on stabilized finite element methods based on the scott zhang projector circumventing the inf sup condition for the stokes problem abstract in this work we propose a stabilized finite element method that permits us to circumvent discrete inf sup conditions e g allowing equal order interpolation the type of method we propose belongs to the family of symmetric stabilization techniques which are based on the introduction of additional terms that penalize the difference between some quantities i e the pressure gradient in the stokes problem and their finite element projections the key feature of the formulation we propose is the definition of the projection to be used a non standard scott zhang projector that is well defined for l 1 functions the resulting method has some appealing features the projector is local and nested meshes or enriched spaces are not required

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

on mixed finite element methods for linear elastodynamics we construct and analyze finite element methods for approximating the equations of linear elastodynamics using mixed elements for the discretization of the spatial variables we consider two different mixed formulations for the problem and analyze semidiscrete and up to fourth order in time fully discrete approximations l 2 optimal order error estimates are proved for the approximations of displacement and stress

a space time finite element method for the wave equation a numerical method is introduced which uses finite elements in time and space simultaneously to solve the wave equation an error analysis for this scheme is presented and the results of several computations this scheme is accurate for time slabs of arbitrary thickness without introducing extra least squares or stabilization terms

on the littlewood paley stein g function we consider semigroups tt which are contractive on lp m for all p e q q and q e 1 2 we give an example on symmetric spaces of the noncompact type which shows that the littlewood paley stein g function associated to the infinitesimal generator of tt may be unbounded on lq m and on l m we prove that variants of the g function are bounded on these lebesgue spaces in this paper we present some new results concerning the lp boundedness of the littlewood paley stein g function suppose that tt is a co semigroup of operators on lp m where m is a a finite measure space and let a e n the littlewood paley stein function ga associated to the given semigroup is defined by the rule ga f 1 ta ttfi2 dt for all f for which the right hand side makes sense in classical cases where tt is the poisson semigroup acting on lp r or lp t the g function is an important tool in fourier analysis for instance it is frequently used to prove multiplier theorems and pointwise convergence results the reader is referred to 1 1 for a survey of its role in a general context the g function was considered by e m stein he proved that if tt is a symmetric diffusion semigroup then ga satisfies the estimate 1 aplifiip liga f iip bp fllp vf e lp m whenever p e 1 oo a proof of this result can be found in 10 a simpler and more general approach via transference techniques is presented in 2 a third proof is in 4 and 5 it is of an entirely different nature in that it is based on the fact that generators of symmetric diffusion semigroups and even more general semigroups 3 have an hx functional calculus one of the interesting applications of g functions is to functional calculus for infinitesimal generators of co semigroups stein proved a marcinkiewicz type result for those as a corollary of the theory he developed for the g function the relationship between g functions and functional calculus for the infinitesimal generator of tt was investigated in 4 and 5 our treatment here owes received by the editors march 7 1994 originally communiated to the proceedings of the ams by j marshall ash 1991 mathematics subject classification primary 42b25 47a60 secondary 43a85

on the consequences of a mihlin hormander functional calculus maximal and square function estimates we prove that the existence of a mihlin hormander functional calculus for an operator l implies the boundedness on l p of both the maximal multiplier operators and the continuous square functions build on spectral multipliers of l the considered multiplier functions are finitely smooth and satisfy an integral condition at infinity in particular multipliers of compact support are admitted

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

application of the natural stress formulation for solving unsteady viscoelastic contraction flows abstract we present a numerical scheme for a previously unexploited formulation of the equations for unsteady viscoelastic flow the formulation aligns the polymer stress along particle paths streamlines utilising the characteristic curves associated with the hyperbolic part of the constitutive equations we illustrate the approach for the oldroyd b model in the benchmark 4 1 contraction for moderate elasticity numbers we show that the scheme is able to accurately capture the re entrant corner singularity for the polymer stresses and the pressure the latter variable being inaccurately determined by schemes using the traditional formulation in terms of cartesian polymer stresses a space step restriction for stability is derived which can be numerically limiting in certain recirculation regions this contrasts with the equivalent space step restriction for the formulation in cartesian stresses which is limiting in flow regions of high velocity gradients for example at sharp corners in contraction flows

