experiments with implicit upwind methods for the euler equations abstract a number of implicit integration schemes for the one dimensional euler equations with conservative upwind spatial differencing are tested on a problem of steady discontinuous flow fastest convergence quadratic for the first order backward euler scheme is obtained with the upwind switching provided by van leer s differentiable split fluxes which easily linearize in time with roe s nondifferentiable split flux differences the iterations may get trapped in a limit cycle this also happens in a second order scheme with split fluxes if the matrix coefficients arising in the implicit time linearization are not properly centered in space the use of second order terms computed from split fluxes degrades the accuracy of the solution especially if these are subjected to a limiter for the sake of monotonicity preservation second order terms computed from the characteristic variables perform best

convergence and superconvergence of staggered discontinuous galerkin methods for the three dimensional maxwell s equations on cartesian grids in this paper a new type of staggered discontinuous galerkin methods for the three dimensional maxwell s equations is developed and analyzed the spatial discretization is based on staggered cartesian grids so that many good properties are obtained first of all our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the gauss law moreover the mass matrices are diagonal thus time marching is explicit and is very efficient our method is high order accurate and the optimal order of convergence is rigorously proved it is also very easy to implement due to its cartesian structure and can be regarded as a generalization of the classical yee s scheme as well as the quadrilateral edge finite elements furthermore a superconvergence result that is the convergence rate is one order higher at interpolation nodes is proved numerical results are shown to confirm our theoretical statements and applications to problems in unbounded domains with the use of pml are presented a comparison of our staggered method and non staggered method is carried out and shows that our method has better accuracy and efficiency

a mixed formulation for the brinkman problem the brinkman model is a unified law governing the flow of a viscous fluid in an inhomogeneous medium where fractures bubbles or channels alternate inside a porous matrix in this work we explore a novel mixed formulation of the brinkman problem based on the hodge decomposition of the vector laplacian introducing the flow s vorticity as an additional unknown this formulation allows for a uniformly stable and conforming discretization by standard finite elements nedelec raviart thomas piecewise discontinuous a priori error estimates for the discretization error in the h rm curl omega h rm div omega l 2 omega norm of the solution which are optimal with respect to the approximation properties of finite element spaces are obtained the theoretical results are illustrated with numerical experiments finally the proposed formulation allows for a scalable block diagonal preconditioner which takes advantage of the auxiliary space algebraic multigrid solvers for h rm curl and h

accurate three dimensional lid driven cavity flow a chebyshev collocation method in space is introduced which allows an accurate calculation of three dimensional lid driven cavity flows the time integration is carried out by an adams bashforth backward euler scheme the accuracy of the method relies on the representation of the solution as a superposition of stationary local asymptotic solutions and a residual flow field this way the most severe discontinuities in the boundary conditions which arise along the lines where moving and stationary walls meet are taken care of analytically and thus do not spoil the numerical part of the solution calculations are carried out for no slip boundary conditions at the cavity end walls as well as for periodic end wall conditions in general the spatial accuracy is better than fifth order for rigid end wall conditions the accuracy is reduced near the end walls to o n 3 2 but recovers in the bulk tabulated data are provided for the most interesting flow properties

a staggered discontinuous galerkin method for the stokes system discontinuous galerkin dg methods are a class of efficient tools for solving fluid flow problems there are in the literature many greatly successful dg methods in this paper a new staggered dg

discontinuous skeletal gradient discretisation methods on polytopal meshes in this work we develop arbitrary order discontinuous skeletal gradient discretisations dsgd on general polytopal meshes discontinuous skeletal refers to the fact that the globally coupled unknowns are broken polynomial on the mesh skeleton the key ingredient is a high order gradient reconstruction composed of two terms i a consistent contribution obtained mimicking an integration by parts formula inside each element and ii a stabilising term for which sufficient design conditions are provided an example of stabilisation that satisfies the design conditions is proposed based on a local lifting of high order residuals on a raviart thomas nedelec subspace we prove that the novel dsgds satisfy coercivity consistency limit conformity and compactness requirements that ensure convergence for a variety of elliptic and parabolic problems links with hybrid high order non conforming mimetic finite difference and non conforming virtual element methods are also studied numerical examples complete the exposition

