convergence and nonlinear stability of the lagrange galerkin method for the navier stokes equations the lagrange galerkin method is a numerical technique for solving convection dominated diffusion problems based on combining a special discretisation of the lagrangian material derivative along particle trajectories with a galerkin finite element method we present optimal error estimates for the lagrange galerkin mixed finite element approximation of the navier stokes equations in a velocity pressure formulation the method is shown to be nonlinearly stable

the finite element method with nodes moving along the characteristics for convection diffusion equations we examine the convergence properties of the finite element method with nodes moving along the characteristics for one dimensional convection diffusion equations for linear elements we demonstrate optimal rates of convergence in thel 2 h 1 andl norms both linear and nonlinear problems are considered

numerical analysis of a finite element volume penalty method we present here some contributions to the numerical analysis of the penalty method in the finite element context we are especially interested in the ability provided by this approach to use cartesian non boundary fitted meshes to solve elliptic problems in complicated domain in the spirit of fictitious domains the initial problem is replaced by a penalized one posed over a simply shaped domain which covers the original one this method relies on two parameters namely h space discretization parameter and varepsilon penalty parameter we propose here a general strategy to estimate the error in both parameters and we present how it can be applied to various situations we pay special attention to a scalar version of the rigid motion constraint for fluid particle flows

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

analysis of an algorithm for the galerkin characteristic method it is shown that the interpretation of the galerkin characteristic method for the scalar advection equation in the framework of particle methods yields a computationally efficient algorithm such an algorithm consists of updating the dependent variable at the grid points by cubic spline interpolation at the feet of the characteristic curves the algorithm is unconditionally stable the error analysis in the maximum norm shows that for sufficiently smooth functions the feet of the characteristic curves are points of high order convergence

convergence of a lagrangian scheme for a compressible naviers stokes model defined on a domain depending on time abstract this paper deals with the numerical resolution of the navier stokes equations defined on a time dependent domain we give an existence result for a fluid structure interaction problem in which the boundary is governed by a thin plate operator we propose to solve the fluid equations with the characteristics method we approach the total derivative with a regularized finite difference scheme and we study the convergence of the discrete problem towards the continuous one

on the second real eigenvalue of nonegative and z matrices abstract we give bounds for the second real eigenvalue of nonegative matrices and z matrices furthermore we establish upper bounds for the maximal spectral radii of principal submatrices of nonnegative matrices using these bounds we prove that our inequality for the second real eigenvalue of the adjacency matrix of a connected regular graph improves a well known bound for the second eigenvalue using cheeger s inequality

characteristic stabilized finite element method for the transient navier stokes equations abstract based on the lowest equal order conforming finite element subspace xh mh i e p1 p1 or q1 q1 elements a characteristic stabilized finite element method for transient navier stokes problem is proposed the proposed method has a number of attractive computational properties parameter free avoiding higher order derivatives or edge based data structures and averting the difficulties caused by trilinear terms existence uniqueness and error estimates of the approximate solution are proved by applying the technique of characteristic finite element method finally a serious of numerical experiments are given to show that this method is highly efficient for transient navier stokes problem

tuning the mesh of a mixed method for the stream function vorticity formulation of the navier stokes equations in this article we study a new mixed method for the stokes and navier stokes equations the method uses two meshes one very fine for and a coarser one for error estimates show that boundary layers do not require to refine the mesh for the stream function as much as for the vorticity when the reynolds number is large we prove estimates and study implementation problems

nonlinear stability and convergence of finite difference methods for the good boussinesq equation the good boussinesq equationu tt u xxxx u xx u 2 xx has recently been found to possess an interesting soliton interaction mechanism in this paper we study the nonlinear stability and the convergence of some simple finite difference schemes for the numerical solution of problems involving the good boussinesq equation numerical experimentas are also reported

time stepping along characteristics with incomplete iteration for a galerkin approximation of miscible displacement in porous media miscible displacement of one incompressible fluid by another in a porous medium is modeled by a nonlinear coupled system of two partial differential equations the pressure equation is elliptic while the concentration equation is parabolic but normally convection dominated a sequential backward difference time stepping scheme is defined it approximates the pressure by a standard galerkin procedure and the concentration by a combination of a galerkin method and the method of characteristics optimal convergence rates in l 2 and h 1 are demonstrated for this scheme and for a modified version in which the algebraic equations at each time step are solved approximately by a limited number of preconditioned conjugate gradient iterations time stepping along the characteristics of the hyperbolic part of the concentration equation is shown to result in smaller time truncation errors than those of standard methods numerical results published elsewhere have confirmed that larger time steps are appropriate