a new family of mixed finite elements for the linear elastodynamic problem we construct and analyze a new family of quadrangular in two dimensions or cubic in three dimensions mixed finite elements for the approximation of elastic wave equations our elements lead to explicit schemes via mass lumping after time discretization including in the case of anisotropic media error estimates are given for these new elements

estimates for functions of the laplace operator on homogeneous trees in this paper we study the heat equation on a homogeneous graph relative to the natural nearest neighbour laplacian we find pointwise estimates for the heat and resolvent kernels and the lp lq mapping properties of the corresponding operators in recent years random walks on graphs have been studied as analogues of diffusions on manifolds see e g t coulhon cn m kanai k1 k2 and n th varopoulos v1 v2 more recently i chavel and e a feldman cf and m m h pang p considered the semigroup e t 0 in this paper we consider the heat semigroup associated to the canonical laplace operator on a homogeneous tree which is perhaps the basic example of a graph where the cardinality of the ball of radius r grows exponentially in r we study the properties of the heat and resolvent operators and discover very close analogies with diffusions on hyperbolic spaces in line with the philosophy of the above mentioned authors a homogeneous tree x of degree q 1 is defined to be a connected graph with no loops in which every vertex is adjacent to q 1 other vertices we denote by d the natural distance on x d x y being the number of edges between the vertices x and y and by l x the lebesgue space with respect to counting measure on x on x the canonical laplace operator l is defined thus lf x f x 1 q 1 y x d x y 1 f y x x it is easily seen to be bounded on l x for every p in 1 and self adjoint on l x denote by 2 l the l x spectrum of l and by p the associated spectral resolution of the identity such that lf 2 l dp f f l x and denote inf 2 l by b2 for in 0 1 and with re 0 the heat operator and the resolvent of l are the operators defined by the formulae htf 2 l e t dp f t r received by the editors october 4 1996 1991 mathematics subject classification primary 43a85 secondary 20e08 43a90 22e35

an analysis of new mixed finite elements for the approximation of wave propagation problems we construct and analyze a new family of rectangular two dimensional or cubic three dimensional mixed finite elements for the approximation of the acoustic wave equations the main advantage of this element is that it permits us to obtain through mass lumping an explicit scheme even in an anisotropic medium nonclassical error estimates are given for this new element

stability convergence and accuracy of stabilized finite element methods for the wave equation in mixed form in this paper we propose two stabilized finite element methods for different functional frameworks of the wave equation in mixed form these stabilized finite element methods are stable for any pair of interpolation spaces of the unknowns the variational forms corresponding to different functional settings are treated in a unified manner through the introduction of length scales related to the unknowns stability and convergence analysis is performed together with numerical experiments it is shown that modifying the length scales allows one to mimic at the discrete level the different functional settings of the continuous problem and influence the stability and accuracy of the resulting methods

finite element approximation of the hyperbolic wave equation in mixed form the purpose of this paper is to present a finite element approximation of the scalar hyperbolic wave equation written in mixed form that is introducing an auxiliary vector field to transform the problem into a first order problem in space and time we explain why the standard galerkin method is inappropriate to solve this problem and propose as alternative a stabilized finite element method that can be cast in the variational multiscale framework the unknown is split into its finite element component and a remainder referred to as subscale as original features of our approach we consider the possibility of letting the subscales to be time dependent and orthogonal to the finite element space the formulation depends on algorithmic parameters whose expression is proposed from a heuristic fourier analysis

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

a priori error estimates for mixed finite element approximations of the acoustic wave equation in this paper we derive optimal a priori l infty l 2 error estimates for mixed finite element displacement formulations of the acoustic wave equation the computational complexity of this approach is equivalent to the traditional mixed finite element formulations of the second order hyperbolic equations in which the primary unknowns are pressure and the gradient of pressure however the displacement formulations with the physical variables of interest displacement and pressure requires less regularity on the displacement

bellman functions and dimension free lp estimates for the riesz transforms in bessel settings abstract in this article we prove dimension free l p boundedness of riesz transforms associated with a bessel differential operator we obtain explicit estimates of the l p norms for the bessel riesz transforms in terms of p establishing a linear behavior with respect to p we use the bellman function technique to prove a bilinear dimension free inequality involving poisson semigroups defined through this bessel operator

