a finite element method using singular functions for the poisson equation corner singularities consider the poisson equation with homogeneous dirichlet boundary conditions on a polygonal domain with one reentrant corner in this paper we develop a new finite element method for the accurate computation of the solution and stress intensity factors it is well known that the solution of such a problem has a singular function representation u w lambda eta s where w in h 2 omega cap h 1 0 omega lambda in cal r and eta are the stress intensity factor and cut off function respectively and s is a known singular function depending only on the reentrant angle by using the dual singular and an extra cut off function we are able to derive a new extraction formula for lambda in terms of w and hence deduce a well posed variational problem for w standard continuous piecewise linear finite element approximation yields o h optimal accuracy for w which in turn implies the same accuracy for u in the h 1 norm we are able only to prove o h 1 frac pi omega error bounds for w and u in the l 2 norm and for lambda in the absolute value where omega is the internal angle

fully discrete stabilized multiphysics finite element method for the polymer gel model abstract in this paper we propose a fully discrete stabilized multiphysics finite element method by the pressure projection method to solve the polymer gel model to reveal the multi physical process of the polymer gel model we reformulate the polymer gel model by introducing a new variable to get a new multi field problem then we propose the fully discrete stabilized multiphysics finite element method using the simplest equal order element of p 1 p 1 p 1 without the discrete inf sup condition to solve the new reformulated system and we prove that the new method not only has no locking phenomenon but also preserves some energy conservation laws also we prove that the stabilized multiphysics finite element method has optimal convergence order finally we give the numerical example to verify the theoretical results

an analysis of some high accuracy finite element methods for hyperbolic problems some high accuracy finite element methods for hyperbolic problems are studied in this paper it is proven that over a finite element mesh the convergence order of linear finite element solutions for both linear and nonlinear equations can be higher than one even if the exact solutions are discontinuous the theoretical tools for the convergence analysis are some superclose error estimates that are also developed in this paper for nonsmooth solutions

pressure stability in fractional step finite element methods for incompressible flows abstract the objective of this paper is to analyze the pressure stability of fractional step finite element methods for incompressible flows that use a pressure poisson equation for the classical first order projection method it is shown that there is a pressure control which depends on the time step size and therefore there is a lower bound for this time step for stability reasons the situation is much worse for a second order scheme in which part of the pressure gradient is kept in the momentum equation the pressure stability in this case is extremely weak to overcome these shortcomings a stabilized fractional step finite element method is also considered and its stability is analyzed some simple numerical examples are presented to support the theoretical results

a modified streamline diffusion method for solving the stationary navier stokes equation recently hughes et al 11 12 proposed new finite element schemes of petrov galerkin type for solving the stokes problem which do not require the discrete version of the ladyshenskaya babuska brezzi condition lbb condition in this paper we derive a conforming finite element method for solving the stationary navier stokes equations which combines the advantages of arbitrary finite element spaces for velocity pressure with the favourable properties of the streamline diffusion method in the case of moderate and high reynolds number

distributed delta 1 coloring in linear in delta time the distributed delta 1 coloring problem is one of the most fundamental and well studied problems in distributed algorithms starting with the work of cole and vishkin in 1986 a long line of gradually improving algorithms has been published the state of the art running time prior to our work is o delta log delta log n due to kuhn and wattenhofer proceedings of the 25 th annual acm symposium on principles of distributed computing denver co 2006 pp 7 15 linial proceedings of the 28 th annual ieee symposium on foundation of computer science los angeles ca 1987 pp 331 335 proved a lower bound of frac 1 2 log n for the problem and szegedy and vishwanathan proceedings of the 25th annual acm symposium on theory of computing san diego ca 1993 pp 201 207 provided a heuristic argument that shows that algorithms from a wide family of locally iterative algorithms are unlikely to achieve a running time smaller than theta delta log delta we present a de

error analysis of some finite element methods for the stokes problem we prove the optimal order of convergence for some two dimensional finite element methods for the stokes equations first we consider methods of the taylor hood type the triangular p3 p2 element and the qk qk 1 i k 2 family of quadrilateral elements then we introduce two new low order methods with piecewise constant approximations for the pressure the analysis is performed using our macroelement technique which is reviewed in a slightly altered form

an error analysis of the finite element method overcoming corner singularities for the stationary stokes problem in this paper we introduce a mixed finite element method to overcome corner singularities of the stationary stokes system on a non convex polygon the reference choi and kweon 2013 says that the velocity field and the pressure function are composed of singular parts and regular remainders near a non convex vertex and furthermore the regular parts are sufficiently smooth on the polygon we use the corner singularity expansion and propose new extraction formulae for coefficients of singularities for the proposed numerical method we first try to find finite element solutions for the regular parts and then calculate approximations of the coefficients by the derived formulae expressed by the remainders and given functions we show error estimates of the approximations for the regular parts and coefficients and give some numerical experiments to confirm the efficiency and reliability of the proposed method

distributed δ 1 coloring in sublogarithmic rounds we give a new randomized distributed algorithm for 1 coloring in the local model running in o log 2o log log n rounds in a graph of maximum degree this implies that the 1 coloring problem is easier than the maximal independent set problem and the maximal matching problem due to their lower bounds of min log n log log n log log log by kuhn moscibroda and wattenhofer podc 04 our algorithm also extends to list coloring where the palette of each node contains 1 colors we extend the set of distributed symmetry breaking techniques by performing a decomposition of graphs into dense and sparse parts