an implicit high order hybridizable discontinuous galerkin method for the incompressible navier stokes equations we present an implicit high order hybridizable discontinuous galerkin method for the steady state and time dependent incompressible navier stokes equations the method is devised by using the discontinuous galerkin discretization for a velocity gradient pressure velocity formulation of the incompressible navier stokes equations with a special choice of the numerical traces the method possesses several unique features which distinguish itself from other discontinuous galerkin methods first it reduces the globally coupled unknowns to the approximate trace of the velocity and the mean of the pressure on element boundaries thereby leading to a significant reduction in the degrees of freedom moreover if the augmented lagrangian method is used to solve the linearized system the globally coupled unknowns become the approximate trace of the velocity only second it provides for smooth viscous dominated problems approximations of the velocity pressure and velocity gradient which converge with the optimal order of k 1 in the l2 norm when polynomials of degree k 0 are used for all components of the approximate solution and third it displays superconvergence properties that allow us to use the above mentioned optimal convergence properties to define an element by element postprocessing scheme to compute a new and better approximate velocity indeed this new approximation is exactly divergence free h div conforming and converges with order k 2 for k 1 and with order 1 for k 0 in the l2 norm moreover a novel and systematic way is proposed for imposing boundary conditions for the stress viscous stress vorticity and pressure which are not naturally associated with the weak formulation of the method this can be done on different parts of the boundary and does not result in the degradation of the optimal order of convergence properties of the method extensive numerical results are presented to demonstrate the convergence and accuracy properties of the method for a wide range of reynolds numbers and for various polynomial degrees

analysis of a pseudostress based mixed finite element method for the brinkman model of porous media flow in this paper we introduce and analyze a new mixed finite element method for the two dimensional brinkman model of porous media flow with mixed boundary conditions we use a dual mixed formulation in which the main unknown is given by the pseudostress the original velocity and pressure unknowns are easily recovered through a simple postprocessing in addition since the neumann boundary condition becomes essential we impose it in a weak sense which yields the introduction of the trace of the fluid velocity over the neumann boundary as the associated lagrange multiplier we apply the babuska brezzi theory to establish sufficient conditions for the well posedness of the resulting continuous and discrete formulations in particular a feasible choice of finite element subspaces is given by raviart thomas elements of order k ge 0 k 0 for the pseudostress and continuous piecewise polynomials of degree k 1 k 1 for the lagrange multiplier we also derive a reliable and efficient residual based a posteriori error estimator for this problem suitable auxiliary problems the continuous inf sup conditions satisfied by the bilinear forms involved a discrete helmholtz decomposition and the local approximation properties of the raviart thomas and clement interpolation operators are the main tools for proving the reliability then helmholtz s decomposition inverse inequalities and the localization technique based on triangle bubble and edge bubble functions are employed to show the efficiency finally several numerical results illustrating the performance and the robustness of the method confirming the theoretical properties of the estimator and showing the behaviour of the associated adaptive algorithm are provided

a new staggered dg method for the brinkman problem robust in the darcy and stokes limits abstract in this paper we propose a novel staggered discontinuous galerkin method for the brinkman problem on general polygonal meshes the proposed method is robust in the stokes and darcy limits in addition hanging nodes can be automatically incorporated in the construction of the method which are desirable features in practical applications there are three unknowns involved in our formulation namely velocity gradient velocity and pressure unlike the original staggered dg formulation proposed for the stokes equations in kim et al 2013 we relax the tangential continuity of velocity and enforce different staggered continuity properties for the three unknowns which is tailored to yield an optimal l 2 error estimates for velocity gradient velocity and pressure independent of the viscosity coefficient moreover by choosing suitable projection superconvergence can be proved for l 2 error of velocity finally several numerical results illustrating the good performances of the proposed method and confirming the theoretical findings are presented

a staggered semi implicit discontinuous galerkin method for the two dimensional incompressible navier stokes equations in this paper we propose a new spatially high order accurate semi implicit discontinuous galerkin dg method for the solution of the two dimensional incompressible navier stokes equations on staggered unstructured curved meshes while the discrete pressure is defined on the primal grid the discrete velocity vector field is defined on an edge based dual grid the flexibility of high order dg methods on curved unstructured meshes allows to discretize even complex physical domains on rather coarse grids formal substitution of the discrete momentum equation into the discrete continuity equation yields one sparse linear equation system with four non zero blocks per element for only one scalar unknown namely the pressure the method is computationally efficient since the resulting system is not only very sparse but also symmetric and positive definite for appropriate boundary conditions furthermore all the volume and surface integrals needed by the scheme presented in this paper depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessor stage which leads to savings in terms of computational effort for the time evolution part in this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step the method is validated for polynomial degrees up to p 3 by solving some typical numerical test problems and comparing the numerical results with available analytical solutions or other numerical and experimental reference data

colouring the rational quantum sphere and the kochen specker theorem we review and extend recent findings of godsil and zaks who published a constructive colouring of the rational unit sphere with the property that for any orthogonal tripod formed by rays extending from the origin of the points of the sphere exactly one ray is red one white and one black they also showed that any consistent colouring of the real sphere requires an additional colour we discuss some of the consequences for the kochen specker theorem

