extended formulations nonnegative factorizations and randomized communication protocols an extended formulation of a polyhedron p p is a linear description of a polyhedron q q together with a linear map pi such that pi q p q p these objects are of fundamental importance in polyhedral combinatorics and optimization theory and the subject of a number of studies yannakakis factorization theorem yannakakis in j comput syst sci 43 3 441 466 1991 provides a surprising connection between extended formulations and communication complexity showing that the smallest size of an extended formulation of p p equals the nonnegative rank of its slack matrix s s moreover yannakakis also shows that the nonnegative rank of s s is at most 2 c 2c where c c is the complexity of any deterministic protocol computing s s in this paper we show that the latter result can be strengthened when we allow protocols to be randomized in particular we prove that the base 2 2 logarithm of the nonnegative rank of any nonnegative matrix equals the minimum complexity of a randomized communication protocol computing the matrix in expectation using yannakakis factorization theorem this implies that the base 2 2 logarithm of the smallest size of an extended formulation of a polytope p p equals the minimum complexity of a randomized communication protocol computing the slack matrix of p p in expectation we show that allowing randomization in the protocol can be crucial for obtaining small extended formulations specifically we prove that for the spanning tree and perfect matching polytopes small variance in the protocol forces large size in the extended formulation

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

on a higher order accurate fully discrete galerkin approximation to the navier stokes equations we consider approximating the solution of the initial and boundary value problem for the navier stokes equations in bounded twoand three dimensional domains using a nonstandard galerkin finite element method for the space discretization and the third order accurate three step backward differentiation method coupled with extrapolation for the nonlinear terms for the time stepping the resulting scheme requires the solution of one linear system per time step plus the solution of five linear systems for the computation of the required initial conditions all these linear systems have the same matrix the resulting approximations of the velocity are shown to have optimal rate of convergence in l2 under suitable restrictions on the discretization parameters of the problem and the size of the solution in an appropriate function space

higher order godunov methods for general systems of hyperbolic conservation laws abstract we describe an extension of higher order godunov methods to general systems of hyperbolic conservation laws this extension allow the method to be applied to problems that are not strictly hyperbolic and exhibit local linear degeneracies in the wave fields the method constructs an approximation of the riemann problem from local wave information a generalization of the engquist osher flux for systems is then used to compute a numerical flux based on this approximation this numerical flux replaces the godunov numerical flux in the algorithm thereby eliminating the need for a global riemann problem solution the additional modifications to the godunov methodology that are needed to treat loss of strict hyperbolicity are described in detail the method is applied to some simple model problems for which the global analytic structure is known the method is also applied to the black oil model for multiphase flow in petroleum reservoirs

two level method and some a priori estimates in unsteady navier stokes calculations a two level method proposed for quasielliptic problems is adapted in this paper to the simulation of unsteady incompressible navier stokes flows the method requires a solution of a nonlinear problem on a coarse grid and a solution of linear symmetric problem on a fine grid the scaling between these two grids is superlinear approximation stability and convergence aspects of a fully discrete scheme are considered stability properties of the two level scheme are compared with those for a commonly used semi implicit scheme some new estimates are also proved for the latter

on the stability of bilinear constant velocity pressure finite elements an analysis of the babuska stability of bilinear constant finite element pairs for viscous flow calculations is given an unstable mode not of the checkerboard type is given for which the stability constant turns out to beo h thus the indicated spaces are not stable in general for numerical calculation

a comparative study of finite element and finite difference methods for cauchy riemann type equations a least squares formulation of the system operatorname div bf u rho operatorname curl bf u zeta is surveyed from the viewpoint of both finite element and finite difference

nonlinear equations and weighted norm inequalities we study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities in particular we obtain explicit criteria for the solvability of the dirichlet problem u v u w u 0 on u 0 on on a regular domain in rn in the superlinear case q 1 the coefficients v w are arbitrary positive measurable functions or measures on we also consider more general nonlinear differential and integral equations and study the spaces of coefficients and solutions naturally associated with these problems as well as the corresponding capacities our characterizations of the existence of positive solutions take into account the interplay between v w and the corresponding green s kernel they are not only sufficient but also necessary and are established without any a priori regularity assumptions on v and w we also obtain sharp two sided estimates of solutions up to the boundary some of our results are new even if v 1 and is a ball or half space the corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so called 3g inequality by an elementary integration by parts argument this also gives a new unified proof for some classical inequalities including the carleson measure theorem for poisson integrals and trace inequalities for riesz potentials and green potentials