an anisotropic pressure stabilized finite element method for incompressible flow problems we consider a pressure stabilized finite element approximation of incompressible flow problems in primitive velocity pressure variables which is based on a projection of the gradient of the discrete pressure onto the space of discrete functions equal order interpolation for the velocity and the pressure can be employed with this formulation the method introduced here is specially developed to be used on anisotropic finite element meshes with large element aspect ratios

a new finite element gradient recovery method superconvergence property this is the first in a series of papers in which a new gradient recovery method is introduced and analyzed it is proved that the method is superconvergent for translation invariant finite element spaces of any order the method maintains the simplicity efficiency and superconvergence properties of the zienkiewicz zhu patch recovery method in addition for uniform triangular meshes the method is superconvergent for the linear element under the chevron pattern and ultraconvergent at element edge centers for the quadratic element under the regular pattern applications of this new gradient recovery technique will be discussed in forthcoming papers

on a multiscale approach to the transient stokes problem dynamic subscales and anisotropic space time discretization abstract in this article we analyze some residual based stabilization techniques for the transient stokes problem when considering anisotropic time space discretizations we define an anisotropic time space discretization as a family of time space partitions that does not satisfy the condition h 2 c t with c uniform with respect to h and t standard residual based stabilization techniques are motivated by a multiscale approach approximating the effect of the subscales onto the large scales one of the approximations is to consider the subscales quasi static neglecting their time derivative it is well known that these techniques are unstable for anisotropic time space discretizations we show that the use of dynamic subscales where the subscales time derivatives are not neglected solves the problem and prove optimal convergence and stability results that are valid for anisotropic time space discretizations also the improvements related to the use of orthogonal subscales are addressed

a posteriori error estimates for the stokes problem an a posteriors error estimate is derived and analyzed for the mini element discretization of the stokes equations the estimate is based on the solution of a local stokes problem in each element of the finite element mesh using spaces of quadratic bump functions for both velocity and pressure errors this results in solving a 9 times 9 system which reduces to two easily invertible 3 times 3 systems comparisons with other estimates based on a petrov galerkin solution are used in this analysis which shows that it provides a reasonable approximation of the actual discretization error numerical experiments clearly show the efficiency of such an estimate in the solution of self adaptive mesh refinement procedures

deterministic distributed vertex coloring in polylogarithmic time consider an n vertex graph g v e of maximum degree and suppose that each vertex v v hosts a processor the processors are allowed to communicate only with their neighbors in g the communication is synchronous that is it proceeds in discrete rounds in the distributed vertex coloring problem the objective is to color g with 1 or slightly more than 1 colors using as few rounds of communication as possible the number of rounds of communication will be henceforth referred to as running time efficient randomized algorithms for this problem are known for more than twenty years alon et al 1986 luby 1986 specifically these algorithms produce a 1 coloring within o log n time with high probability on the other hand the best known deterministic algorithm that requires polylogarithmic time employs o 2 colors this algorithm was devised in a seminal focs 87 paper by linial 1987 its running time is o log n in the same article linial asked whether one can color with significantly less than 2 colors in deterministic polylogarithmic time by now this question of linial became one of the most central long standing open questions in this area in this article we answer this question in the affirmative and devise a deterministic algorithm that employs 1 o 1 colors and runs in polylogarithmic time specifically the running time of our algorithm is o f log log n for an arbitrarily slow growing function f 1 we can also produce an o 1 coloring in o log log n time for an arbitrarily small constant 0 and an o coloring in o e log n time for an arbitrarily small constant e 0 our results are in fact far more general than this in particular for a graph of arboricity a our algorithm produces an o a1 coloring for an arbitrarily small constant 0 in time o log a log n

locality in distributed graph algorithms this paper concerns a number of algorithmic problems on graphs and how they may be solved in a distributed fashion the computational model is such that each node of the graph is occupied by a processor which has its own id processors are restricted to collecting data from others which are at a distance at most t away from them in t time units but are otherwise computationally unbounded this model focuses on the issue of locality in distributed processing namely to what extent a global solution to a computational problem can be obtained from locally available data three results are proved within this model a 3 coloring of an n cycle requires time omega log n this bound is tight by previous work of cole and vishkin any algorithm for coloring the d regular tree of radius r which runs for time at most 2r 3 requires at least omega sqrt d colors in an n vertex graph of largest degree delta an o delta 2 coloring may be found in time o log n

for the stationary compressible viscous navier stokes equations with no slip condition on a convex polygon abstract our concern is with existence and regularity of the stationary compressible viscous navier stokes equations with no slip condition on convex polygonal domains note that u p 0 c c a constant is the eigenpair for the singular value 1 of the stokes problem on the convex sector it is shown that except the pair 0 c the leading order of the corner singularities for the nonlinear equations is the same as that of the stokes problem we split the leading corner singularity from the solution and show an increased regularity for the remainder as a consequence the pressure solution changes the sign at the convex corner and its derivatives blow up

