subdivision based mixed methods for isogeometric analysis of linear and nonlinear nearly incompressible materials this paper addresses the use of isogeometric analysis to solve solid mechanics problems involving nearly incompressible materials the present work is focused on extension of two field mixed variational formulations in both small and large strains to isogeometric analysis inf sup stable displacement pressure combinations for mixed formulations are developed based on the subdivision property of nurbs stability and convergence properties of the proposed displacement pressure combinations are illustrated by computing numerical inf sup constants and error norms the performance of the proposed formulations is assessed by studying several benchmark examples involving nearly incompressible and incompressible elastic and elasto plastic materials in both small and large strain regime

optimal error bounds for two grid schemes applied to the navier stokes equations abstract we consider two grid mixed finite element schemes for the spatial discretization of the incompressible navier stokes equations a standard mixed finite element method is applied over the coarse grid to approximate the nonlinear navier stokes equations while a linear evolutionary problem is solved over the fine grid the previously computed galerkin approximation to the velocity is used to linearize the convective term for the analysis we take into account the lack of regularity of the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data optimal error bounds are obtained

uniform preconditioners for a parameter dependent saddle point problem with application to generalized stokes interface equations we consider an abstract parameter dependent saddle point problem and present a general framework for analyzing robust schur complement preconditioners the abstract analysis is applied to a generalized stokes problem which yields robustness of the cahouet chabard preconditioner motivated by models for two phase incompressible flows we consider a generalized stokes interface problem application of the general theory results in a new schur complement preconditioner for this class of problems the robustness of this preconditioner with respect to several parameters is treated results of numerical experiments are given that illustrate robustness properties of the preconditioner

busemann s intersection inequality in hyperbolic and spherical spaces abstract busemann s intersection inequality asserts that the only maximizers of the integral s n 1 k n d among all convex bodies of a fixed volume in r n are centered ellipsoids we study this question in the hyperbolic and spherical spaces as well as general measure spaces

a unified mixed finite element approximations of the stokes darcy coupled problem abstract in this paper we develop and analyze a unified approximation of the velocity pressure pair for the stokes darcy coupled problem in a plane domain it is well known that stable finite element approximations for the stokes problem may not be appropriate for darcy problem and for the coupling of fluid flow modeled by the stokes equations with porous media flow modeled by the darcy equation and therefore different spaces are commonly used for the discretizations of the darcy and the stokes problems in this work we proposed a modification of the darcy problem which allows us to apply the classical mini element to the whole coupled stokes darcy problem the proposed method is probably one of the cheapest method for continuous approximation of the coupled system has optimal accuracy with respect to solution regularity and has simple and straightforward implementations numerical experiments are also presented which confirm the excellent stability and accuracy of our method

solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements a few years ago costabel and dauge proposed a variational setting which allows one to solve numerically the time harmonic maxwell equations in 3d polyhedral geometries with the help of a continuous approximation of the electromagnetic field in order to remove spurious eigenmodes their method required a parameterization of the variational formulation in order to avoid this difficulty we use a mixed variational setting instead of the parameterization which allows us to handle the divergence free constraint on the field in a straightforward manner the numerical analysis of the method is carried out and numerical examples are provided to show the efficiency of our approach

a new nonconforming galerkin scheme for the stokes problem partially circumventing the discrete babǔska brezzi condition abstract in this paper we introduce a new stable nonconforming galerkin scheme for the interior stokes problem our approach uses a dirichllety mapping d from the pressure space q l 0 2 into the velocity space h h 0 1 2 in order to make a suitable change of variables in the original mixed formulation this transformation yields an equivalent variational formulation whose associated bilinear form is strongly coercive on the product space h q we then replace the mapping d by its finite element approximation on a subspace h h of h and define the corresponding discrete scheme on a finite element subspace h h q h of h q the leads to a nonconforming galerkin method whose bilinear form is uniformly strongly coercive on any family h h q h h h 0 provided that h o h consequently although h o h implies the satisfaction of the discrete bab ska brezzi condition between h h and q h it turns out that this stability condition is not needed between the approximating spaces h h and q h therefore all the combinations of velocity and pressure subspaces with continuous lagrange elements including certainly equal order interpolations become stable in addition the cea estimate is valid for all h h 0 and hence the usual rates of convergence for the error are also derived finally we discuss the matrix formulation of the algorithm and show that the resulting galerkin equations reduce to two uncoupled linear systems whose stiffness matrices are positive definite

