an efficient time splitting approximation of the navier stokes equations with lps modeling abstract in this work the solution of the navier stokes equations nse is addressed by a finite element fe local projection stabilization lps method combined with an efficient time splitting approximation strategy the focus is on the high order term by term stabilization method that has one level in the sense that it is defined on a single mesh and in which the projection stabilized structure of standard lps methods is replaced by an alternative interpolation stabilized structure the main contribution is on numerically analyzing for the cited lps method the proposed time approximation via stable velocity pressure segregation using semi implicit backward differentiation formulas bdf an overview about theoretical results from the numerical analysis of the proposed method is given by highlighting the used mathematical tools numerical studies support the analytical results and illustrate the potential of the method for the efficient and accurate simulation of turbulent flows on relatively coarse grids smooth unsteady flows are simulated with optimal order of accuracy

collision in a cross shaped domain a steady 2d navier stokes example demonstrating the importance of mass conservation in cfd abstract in the numerical simulation of the incompressible navier stokes equations different numerical instabilities can occur while instability in the discrete velocity due to dominant convection and instability in the discrete pressure due to a vanishing discrete ladyzhenskaya babuska brezzi lbb constant are well known instability in the discrete velocity due to a poor mass conservation at high reynolds numbers sometimes seems to be underestimated at least when using conforming galerkin mixed finite element methods like the taylor hood element the classical grad div stabilization for enhancing discrete mass conservation is often neglected in practical computations though simple academic flow problems showing the importance of mass conservation are well known these examples differ from practically relevant ones since specially designed force vectors are prescribed therefore we present a simple steady navier stokes problem in two space dimensions at reynolds number 1024 a colliding flow in a cross shaped domain where the instability of poor mass conservation is studied in detail and where no force vector is prescribed

stabilized finite element schemes with lbb stable elements for incompressible flows we study stabilized fe approximations of supg type to the incompressible navier stokes problem revisiting the analysis for the linearized model we show that for conforming lbb stable elements the design of the stabilization parameters for many practical flows differs from that commonly suggested in literature and initially designed for the case of equal order approximation then we analyze a reduced supg scheme often used in practice for lbb stable elements to provide the reduced scheme with appropriate stability estimates we introduce a modified lbb condition which is proved for a family of fe approximations the analysis is given for the linearized equations numerical experiments for some linear and nonlinear benchmark problems support the theoretical results

a right inverse for the divergence operator in spaces of piecewise polynomials in the first part of this paper we study in detail the properties of the divergence operator acting on continuous piecewise polynomials on some fixed triangulation more specifically we characterize the range and prove the existence of a maximal right inverse whose norm grows at most algebraically with the degree of the piecewise polynomials in the last part of this paper we apply these results to thep version of the finite element method for a nearly incompressible material with homogeneous dirichlet boundary conditions we show that thep version maintains optimal convergence rates in the limit as the poisson ratio approaches 1 2 this fact eliminates the need for any reduced integration such as customarily used in connection with the more standardh version of the finite element method

enforcing energy helicity and strong mass conservation in finite element computations for incompressible navier stokes simulations abstract we study a finite element scheme for the 3d navier stokes equations nse that globally conserves energy and helicity and through the use of scott vogelius elements enforces pointwise the solenoidal constraints for velocity and vorticity a complete numerical analysis is given including proofs for conservation laws unconditional stability and optimal convergence we also show the method can be efficiently computed by exploiting a connection between this method its associated penalty method and the method arising from using grad div stabilized taylor hood elements finally we give numerical examples which verify the theory and demonstrate the effectiveness of the scheme

grad div stablilization for stokes equations in this paper a stabilizing augmented lagrangian technique for the stokes equations is studied the method is consistent and hence does not change the continuous solution we show that this stabilization improves the well posedness of the continuous problem for small values of the viscosity coefficient we analyze the influence of this stabilization on the accuracy of the finite element solution and on the convergence properties of the inexact uzawa method

on the role of the helmholtz decomposition in mixed methods for incompressible flows and a new variational crime abstract according to the helmholtz decomposition the irrotational parts of the momentum balance equations of the incompressible navier stokes equations are balanced by the pressure gradient unfortunately nearly all mixed methods for incompressible flows violate this fundamental property resulting in the well known numerical instability of poor mass conservation the origin of this problem is the lack of l 2 orthogonality between discretely divergence free velocities and irrotational vector fields therefore a new variational crime for the nonconforming crouzeix raviart element is proposed where divergence free lowest order raviart thomas velocity reconstructions reestablish l 2 orthogonality this approach allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergence free flow solvers in the stokes case optimal a priori error estimates for the velocity gradients and the pressure are derived moreover the discrete velocity is independent of the continuous pressure several detailed linear and nonlinear numerical examples illustrate the theoretical findings