achieve higher efficiency at maximum power with finite time quantum otto cycle the optimization of heat engines was intensively explored to achieve higher efficiency while maintaining the output power however most investigations were limited to a few finite time cycles e g the carnot like cycle due to the complexity of the finite time thermodynamics in this paper we propose a class of finite time engine with quantum otto cycle and demonstrate a higher achievable efficiency at maximum power the current model can be widely utilized benefitting from the general c 2 scaling of extra work for a finite time adiabatic process with long control time we apply the adiabatic perturbation method to the quantum piston model and calculate the efficiency at maximum power which is validated with an exact solution

multigrid methods for the computation of singular solutions and stress intensity factors i corner singularities we consider the poisson equation u f with homogeneous dirichlet boundary condition on a two dimensional polygonal domain with cracks multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed the convergence rate for the stress intensity factors is mathcal o h 3 2 in whenfel2 and mathcal o h 2 in whenfeh1 the convergence rate in the energy norm is mathcal o h 1 in in the first case and mathcal o h in the second case the costs of these multigrid methods are proportional to the number of elements in the triangulation the general case wherefehm is also discussed

viscous flow in domains with corners numerical artifacts their origin and removal abstract viscous flows in domains with boundaries forming two dimensional corners are considered we examine the case where on each side of the corner the boundary condition for the tangential velocity is formulated in terms of stress it is shown that computing such flows numerically by straightforwardly applying well tested algorithms and numerical codes based on their use such as comsol multiphysics can lead to spurious multivaluedness and mesh dependence in the distribution of the fluid s pressure the origin of this difficulty is that near a corner formed by smooth parts of the boundary in addition to the solution of the formulated inhomogeneous problem there also exists an eigensolution for obtuse corner angles this eigensolution a becomes dominant and b has a singular radial derivative of velocity at the corner despite the bulk pressure in the eigensolution being constant when the derivatives of the velocity are singular numerical errors in the velocities calculation near the corner give rise to pressure spikes whose magnitude increases as the mesh is refined a method is developed that uses the knowledge about the eigensolution to remove the artifacts in the pressure distribution the method is first explained in the simple case of a stokes flow in a corner region and then generalized for the navier stokes equations applied to describe steady and unsteady free surface flows encountered in problems of dynamic wetting

an absolutely stabilized finite element method for the stokes problem on presente une formulation a elements finis absolument stabilisee pour le probleme de stokes on etablit des estimations d erreurs optimales dans la norme l 2 pour l approximation des champs de vitesses et de pressions

stabilized mixed methods for the stokes problem the solution of the stokes problem is approximated by three stabilized mixed methods one due to hughes balestra and franca and the other two being variants of this procedure in each case the bilinear form associated with the saddle point problem of the standard mixed formulation is modified to become coercive over the finite element space error estimates are derived for each procedure

a unified stabilized method for stokes and darcy s equations we use the lowest possible approximation order piecewise linear continuous velocities and piecewise constant pressures to compute solutions to stokes equation and darcy s equation applying an edge stabilization term to avoid locking we prove that the formulation satisfies the discrete inf sup condition we prove an optimal a priori error estimate for both problems the formulation is then extended to the coupled case using a nitsche type weak formulation allowing for different meshes in the two subdomains finally we present some numerical examples verifying the theoretical predictions and showing the flexibility of the coupled approach

a fully discrete stabilized finite element method for the time dependent navier stokes equations in this article we consider a fully discrete stabilized finite element method based on two local gauss integrations for the two dimensional time dependent navier stokes equations it focuses on the lowest equal order velocity pressure pairs unlike the other stabilized method the present approach does not require specification of a stabilization parameter or calculation of higher order derivatives and always leads to a symmetric linear system the euler semi implicit scheme is used for the time discretization it is shown that the proposed fully discrete stabilized finite element method results in the optimal order bounds for the velocity and pressure

local projection stabilized galerkin approximations for the generalized stokes problem we analyze pressure stabilized finite element methods for the solution of the generalized stokes problem and investigate their stability and convergence properties an important feature of the method is that the pressure gradient unknowns can be eliminated locally thus leading to a decoupled system of equations although stability of the method has been established for the homogeneous stokes equations the proof given here is based on the existence of a special interpolant with additional orthogonal property with respect to the projection space this makes it a lot simpler and more attractive the resulting stabilized method is shown to lead to optimal rates of convergence for both velocity and pressure approximations

a finite element formulation for the stokes problem allowing equal velocity pressure interpolation abstract in this paper we study a variational formulation of the stokes problem that accommodates the use of equal velocity pressure finite element interpolations the motivation of this method relies on the analysis of a class of fractional step methods for the navier stokes equations for which it is known that equal interpolations yield good numerical results the reason for this turns out to be the difference between two discrete laplacian operators computed in a different manner the formulation of the stokes problem considered here aims to reproduce this effect from the analysis of the finite element approximation of the problem we obtain stability and optimal error estimates using velocity pressure interpolations satisfying a compatibility condition much weaker than the inf sup condition of the standard formulation in particular this condition is fulfilled by the most common equal order interpolations

improved distributed algorithms for coloring interval graphs with application to multicoloring trees we give a distributed 1 epsilon approximation algorithm for the minimum vertex coloring problem on interval graphs which runs in the mathcal local model and operates in mathrm o frac 1 epsilon log n rounds if nodes are aware of their interval representations then the algorithm can be adapted to the mathcal congest model using the same number of rounds prior to this work only constant factor approximations using mathrm o log n rounds were known 12 linial s ring coloring lower bound implies that the dependency on log n cannot be improved we further prove that the dependency on frac 1 epsilon is also optimal