k theorie bivariante pour les algebres de banach et conjecture de baum connes je suis parvenu dans ma these a construire une k theorie bivariante pour les algebres de banach cela m a permis de demontrer la conjecture de baum connes pour les groupes de lie semi simples et les groupes reductifs p adiques ainsi que pour les sous groupes discrets de type fini de sl 3 f avec f une extension finie de q p et pour les sous groupes discrets cocompacts de sp n 1 sl 3 r et sl 3 c j ai aussi demontre une conjecture analogue a la conjecture de baum connes pour les algebres l 1 et plus generalement toutes les bonnes completions des sous groupes fermes des groupes de lie semi simples et des groupes reductifs p adiques

density conditions for quantum propositions as has already been pointed out by birkhoff and von neumann quantum logic can be formulated in terms of projective geometry in three dimensional hilbert space elementary logical propositions are associated with one dimensional subspaces corresponding to points of the projective plane it is shown that starting with three such propositions corresponding to some basis u v w successive application of the binary logical operation x y x y generates a set of elementary propositions which is countable infinite and dense in the projective plane if and only if no vector of the basis u v w is orthogonal to the other ones

a hybrid high order method for the incompressible navier stokes problem robust for large irrotational body forces abstract we develop a novel hybrid high order method for the incompressible navier stokes problem robust for large irrotational body forces the key ingredients of the method are discrete versions of the body force and convective contributions in the momentum equation formulated in terms of a globally divergence free velocity reconstruction two key properties are mimicked at the discrete level namely the invariance of the velocity with respect to irrotational body forces and the non dissipativity of the convective term a full convergence analysis is carried out showing optimal orders of convergence under a smallness condition involving only the solenoidal part of the body force the performance of the method is illustrated by a complete panel of numerical tests including comparisons that highlight the benefits with respect to more standard formulations

greechie diagrams nonexistence of measures in quantum logics and kochen specker type constructions we use greechie diagrams to construct finite orthomodular lattices realizable in the orthomodular lattice of subspaces in a three dimensional hilbert space such that the set of two valued states is not large i e full separating unital nonempty resp we discuss the number of elements of such orthomodular lattices of their sets of ortho generators and of their subsets that do not admit a large set of two valued states we show connections with other results of this type

on the algebraic k and l theory of word hyperbolic groups in this paper the assembly maps in algebraic k and l theory for the family of finite subgroups are proven to be split injections for word hyperbolic groups this is done by analyzing the compactification of the rips complex by the boundary of a word hyperbolic group

a stable numerical algorithm for the brinkman equations by weak galerkin finite element methods this paper presents a stable numerical algorithm for the brinkman equations by using weak galerkin wg finite element methods the brinkman equations can be viewed mathematically as a combination of the stokes and darcy equations which model fluid flow in a multi physics environment such as flow in complex porous media with a permeability coefficient highly varying in the simulation domain in such applications the flow is dominated by darcy in some regions and by stokes in others it is well known that the usual stokes stable elements do not work well for darcy flow and vice versa the challenge of this study is on the design of numerical schemes which are stable for both the stokes and the darcy equations this paper shows that the wg finite element method is capable of meeting this challenge by providing a numerical scheme that is stable and accurate for both darcy and the stokes dominated flows error estimates of optimal order are established for the corresponding wg finite element solutions the paper also presents some numerical experiments that demonstrate the robustness reliability flexibility and accuracy of the wg method for the brinkman equations

optimal discontinuous galerkin methods for wave propagation we have developed and analyzed a new class of discontinuous galerkin methods dg which can be seen as a compromise between standard dg and the finite element fe method in the way that it is explicit like standard dg and energy conserving like fe in the literature there are many methods that achieve some of the goals of explicit time marching unstructured grid energy conservation and optimal higher order accuracy but as far as we know only our new algorithms satisfy all the conditions we propose a new stability requirement for our dg the stability analysis is based on the careful selection of the two fe spaces which verify the new stability condition the convergence rate is optimal with respect to the order of the polynomials in the fe spaces moreover the convergence is described by a series of numerical experiments

equal order finite elements with local projection stabilization for the darcy brinkman equations abstract for the darcy brinkman equations which model porous media flow we present an equal order h1 conforming finite element method for approximating velocity and pressure based on a local projection stabilization technique the method is stable and accurate uniformly with respect to the coefficients of the viscosity and the zeroth order term in the momentum equation we prove a priori error estimates in a mesh dependent norm as well as in the l2 norm for velocity and pressure in particular we obtain optimal order of convergence in l2 for the pressure in the darcy case with vanishing viscosity and for the velocity in the general case with a positive viscosity coefficient numerical results for different values of the coefficients in the darcy brinkman model are presented which confirm the theoretical results and indicate nearly optimal order also in cases which are not covered by the theory

a pressure correction scheme for convection dominated incompressible flows with discontinuous velocity and continuous pressure in this work we present a pressure correction scheme for the incompressible navier stokes equations combining a discontinuous galerkin approximation for the velocity and a standard continuous galerkin approximation for the pressure the main interest of pressure correction algorithms is the reduced computational cost compared to monolithic strategies in this work we show how a proper discretization of the decoupled momentum equation can render this method suitable to simulate high reynolds regimes the proposed spatial velocity pressure approximation is lbb stable for equal polynomial orders and it allows adaptive p refinement for velocity and global p refinement for pressure the method is validated against a large set of classical two and three dimensional test cases covering a wide range of reynolds numbers in which it proves effective both in terms of accuracy and computational cost