approximate solution methods for linear stochastic difference equations i moments the cumulant expansion for linear stochastic differential equations is extended to the case of linear stochastic difference equations we consider a vector difference equation which contains a deterministic matrix a0 and a random perturbation matrix a1 t the expansion proceeds in powers of c where c is the correlation time of the fluctuations in a1 t and a measure for their strength compared to the differential case additional cumulants occur in the expansion moreover one has to distinguish between a nonsingular and a singular a0 we also discuss a limiting situation in which the stochastic difference equation can be replaced by a stochastic differential equation the derivation is not restricted to the case where in the limit the stochastic parameters in the difference equation are replaced by white noise

regular article direct simulations of 2d fluid particle flows in biperiodic domains we propose a method to simulate the motion of 2d rigid particles in a viscous incompressible fluid within the arbitrary lagrangian eulerian framework momentum equations for both the fluid and the particles are discretized and a coupled variational formulation is established by introducing an appropriate finite element approximation a symmetric linear system is obtained this system is solved by an inexact uzawa algorithm the main interest of such simulations lies in the average behaviour of a high number of particles we therefore introduced a biperiodic formulation of the problem which makes it possible to represent many body mixtures at a reasonable computational cost in order to model realistic situations an extra term must be added to the pressure this extra term is shown to be the lagrange multiplier associated with the vertical volume conservation constraint we developed an appropriate unstructured mesh generator so that the biperiodicity of the fields can be treated in a strong sense the question of particle contact is addressed and a simple technique to overcome numerical problems is proposed the influence of periodic lengths is investigated through different simulations the same case is simulated for different sizes of the window and the behaviour of some space averaged quantities is studied

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

analysis of mixed finite elements methods for the stokes problem a unified approach we develop a method for the analysis of mixed finite element methods for the stokes problem in the velocity pressure formulation a technical macroelement condition which is sufficient for the classical babuska brezzi inequality to be valid is introduced using this condition we are able to verify the stability and optimal order of convergence of several known mixed finite element methods

tight estimates for convergence of some non stationary consensus algorithms the present paper is devoted to estimating the speed of convergence towards consensus for a general class of discrete time multi agent systems in the systems considered here both the topology of the interconnection graph and the weight of the arcs are allowed to vary as a function of time under the hypothesis that some spanning tree structure is preserved along time and that some non zero minimal weight of the information transfer along this tree is guaranteed an estimate of the contraction rate is given the latter is expressed explicitly as the spectral radius of some matrix depending upon the tree depth and the lower bounds on the weights

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

finite element error estimation for quasi newtonian fluid structure interaction problems we consider a monolithic scheme for fluid structure interaction problems involving an incompressible quasi newtonian fluid the monolithic formulation is obtained using the arbitrary lagrangian eulerian ale method with matching conditions at the fluid structure interface the stability and error analysis are performed for the finite element approximation finally some numerical experiments that confirm the theoretical analysis are presented

a galerkin characteristic unified finite element method for moving thermal fronts in porous media abstract we investigate the performance of a unified finite element method for the numerical solution of moving fronts in porous media under non isothermal flow conditions the governing equations consist of coupling the darcy equation for the pressure to two convection diffusion reaction equations for the temperature and depth of conversion the aim is to develop a non oscillatory unified galerkin characteristic method for efficient simulation of moving fronts in porous media the method is based on combining the modified method of characteristics with a galerkin finite element discretization of the governing equations the main feature of the proposed unified finite element method is that the same finite element space is used for all solutions to the problem including the pressure velocity temperature and concentration analysis of convergence and stability is also presented in this study and error estimates in the l 2 norm are established for the numerical solutions in addition due to the lagrangian treatment of convection terms the standard courant friedrichs lewy condition is relaxed and the time truncation errors are reduced in the diffusion reaction part we verify the method for the benchmark problem of moving fronts around an array of cylinders the numerical results obtained demonstrate the ability of the proposed method to capture the main flow features

global weak solutions for the two dimensional motion of several rigid bodies in an incompressible viscous fluid we consider the two dimensional motion of several non homogeneous rigid bodies immersed in an incompressible non homogeneous viscous fluid the fluid and the rigid bodies are contained in a fixed open bounded set of r 2 the motion of the fluid is governed by the navier stokes equations for incompressible fluids and the standard conservation laws of linear and angular momentum rule the dynamics of the rigid bodies the time variation of the fluid domain due to the motion of the rigid bodies is not known a priori so we deal with a free boundary value problem the main novelty here is the demonstration of the global existence of weak solutions for this problem more precisely the global character of the solutions we obtain is due to the fact that we do not need any assumption concerning the lack of collisions between several rigid bodies or between a rigid body and the boundary we give estimates of the velocity of the bodies when their mutual distance or the distance to the boundary tends to zero