convergence analysis of a residual local projection finite element method for the navier stokes equations this work presents and analyzes a new residual local projection stabilized finite element method relp for the nonlinear incompressible navier stokes equations stokes problems defined elementwise drive the construction of the residual based terms which make the present method stable for the finite element pairs mathbb p 1 mathbb p l l 0 1 numerical upwinding is incorporated through an extra control on the advective derivative and on the residual of the divergence equation well posedness of the discrete problem as well as optimal error estimates in natural norms are proved under standard assumptions next a divergence free velocity field is provided by a simple postprocessing of the computed velocity and pressure using the lowest order raviart thomas basis functions this updated velocity is proved to converge optimally to the exact solution numerics assess the theoretical results and validate the relp method

analysis of a pressure stabilized finite element approximation of the stationary navier stokes equations the purpose of this paper is to analyze a finite element approximation of the stationary navier stokes equations that allows the use of equal velocity pressure interpolation the idea is to introduce as unknown of the discrete problem the projection of the pressure gradient multiplied by suitable algorithmic parameters onto the space of continuous vector fields the difference between these two vectors pressure gradient and projection is introduced in the continuity equation the resulting formulation is shown to be stable and optimally convergent both in a norm associated to the problem and in the l 2 norm for both velocities and pressure this is proved first for the stokes problem and then it is extended to the nonlinear case all the analysis relies on an inf sup condition that is much weaker than for the standard galerkin approximation in spite of the fact that the present method is only a minor modification of this

an anisotropic gls stabilized finite element method for incompressible flow problems abstract we consider a stabilized finite element approximation of incompressible flow problems in primitive velocity pressure variables which is based on a gls formulation equal order interpolation for the velocity and the pressure can be employed with this formulation the method introduced here is specially developed to be used on anisotropic finite element meshes with large element aspect ratios rather than using a single stabilization parameter within each element the jacobian matrix of the affine transformation from the reference element is employed for stabilization purposes stability and anisotropic error estimates in a suitable mesh dependent norm are proved for the linear stokes problem numerical results confirm the accuracy and robustness of the proposed method compared to similar previous stabilized schemes

distributed coloring algorithms for triangle free graphs vertex coloring is a central concept in graph theory and an important symmetry breaking primitive in distributed computing whereas degree graphs may require palettes of 1 colors in the worst case it is well known that the chromatic number of many natural graph classes can be much smaller in this paper we give new distributed algorithms to find k coloring in graphs of girth 4 triangle free graphs girth 5 and trees the parameter k can be at most 1 4 o 1 ln in triangle free graphs and at most 1 o 1 ln in girth 5 graphs and trees where o 1 is a function of specifically for sufficiently large we can find such a coloring in o k log n time moreover for any we can compute such colorings in roughly logarithmic time for triangle free and girth 5 graphs and in o log log log n time on trees as a byproduct our algorithm shows that the chromatic number of triangle free graphs is at most 4 o 1 ln which improves on jamall s recent bound of 67 o 1 ln finally we show that 1 coloring for triangle free graphs can be obtained in sublogarithmic time for any

a finite element method for singular solutions of the navier stokes equations on a non convex polygon it is shown in choi and kweon 2013 that a solution of the navier stokes equations with no slip boundary condition on a non convex polygon can be written as u p c 1 1 1 c 2 2 2 u r p r near each non convex vertex where u r p r h 2 h 1 i i are corner singularity functions for the stokes problem with no slip condition and c i r are coefficients which are called the stress intensity factors we design a finite element method to approximate the coefficients c i and the regular part u r p r show the unique existence of the approximations and derive their error estimates some numerical examples are given confirming convergence rates for the approximations

a posteriori error estimators for the stokes equations we present two a posteriori error estimators for the mini element discretization of the stokes equations one is based on a suitable evaluation of the residual of the finite element solution the other one is based on the solution of suitable local stokes problems involving the residual of the finite element solution both estimators are globally upper and locally lower bounds for the error of the finite element discretization numerical examples show their efficiency both in estimating the error and in controlling an automatic self adaptive mesh refinement process the methods presented here can easily be generalized to the navier stokes equations and to other discretization schemes

approximation of the incompressible navier stokes equations using orthogonal subscale stabilization and pressure segregation on anisotropic finite element meshes abstract this paper describes a finite element model to solve the incompressible navier stokes equations based on the stabilization with orthogonal subscales and a pressure segregation the former consists of adding a least square form of the component orthogonal to the finite element space of the convective and pressure gradient terms this allows to deal with convection dominated flows and to use equal velocity pressure interpolation the pressure segregation is inspired in fractional step schemes although the converged solution corresponds to that of a monolithic time integration likewise we put special emphasis on the use of anisotropic grids in particular we describe some possible choices for the calculation of the element length that appears in the stabilization parameters and check their behavior in two classical numerical examples the preconditioning strategy used to solve the resulting algebraic system for very anisotropic meshes is also briefly described

regularity of solutions for the navier stokes system of incompressible flows on a polygon abstract we are concerned with singularities and regularities of solutions for the navier stokes system of incompressible flows on a polygonal domain with a concave vertex we subtract the corner singularities by the stokes operator from the solution velocity and pressure functions of the system it is shown that the stress intensity factors are functions of time variable belong to a fractional sobolev space on the time interval and can be expressed in terms of given data an increased regularity for the remainder is obtained

time dependent subscales in the stabilized finite element approximation of incompressible flow problems abstract in this paper we analyze a stabilized finite element approximation for the incompressible navier stokes equations based on the subgrid scale concept the essential point is that we explore the properties of the discrete formulation that results allowing the subgrid scales to depend on time this apparently natural idea avoids several inconsistencies of previous formulations and also opens the door to generalizations