numerical aspects in modeling high deborah number flow and elastic instability investigating highly nonlinear viscoelastic flow in 2d domain we explore problem as well as property possibly inherent in the streamline upwinding technique supg and then present various results of elastic instability the mathematically stable leonov model written in tensor logarithmic formulation is employed in the framework of finite element method for spatial discretization of several representative problem domains for enhancement of computation speed decoupled integration scheme is applied for shear thinning and boger type fluids from the analysis of 4 1 contraction flow at low and moderate values of the deborah number de the solution with supg method does not show noticeable difference from the one by the computation without upwinding on the other hand in the flow regime of high de especially in the state of elastic instability the supg significantly distorts the flow field and the result differs considerably from the solution acquired straightforwardly when the strength of elastic flow and thus the nonlinearity further increase the computational scheme with upwinding fails to converge and evolutionary solution does not become available any more all this result suggests that extreme care has to be taken on occasions where upwinding is applied and one has to first of all prove validity of this algorithm in the case of high nonlinearity on the contrary the straightforward computation with no upwinding can efficiently model representative phenomena of elastic instability in such benchmark problems as 4 1 contraction flow flow over a circular cylinder and flow over asymmetric array of cylinders asymmetry of the flow field occurring in the symmetric domain enhanced spatial and temporal fluctuation of dynamic variables and flow effects caused by extension hardening are properly described in this study

first second and third order fractional step methods for the three field viscoelastic flow problem in this paper three different fractional step methods are designed for the three field viscoelastic flow problem whose variables are velocity pressure and elastic stress the starting point of our methods is the same as for classical pressure segregation algorithms used in the newtonian incompressible navier stokes problem these methods can be understood as an inexact lu block factorization of the original system matrix of the fully discrete problem and are designed at the pure algebraic level the final schemes allow one to solve the problem in a fully decoupled form where each equation for velocity pressure and elastic stress is solved separately the first order scheme is obtained from a straightforward segregation of pressure and elastic stress in the momentum equation whereas the key point for the second order scheme is a first order extrapolation of these variables the third order fractional step method relies on yosida s scheme referring to the spatial discretization either the galerkin method or a stabilized finite element formulation can be used we describe the fractional step methods first assuming the former and then we explain the modifications introduced by the stabilized formulation we employ and that has been proposed in a previous work this discretization in space shows very good stability permitting in particular the use of equal interpolation for all variables

sobolev algebras on lie groups and riemannian manifolds we prove that on any connected unimodular lie group g the space l p g l g where l p g is the sobolev space of order 0 associated with a sublaplacian is an algebra under pointwise product this generalizes results due to strichartz in the euclidean case to bohnke in the case of stratified groups and others a global version of this fact holds for groups with polynomial growth we give similar results for riemannian manifolds with ricci curvature bounded from below respectively nonnegative

harmonic analysis on semigroups wherever t 0 and 1 p so e m stein st 3 developed a littlewoodpaley theory for such semigroups with some additional hypotheses r r coifman r rochberg and g weiss crw pointed out that some of stein s estimates could be obtained more simply by the use of transference techniques which were developed by coifman and weiss cw our object is to push these transference methods to present an alternative and simpler approach to some of the principal results of stein s book we shall also use these results to prove estimates for the wave equation

besov and triebel lizorkin spaces on lie groups in this paper we develop a theory of besov and triebel lizorkin spaces on general noncompact connected lie groups endowed with a sub riemannian structure such spaces are defined by means of hypoelliptic sub laplacians with drift and endowed with a measure whose density with respect to a right haar measure is a continuous positive character of the group we prove several equivalent characterizations of their norms we establish comparison results also involving sobolev spaces of recent introduction and investigate their complex interpolation and algebra properties

unified stabilized finite element formulations for the stokes and the darcy problems in this paper we propose stabilized finite element methods for both stokes and darcy s problems that accommodate any interpolation of velocities and pressures apart from the interest of this fact the important issue is that we are able to deal with both problems at the same time in a completely unified manner in spite of the fact that the functional setting is different concerning the stabilization formulation we discuss the effect of the choice of the length scale appearing in the expression of the stabilization parameters both in what refers to stability and to accuracy this choice is shown to be crucial in the case of darcy s problem as an additional feature of this work we treat two types of stabilized formulations showing that they have a very similar behavior