inhomogeneous linear random differential equations with mutual correlations between multiplicative additive and initial value terms abstract the cumulant expansion for linear stochastic differential equations is extended to the general case in which the coefficient matrix the inhomogeneous part and the initial condition are all random and moreover statistically interdependent the expansion now involves not only the autocorrelation functions of the coefficient matrix as in the homogeneous case but also crosscorrelation functions of the coefficient matrix with the inhomogeneous part and with the initial value term as a first illustration we consider an exactly solvable stochastic differential equation with initial correlations and compare the exact solution with that of the cumulant expansion secondly we show in general how the method can be used for the calculation of second moments and treat the harmonic oscillator with random frequency and driving term as an example

a second order characteristics finite element scheme for natural convection problems in this paper a second order characteristics finite element scheme is applied to the numerical solution of natural convection problems firstly after recalling the mathematical model a second order time discretization of the material time derivative is introduced next fully discretized schemes are proposed by using finite element methods numerical results for the two dimensional problem of buoyancy driven flow in a square cavity with differentially heated side walls are given and compared with a reference solution

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

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

a second order in time local projection stabilized lagrange galerkin method for navier stokes equations at high reynolds numbers we present a stabilized backward difference formula of order 2 lagrange galerkin method to integrate the incompressible navier stokes equations at high reynolds numbers re the stabilization of the conventional lagrange galerkin method is done via a local projection technique for inf sup stable finite elements we prove that for a finite time t the a priori error estimate for velocity in a mesh dependent norm is o h m t 2 whereas the error for pressure in the l 2 l 2 d norm is o h m t 2 with error constants that are independent of the re 1 here m denotes the degree of the polynomials of the velocity finite element space the size of the stabilization parameters is calculated from the velocity error estimate in a way that the error is optimal when the solution is sufficiently smooth numerical examples at high reynolds numbers show the robustness of our method

a finite element semi lagrangian method with l2 interpolation summary high order accurate methods for convection dominated problems have the potential to reduce the computational effort required for a given order of solution accuracy the state of the art in this field is more advanced for eulerian methods than for semi lagrangian slag methods in this paper we introduce a new slag method that is based on combining the modified method of characteristics with a high order interpolating procedure the method employs the finite element method on triangular meshes for the spatial discretization an l2 interpolation procedure is developed by tracking the feet of the characteristic lines from the integration nodes numerical results are illustrated for a linear advection diffusion equation with known analytical solution and for the viscous burgers equation the computed results support our expectations for a robust and highly accurate finite element slag method copyright 2012 john wiley sons ltd

a parallel adaptive barotropic model of the atmosphere the parallel adaptive model plasma has been developed for modeling a barotropic atmosphere this model adapts the computational grid at every time step according to a physical error indicator thus compared to uniform grid experiments the number of grid points is reduced significantly at the same time the error increases only slightly when comparing with uniform grid solutions for the discretization of the underlying spherical shallow water equations a lagrange galerkin method is used the unstructured triangular grid is maintained by the grid generator amatos and the large linear systems are solved by the parallel solver interface fossi experimental convergence is shown by means of steady state and unsteady analytical solutions plasma yields satisfactory results for quasi standard experiments that is the rossby haurwitz wave and zonal flows over an isolated mountain

unconditional stability and error estimates of the modified characteristics fems for the time dependent incompressible mhd equations abstract this paper focuses on the unconditional stability and convergence of characteristics type methods for the time dependent incompressible mhd equations for this purpose we introduce a new characteristics time discrete system the optimal error estimates in l 2 and h 1 norms for the typical modified characteristics finite element method unconditionally can be deduced while the whole previous works require certain time step restrictions some numerical experiments document performance of the characteristics type methods for the time dependent incompressible mhd equations

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 modified lagrange galerkin method for a fluid rigid system with discontinuous density in this paper we propose a new characteristics method for the discretization of the two dimensional fluid rigid body problem in the case where the densities of the fluid and the solid are different the method is based on a global weak formulation involving only terms defined on the whole fluid rigid domain to take into account the material derivative we construct a special characteristic function which maps the approximate rigid body at the k 1 th discrete time level into the approximate rigid body at k th time convergence results are proved for both semi discrete and fully discrete schemes

evolution galerkin methods and their supraconvergence evolution galerkin methods for partial differential equations of the form u t l u 0 are characterised by i the use of some form of approximation to the corresponding evolution operator e t and ii projection onto an approximation space s h to obtain u n in this paper we concentrate on characteristic galerkin and lagrange galerkin methods to derive basic error estimates for multidimensional convection problems methods covered include those using recovery techniques to improve accuracy many schemes exhibit a supraconvergence phenomenon and a general technique for its analysis is given together with a number of particular examples