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

pressure stabilization of finite element approximations of time dependent incompressible flow problems in this paper we develop and analyze pressure stabilized finite element methods for the solution of the transient stokes problem which is a linear model problem of transient incompressible flow a model for the bubble enrichment is proposed to stabilize the numerical solution and an implicit backward euler method is considered for the time approximation different methods are obtained depending on whether bubbles are allowed to evolve with time or they are considered quasi static stability estimates are provided for the two methods considered both for the velocity and the pressure solutions numerical results obtained on test cases are also given which confirm the theoretical stability results

continuous interior penalty finite element method for oseen s equations in this paper we present an extension of the continuous interior penalty method of douglas and dupont interior penalty procedures for elliptic and parabolic galerkin methods in computing methods in applied sciences lecture notes in phys 58 springer verlag berlin 1976 pp 207 216 to oseen s equations the method consists of a stabilized galerkin formulation using equal order interpolation for pressure and velocity to counter instabilities due to the pressure velocity coupling or due to a high local reynolds number we add a stabilization term giving l2 control of the jump of the gradient over element faces edges in two dimensions to the standard galerkin formulation boundary conditions are imposed in a weak sense using a consistent penalty formulation due to nitsche we prove energy type a priori error estimates independent of the local reynolds number and give some numerical examples recovering the theoretical results

lp theory for stokes and navier stokes equations with navier boundary condition abstract this work was intended as an attempt at studying stationary stokes and navier stokes problem with navier boundary conditions 1 3 we wish to investigate some results of existence uniqueness and regularity of solutions in hilbert case and in l p theory

error analysis of some galerkin least squares methods for the elasticity equations the recent technique of stabilizing mixed finite element methods by augmenting the galerkin formulation with least squares terms calculated separately on each element is considered the error analysis is performed in a unified manner yielding improved results for some methods introduced earlier in addition a new formulation is introduced and analyzed method is used within two different contexts the first is in connection with the plain strain or three dimensional problems near the incompressible limit for which the poisson ratio equals one half it is well known that for these problems a standard lower order displacement finite element method will break down by now it is also well known how a working method can be obtained a new unknown the pressure is introduced and discretized with a finite element space different from that used for the displacements the finite element method obtained falls into the class of saddlepoint problems for which there is a general theory developed by brezzi and babuska 5 7 the application of this theory shows that the method converges optimally if the discrete spaces for the displacement and pressure satisfy a stability condition for a particular choice of displacement and pressure approximations it can be a difficult task to verify this babuska brezzi or inf sup condition during the last decade this problem has been intensively studied and the performances of many combinations are now known cf 18 and 23 26 for some recent extensions in addition there exists a general technique utilizing bubble functions by which a stable method can

supercloseness and superconvergence of stabilized low order finite element discretizations of the stokes problem the supercloseness and superconvergence properties of stabilized finite element methods applied to the stokes problem are studied we consider consistent residual based stabilization methods as well as inconsistent local projection type stabilizations moreover we are able to show the supercloseness of the linear part of the mini element solution which has been previously observed in practical computations the results on supercloseness hold on three directional triangular axiparallel rectangular and brick type meshes respectively but extensions to more general meshes are also discussed applying an appropriate postprocess to the computed solution we establish superconvergence results numerical examples illustrate the theoretical predictions

a mathematical theorem as the basis for the second law thomson s formulation applied to equilibrium abstract there are several formulations of the second law and they may in principle have different domains of validity here a simple mathematical theorem is proven which serves as the most general basis for the second law namely the thomson formulation cyclic changes cost energy applied to equilibrium this formulation of the second law is a property akin to particle conservation normalization of the wave function it has been strictly proven for a canonical ensemble and made plausible for a micro canonical ensemble as the derivation does not assume time inversion invariance it is applicable to situations where persistent currents occur this clear cut derivation allows to revive the no perpetuum mobile in equilibrium formulation of the second law and to criticize some assumptions which are widespread in literature the result puts recent results devoted to foundations and limitations of the second law in proper perspective and structurizes this relatively new field of research

solution methods for the poisson equation with corner singularities numerical results in z cai and s kim siam j numer anal 39 2001 pp 286 299 we developed a new finite element method using singular functions for the poisson equation on a polygonal domain with re entrant angles such a method first computes the regular part of the solution then the stress intensity factor and finally the solution itself this paper studies solution methods for solving the system of linear equations arising from the discretization and focuses on numerical results including the finite element accuracy and the multigrid performance

coefficients of the singularities for elliptic boundary value problems on domains with conical points iii finite element methods on polygonal domains in the two first parts of this work rairo model math anal numer 24 1990 pp 27e52 rairo model math anal numer 24 1990 pp 343 367 formulas giving the coefficients arising in the singular expansion of the solutions of elliptic boundary value problems on nonsmooth domains are investigated now for the case of homogeneous strongly elliptic operators with constant coefficients on polygonal domains the solution of such problems by the finite element method is considered in order to approximate the solution or the coefficients different methods are used based on the expressions of the coefficients that were obtained in the first two parts the dual singular function method is also generalized