nonsymmetric macdonald polynomials and demazure characters we establish a connection between a specialization of the nonsymmetric macdonald polynomials and the demazure characters of the corresponding affine kac moody algebra this allows us to obtain a representation theoretical interpretation of the coefficients of the expansion of the specialized symmetric macdonald polynomials in the basis formed by the irreducible characters of the associated finite lie algebra

on rosenbrock methods for the time integration of nearly incompressible materials and their usage for nonlinear model reduction this paper deals with the time integration and nonlinear model reduction of nearly incompressible materials that have been discretized in space by mixed finite elements we analyze the structure of the equations of motion and show that a differential algebraic system of index 1 with a singular perturbation term needs to be solved in the limit case the index may jump to index 3 and thus renders the time integration into a difficult problem for the time integration we apply rosenbrock methods and study their convergence behavior for a test problem which highlights the importance of the well known scholz conditions for this problem class numerical tests demonstrate that such linear implicit methods are an attractive alternative to established time integration methods in structural dynamics we also combine the simulation of nonlinear materials with a model reduction step in order to prepare the ground for including such finite element structures as components in complex vehicle dynamics applications

a stabilised immersed boundary method on hierarchical b spline grids for fluid rigid body interaction with solid solid contact abstract an accurate efficient and robust numerical scheme is presented for the simulation of the interaction between flexibly supported rigid bodies and incompressible fluid flow with topology changes and solid solid contact the solution of the incompressible navier stokes equations is approximated by employing a stabilised formulation on cartesian grids discretised with hierarchical b splines the solid is modelled as a rigid body and represented by linear segments along its boundary kinematic conditions along the fluid rigid body interface are enforced weakly using nitsche s method while ghost penalty operators are employed to avoid excessive ill conditioning of the system matrix arising from small cut cells a staggered scheme is used for resolving the coupled fluid rigid body interaction the contact between moving or moving and fixed solid bodies is modelled with lagrange multipliers the excellent performance and wide range of applicability of the proposed scheme are demonstrated in a number of benchmark tests as well as industrially relevant model problems the examples cover the galloping phenomena particulate flow hydraulic check valves and a model turbine

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

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

accuracy enhancement for non isoparametric finite element simulations in curved domains application to fluid flow abstract among a few known techniques the isoparametric version of the finite element method for meshes consisting of curved triangles or tetrahedra is the one most widely employed to solve pdes with essential conditions prescribed on curved boundaries it allows to recover optimal approximation properties that hold for elements of order greater than one in the energy norm for polytopic domains however besides a geometric complexity this technique requires the manipulation of rational functions and the use of numerical integration we consider a simple alternative to deal with dirichlet boundary conditions that bypasses these drawbacks without eroding qualitative approximation properties in the present work we first recall the main principle this technique is based upon by taking as a model the solution of the poisson equation with quadratic lagrange finite elements then we show that it extends very naturally to viscous incompressible flow problems although the technique applies to any higher order velocity pressure pairing as an illustration a thorough study thereof is conducted in the framework of the stokes system solved by the classical taylor hood method

postprocessing finite element methods for the navier stokes equations the fully discrete case an accuracy enhancing postprocessing technique for finite element discretizations of the navier stokes equations is analyzed the technique had been previously analyzed only for semidiscretizations and fully discrete methods are addressed in the present paper we show that the increased spatial accuracy of the postprocessing procedure is not affected by the errors arising from any convergent time stepping procedure further refined bounds are obtained when the time stepping procedure is either the backward euler method or the two step backward differentiation formula

three dimensional finite element methods for the stokes problem the tetrahedral finite element approximation of the stokes problem is analyzed by means of polynomials piecewise of degree k 1 for the velocity and continuous piecewise of degree k for the pressure a stability result is given for every k ge 1

constant term identities extending the q dyson theorem andrews 1 has conjectured that the constant term in a certain product is equal to a q multinomial coefficient this conjecture is a q analogue of dyson s conjecture 5 and has been proved combinatorically by zeilberger and bressoud 15 in this paper we give a combinatorial proof of a master theorem that the constant term in a similar product computed over the edges of a nontransitive tournament is zero many constant terms are evaluated as consequences of this master theorem including andrews q dyson theorem in two ways one of which is a q analogue of good s 6 recursive proof

inf sup stable non conforming finite elements of arbitrary order on triangles we introduce a family of scalar non conforming finite elements of arbitrary order k 1 with respect to the h1 norm on triangles their vector valued version generates together with a discontinuous pressure approximation of order k 1 an inf sup stable finite element pair of order k for the stokes problem in the energy norm for k 1 the well known crouzeix raviart element is recovered