pressure robustness and discrete helmholtz projectors in mixed finite element methods for the incompressible navier stokes equations abstract recently it was understood how to repair a certain l 2 orthogonality of discretely divergence free vector fields and gradient fields such that the velocity error of inf sup stable discretizations for the incompressible stokes equations becomes pressure independent these new pressure robust stokes discretizations deliver a small velocity error whenever the continuous velocity field can be well approximated on a given grid on the contrary classical inf sup stable stokes discretizations can guarantee a small velocity error only when both the velocity and the pressure field can be approximated well simultaneously in this contribution pressure robustness is extended to the time dependent navier stokes equations in particular steady and time dependent potential flows are shown to build an entire class of benchmarks where pressure robust discretizations can outperform classical approaches significantly speedups will be explained by a new theoretical concept the discrete helmholtz projector of an inf sup stable discretization moreover different discrete nonlinear convection terms are discussed and skew symmetric pressure robust discretizations are proposed

automorphic forms with anisotropic periods on a unitary group let g be an algebraic group over a global field f with ring of adeles denote by z the center of g and by c an algebraic subgroup of g over f such that the cycle c f c has finite volume fix unitary characters x z z f 1 unit circle in and x c f c 1 and denote by x g f g a cusp form in the cuspidal representation of g whose central character is by a cuspidal representation we mean an irreducible one we say that is c cyclic if it has a nonzero c period pc c f c c c dc the overbar indicates complex conjugation studies of cyclic automorphic forms have applications to special values of l functions waldspurger w1 w2 jacquet j1 j2 lifting problems f and studies of cohomology of symmetric spaces in particular the tate conjecture on algebraic cycles on some shimura surfaces fh the purpose of this paper inspired by the applications to the tate conjecture of fh is to compare the notion of cyclicity by c with cyclicity by an inner form of c we let g be the quasi split unitary group u 2 1 u 2 1ye f in three variables defined by means of a quadratic separable extension e f of global fields the subgroup c is taken to be the

segregated runge kutta time integration of convection stabilized mixed finite element schemes for wall unresolved les of incompressible flows in this work we develop a high performance numerical framework for the large eddy simulation les of incompressible flows the spatial discretization of the nonlinear system is carried out using mixed finite element fe schemes supplemented with symmetric projection stabilization of the convective term and a penalty term for the divergence constraint these additional terms introduced at the discrete level have been proved to act as implicit les models in order to perform meaningful wall unresolved simulations we consider a weak imposition of the boundary conditions using a nitsche s type scheme where the tangential component penalty term is designed to act as a wall law next segregated runge kutta srk schemes recently proposed by the authors for laminar flow problems are applied to the les simulation of turbulent flows by the introduction of a penalty term on the trace of the acceleration these methods exhibit excellent stability properties for both implicit and explicit treatment of the convective terms srk schemes are excellent for large scale simulations since they reduce the computational cost of the linear system solves by splitting velocity and pressure computations at the time integration level leading to two uncoupled systems the pressure system is a darcy type problem that can easily be preconditioned using a traditional block preconditioning scheme that only requires a poisson solver at the end only coercive systems have to be solved which can be effectively preconditioned by multilevel domain decomposition schemes which are both optimal and scalable the framework is applied to the taylor green and turbulent channel flow benchmarks in order to prove the accuracy of the convection stabilized mixed fes as les models and srk time integrators the scalability of the preconditioning techniques in space only has also been proven for one step of the srk scheme for the taylor green flow using uniform meshes moreover a turbulent flow around a naca profile is solved to show the applicability of the proposed algorithms for a realistic problem

edge stabilization for galerkin approximations of convection diffusion reaction problems n this paper we recall a stabilization technique for finite element methods for convection diffusion reaction equations originally proposed by j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 207 216 lecture notes in phys 58 springer berlin 1976 mr0440955 55 13823 the method uses least square stabilization of the gradient jumps across element boundaries we prove that the method is stable in the hyperbolic limit and prove optimal a priori error estimates we address the question of monotonicity of discrete solutions and present some numerical examples illustrating the theoretical results