local projection stabilization of equal order interpolation applied to the stokes problem the local projection stabilization allows us to circumvent the babuska brezzi condition and to use equal order interpolation for discretizing the stokes problem the projection is usually done in a two level approach by projecting the pressure gradient onto a discontinuous finite element space living on a patch of elements we propose a new local projection stabilization method based on possibly enriched finite element spaces and discontinuous projection spaces defined on the same mesh optimal order of convergence is shown for pairs of approximation and projection spaces satisfying a certain inf sup condition examples are enriched simplicial finite elements and standard quadrilateral hexahedral elements the new approach overcomes the problem of an increasing discretization stencil and thus is simple to implement in existing computer codes numerical tests confirm the theoretical convergence results which are robust with respect to the user chosen stabilization parameter

a new family of stable elements for nearly incompressible elasticity based on a mixed petrov galerkin finite element formulation adding to the classical hellinger reissner formulation another residual form of the equilibrium equation a new petrov galerkin finite element method is derived it fits within the framework of a mixed finite element method and is proved to be stable for rather general combinations of stress and displacement interpolations including equal order discontinuous stress and continuous displacement interpolations which are unstable within the galerkin approach error estimates are presented using the babuska brezzi theory and numerical results confirm these estimates as well as the good accuracy and stability of the method

a distributed 2 e approximation for vertex cover in o log δ e log log δ rounds we present a simple deterministic distributed 2 e approximation algorithm for minimum weight vertex cover which completes in o log elog log rounds where is the maximum degree in the graph for any e 0 that is at most o 1 for a constant e this implies a constant approximation in o log log log rounds which contradicts the lower bound of kmw10

superconvergence of finite element approximations for the stokes problem by projection methods this paper derives a general superconvergence result for finite element approximations of the stokes problem by using projection methods proposed and analyzed recently by wang j math study 33 2000 pp 229 243 for the standard galerkin method the superconvergence result is based on some regularity assumption for the stokes problem and is applicable to any finite element method with regular but nonuniform partitions the method is proved to give a convergent scheme for certain finite element spaces which fail to satisfy the well known uniform inf sup condition of brezzi and babuska

stabilization of low order cross grid pkql mixed finite elements abstract in this paper we analyze a low order family of mixed finite element methods for the numerical solution of the stokes problem and a second order elliptic problem in two space dimensions in these schemes the pressure is interpolated on a mesh of rectangular elements while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals for the lowest order p 1 q 0 a global spurious pressure mode is shown to exist and so this element as p 1 q 1 case analyzed in armentano and blasco 2010 is unstable however following the ideas given in bochev et al 2006 a simple stabilization procedure can be applied when we approximate the solution of the stokes problem such that the new p 1 q 0 and p 1 q 1 methods are unconditionally stable and achieve optimal accuracy with respect to solution regularity with simple and straightforward implementations moreover we analyze the application of our p 1 q 1 element to the mixed formulation of the elliptic problem in this case by introducing the modified mixed weak form proposed in brezzi et al 1993 optimal order of accuracy can be obtained with our stabilized p 1 q 1 elements numerical results are also presented which confirm the existence of the spurious pressure mode for the p 1 q 0 element and the excellent stability and accuracy of the new stabilized methods

the locality of distributed symmetry breaking symmetry breaking problems are among the most well studied in the field of distributed computing and yet the most fundamental questions about their complexity remain open in this article we work in the local model where the input graph and underlying distributed network are identical and study the randomized complexity of four fundamental symmetry breaking problems on graphs computing miss maximal independent sets maximal matchings vertex colorings and ruling sets a small sample of our results includes the following an mis algorithm running in o log2 2o log log n time where is the maximum degree this is the first mis algorithm to improve on the 1986 algorithms of luby and alon babai and itai when log n l l 2 log n and comes close to the log log log lower bound of kuhn moscibroda and wattenhofer a maximal matching algorithm running in o log log 4log n time this is the first significant improvement to the 1986 algorithm of israeli and itai moreover its dependence on is nearly optimal a 1 coloring algorithm requiring o log 2o log log n time improving on an o log log n time algorithm of schneider and wattenhofer a method for reducing symmetry breaking problems in low arboricity degeneracy graphs to low degree graphs roughly speaking the arboricity or degeneracy of a graph bounds the density of any subgraph corollaries of this reduction include an o log n time maximal matching algorithm for graphs with arboricity up to 2 log n and an o log 2 3n time mis algorithm for graphs with arboricity up to 2 log n 1 3 each of our algorithms is based on a simple but powerful technique for reducing a randomized symmetry breaking task to a corresponding deterministic one on a poly log n size graph

edge stabilization for the generalized stokes problem a continuous interior penalty method in this note we introduce and analyze a stabilized finite element method for the generalized stokes equation stability is obtained by adding a least squares penalization of the gradient jumps acro

the optimal convergence rate of a c1 finite element method for non smooth domains we establish optimal up to arbitrary e 0 convergence rates for a finite element formulation of a model second order elliptic boundary value problem in a weightedh 2 sobolev space with 5th degree argyris elements this formulation arises while generalizing to the case of non smooth domains an unconditionally stable scheme developed by liu et al j g liu j liu r l pego stability and convergence of efficient navier stokes solvers via a commutator estimate comm pure appl math 60 2007 pp 1443 1487 for the navier stokes equations we prove the optimality for both quasiuniform and graded mesh refinements and provide numerical results that agree with our theoretical predictions