incompressible navier stokes solutions by a triangular spectral p element projection method the application of the fractional step projection method recently proposed by guermond and quartapelle to the numerical approximation of unsteady navier stokes solutions by means of a spectral p element method is considered in particular we illustrate the second order pressure correction technique and evaluate its accuracy properties in some test cases stability with respect to the compatibility condition between the approximation spaces for velocity and pressure is also addressed the high spectral accuracy in space and the second order accuracy in time are verified by two simple test cases with analytical solution a more interesting problem is solved showing the ability of the method to produce very accurate results also for problems in complex geometries

a mixed finite element method for nearly incompressible multiple network poroelasticity in this paper we present and analyze a new mixed finite element formulation of a general family of quasi static multiple network poroelasticity mpet equations the mpet equations describe flow a

analysis of finite element approximation of evolution problems in mixed form this paper deals with the finite element approximation of evolution problems in mixed form following d boffi f brezzi and l gastaldi math comp 69 2000 pp 121 140 we handle separately two types of problems a model for the first case is the heat equation in mixed form while the time dependent stokes problem fits within the second one for either case we give sufficient conditions for a good approximation in the natural functional spaces the results are not obvious in the first situation in this case the well known conditions for the well posedness and convergence of the corresponding steady problem are not sufficient for the good approximation of the time dependent problem this issue is demonstrated with a numerical counter example and justified analytically

the postprocessed mixed finite element method for the navier stokes equations a postprocessing technique for mixed finite element methods for the incompressible navier stokes equations is studied the technique was earlier developed for spectral and standard finite element methods for dissipative partial differential equations the postprocessing amounts to solving a stokes problem on a finer grid or higher order space once the time integration on the coarser mesh is completed the analysis presented here shows that this technique increases the convergence rate of both the velocity and the pressure approximations numerical experiments are presented that confirm both this increase in the convergence rate and the corresponding improvement in computational efficiency

analysis of a conforming finite element method for the boussinesq problem with temperature dependent parameters abstract in this paper we analyze a conforming finite element method for the numerical simulation of non isothermal incompressible fluid flows subject to a heat source modeled by a generalized boussinesq problem with temperature dependent parameters we consider the standard velocity pressure formulation for the fluid flow equations which is coupled with a primal mixed scheme for the convection diffusion equation modeling the temperature in this way the unknowns of the resulting formulation are given by the velocity the pressure the temperature and the normal derivative of the latter on the boundary hence assuming standard hypotheses on the discrete spaces we prove existence and stability of solutions of the associated galerkin scheme and derive the corresponding cea s estimate for small and smooth solutions in particular any pair of stable stokes elements such as hood taylor elements for the fluid flow variables continuous piecewise polynomials of degree k 1 for the temperature and piecewise polynomials of degree k for the boundary unknown becomes feasible choices of finite element subspaces finally we derive optimal a priori error estimates and provide several numerical results illustrating the performance of the conforming method and confirming the theoretical rates of convergence

the postprocessed mixed finite element method for the navier stokes equations refined error bounds a postprocessing technique for mixed finite element methods for the incompressible navier stokes equations is analyzed the postprocess which amounts to solving a linear stokes problem is shown to increase the order of convergence of the method to which it is applied by one unit times the logarithm of the mesh diameter in proving the error bounds some superconvergence results are also obtained contrary to previous analysis of the postprocessing technique in the present paper we take into account the loss of regularity suffered by the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data as in h g heywood and r rannacher siam j numer anal 25 1988 pp 489 512 where the same hypothesis is assumed no better than fifth order convergence is achieved

adaptive inexact iterative algorithms based on polynomial degree robust a posteriori estimates for the stokes problem in this paper we develop adaptive inexact versions of iterative algorithms applied to finite element discretizations of the linear stokes problem we base our developments on an equilibrated stress a posteriori error estimate distinguishing the different error components namely the discretization error component the inner algebraic solver error component and possibly the outer algebraic solver error component for algorithms of the uzawa type we prove that our estimate gives a guaranteed upper bound on the total error as well as a polynomial degree robust local efficiency and this on each step of the employed iterative algorithm our adaptive algorithms stop the iterations when the corresponding error components do not have a significant influence on the total error the developed framework covers all standard conforming and conforming stabilized finite element methods on simplicial and rectangular parallelepipeds meshes in two or three space dimensions and an arbitrary algebraic solver implementation into the freefem programming language is invoked and numerical examples showcase the performance of our a posteriori estimates and of the proposed adaptive strategies as example we choose here the unpreconditioned and preconditioned uzawa algorithm and the preconditioned minimum residual minres algorithm in combination with the taylor hood discretization

