parameters respectively and alpha in 0 2 is any fixed constant the results contain a compactness method based on a new interpolation inequality for step functions

lineability and algebrability of pathological phenomena in analysis we show that in analysis many pathological phenomena occur more often than one could expect that is in a linear or algebraic way we show this by means of the construction of large algebraic structures infinite dimensional vector spaces or infinitely generated algebras enjoying some special or pathological properties

maximal spaceability in sequence spaces abstract in this paper we introduce a new technique to prove the existence of closed subspaces of maximal dimension inside sets of topological vector sequence spaces the results we prove cover some sequence spaces not studied before in the context of spaceability and settle some questions on classical sequence spaces that remained open