opinion dynamics with decaying confidence application to community detection in graphs we study a class of discrete time multi agent systems modeling opinion dynamics with decaying confidence we consider a network of agents where each agent has an opinion at each time step the agents exchange their opinion with their neighbors and update it by taking into account only the opinions that differ from their own less than some confidence bound this confidence bound is decaying an agent gives repetitively confidence only to its neighbors that approach sufficiently fast its opinion essentially the agents try to reach an agreement with the constraint that it has to be approached no slower than a prescribed convergence rate under that constraint global consensus may not be achieved and only local agreements may be reached the agents reaching a local agreement form communities inside the network in this paper we analyze this opinion dynamics model we show that communities correspond to asymptotically connected components of the network and give an algebraic characterization of communities in terms of eigenvalues of the matrix defining the collective dynamics finally we apply our opinion dynamics model to address the problem of community detection in graphs we propose a new formulation of the community detection problem based on eigenvalues of normalized laplacian matrix of graphs and show that this problem can be solved using our opinion dynamics model we consider three examples of networks and compare the communities we detect with those obtained by existing algorithms based on modularity optimization we show that our opinion dynamics model not only provides an appealing approach to community detection but that it is also effective

convergence of a finite element ale method for the stokes equations in a domain depending on time we consider the approximation of the unsteady stokes equations in a time dependent domain when the motion of the domain is given more precisely we apply the finite element method to an arbitrary lagrangian eulerian ale formulation of the system our main results state the convergence of the solutions of the semi discretized with respect to the space variable and of the fully discrete problems towards the solutions of the stokes system

evolution galerkin methods for hyperbolic systems in two space dimensions the subject of the paper is the analysis of three new evolution galerkin schemes for a system of hyperbolic equations and particularly for the wave equation system the aim is to construct methods which take into account all of the infinitely many directions of propagation of bicharacteristics the main idea of the evolution galerkin methods is the following the initial function is evolved using the characteristic cone and then projected onto a finite element space a numerical comparison is given of the new methods with already existing methods both those based on the use of bicharacteristics as well as commonly used finite difference and finite volume methods we discuss the stability properties of the schemes and derive error estimates

a spectral method of characteristics for hyperbolic problems a new numerical technique is introduced for first order hyperbolic equations based on combining the spectral method with explicit timestepping along the characteristics which permits the use of large timesteps at no cost in accuracy the method is unconditionally stable and spectrally accurate the theoretical results are illustrated by numerical experiments

a fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies application to particulate flow abstract in this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies the simulation methods rest essentially on the combination of a lagrange multiplier based fictitious domain methods which allow the fluid flow computations to be done in a fixed flow region b finite element approximations of the navier stokes equations occurring in the global model c time discretizations by operator splitting schemes in order to treat optimally the various operators present in the model the above methodology is particularly well suited to the direct numerical simulation of particulate flow such as the flow of mixtures of rigid solid particles and incompressible viscous fluids possibly non newtonian we conclude this article with the presentation of the results of various numerical experiments including the simulation of store separation for rigid airfoils and of sedimentation and fluidization phenomena in two and three dimensions

a semi lagrangian high order method for navier stokes equations we present a semi lagrangian method for advection diffusion and incompressible navier stokes equations the focus is on constructing stable schemes of second order temporal accuracy as this is a crucial element for the successful application of semi lagrangian methods to turbulence simulations we implement the method in the context of unstructured spectral hp element discretization which allows for efficient search interpolation procedures as well as for illumination of the nonmonotonic behavior of the temporal advection error of the form o tk xp 1 t we present numerical results that validate this error estimate for the advection diffusion equation and we document that such estimate is also valid for the navier stokes equations at moderate or high reynolds number two and three dimensional laminar and transitional flow simulations suggest that semi lagrangian schemes are more efficient than their eulerian counterparts for high order discretizations on nonuniform grids

convergence analysis of the q1 finite element method for elliptic problems with non boundary fitted meshes summary the aim of this paper is to derive a priori error estimates when the mesh does not fit the original domain s boundary this problematic of the last century e g the finite difference methodology returns to topical studies with the huge development of domain embedding fictitious domain or cartesiangrid methods these methods use regular structured meshes most often cartesian for non aligned domains although non boundary fitted approaches become more and more applied very few studies are devoted to theoretical error estimates in this paper the convergence of a q1 nonconforming finite element method is analyzed for secondorder elliptic problems with dirichlet robin or neumann boundary conditions the finite element method uses standard q1 rectangular finite elements as the finite element approximate space is not contained in the original solution space this method is referred to as nonconforming a stairstep boundary defined from the cartesian mesh approximates the original domain s boundary the convergence analysis of the finite element method for such a kind of non boundary fitted stairstepped approximation is no treated in the literature the study of dirichlet problems is based on similar techniques as those classically used with boundary fitted linear triangular finite elements the estimates obtained for robin problems are novel and use some more technical arguments the rate of convergence is proved to be in o h 1 2 for the h 1 norm for all general boundary conditions and classical duality arguments allow to obtain an o h error estimate in the l 2 norm for dirichlet problems numerical results obtained with fictitious domain techniques that impose original boundary conditions on a non boundary fitted approximate immersed interface are presented these results confirm the theoretical rates of convergence copyright c 2007 john wiley sons ltd