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

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

an arbitrary order scheme on generic meshes for miscible displacements in porous media we design analyze and implement an arbitrary order scheme applicable to generic meshes for a coupled elliptic parabolic pde system describing miscible displacement in porous media the discretiza

the nonconforming virtual element method for the stokes equations we present the nonconforming virtual element method vem for the numerical approximation of velocity and pressure in the steady stokes problem the pressure is approximated using discontinuous piecewise polynomials while each component of the velocity is approximated using the nonconforming virtual element space on each mesh element the local virtual space contains the space of polynomials of up to a given degree plus suitable nonpolynomial functions the virtual element functions are implicitly defined as the solution of local poisson problems with polynomial neumann boundary conditions as typical in vem approaches the explicit evaluation of the non polynomial functions is not required this approach makes it possible to construct nonconforming virtual spaces for any polynomial degree regardless of the parity for two and three dimensional problems and for meshes with very general polygonal and polyhedral elements we show that the nonconforming vem is inf sup stable and establish optimal a pri

hybridizable discontinuous galerkin projection methods for navier stokes and boussinesq equations schemes for the incompressible navier stokes and boussinesq equations are formulated and derived combining the novel hybridizable discontinuous galerkin hdg method a projection method and implicit explicit runge kutta imex rk time integration schemes we employ an incremental pressure correction and develop the corresponding hdg finite element discretization including consistent edge space fluxes for the velocity predictor and pressure correction we then derive the proper forms of the element local and hdg edge space final corrections for both velocity and pressure including the hdg rotational correction we also find and explain a consistency relation between the hdg stability parameters of the pressure correction and velocity predictor we discuss and illustrate the effects of the time splitting error we then detail how to incorporate the hdg projection method time split within standard imex rk time stepping schemes our high order hdg projection schemes are implemented for arbitrary mixed element unstructured grids with both straight sided and curved meshes in particular we provide a quadrature free integration method for a nodal basis that is consistent with the hdg method to prevent numerical oscillations we develop a selective nodal limiting approach its applications show that it can stabilize high order schemes while retaining high order accuracy in regions where the solution is sufficiently smooth we perform spatial and temporal convergence studies to evaluate the properties of our integration and selective limiting schemes and to verify that our solvers are properly formulated and implemented to complete these studies and to illustrate a range of properties for our new schemes we employ an unsteady tracer advection benchmark a manufactured solution for the steady diffusion and stokes equations and a standard lock exchange boussinesq problem

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

virtual elements for the navier stokes problem on polygonal meshes a family of virtual element methods for the two dimensional navier stokes equations is proposed and analyzed the schemes provide a discrete velocity field which is pointwise divergence free a ri

staggered discontinuous galerkin methods for the helmholtz equation with large wave number abstract in this paper we investigate staggered discontinuous galerkin method for the helmholtz equation with large wave number on general polygonal meshes the method is highly flexible by allowing rough grids such as the trapezoidal grids and highly distorted grids and at the same time is numerical flux free furthermore it allows hanging nodes which can be simply treated as additional vertices by exploiting a modified duality argument the stability and convergence can be proved under the condition that h c 0 1 1 m 1 where m is the polynomial order is the wave number h is the mesh size and c 0 is a positive constant independent of h error estimates for both the scalar and vector variables in l 2 norm are established several numerical experiments are tested to verify our theoretical results and to present the capability of our method for capturing singular solutions

a nonconforming high order method for the biot problem on general meshes in this work we introduce a novel algorithm for the biot problem based on a hybrid high order discretization of the mechanics and a symmetric weighted interior penalty discretization of the flow the method has several assets including in particular the support of general polyhedral meshes and arbitrary space approximation order our analysis delivers stability and error estimates that hold also when the specific storage coefficient vanishes and shows that the constants have only a mild dependence on the heterogeneity of the permeability coefficient numerical tests demonstrating the performance of the method are provided

algebraic partial boolean algebras partial boolean algebras first studied by kochen and specker in the 1960s provide the structure for bell kochen specker theorems which deny the existence of non contextual hidden variable theories in this paper we study partial boolean algebras which are algebraic in the sense that their elements have coordinates in an algebraic number field several of these algebras have been discussed recently in a debate on the validity of bell kochen specker theorems in the context of finite precision measurements the main result of this paper is that every algebraic finitely generated partial boolean algebra b t is finite when the underlying space is three dimensional answering a question of kochen and showing that conway and kochen s infinite algebraic partial boolean algebra has minimum dimension this result contrasts the existence of an infinite non algebraic b t generated by eight elements in an abstract orthomodular lattice of height 3 we then initiate a study of higher dimensional algebraic partial boolean algebras first we describe a restriction on the determinants of the elements of b t that are generated by a given set t we then show that when the generating set t consists of the rays spanning the minimal vectors in a real irreducible root lattice b t is infinite just if that root lattice has an a5 sublattice finally we characterize the rays of b t when t consists of the rays spanning the minimal vectors of the root lattice e8