finding weighted k truss communities in large networks community search is a fundamental problem in social network mining which has attracted much attention in recent years however most previous community models only consider the link structure and ignore the link weights of the community which may miss some useful properties of the community in this paper we propose a novel community model called weightedk truss community based on the concept of k truss the proposed model takes the edge weight into consideration thus can better characterize the properties of a community based on the new community model we design a bfs based online search algorithm to find the top rweightedk truss communities in o m1 5 time where m denotes the number of edges in a network to speed up the online search algorithm we devise a space efficient index structure namely kep index to support efficient community search we propose two algorithms to construct the index structure in an offline manner based on kep index the time complexity for finding the top rweightedk truss communities is linear to the size of these communities thus it is optimal we conduct extensive experiments on six large real world networks as well as a case study over a co authorship network the results demonstrate the efficiency and effectiveness of the proposed community model and algorithms

a discontinuous skeletal method for the viscosity dependent stokes problem we devise and analyze arbitrary order nonconforming methods for the discretization of the viscosity dependent stokes equations on simplicial meshes we keep track explicitly of the viscosity and aim at pressure robust schemes that can deal with the practically relevant case of body forces with large curl free part in a way that the discrete velocity error is not spoiled by large pressures the method is inspired from the recent hybrid high order hho methods for linear elasticity after elimination of the auxiliary variables by static condensation the linear system to be solved involves only discrete face based velocities which are polynomials of degree k ge 0 and cell wise constant pressures our main result is a pressure independent energy error estimate on the velocity of order k 1 the main ingredient to achieve pressure independence is the use of a divergence preserving velocity reconstruction operator in the discretization of the body forces we also prove an l 2 pressure estimate of order k 1 and an l 2 velocity estimate of order k 2 the latter under elliptic regularity the local mass and momentum conservation properties of the discretization are also established finally two and three dimensional numerical results are presented to support the analysis

short note new connections between finite element formulations of the navier stokes equations we show the velocity solutions to the convective skew symmetric and rotational galerkin finite element formulations of the navier stokes equations are identical if scott vogelius elements are used and thus all three formulations will be the same pointwise divergence free solution velocity a connection is then established between the formulations for grad div stabilized taylor hood elements under mild restrictions the formulations velocity solutions converge to each other and to the scott vogelius solution as the stabilization parameter tends to infinity thus the benefits of using scott vogelius elements can be obtained with the less expensive taylor hood elements and moreover the benefits of all the formulations can be retained if the rotational formulation is used numerical examples are provided that confirm the theory

stabilized finite element methods for the generalized oseen problem the numerical solution of the non stationary incompressible navierstokes model can be split into linearized auxiliary problems of oseen type we present in a unique way different stabilization techniques of finite element schemes on isotropic meshes first we describe the state of the art for the classical residual based supg pspg method then we discuss recent symmetric stabilization techniques which avoid some drawbacks of the classical method these methods are closely related to the concept of variational multiscale methods which seems to provide a new approach to large eddy simulation finally we give a critical comparison of these methods

on twisted lifting if a is a generator of the galois group of a finite cyclic extension e f of local or global fields and e is a character of ce ex or ex ax whose restriction to cf has order n then the irreducible admissible or automorphic representations 1 cf denotes the multiplicative group fx if f is local and the idele class group fx a x if f is global put g gl and fix a character e of ce whose restriction to cf is of order n it determines a character e of g e or g e g ae by e g e det g the restriction of e to cf determines by class field theory a cyclic extension fe of f of degree n fix a generator a of the galois group gal e f denote by r g the set of equivalence classes of irreducible admissible or automorphic representations of g e or g ae then a acts on r g by v g vr a g g in g e or g a and we denote by re g the set of vr in r g with v n e theorem fftere ex s i a natural bijection between are g and the set of gal f f orbits of characters on nl fcl where l fee in the case e f this is due to kazhdan k in the local case and if it is cuspidal with two supercuspidal components kazhdan s proof is remarkably elegant com plete local and global results are derived from the study of the unramified case only our proof follows kazhdan s and uses kottwitz s base change lemma ko and arthur s explicit trace formula a we also show lemma 15 1 that the above 77 are induced ip r from a representation t t t e t e v e where e e 1 and t is a super cuspidal representation of gl w e n mm f is a parabolic subgroup with levi component gl m m then if l e n fe is an extension of f of degree we must have that divides m corollary 16 2 and m l is prime to e proposition 17 1 for example in the special case where e fe each it of the above type is induced from a character of a borel subgroup hence it is not in the discrete series we proved this result in f when e fe is a quadratic extension of f using the trace formula twisted as here in fact this motivated us to look for a twisted analogue of kazhdan s lifting further description of our vr is given in 16 and 17