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

sobolev spaces on lie groups embedding theorems and algebra properties abstract let g be a noncompact connected lie group denote with a right haar measure and choose a family of linearly independent left invariant vector fields x on g satisfying hormander s condition let be a positive character of g and consider the measure whose density with respect to is in this paper we introduce sobolev spaces l p adapted to x and 1 p 0 and study embedding theorems and algebra properties of these spaces as an application we prove local well posedness and regularity results of solutions of some nonlinear heat and schrodinger equations on the group

a numerical study of the kernel conformation transformation for transient viscoelastic fluid flows this work presents a numerical application of a generic conformation tensor transformation for simulating highly elastic flows of non newtonian fluids typically observed in computational rheology in the kernel conformation framework 14 the conformation tensor constitutive law for a viscoelastic fluid is transformed introducing a generic tensor transformation function the numerical stability of the application of the kernel conformation for highly elastic flows is ultimately related with the specific kernel function used in the matrix transformation but also to the existence of singularities introduced either by flow geometry or by the characteristics of the constitutive equation in this work we implement this methodology in a free surface marker and cell discretization methodology implemented in a finite differences method the main contributions of this work are two fold on one hand we demonstrate the accuracy of this kernel conformation formulation using a finite differences method and free surfaces on the other hand we assess the numerical efficiency of specific kernel functions at high weissenberg number flows the numerical study considers different viscoelastic fluid flow problems including the poiseuille flow in a channel the lid driven cavity flow and the die swell free surface flow the numerical results demonstrate the adequacy of this methodology for high weissenberg number flows using the oldroyd b model we analyze a generic conformation tensor transformation for simulating the hwnp the numerical scheme is constructed in the context of finite differences the numerical study considers transient benchmarks in computational rheology complex flows are solved including the die swell free surface problem

stabilized finite element approximation of transient incompressible flows using orthogonal subscales abstract in this paper we present a stabilized finite element method to solve the transient navier stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales the latter are approximately accounted for so as to end up with a stable finite element problem which in particular allows to deal with convection dominated flows and the use of equal velocity pressure interpolations three main issues are addressed the first is a method to estimate the behavior of the stabilization parameters based on a fourier analysis of the problem for the subscales secondly the way to deal with transient problems discretized using a finite difference scheme is discussed finally the treatment of the nonlinear term is also analyzed a very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space in the transient case this simplifies considerably the numerical scheme

stabilization of incompressibility and convection through orthogonal sub scales in finite element methods two apparently different forms of dealing with the numerical instability due to the incompressibility constraint of the stokes problem are analyzed in this paper the first of them is the stabilization through the pressure gradient projection which consists of adding a certain least squares form of the difference between the pressure gradient and its l2 projection onto the discrete velocity space in the variational equations of the problem the second is a sub grid scale method whose stabilization effect is very similar to that of the galerkin least squares gls method for the stokes problem it is shown here that the first method can also be recast in the framework of sub grid scale methods with a particular choice for the space of sub scales this leads to a new stabilization procedure whose applicability to stabilize convection is also studied in this paper

numerical approach to simulating turbulent flow of a viscoelastic polymer solution in this paper we present two new numerical algorithms for updating the equations of motion for a viscoelastic fluid that can be described by the finite extensible nonlinear elastic polymer model with the closure proposed by peterlin so called fene p model in a transient calculation in particular our algorithms address two difficulties found in earlier formulations first the polymer extension represented by the trace of the conformation tensor can numerically exceed the finite extensible length causing the restoring spring force to change sign and the calculation to rapidly diverge in our formulations we have redefined the conformation tensor so that this possibility no longer exists secondly the conformation tensor must remain symmetric and positive definite at all times for the calculation to remain stable the accumulation of numerical errors can cause loss of this property leading to the growth of hadamard instabilities j non newtonian fluid mech 60 1995 53 we present two matrix decompositions that enable us to construct the conformational tensor in a manner that ensures positive definiteness numerical tests of the new algorithms show significant departures from other approaches that rely on filtering to remove the instabilities

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100