on conforming mixed finite element methods for incompressible viscous flow problems abstract the asymptotic rates of convergence for approximate solutions of linearizations of the stationary navier stokes equations are computationally determined for some specific choices of conforming finite element spaces these rates are computed for norms of physical interest and are compared to available theoretical estimates it is shown that equivalent rates of convergence are achieved by algorithms which differ greatly in their computer storage and time requirements the solution of the discrete system of equations resulting from the finite element discretization is also discussed

a finite element method for diffusion dominated unsteady viscous flows a general conforming finite element scheme for computing viscous flows is presented which is of second order accuracy in space and time viscous terms are treated implicitly and advection terms are treated explicitly in the time marching segment of the algorithm a method for solving the algebraic equations at each time step is given the method is demonstrated on two test problems one of them being a plane vortex flow for which asymptotic methods are used to obtain suitable numerical boundary conditions at each time step

on mixed finite element methods for first order elliptic systems a physically based duality theory for first order elliptic systems is shown to be of central importance in connection with the galerkin finite element solution of these systems using this theory in conjunction with a certain hypothesis concerning approximation spaces optimal error estimates for galerkin type approximations are demonstrated an example of a grid which satisfies the hypothesis is given and numerical examples which illustrate the theory are provided

a numerical study of the two dimensional navier stokes equations in vorticity velocity variables abstract solution methods for compact finite difference schemes are applied to the vorticity velocity form of the two dimensional unsteady navier stokes equations numerical experiments for stagnation point and driven cavity flows are described

stabilized finite element method based on the crank nicolson extrapolation scheme for the time dependent navier stokes equations this paper provides an error analysis for the crank nicolson extrapolation scheme of time discretization applied to the spatially discrete stabilized finite element approximation of the two dimensional time dependent navier stokes problem where the finite element space pair x h m h for the approximation u n h p n h of the velocity u and the pressure p is constructed by the low order finite element the q 1 p 0 quadrilateral element or the p 1 p 0 triangle element with mesh size h error estimates of the numerical solution u n h p n h to the exact solution u t n p t n with t n 0 t are derived

on the convergence of a time discretization scheme for the navier stokes equations version linearisee du schema d euler implicite utilisee pour l approximation des solutions des equations de navier stokes dans un domaine bidimensionnel

galerkin runge kutta discretizations for semilinear parabolic equations a new class of fully discrete galerkin runge kutta methods is constructed and analyzed for semilinear parabolic initial boundary value problems unlike any classical counterpart this class offers arbitrarily high order convergence without suffering from what has been called order reduction in support of this claim error estimates are proved and computational results are presented furthermore it is noted that special runge kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method

existence and nonexistence of global solutions forut δu a x upinrd abstract we study nonnegative solutions of the equation u t 2 u a x u p in r d t 0 under the assumption that a x 0 is on the order x m for m 2 or that 0 a x c x 2 extending the classical result of fujita and more recent results of bandle and levine and of levine and meier we find a critical exponent p p m d such that if 1 p p then there exist no solutions that are global in time while if p p then there exist both global and nonglobal solutions

unconditional convergence and optimal l2 error estimates of the crank nicolson extrapolation fem for the nonstationary navier stokes equations abstract in this paper we study stability and convergence of fully discrete finite element method on large timestep which used crank nicolson extrapolation scheme for the nonstationary navier stokes equations this approach bases on a finite element approximation for the space discretization and the crank nicolson extrapolation scheme for the time discretization it reduces nonlinear equations to linear equations thus can greatly increase the computational efficiency we prove that this method is unconditionally stable and unconditionally convergent moreover taking the negative norm technique we derive the l 2 h 1 unconditionally optimal error estimates for the velocity and the l 2 unconditionally optimal error estimate for the pressure also numerical simulations on unconditional l 2 stability and convergent rates of this method are shown