a discontinuous skeletal method for advection diffusion reaction on general meshes we design and analyze an approximation method for advection diffusion reaction equa tions where the generalized degrees of freedom are polynomials of order k 0 at mesh faces the method hinges on local discrete reconstruction operators for the diffusive and advective derivatives and a weak enforcement of boundary conditions fairly general meshes with poly topal and nonmatching cells are supported arbitrary polynomial orders can be considered including the case k 0 which is closely related to mimetic finite difference mixed hybrid finite volume methods the error analysis covers the full range of peclet numbers including the delicate case of local degeneracy where diffusion vanishes on a strict subset of the domain computational costs remain moderate since the use of face unknowns leads to a compact stencil with reduced communications numerical results are presented

a discontinuous galerkin method with nonoverlapping domain decomposition for the stokes and navier stokes problems a family of discontinuous galerkin finite element methods is formulated and analyzed for stokes and navier stokes problems an inf sup condition is established as well as optimal energy estimates for the velocity and l 2 estimates for the pressure in addition it is shown that the method can treat a finite number of nonoverlapping domains with nonmatching grids at interfaces

analysis of an sdg method for the incompressible navier stokes equations in this paper we analyze a staggered discontinuous galerkin sdg method for the incompressible navier stokes equations the method is based on a novel splitting of the nonlinear convection term and results in a skew symmetric discretization of it as a result the sdg discretization has a better conservation property and numerical stability property the aim of this paper is to develop a mathematical theory for this method in particular we will show that the sdg method is well posed and has an optimal rate of convergence a superconvergence result will also be shown

kochen specker theorem for finite precision spin one measurements unsharp spin one observables arise from the fact that a residual uncertainty about the actual orientation of the measurement device remains if the uncertainty is below a certain level and if the distribution of measurement errors is covariant under rotations a kochen specker theorem for the unsharp spin observables follows there are finite sets of directions such that not all the unsharp spin observables in these directions can consistently be assigned approximate truth values in a noncontextual way

noncontextual hidden variables and physical measurements meyer phys rev lett 83 3751 1999 recently showed that the kochen specker theorem which demonstrates the impossibility of a deterministic hidden variable description of ideal spin measurements on a spin 1 particle can be effectively nullified if only finite precision measurements are considered we generalize this result one can ascribe consistent outcomes to a dense subset of the set of projection valued measurements or to a dense subset of the set of positive operator valued measurements on any finite dimensional system hence no kochen specker like contradiction can rule out hidden variable theories indistinguishable from quantum theory by finite precision measurements in either class

a hybrid high order method for the incompressible navier stokes equations based on temam s device abstract in this work we propose a novel hybrid high order method for the incompressible navier stokes equations based on a formulation of the convective term including temam s device for stability the proposed method has several advantageous features it supports arbitrary approximation orders on general meshes including polyhedral elements and non matching interfaces it is inf sup stable it is locally conservative it supports both the weak and strong enforcement of velocity boundary conditions it is amenable to efficient computer implementations where a large subset of the unknowns is eliminated by solving local problems inside each element particular care is devoted to the design of the convective trilinear form which mimicks at the discrete level the non dissipation property of the continuous one the possibility to add a convective stabilisation term is also contemplated and a formulation covering various classical options is discussed the proposed method is theoretically analysed and an energy error estimate in h k 1 with h denoting the meshsize is proved under the usual data smallness assumption a thorough numerical validation on two and three dimensional test cases is provided both to confirm the theoretical convergence rates and to assess the method in more physical configurations including in particular the well known two and three dimensional lid driven cavity problems

a hybrid high order locking free method for linear elasticity on general meshes abstract we devise an arbitrary order locking free method for linear elasticity the method relies on a pure displacement primal formulation and leads to a symmetric positive definite system matrix with compact stencil the degrees of freedom are vector valued polynomials of arbitrary order k 1 on the mesh faces so that in three space dimensions the lowest order scheme only requires 9 degrees of freedom per mesh face the method can be deployed on general polyhedral meshes the key idea is to reconstruct the symmetric gradient and divergence inside each mesh cell in terms of the degrees of freedom by solving inexpensive local problems the discrete problem is assembled cell wise using these operators and a high order stabilization bilinear form locking free error estimates are derived for the energy norm and for the l 2 norm of the displacement with optimal convergence rates of order k 1 and k 2 respectively for smooth solutions on general meshes the theoretical results are confirmed numerically and the cpu cost is evaluated on both standard and polygonal meshes