numerical method of characteristics for one dimensional blood flow mathematical modeling at the level of the full cardiovascular system requires the numerical approximation of solutions to a one dimensional nonlinear hyperbolic system describing flow in a single vessel this model is often simulated by computationally intensive methods like finite elements and discontinuous galerkin while some recent applications require more efficient approaches e g for real time clinical decision support phenomena occurring over multiple cardiac cycles iterative solutions to optimization inverse problems and uncertainty quantification further the high speed of pressure waves in blood vessels greatly restricts the time step needed for stability in explicit schemes we address both cost and stability by presenting an efficient and unconditionally stable method for approximating solutions to diagonal nonlinear hyperbolic systems theoretical analysis of the algorithm is given along with a comparison of our method to a discontinuous galerkin implementation lastly we demonstrate the utility of the proposed method by implementing it on small and large arterial networks of vessels whose elastic and geometrical parameters are physiologically relevant

arbitrary lagrangian eulerian method for navier stokes equations with moving boundaries this article deals with the numerical approximation of navier stokes equations in a domain with moving boundaries using the arbitrary lagrangian eulerian ale method we transform the problem from a moving domain to a fixed reference domain through an artificial domain velocity then we apply the characteristic method to solve the navier stokes equation in the fixed domain suitable boundary conditions are used for the interfaces three examples are provided to show the efficiency of the present method

a distributed lagrange multiplier fictitious domain method for the simulation of flow around moving rigid bodies application to particulate flow in this article we discuss the application of a lagrange multiplier based fictitious domain method to the numerical simulation of incompressible viscous flow modeled by the navier stokes equations around moving rigid bodies the rigid body motion is due to hydrodynamical forces and gravity the solution method combines finite element approximations time discretization by operators splitting and conjugate gradient algorithms for the solution of the linearly constrained quadratic minimization problems coming from the splitting method we conclude this article by the presentation of numerical results concerning the simulation of an incompressible viscous flow around a naca0012 airfoil with a fixed center but free to rotate then the sedimentation of circular cylinders in 2 d channels and finally the sedimentation of spherical balls in cylinders with square cross sections 2000 elsevier science s a all rights reserved

convergence of the lagrange galerkin method for the equations modelling the motion of a fluid rigid system in this paper we consider a lagrange galerkin scheme to approximate a two dimensional fluid rigid body problem the equations of the system are the navier stokes equations in the fluid part coupled with ordinary differential equations for the dynamics of the rigid body in this problem the equations of the fluid are written in a domain whose variation is one of the unknowns we introduce a numerical method based on the use of characteristics and on finite elements with a fixed mesh our main result asserts the convergence of this scheme

some additional remarks on the cumulant expansion for linear stochastic differential equations we summarize our previous results on cumular expasions for linear stochastic differential equations with correlated multipliclative and additive noise the application of the general formulas to equations with statistically independent multiplicative and additive noise is reconsidered in detail because our earlier results for this case are only valid in special situations moreover a second partially time ordered cumulant is introduced which is appropriate for the calculation of multi time averages of the solution process and which differs from the one to be used for single time averages by the two supplements just mentioned we complete our earlier investigations on the subject

convergence speed of unsteady distributed consensus decay estimate along the settling spanning trees results for estimating the convergence rate of nonstationary distributed consensus algorithms are provided on the basis of qualitative mainly topological as well as basic quantitative information lower bounds on the matrix entries the results appear to be tight in a number of instances and are illustrated through simple as well as more sophisticated examples the main idea is to follow propagation of information along certain spanning trees which arise in the communication graph

convergence rates for markov chains this is an expository paper that presents various ideas related to nonasymptotic rates of convergence for markov chains such rates are of great importance for stochastic algorithms that are widely used in statistics and in computer science they also have applications to analysis of card shuffling and other areas in this paper we attempt to describe various mathematical techniques that have been used to bound such rates of convergence in particular we describe eigenvalue analysis random walks on groups coupling and minorization conditions connections are made to modern areas of research wherever possible elements of linear algebra probability theory group theory and measure theory are used but efforts are made to keep the presentation elementary and accessible