stabilized continuous and discontinuous galerkin techniques for darcy flow we design stabilized methods based on the variational multiscale decomposition of darcy s problem a model for the subscales is designed by using a heuristic fourier analysis this model involves a characteristic length scale that can go from the element size to the diameter of the domain leading to stabilized methods with different stability and convergence properties these sta bilized methods mimic the different possible functional settings of the continuous problem the optimal method depends on the velocity and pressure approximation order they also involve a subgrid projector that can be either the identity when applied to finite element residuals or can have an image orthogonal to the finite element space in pa rticular we have designed a new stabilized method that allows the use of piecewise constant pressures we consider a general setting in which velocity and pressure can be approximated by either continuous or discontinuous approximations all these methods have been analyzed proving stability and convergence results in some cases duality arguments have been used to obtain error bounds in the l 2 norm

penalty finite element method for the navier stokes equations abstract we present an analysis of a penalty formulation of the stationary navier stokes equations for an incompressible fluid subject to restrictions on the viscosity and prescribed body force it is shown that there exists a unique solution to this penalty problem the solution to the penalty problem is shown to converge to the solution of the navier stokes problem as o e where e 0 is the penalty parameter existence uniqueness and stability properties for the approximate problem are then developed and we derive estimates for finite element approximation of the penalized navier stokes problem presented here numerical studies are conducted to examine rates of convergence and sample numerical results presented for test cases

a nonconforming finite element method for the stationary navier stokes equations approximations to solutions of the inhomogeneous boundary value problem for the navier stokes equations are constructed via a nonstandard finite element method the velocity field is approximated using piecewise solenoidal functions that are totally discontinuous across interelement boundaries but which are pointwise divergence free on each element the pressure is approximated by c0 functions optimal rates of convergence results are obtained requiring only local quasi uniformity assumptions on the meshes

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

continuous interior penalty finite element method for the time dependent navier stokes equations space discretization and convergence this paper focuses on the numerical analysis of a finite element method with stabilization for the unsteady incompressible navier stokes equations incompressibility and convective effects are both stabilized adding an interior penalty term giving l 2 control of the jump of the gradient of the approximate solution over the internal faces using continuous equal order finite elements for both velocities and pressures in a space semi discretized formulation we prove convergence of the approximate solution the error estimates hold irrespective of the reynolds number and hence also for the incompressible euler equations provided the exact solution is smooth

coefficients of singularities and mixed methods for the mixed dirichlet neumann problem for the stokes operator on a polygon abstract the behaviour of the weak solution of the stokes problem on a polygon is considered with emphasis on the maximal regularity of the solution and on global formulae for the coefficients of singularities this regularity leads to a slow convergent mixed finite element method of fractional order less than one while the use of the above formulae provides better approximations for the solution and for the coefficients

a simple parallel algorithm for the maximal independent set problem two basic design strategies are used to develop a very simple and fast parallel algorithms for the maximal independent set mis problem the first strategy consists of assigning identical copies o

an l p theory for the n dimensional stationary compressible navier stokes equations and the incompressible limit for compressible fluids the equilibrium solutions in this paper we study the system 1 1 1 3 which describes the stationary motion of a given amount of a compressible heat conducting viscous fluid in a bounded domain ofr n n 2 hereu x is the velocity field x is the density of the fluid x is the absolute temperature f x andh x are the assigned external force field and heat sources per unit mass andp is the pressure in the physically significant case one hasg 0 we prove that for small data f g h there exists a unique solution u of problem 1 1 1 3 1 in a neighborhood of 0 m 0 for arbitrarily large data the stationary solution does not exist in general see sect 5 moreover we prove that for barotropic flows the stationary solution of the navier stokes equations 1 8 is the incompressible limit of the stationary solutions of the compressible navier stokes equations 1 7 as the mach number becomes small finally in sect 5 we will study the equilibrium solutions for system 4 1 for a more detailed explanation see the introduction

a finite volume method for stokes problems on quadrilateral meshes abstract we present a finite volume method for stokes problems using the isoparametric q 1 q 0 element pair on quadrilateral meshes to offset the lack of the i n f s u p condition a jump term of discrete pressure stabilizing term is added to the continuity approximation equation thus we establish a stabilized finite volume scheme on quadrilateral meshes then based on some superclose estimates we derive the optimal error estimates in the h 1 and l 2 norms for velocity and in the l 2 norm for pressure respectively numerical examples are provided to illustrate our theoretical analysis we emphasize that our work is the first time to propose and analyze a finite volume method for stoke problems using isoparametric elements on quadrilateral meshes

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

stable and unstable cross grid pkql mixed finite elements for the stokes problem in this paper we develop and analyze a family of mixed finite element methods for the numerical solution of the stokes problem in two space dimensions in these schemes the pressure is interpolated on a mesh of rectangular elements while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals continuous interpolations of degrees k for the velocity and l for the pressure are considered so the new finite elements are called cross grid p kq l a stability analysis of these approximations is provided based on the macroelement technique of stenberg the lowest order p 1q 1 and p 2q 1 cases are analyzed in detail in the first case a global spurious pressure mode is shown to exist so this element is unstable in the second case however stability is rigorously proved numerical results obtained in these two cases are also presented which confirm the existence of the spurious pressure mode for the p 1q 1 element and the stability of the p 2q 1 element