a priori and a posteriori error analysis of a mixed scheme for the brinkman problem this paper deals with the analysis of new mixed finite element methods for the brinkman equations formulated in terms of velocity vorticity and pressure employing the babuska brezzi theory it is proved that the resulting continuous and discrete variational formulations are well posed in particular we show that raviart thomas elements of order k ge 0 k 0 for the approximation of the velocity field piecewise continuous polynomials of degree k 1 k 1 for the vorticity and piecewise polynomials of degree k for the pressure yield unique solvability of the discrete problem on the other hand we also show that families of finite elements based on brezzi douglas marini elements of order k 1 k 1 for the approximation of velocity piecewise continuous polynomials of degree k 2 k 2 for the vorticity and piecewise polynomials of degree k for the pressure ensure the well posedness of the associated galerkin scheme we note that these families provide exactly divergence free approximations of the velocity field we establish a priori error estimates in the natural norms with constants independent of the viscosity and we carry out the reliability and efficiency analysis of a residual based a posteriori error estimator finally we report several numerical experiments illustrating the behaviour of the proposed schemes and confirming our theoretical results on unstructured meshes additional examples of cases not covered by our theory are also presented

geometry of pseudodifferential algebra bundles and fourier integral operators we study the geometry and topology of filtered algebra bundles z over a smooth manifold x with typical fiber z z v the algebra of classical pseudodifferential operators acting on smooth sections of a vector bundle v over the compact manifold z and of integral order first a theorem of duistermaat and singer is generalized to the assertion that the group of projective invertible fourier integral operators pg fc z v is precisely the automorphism group of the filtered algebra of pseudodifferential operators we replace some of the arguments in their work by microlocal ones thereby removing the topological assumption we define a natural class of connections and b fields on the principal bundle to which z is associated and obtain a de rham representative of the dixmier douady class in terms of the outer derivation on the lie algebra and the residue trace of guillemin and wodzicki the resulting formula only depends on the formal symbol algebra z examples of pseudodifferential algebra bundles are given that are not associated to a finite dimensional fiber bundle over x

an artificial compressibility flux for the discontinuous galerkin solution of the incompressible navier stokes equations discontinuous galerkin dg methods have proved to be well suited for the construction of robust high order numerical schemes on unstructured and possibly nonconforming grids for a variety of problems their application to the incompressible navier stokes ins equations has also been recently considered although the subject is far from being fully explored in this work we propose a new approach for the dg numerical solution of the ins equations written in conservation form the inviscid numerical fluxes both in the continuity and in the momentum equation are computed using the values of velocity and pressure provided by the exact solution of the riemann problem associated with a local artificial compressibility perturbation of the equations unlike in most of the existing methods artificial compressibility is here introduced only at the interface flux level therefore resulting in a consistent discretization of the ins equations irrespectively of the amount of artificial compressibility introduced the discretization of the viscous term follows the well established dg scheme named br2 the performance and the accuracy of the method are demonstrated by computing the kovasznay flow and the two dimensional lid driven cavity flow for a wide range of reynolds numbers and for various degrees of polynomial approximation

equivariant lefschetz maps for simplicial complexes and smooth manifolds let x be a locally compact space with a continuous proper action of a locally compact group g assuming that x satisfies a certain kind of duality in equivariant bivariant kasparov theory we can enrich the classical construction of lefschetz numbers for self maps to an equivariant k homology class we compute the lefschetz invariants for self maps of finite dimensional simplicial complexes and smooth manifolds the resulting invariants are independent of the extra structure used to compute them since smooth manifolds can be triangulated we get two formulas for the same lefschetz invariant in this case the resulting identity is closely related to the equivariant lefschetz fixed point theorem of luck and rosenberg

a locally conservative ldg method for the incompressible navier stokes equations in this paper a new local discontinuous galerkin method for the incompressible stationary navier stokes equations is proposed and analyzed four important features render this method unique its stability its local conservativity its high order accuracy and the exact satisfaction of the incompressibility constraint although the method uses completely discontinuous approximations a globally divergence free approximate velocity in h div is obtained by simple element by element post processing optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge this procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible navier stokes equations by solving a sequence of oseen problems numerical results are shown which verify the theoretical rates of convergence they also confirm the independence of the number of fixed point iterations with respect to the discretization parameters finally they show that the method works well for a wide range of reynolds numbers