a spectral lagrange galerkin method for convection dominated diffusion problems abstract the spectral lagrange galerkin method is a numerical technique for time dependent convection diffusion problems based on combining a lagrangian formulation of the equations with the spectral method the resulting scheme can be shown to be unconditionally stable for linear advection diffusion equations and optimal convergence rates may be obtained both the spectral accuracy and the efficiency of the scheme are examined theoretically and by various numerical experiments and extensions of the method to give high accuracy in time are presented

a finite element semi lagrangian explicit runge kutta chebyshev method for convection dominated reaction diffusion problems explicit runge kutta chebyshev methods have proved to be efficient for reaction diffusion problems of moderate stiffness in this paper we extend such an efficiency to convection dominated reaction diffusion problems by giving a formulation of these methods in a semi lagrangian framework using c0 finite elements of degree m 2 as the space discretization method we also study the convergence in the l2 norm of the methods proposed in this paper

pseudospectral method for the good boussinesq equation we prove the nonlinear stability and convergence of a fully discrete pseudospectral scheme for the good boussinesq equation utt uxxxx uxx u2 xx numerical comparisons with finite difference schemes are also reported

analysis of some finite elements for the stokes problem we study some finite elements which are used in the approximation of the stokes problem so as to obtain error estimates of optimal order resume nous etudions deux elements finis utilises pour l approximation du probleme de stokes et obtenons des estimations d erreur d ordre optimal

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

a characteristic finite element algorithm for the 3 d navier stokes equations using unstructured grids abstract this paper discusses a finite element algorithm for solution of the 3 d navier stokes equations that is based on the method of characteristics the algorithm which is based on a time slitting of the convective terms from the generalized stokes problem comprises a point wise characteristic integration of the convective terms and uzawa integration of the generalized stokes problem the most costly part of the characteristic integration scheme is the search for an element containing the foot of a given characteristic the present study uses a very efficient algorithm for element searching which is particularly useful for complex geometries the implementation of the algorithm is discussed in the paper and the resulting code s performance is compared to that of a code based on a more conventional fourth order runge kutta convection integration a variety of flow problems are tested including an analytical solution of the navier stokes equations flow in a lid driven cavity flow over a backward facing step and flow in an end to side anastomoses in all cases the proposed algorithm demonstrated performance superior to that of a conventional explicit treatment of the convective terms

a level set formulation of eulerian interface capturing methods for incompressible fluid flows a level set formulation is derived for incompressible immiscible navier stokes equations separated by a free surface the interface is identified as the zero level set of a smooth function eulerian finite difference methods based on this level set formulation are proposed these methods are robust and efficient and are capable of computing interface singularities such as merging and reconnection numerical experiments are presented to demonstrate the effectiveness of the methods

a high order characteristics method for the incompressible navier stokes equations abstract we analyze a high order characteristics method for the navier stokes equations we focus on the cases of the first second and third order in time schemes with finite element spatial discretization a numerical comparison between the first and second order schemes is done for steady or transient states flows

unconditional stability and error estimates of modified characteristics fems for the navier stokes equations the paper is concerned with the unconditional stability and convergence of characteristics type methods for the time dependent navier stokes equations we present optimal error estimates in l 2 l2 and h 1 h1 norms for a typical modified characteristics finite element method unconditionally while all previous works require certain time step restrictions the analysis is based on an iterated characteristic time discrete system with which the error function is split into a temporal error and a spatial error with a rigorous analysis to the characteristic time discrete system we prove that the difference between the numerical solution and the solution of the time discrete system is tau independent where tau denotes the time stepsize thus numerical solution in w 1 infty w1 is bounded and optimal error estimates can be obtained in a traditional way numerical results confirm our analysis and show clearly the unconditional stability and convergence of the modified characteristics finite element method for the time dependent navier stokes equations the approach used in this paper can be easily extended to many other characteristics based methods

convergence speed in distributed consensus and averaging we study the convergence speed of distributed iterative algorithms for the consensus and averaging problems with emphasis on the latter we first consider the case of a fixed communication topology we show that a simple adaptation of a consensus algorithm leads to an averaging algorithm we prove lower bounds on the worst case convergence time for various classes of linear time invariant distributed consensus methods and provide an algorithm that essentially matches those lower bounds we then consider the case of a time varying topology and provide a polynomial time averaging algorithm

on the relationship between semi lagrangian and lagrange galerkin schemes following a previous result stating their equivalence under constant advection speed semi lagrangian and lagrange galerkin schemes are compared in this paper in the situation of variable coefficient advection equations once known that semi lagrangian schemes can be proved to be equivalent to area weighted lagrange galerkin schemes via a suitable definition of the basis functions we will further prove that area weighted lagrange galerkin schemes represent a small more precisely an o delta t perturbation of exact lagrange galerkin schemes this equivalence implies a general result of stability for semi lagrangian schemes