the euler implicit explicit scheme for the 2d time dependent navier stokes equations with smooth or non smooth initial data this paper considers the stability and convergence results for the euler implicit explicit scheme applied to the spatially discretized two dimensional 2d time dependent navier stokes equations a galerkin finite element spatial discretization is assumed and the temporal treatment is implicit explict scheme which is implicit for the linear terms and explicit for the nonlinear term here the stability condition depends on the smoothness of the initial data u 0 h i e the time step condition is c 0 in the case of a 2 log h c 0 in the case of a 1 and th 2 c 0 in the case of a 0 for mesh size h and some positive constant c 0 we provide the h 2 stability of the scheme under the stability condition with a 0 1 2 and obtain the optimal h 1 l 2 error estimate of the numerical velocity and the optimal l 2 error estimate of the numerical pressure under the stability condition with a 1 2

driven cavity flows by efficient numerical techniques abstract efficient and reliable numerical techniques of high order accuracy are presented for solving problems of steady viscous incompressible flow in the plane and are used to obtain accurate solutions for the driven cavity a solution is obtained at reynolds number 10 000 on a 180 180 grid the numerical methods combine an efficient linear system solver an adaptive newton like method for nonlinear systems and a continuation procedure for following a branch of solutions over a range of reynolds numbers

existence uniqueness and approximation for generalized saddle point problems extensions of the well known results of brezzi on saddle point problems are presented the class of problems is generalized to include the unsymmetric case and the known stability and approximation results are strengthened and applied to the generalized problem an application to the solution of certain exterior interface problems is indicated

convergence analysis for a higher order scheme for the time dependent navier stokes equations abstract in this paper we discuss a third order fully discrete scheme for the time dependent navier stokes equations we prove that the third order temporal discrete scheme proposed by baker et al in 1 is convergent under a weaker condition some numerical experiments are shown to confirm the efficiency of the higher order scheme

convergence of three iterative methods based on the finite element discretization for the stationary navier stokes equations abstract this paper considers three iterative methods for solving the stationary navier stokes equations iterative method i consists in solving the stationary stokes equations iterative method ii consists in solving the stationary linearized navier stokes equations and iterative method iii consists in solving the stationary oseen equations under the finite element discretization respectively at each iterative step also we discuss the stability and convergence of three iterative methods the iterative methods i and ii are stability and convergence under the strong uniqueness conditions where the iterative method ii is the second order convergence furthermore the iterative method iii is uncondition stability and convergence under the uniqueness condition finally some numerical tests show that the efficiency of the theoretical analysis

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

compact finite difference schemes for mixed initial boundary value problems this paper discusses a class of compact second order accurate finite difference equations for mixed initial boundary value problems for hyperbolic and convective diffusion equations convergence is proved by means of energy arguments and both types of equations are solved by similar algorithms for hyperbolic equations an extension of the lax wendroff method is described which incorporates dissipative boundary conditions upwind downwind differencing techniques arise as the formal hyperbolic limit of the convective diffusion equation finally a finite difference chain rule transforms the schemes from rectangular to quadrilateral subdomains

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

geometric arguments yield better bounds for threshold circuits and distributed computing we consider the following norm for boolean functions 0 1 n 0 1 n 0 1 max m v v r2n v 1 here m denotes the 2n 2n matrix obtained by setting m x y 1 if x y 1 1 if x y 0 further we refer to the usual product of real matrices and vectors and denote by v the euclidean norm of real vectors v in this paper it will be shown by geometric arguments that for each function 0 1 n 0 1 n 0 1 the following is true the length of each probabilistic communication protocol which computes with error bounded by 12 1s s n can be estimated from below by 14 n log2 log2 s 2 the number of edges of any threshold circuit of depth two computing cannot be smaller than 2n 1 these results yield better lower bounds on probabilistic communication complexity as well as on the complexity of threshold circuits of depth two further characterizing by the eigenvalues of m we obtain a method to construct iteratively functions which are hard to compute in the above models this implies lower bounds for decision problems that previous probabilistic techniques from yao 1983 halstenberg and reischuk 1988 and hajnal et al 1987 cannot be applied to