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

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

a robust finite element method for darcy stokes flow finite element methods for a family of systems of singular perturbation problems of a saddle point structure are discussed the system is approximately a linear stokes problem when the perturbation parameter is large while it degenerates to a mixed formulation of poisson s equation as the perturbation parameter tends to zero it is established basically by numerical experiments that most of the proposed finite element methods for stokes problem or the mixed poisson s system are not well behaved uniformly in the perturbation parameter this is used as the motivation for introducing a new robust finite element which exhibits this property

euler characteristics and gysin sequences for group actions on boundaries let g be a locally compact group let x be a universal proper g space and let open image in new window be a g equivariant compactification of x that is h equivariantly contractible for each compact subgroup open image in new window let open image in new window assuming the baum connes conjecture for g with coefficients open image in new window and c x we construct an exact sequence that computes the map on k theory induced by the embedding open image in new window this exact sequence involves the equivariant euler characteristic of x which we study using an abstract notion of poincare duality in bivariant k theory as a consequence if g is torsion free and the euler characteristic open image in new window is non zero then the unit element of open image in new window is a torsion element of order open image in new window furthermore we get a new proof of a theorem of luck and rosenberg concerning the class of the de rham operator in equivariant k homology

groups acting properly on bolic spaces and the novikov conjecture we introduce a class of metric spaces which we call bolic they include hyperbolic spaces simply connected complete manifolds of nonpositive curvature euclidean buildings etc we prove the novikov conjecture on higher signatures for any discrete group which admits a proper isometric action on a bolic weakly geodesic metric space of bounded geometry

kochen specker vectors some reference citations in the text of this paper are incorrect and should be amended as listed below the reference list is correct as published page 1579 line 5 23 34 38 39 should read 23 34 35 36 page 1579 line 9 40 should read 37 page 1580 line 8 31 33 35 should read 31 33 38 page 1580 line 14 36 should read 39 page 1580 line 36 36 should read 39 page 1580 line 38 37 should read 40 page 1580 line 42 38 should read 35 page 1581 line 3 of caption 33 35 should read 33 38 page 1583 line 12 41 should read 42 page 1587 line 9 39 should read 36 page 1587 line 11 39 should read 36 page 1587 line 18 38 39 should read 35 36 page 1587 line 37 39 should read 36 page 1589 line 18 39 46 47 should read 36 46 47 page 1590 line 2 39 should read 36 age 1590 line 3 39 should read 36

bivariant k theory via correspondences abstract we use correspondences to define a purely topological equivariant bivariant k theory for spaces with a proper groupoid action our notion of correspondence differs slightly from that of connes and skandalis our construction uses no special features of equivariant k theory to highlight this we construct bivariant extensions for arbitrary equivariant multiplicative cohomology theories we formulate necessary and sufficient conditions for certain duality isomorphisms in the topological bivariant k theory and verify these conditions in some cases including smooth manifolds with a smooth cocompact action of a lie group one of these duality isomorphisms reduces bivariant k theory to k theory with support conditions since similar duality isomorphisms exist in kasparov theory the topological and analytic bivariant k theories agree if there is such a duality isomorphism

a high order mimetic method on unstructured polyhedral meshes for the diffusion equation abstract we present a new family of mimetic finite difference schemes for solving elliptic partial differential equations in the primal form on unstructured polyhedral meshes these mimetic discretizations are built to satisfy local consistency and stability conditions the consistency condition is an exactness property i e the mimetic schemes are exact when the solution is a polynomial of an assigned degree the stability condition ensures the well posedness of the method the degrees of freedom are the solution moments on mesh faces and inside mesh cells higher order schemes are built using higher order moments the developed schemes are verified numerically on diffusion problems with constant and spatially variable possibly discontinuous tensorial coefficients

the l 2 index theorem for homogeneous spaces of lie groups the l2 index theorem for covering spaces of atiyah 3 and singer 20 asserts that given a discrete group f acting smoothly and freely on a manifold m with compact quotient m f m and an elliptic differential operator d on m which is f invariant and thus descends to an elliptic differential operator d on the compact manifold m then the p index indrd dimrkerd dimrkerd of d coincides with the ordinary index ind d dim ker d dim ker d of d here ker d is the space of l2 solutions of du 0 and dim r denotes the dimension function corresponding to the trace on the commutant of f acting on the hilbert space of l2 sections over m naturally associated to p the importance of this theorem lies in the fact that indd 0 implies the existence of nontrivial l2 solutions for the equation du 0 and as such it was used in a crucial way by atiyah and schmid 5 to construct explicit realizations of the discrete series representations for semisimple lie groups indeed if g is a lie group which possesses a discrete torsion free cocompact subgroup f and h is a compact subgroup of g then the l2 index theorem applied to the covering space m g h of m f g h combined with the index formula of atiyah singer 6 yields existence results for l2 solutions of g invariant elliptic equations on the homogeneous space g h it is relevant to note that with d denoting this time a g invariant elliptic differential operator on g h the ratio between the f index of d and the covolume of f gives a number independent of f this real number indg d can be in fact intrinsically defined as the difference of the two formal degrees corresponding to the representations of g on ker d and ker d respectively and hence makes sense for any unimodular lie group g even when it has no discrete cocompact subgroups which outside the semisimple case is the generic