on the design of algebraic flux correction schemes for quadratic finite elements abstract a fully algebraic approach to the design of nonlinear high resolution schemes is revisited and extended to quadratic finite elements the matrices resulting from a standard galerkin discretization are modified so as to satisfy sufficient conditions of the discrete maximum principle for nodal values in order to provide mass conservation the perturbation terms are assembled from skew symmetric internodal fluxes which are redefined as a combination of first and second order divided differences the new approach to the construction of artificial diffusion operators is combined with a node oriented limiting strategy the resulting algorithm is applied to p 1 and p 2 approximations of stationary convection diffusion equations in 1d 2d

grad div stabilization and subgrid pressure models for the incompressible navier stokes equations in this paper the grad div stabilization for the incompressible navier stokes finite element approximations is considered from two different viewpoints i as a variational multiscale approach for the pressure subgrid modeling and ii as a stabilization procedure of least square type some new error estimates for the linearized problem with the grad div stabilization are proved with the help of norms induced by the pressure schur complement operator we discuss the stabilization parameter choice arising in the frameworks of least square and multiscale methods and consider assumptions which allow to relate both approaches

on spurious oscillations at layers diminishing sold methods for convection diffusion equations part i a review abstract an unwelcome feature of the popular streamline upwind petrov galerkin supg stabilization of convection dominated convection diffusion equations is the presence of spurious oscillations at layers since the mid of the 1980s a number of methods have been proposed to remove or at least to diminish these oscillations without leading to excessive smearing of the layers the paper gives a review and state of the art of these methods discusses their derivation proposes some alternative choices of parameters in the methods and categorizes them some numerical studies which supplement this review provide a first insight into the advantages and drawbacks of the methods

a finite element variational multiscale method for the navier stokes equations this paper presents a variational multiscale method vms for the incompressible navier stokes equations which is defined by a large scale space lh for the velocity deformation tensor and a turbulent viscosity nu t the connection of this method to the standard formulation of a vms is explained the conditions on lh under which the vms can be implemented easily and efficiently into an existing finite element code for solving the navier stokes equations are studied numerical tests with the smagorinsky large eddy simulation model for nu t are presented

unitary quasi lifting applications let tj 3 be the quasi split unitary group in three variables defined using a quadratic extension e f of number fields complete local and global results are obtained for the o endo unstable lifting from u 2 to gl 3 e this is used to establish quasi endo lifting for automorphic forms from u 2 to u 3 by means of base change from uf3 to gl 3 e base change quasi lifting is also proven continuing the work of i the exposition is elementary and uses only a simple form of an identity of trace formulas and base change transfer of orbital integrals of spherical functions

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

on the convergence rate of grad div stabilized taylor hood to scott vogelius solutions for incompressible flow problems abstract it was recently proven in case et al 2010 2 that under mild restrictions grad div stabilized taylor hood solutions of navier stokes problems converge to the scott vogelius solution of that same problem however even though the analytical rate was only shown to be 1 2 where is the stabilization parameter the computational results suggest the rate may be improvable to 1 we prove herein the analytical rate is indeed 1 and extend the result to other incompressible flow problems including leray and mhd numerical results are given that verify the theory

pressure separation a technique for improving the velocity error in finite element discretisations of the navier stokes equations abstract this paper presents a technique to improve the velocity error in finite element solutions of the steady state navier stokes equations this technique is called pressure separation it relies upon subtracting the gradient of an appropriate approximation of the pressure on both sides of the navier stokes equations with this the finite element error estimate can be improved in the case of higher reynolds numbers for practical reasons the pressure separation can be applied above all for finite element discretisations of the navier stokes equations with piecewise constant pressure this paper presents a computational study of five ways to compute an appropriate approximation of the pressure these ways are assessed on two and three dimensional examples they are compared with respect to the error reduction in the discrete velocity and the computational overhead

an analysis of thep version of the finite element method for nearly incompressible materials in this paper we analyze the behavior of the so calledp version of the finite element method when applied to the equations of plane strain linear elasticity we establish optimal rate error estimates that are uniformly valid independent of the value of the poisson ratio v in the interval 0 1 2 this shows that thep versiondoes not exhibit the degeneracy phenomenon which has led to the use of various only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material

analysis of the pressure stabilized petrov galerkin method for the evolutionary stokes equations avoiding time step restrictions optimal error estimates for the pressure stabilized petrov galerkin method for the evolutionary stokes equations are proved in the case of regular solutions without restriction on the length of the time step these results clarify the instability of the discrete pressure for small time steps reported in the literature first the limit situation of the continuous in time discretization is considered and second the numerical analysis for the backward euler scheme is carried out the main results are applicable to higher order finite elements the analytical results are strongly based on an appropriate approach for computing the initial velocity which is suggested in this paper numerical studies confirm the theoretical results showing in particular that this instability does not occur for the proposed initial condition