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

a unified numerical treatment of the wave equation and the cauchy riemann equations a unified second order accurate finite difference approach to treating the wave equation and the cauchy riemann equations is developed the procedure is motivated by an analysis of properties of weak solutions

a mass conserving vorticity velocity formulation with application to nonreacting and reacting flows in a commonly implemented version of the vorticity velocity formulation the governing equations for the fluid dynamics are expressed as two poisson like velocity equations together with the vorticity transport equation however for some flows with large vorticity gradients spurious mass loss or gain can be observed in order to conserve mass a modification to the vorticity velocity formulation is proposed involving the substitution of the kinematic definition of vorticity in certain terms of the fluid dynamic equations this modified formulation results in a broader computational stencil when the equations are in a second order accurate discretized form and a stronger coupling between the predicted vorticity and the curl of the predicted velocity field the resulting system of elliptic equations which includes the energy and species transport equations for the reacting flow case is discretized with finite differences on a nonstaggered grid and is then solved using newton s method both the unmodified and modified vorticity velocity formulations are applied to two problems with high vorticity gradients 1 incompressible axisymmetric fluid flow through a suddenly expanding pipe and 2 a confined axisymmetric laminar flame with detailed chemistry and multicomponent transport generated on a burner whose inner tube extends above the burner surface the modified formulation effectively eliminates the spurious mass loss in the two test cases to within an acceptable tolerance the two cases demonstrate the broader range of applicability of the modified formulation as compared with the unmodified formulation

the log approximate rank conjecture is false we construct a simple and total boolean function f e f c xor on 2n variables that has only o sn spectral norm o n2 approximate rank and o n2 5 approximate nonnegative rank we show it has polynomially large randomized bounded error communication complexity of sn this yields the first exponential gap between the logarithm of the approximate rank and randomized communication complexity for total functions thus f witnesses a refutation of the log approximate rank conjecture that was posed by lee and shraibman as a very natural analogue for randomized communication of the still unresolved log rank conjecture for deterministic communication the best known previous gap for any total function between the two measures is a recent 4th power separation by goos et al additionally our function f refutes grolmusz s conjecture and a variant of the log approximate nonnegative rank conjecture suggested recently by kol et al both of which are implied by the log approximate rank conjecture the complement of f has exponentially large approximate nonnegative rank this answers a question of lee 32 showing that approximate nonnegative rank can be exponentially larger than approximate rank the inner function f also falsifies a conjecture about parity measures of boolean functions made by tsang et al the latter conjecture implied the log rank conjecture for xor functions we are pleased to note that shortly after we published our results two independent groups of researchers anshu et al and sinha and de wolf used our function f to prove that the quantum log rank conjecture is also false by showing that f has n1 6 quantum communication complexity

compactness properties of the dg and cg time stepping schemes for parabolic equations for a broad class of parabolic equations it is shown that numerical solutions computed using the discontinuous galerkin or the continuous galerkin time stepping schemes of arbitrary order will inherit the compactness properties of the underlying equation convergence of numerical schemes for a phase field approximation of the flow of two fluids with surface tension is presented to illustrate these results

the critical fujita exponent for the fast diffusion equation with potential abstract this paper studies the cauchy problem for positive solutions of the fast diffusion equation with source and quadratically decaying potential u t u m v x u m u p in r n 0 t where 1 2 m n m 1 p 1 n 2 v x x 2 with 0 as x and is the positive root of m m n 2 0 we obtain the critical fujita exponent p c m 2 m n to the problem in the sense that every nontrivial solution blows up in finite time when 1 p p c and there are both global and non global solutions if p p c

the fujita exponent for semilinear heat equations with quadratically decaying potential or in an exterior domain consider the equation 0 1 ut u vu aupin rn 0 t u x 0 x 0in rn where p 1 n 2 t 0 v x x 2 as x for some 0 and a x is on the order x m as x for some m a solution to the above equation is called global if t under some additional technical conditions we calculate a critical exponent p such that global solutions exist for p p while for 1 p p all solutions blow up in finite time we also show that when v 0 the blow up global solution dichotomy for 0 1 coincides with that for the corresponding problem in an exterior domain with the dirichlet boundary condition including the case in which p is equal to the critical exponent