modified lagrange galerkin methods of first and second order in time for convection diffusion problems we introduce modified lagrange galerkin mlg methods of order one and two with respect to time to integrate convection diffusion equations as numerical tests show the new methods are more efficient but maintaining the same order of convergence than the conventional lagrange galerkin lg methods when they are used with either p 1 or p 2 finite elements the error analysis reveals that 1 when the problem is diffusion dominated the convergence of the modified lg methods is of the form o h m 1 h 2 t q q 1 or 2 and m being the degree of the polynomials of the finite elements 2 when the problem is convection dominated and the time step t is large enough the convergence is of the form o frac h m 1 delta t h 2 delta t q 3 as in case 2 but with t small then the order of convergence is now o h m h 2 t q 4 when the problem is convection dominated the convergence is uniform with respect to the diffusion parameter x t so that when 0 and the forcing term is also equal to zero the error tends to that of the pure convection problem our error analysis shows that the conventional lg methods exhibit the same error behavior as the mlg methods but without the term h 2 numerical experiments support these theoretical results

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

a fictitious domain method for external incompressible viscous flow modeled by navier stokes equations abstract in this article we discuss a fictitious domain method for the numerical solutions of three dimensional elliptic problems with dirichlet boundary conditions and also of the navier stokes equations modeling incompressible viscous flow the methodology for the navier stokes equations described here takes a systematic advantage of time discretization by operator splitting in order to treat separately advection imbedding and incompressibility due to the decoupling fast elliptic solvers can be used to treat the incompressibility condition even if the original problem is taking place on a nonregular geometry the resulting methodology is applied to two dimensional unsteady external incompressible viscous flow problems and three dimensional stokes problems

unconditional convergence and optimal error estimates of a galerkin mixed fem for incompressible miscible flow in porous media in this paper we study the unconditional convergence and error estimates of a galerkin mixed fem with the linearized semi implicit euler scheme for the equations of incompressible miscible flow in porous media we prove that the optimal l 2 error estimates hold without any time step convergence conditions while all previous works require certain time step restrictions our theoretical results provide a new understanding on commonly used linearized schemes the proof is based on a splitting of the error into two parts the error from the time discretization of the pdes and the error from the finite element discretization of corresponding time discrete pdes the approach used in this paper can be applied to more general nonlinear parabolic systems and many other linearized semi implicit time discretizations

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

fully discrete error estimation for a quasi newtonian fluid structure interaction problem we consider an incompressible quasi newtonian fluid structure interaction fsi problem formulated in a monolithic framework where the matching conditions at the moving interface are satisfied the fully discretized fsi system is discussed with detailed analysis for the stability and error estimate as well as numerical experiments that confirm the theoretical results

degree fluctuations and the convergence time of consensus algorithms we consider a consensus algorithm in which every node in a sequence of undirected b connected graphs assigns equal weight to each of its neighbors under the assumption that the degree of each node is fixed except for times when the node has no connections to other nodes we show that consensus is achieved within a given accuracy e on n nodes in time b 4n3 bln 2n e because there is a direct relation between consensus algorithms in time varying environments and in homogeneous random walks our result also translates into a general statement on such random walks moreover we give a simple proof of a result of cao spielman and morse that the worst case convergence time becomes exponentially large in the number of nodes n under slight relaxation of the degree constancy assumption

on the collatz wielandt numbers and the local spectral radius of a nonnegative matrix let t be a nonnegative linear continuous operator in a partially ordered banach space e and let x be a nonnegative nonzero vector in e we establish relations between the local spectral radius rt x and the upper and lower collatz wielandt numbers rt x and rt x examples will show that the situation in an infinite dimensional space fe can be much more complicated than in the classical case of rn with positive cone rn we also present in certain situations sufficient and or necessary conditions for both sequences rt tnx n and rt tnx n to converge to rt x the significance of some conditions in the infinite dimensional case is demonstrated and we make several comments on the earlier literature of the subject

multistep characteristic method for incompressible flow in porous media in this paper we present a multistep difference scheme for the problem of miscible displacement of incompressible fluid flow in porous media the discretization involves a three level time scheme based on the characteristic method and a five point finite difference scheme for space discretization we prove that the convergence is of order o h 2 dt 2 which is in contrast to the convergence of order o h dt proved for a singlestep characteristic with the same space discretization numerical experiments demonstrate the stability and second order convergence of the scheme

numerical analysis of a characteristic stabilized finite element method for the time dependent navier stokes equations with nonlinear slip boundary conditions abstract based on a characteristic method this work is concerned with a finite element approximation to the time dependent navier stokes equations with nonlinear slip boundary conditions since this slip boundary condition of friction type contains a subdifferential property its continuous variational problem is formulated as an inequality which can turn into an equality problem by using a powerful regularized method then a fully discrete characteristic scheme under the stabilized lower order finite element pairs is proposed for the equality problem optimal error estimates for velocity and pressure are derived under the corresponding l 2 h 1 norms finally a smooth problem test is reported to demonstrate the theoretically predicted convergence order and the expected slip phenomena and the simulation of a bifurcated blood flow model is displayed to illustrate the efficiency of the proposed method