low rank solvers for unsteady stokes brinkman optimal control problem with random data abstract we consider the numerical simulation of an optimal control problem constrained by the unsteady stokes brinkman equation involving random data more precisely we treat the state the control the target or the desired state as well as the viscosity as analytic functions depending on uncertain parameters this allows for a simultaneous generalized polynomial chaos approximation of these random functions in the stochastic galerkin finite element method discretization of the model the discrete problem yields a prohibitively high dimensional saddle point system with kronecker product structure we develop a new alternating iterative tensor method for an efficient reduction of this system by the low rank tensor train representation besides we propose and analyze a robust schur complement based preconditioner for the solution of the saddle point system the performance of our approach is illustrated with extensive numerical experiments based on two and three dimensional examples where the full problem size exceeds one billion degrees of freedom the developed tensor train scheme reduces the solution storage by two three orders of magnitude depending on discretization parameters

a posteriori error estimates for the virtual element method an posteriori error analysis for the virtual element method vem applied to general elliptic problems is presented the resulting error estimator is of residual type and applies on very general polygonal polyhedral meshes the estimator is fully computable as it relies only on quantities available from the vem solution namely its degrees of freedom and element wise polynomial projection upper and lower bounds of the error estimator with respect to the vem approximation error are proven the error estimator is used to drive adaptive mesh refinement in a number of test problems mesh adaptation is particularly simple to implement since elements with consecutive co planar edges faces are allowed and therefore locally adapted meshes do not require any local mesh post processing

a hybrid high order discretisation of the brinkman problem robust in the darcy and stokes limits abstract in this work we develop and analyse a novel hybrid high order discretisation of the brinkman problem the method hinges on hybrid discrete velocity unknowns at faces and elements and on discontinuous pressures based on the discrete unknowns we reconstruct inside each element a stokes velocity one degree higher than face unknowns and a darcy velocity in the raviart thomas nedelec space these reconstructed velocities are respectively used to formulate the discrete versions of the stokes and darcy terms in the momentum equation along with suitably designed penalty contributions the proposed construction is tailored to yield optimal error estimates that are robust throughout the entire spectrum of local stokes or darcy dominated regimes as identified by a dimensionless number which can be interpreted as a friction coefficient the singular limit corresponding to the darcy equation is also fully supported by the method numerical examples corroborate the theoretical results this paper also contains two contributions whose interest goes beyond the specific method and application treated in this work an investigation of the dependence of the constant in the second korn inequality on star shaped domains and its application to the study of the approximation properties of the strain projector in general sobolev seminorms

optimal discontinuous galerkin methods for the acoustic wave equation in higher dimensions in this paper we develop and analyze a new class of discontinuous galerkin dg methods for the acoustic wave equation in mixed form traditional mixed finite element fe methods produce energy conserving schemes but these schemes are implicit making the time stepping inefficient standard dg methods give explicit schemes but these approaches are typically dissipative or suboptimally convergent depending on the choice of numerical fluxes our new method can be seen as a compromise between these two kinds of techniques in the way that it is both explicit and energy conserving locally and globally moreover it can be seen as a generalized version of the raviart thomas fe method and the finite volume method stability and convergence of the new method are rigorously analyzed and we have shown that the method is optimally convergent furthermore in order to apply the new method for unbounded domains we apply our new method with the first order absorbing boundary condition the stability of the resulting numerical scheme is analyzed

e theory and kk theory for groups which act properly and isometrically on hilbert space a good deal of research in c algebra k theory in recent years has been devoted to the baum connes conjecture 3 which proposes a formula for the k theory of group c algebras that blends group homology with the representation theory of compact subgroups the conjecture has brought c algebra theory into close contact with manifold theory through its obvious similarity to the borel conjecture of surgery theory 9 31 and its links with the theory of positive scalar curvature 27 in addition there are points of contact with harmonic analysis particularly the tempered representation theory of semisimple groups although the proper relation between the baum connes conjecture and representation theory is not well understood the conjecture is most easily formulated for groups which are discrete and torsion free for such a group g there is a natural homomorphism

discrete functional analysis tools for discontinuous galerkin methods with application to the incompressible navier stokes equations two discrete functional analysis tools are established for spaces of piecewise polynomial functions on general meshes i a discrete counterpart of the continuous sobolev embeddings in both hilbertian and non hilbertian settings ii a compactness result for bounded sequences in a suitable discontinuous galerkin norm together with a weak convergence property for some discrete gradients the proofs rely on techniques inspired by the finite volume literature which differ from those commonly used in finite element analysis the discrete functional analysis tools are used to prove the convergence of discontinuous galerkin approximations of the steady incompressible navier stokes equations two discrete convective trilinear forms are proposed a non conservative one relying on temam s device to control the kinetic energy balance and a conservative one based on a nonstandard modification of the pressure