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

stabilization of low order mixed finite elements for the stokes equations we present a new family of stabilized methods for the stokes problem the focus of the paper is on the lowest order velocity pressure pairs while not lbb compliant their simplicity and attractive computational properties make these pairs a popular choice in engineering practice our stabilization approach is motivated by terms that characterize the lbb deficiency of the unstable spaces the stabilized methods are defined by using these terms to modify the saddle point lagrangian associated with the stokes equations the new stabilized methods offer a number of attractive computational properties in contrast to other stabilization procedures they are parameter free do not require calculation of higher order derivatives or edge based data structures and always lead to symmetric linear systems furthermore the new methods are unconditionally stable achieve optimal accuracy with respect to solution regularity and have simple and straightforward implementations we present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods

analysis of a streamline diffusion finite element method for the stokes and navier stokes equations for the stokes equations with convection and the incompressible navier stokes equations the authors analyze a streamline diffusion finite element method that is capable of balancing both the convection and the pressure thus allowing the use of arbitrary pairs of velocity pressure spaces for the linear problem the authors obtain for all mesh peclet numbers optimal error estimates in natural norms including in particular the l 2 norm of the pressure the same holds for the nonlinear problem which close to a regular branch of solutions i e the linearized operator is an isomorphism of the norm of the inverse of which still depends on the reynolds number consequently the dependence of the error constants on the reynolds number is not completely resolved in this case

symmetry breaking depending on the chromatic number or the neighborhood growth we deterministically compute a d 1 coloring and a maximal independent set mis in time o d 1 2 q 1 h log n for d 1 email protected email protected 1 i h where d j is defined as the maximal number of nodes within distance j for a node and d email protected 1 our greedy coloring and mis algorithms improve the state of the art algorithms running in o d log n for a large class of graphs i e graphs of moderate neighborhood growth with h 36 we also state and analyze a randomized coloring algorithm in terms of the chromatic number the run time and the used colors our algorithm runs in time o email protected log n for email protected email protected log 1 1 l o g nn and email protected o d log 1 1 l o g nn for graphs of polylogarithmic chromatic number the analysis reveals an exponential gap compared to the fastest d 1 coloring algorithm running in time o email protected logn the algorithm works without knowledge of g and uses less than d colors i e 1 1 o g d with high probability to the best of our knowledge this is the first distributed algorithm for such general graphs taking the chromatic number g into account we also improve on the state of the art deterministic computation of 2 c ruling sets

accurate solution of the thomasfermi equation using the fractional order of rational chebyshev functions in this paper the nonlinear singular thomasfermi differential equation for neutral atoms is solved using the fractional order of rational chebyshev orthogonal functions frcs of the first kind ftn t l on a semi infinite domain where l is an arbitrary numerical parameter first using the quasilinearization method the equation be converted into a sequence of linear ordinary differential equations ldes and then these ldes are solved using the frcs collocation method using 300 collocation points we have obtained a very good approximation solution and the value of the initial slope y 0 1 5880710226113753127186845094239501095 highly accurate to 37 decimal places

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

analysis of velocity flux first order system least squares principles for the navier stokes equations part i this paper develops a least squares approach to the solution of the incompressible navier stokes equations in primitive variables as with our earlier work on stokes equations we recast the navier stokes equations as a first order system by introducing a velocity flux variable and associated curl and trace equations we show that a least squares principle based on l2 norms applied to this system yields optimal discretization error estimates in the h1 norm in each variable including the velocity flux an analogous principle based on the use of an h 1 norm for the reduced system with no curl or trace constraints is shown to yield similar estimates but now in the l2 norm for velocity flux and pressure although the h 1 least squares principle does not allow practical implementation these results are critical to the analysis of a practical least squares method for the reduced system based on a discrete equivalent of the negative norm a practical method of this type is the subject of a companion paper finally we establish optimal multigrid convergence estimates for the algebraic system resulting from the l2 norm approach

construction and optimization of a quantum analog of the carnot cycle the quantum analog of carnot cycles in few particle systems consists of two quantum adiabatic steps and two isothermal steps this construction is formally justified by use of a minimum work principle it is then shown using minimal assumptions of work or heat in nanoscale systems that the heat to work efficiency of such quantum heat engine cycles can be further optimized via two conditions regarding the expectation value of some generalized force operators evaluated at equilibrium states in general the optimized efficiency is system specific lower than the carnot efficiency and dependent upon both temperatures of the cold and hot reservoirs simple computational examples are used to illustrate our theory the results should be an important guide towards the design of favorable working conditions of a realistic quantum heat engine

minimal work principle proof and counterexamples the minimal work principle states that work done on a thermally isolated equilibrium system is minimal for adiabatically slow reversible realization of a given process this principle one of the formulations of the second law is studied here for finite possibly large quantum systems interacting with macroscopic sources of work it is shown to be valid as long as the adiabatic energy levels do not cross if level crossing does occur counter examples are discussed showing that the minimal work principle can be violated and that optimal processes are neither adiabatically slow nor reversible the results are corroborated by an exactly solvable model