consensus of multi agent systems with unbounded time varying delays in this note the consensus problem with infinite time varying delays for linearly coupled static network is investigated the delay affects only the off diagonal terms in continuous time equations at first we define an effective consensus ability index then by using the graph theory and a new concept of consensus we prove that under some mild conditions the network can realize consensus an example is given to show the validity of obtained results

a space time galerkin least squares finite element formulation of the navier stokes equations for moving domain problems a space time galerkin least squares finite element formulation of the navier stokes equations is presented for the analysis of free surface flows moving spatial configurations and deforming fluid structure interfaces the variational equation is based on the time discontinuous galerkin method employing the physical entropy variables the space time elements are oriented in time to accommodate the spatial deformations if the elements are oriented along the particle paths the formulation is lagrangian and if they are fixed in time it is eulerian consequently this formulation is analogous to the arbitrary lagrangian eulerian ale technique a novel mesh rezoning strategy is presented to orient the elements in time and adapt the fluid mesh to the changing spatial configuration numerical results are presented to show the performance of the method

a second order characteristic finite element scheme for convection diffusion problems a new characteristic finite element scheme is presented for it is of second order accuracy in time increment symmetric and unconditionally stable optimal error estimates are proved in the framework of l 2 theory numerical results are presented for two examples which show the advantage of the scheme

approximate solution methods for linear stochastic difference equations ii inhomogeneous equations multi time averages biological application abstract the cumulant expansion for linear stochastic difference equations introduced in part i is applied to the general case where the equation contains multiplicative additive and initial value terms which are all random and statistically interdependent also the two time correlation functions of the solution are discussed finally the expansion for the probability density functions is studied in the case that the coefficient matrix constitutes a markov process an exact equation for the joint probability density function of the solution of the difference equation and the random coefficient matrix is derived from this equation the moments of the solution can be obtained in a simple way as an application we consider the growth of a biological population with two age classes in a random environment which itself is modelled by a two state markov chain the exact results are compared with those of the cumulant expansion and with previous findings of tuljapurkar

a cumulant expansion for the time correlation functions of solutions to linear stochastic differential equations it is shown that the cumulant expansion for linear stochastic differential equations hitherto used to compute one time averages of the solution process is also capable of yielding the two time correlation and probability density functions the general case with a coefficient matrix an inhomogeneous part and an initial condition which are all random and mutually correlated is discussed two examples are given the latter of which treats the harmonic oscillator with stochastic frequency and driving term studied before finally we investigate the relation of our method with the so called smoothing method

an initial and boundary value problem modeling of fish like swimming in this paper we consider an initial and boundary value problem that models the self propelled motion of solids in a bidimensional viscous incompressible fluid the self propelling mechanism consisting of appropriate deformations of the solids is a simplified model of the propulsion mechanism of fish like swimmers the governing equations consist of the navier stokes equations for the fluid coupled to newton s laws for the solids since we consider the case in which the fluid solid system fills a bounded domain we have to tackle a free boundary value problem the main theoretical result in the paper asserts the global existence and uniqueness up to possible contacts of strong solutions of this problem the second novel result of this work is the provision of a numerical method for the fluid solid system this method provides a simulation of the simultaneous displacement of several swimmers and is tested on several examples

uniform estimates for characteristics mixed finite method for transient advection dominated diffusion problems in two dimensional space we prove uniform optimal order error estimates for the characteristics mixed finite element for advection diffusion equations in the proof the interpolation operators are used instead of the mixed elliptic projections the generic constants of the estimates do not explicitly depend on e but depend linearly on sobolev norms of the true solution we prove uniform error estimates for the characteristics mixed finite element method for two dimensional transient convection dominated diffusion equations in the sense that the generic constants in the error estimates depend linearly on certain sobolev norms of the true solution but not on the scaling diffusion parameter e numerical experiments are presented to confirm our theoretical findings

stability of multiagent systems with time dependent communication links we study a simple but compelling model of network of agents interacting via time dependent communication links the model finds application in a variety of fields including synchronization swarming and distributed decision making in the model each agent updates his current state based upon the current information received from neighboring agents necessary and or sufficient conditions for the convergence of the individual agents states to a common value are presented thereby extending recent results reported in the literature the stability analysis is based upon a blend of graph theoretic and system theoretic tools with the notion of convexity playing a central role the analysis is integrated within a formal framework of set valued lyapunov theory which may be of independent interest among others it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence even for the simple models discussed in the present paper