stability of implicit explicit backward difference formulas for nonlinear parabolic equations we analyze stability properties of backward difference formulas bdf methods of order up to 5 for linear parabolic equations as well as of implicit explicit bdf methods for nonlinear parabolic equations by energy techniques time dependent norms also play a key role in the analysis

a compact monotonic discretization scheme for solving second order vorticity velocity equations abstract this paper presents a numerical method for solving the steady state navier stokes equations for incompressible fluid flows using velocities and vorticity as working variables the method involves solving a second order differential equation for the velocity and a convection diffusion equation for the vorticity in cartesian grids the key to the success of the numerical simulation of this class of flow equations depends largely on proper simulation of vorticity transport equation subject to proper boundary vorticity in this paper we present a monotonic advection diffusion multi dimensional scheme and a theoretically rigorous implementation of vorticity boundary conditions while the derivation of the proposed integral vorticity boundary condition is more elaborate and is more difficult to solve than conventional local approaches the present approach offers significant advantages in this study both lid driven and backward facing step problems have been selected for comparison and validation purposes

on conforming finite element methods for the inhomogeneous stationary navier stokes equations we consider the stationary navier stokes equations written in terms of the primitive variables in the case where both the partial differential equations and boundary conditions are inhomogeneous under certain conditions on the data the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed a conforming finite element method is presented and optimal estimates for the error of the approximate solution are proved in addition the convergence properties of iterative methods for the solution of the discrete nonlinear algebraic systems resulting from the finite element algorithm are given numerical examples using an efficient choice of finite element spaces are also provided

unconditional optimal error estimates for bdf2 fem for a nonlinear schrodinger equation abstract this paper analyzes unconditional optimal error estimates for a 2 step backward differentiation formula bdf2 method for a nonlinear schrodinger equation in the analysis we split an error estimate into two parts one from the spatial discretization and the other from the temporal discretization we present the boundedness of the solution of the time discrete system in the certain strong norms and the error estimates for time discretization by these boundedness and temporal error estimates we obtain the l 2 error estimates without any conditions on a time step size numerical experiments are provided to validate our analysis and check the efficiency of our method

on the critical exponent for reaction diffusion equations in this paper we study the initial boundary value problem for u t u h t u p with homogeneous dirichlet boundary conditions where h t t q for large t let s sup s positive solutions w of u t u such that mathop lim sup limits t to infty t s parallel w cdot t parallel infty infty then for p 1 q 1 s we show if p p there are global positive solutions that decay to zero uniformly for t if 1 p p then all nontrivial solutions blow up in finite time we determine p for some conical domains in r 2 and r 3 a similar result is derived for a bounded domain if h t e t for large t

a priori estimates and the representation formula for all positive solutions to a semilinear parabolic problem we establish some a priori estimates and a representation formula for all classical positive solutions to a semilinear parabolic problem up to now some uncertainties about the formula have been circulating

stability and convergence of the crank nicolson adams bashforth scheme for the time dependent navier stokes equations in this paper we study the stability and convergence of the crank nicolson adams bashforth scheme for the two dimensional nonstationary navier stokes equations a finite element method is applied for the spatial approximation of the velocity and pressure the time discretization is based on the crank nicolson scheme for the linear term and the explicit adams bashforth scheme for the nonlinear term moreover we present optimal error estimates and prove that the scheme is almost unconditionally stable and convergent i e stable and convergent when the time step is less than or equal to a constant

numerics and hydrodynamic stability toward error control in computational fluid dynamics we critically review the available error analysis in computational fluid dynamics cfd and come to the conclusion that the existing error estimates are meaningless in most cases of interest we propose a new approach to error analysis in cfd aiming at reliable and efficient adaptive quantitative error control this is based on a precise analysis of hydrodynamic stability coupled with galerkin orthogonality we prove a priori and a posteriori type error estimates in a model case for pipe flow formulate corresponding adaptive algorithms and discuss the potential of this approach for adaptive error control in cfd

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