a new approach to constant term identities and selberg type integrals abstract selberg type integrals that can be turned into constant term identities for laurent polynomials arise naturally in conjunction with random matrix models in statistical mechanics built on a recent idea of karasev and petrov we develop a general interpolation based method that is powerful enough to establish many such identities in a simple manner the main consequence is the proof of a conjecture of forrester related to the calogero sutherland model in fact we prove a more general theorem which includes aomoto s constant term identity at the same time we also demonstrate the relevance of the method in additive combinatorics

a new family of projection schemes for the incompressible navier stokes equations with control of high frequency damping abstract a simple spatially discrete model problem consisting of mass points and dash pots is presented which allows for the assessment of the properties of different projection schemes for the solution of the incompressible navier stokes equations in particular the temporal accuracy the stability and the numerical damping are investigated the present study suggests that it is not possible to formulate a second order accurate projection pressure correction scheme which possesses any high frequency damping motivated by this observation two new families of projection schemes are proposed which are developed from the generalised midpoint rule and from the generalised method respectively and offer control over high frequency damping both schemes are investigated in detail on the basis of the model problem and subsequently implemented in the context of a finite element formulation for the incompressible navier stokes equations comprehensive numerical studies of the flow in a lid driven cavity and the flow around a cylinder are presented the observations made are in agreement with the conclusions drawn from the model problem

stability of higher order hood taylor methods the stability of a higher order hood taylor method for the approximation of the stationary stokes equations using continuous piecewise polynomials of degree 3 to approximate velocities and continuo

mathematical analysis and numerical approximation of a general linearized poro hyperelastic model abstract we describe the behavior of a deformable porous material by means of a poro hyperelastic model that has been previously proposed in chapelle and moireau 2014 under general assumptions for mass and momentum balance and isothermal conditions for a two component mixture of fluid and solid phases in particular we address here a linearized version of the model based on the assumption of small displacements we consider the mathematical analysis and the numerical approximation of the problem more precisely we carry out firstly the well posedness analysis of the model then we propose a numerical discretization scheme based on finite differences in time and finite elements for the spatial approximation stability and numerical error estimates are proved particular attention is dedicated to the study of the saddle point structure of the problem that turns out to be interesting because velocities of the fluid phase and of the solid phase are combined into a single quasi incompressibility constraint our analysis provides guidelines to select the componentwise polynomial degree of approximation of fluid velocity solid displacement and pressure to obtain a stable and robust discretization based on taylor hood type finite element spaces interestingly we show how this choice depends on the porosity of the mixture i e the volume fraction of the fluid phase

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

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

a posteriori error estimations for mixed finite element approximations to the navier stokes equations a posteriori estimates for mixed finite element discretizations of the navier stokes equations are derived we show that the task of estimating the error in the evolutionary navier stokes equations can be reduced to the estimation of the error in a steady stokes problem as a consequence any available procedure to estimate the error in a stokes problem can be used to estimate the error in the nonlinear evolutionary problem a practical procedure to estimate the error based on the so called postprocessed approximation is also considered both the semidiscrete in space and the fully discrete cases are analyzed some numerical experiments are provided

constant term identities and poincare polynomials in 1982 macdonald published his now famous constant term conjectures for classical root systems this paper begins with the almost trivial observation that macdonald s constant term identities admit an extra set of free parameters thereby linking them to poincare polynomials we then exploit these extra degrees of freedom in the case of type a to give the first proof of kadell s orthogonality conjecture a symmetric function generalisation of the q dyson conjecture or zeilberger bressoud theorem key ingredients in our proof of kadell s orthogonality conjecture are mul tivariable lagrange interpolation the scalar product for demazure characters and 0 1 matrices 2015 american mathematical society

the busemann petty problem in hyperbolic and spherical spaces abstract the busemann petty problem asks whether origin symmetric convex bodies in r n with smaller central hyperplane sections necessarily have smaller n dimensional volume it is known that the answer to this problem is affirmative if n 4 and negative if n 5 we study this problem in hyperbolic and spherical spaces

stable q k q k 1 mixed finite elements with discontinuous pressure mixed finite elements play a central role in many important cfd applications involving stokes solvers and alike a natural mixed finite element for the stokes equations is the q k q k 1 element on rectangular grids by which the velocity is approximated by continuous polynomials of separated degree k and the pressure is approximated by discontinuous polynomials of separated degree k 1 such an element is however not stable we propose in this paper three modified q k q k 1 elements with certain element wise divergence free property of velocity where the pressure space is slightly restricted to subspaces in c 1 q k 1 yet the optimal order of approximation is still retained the stability and approximation analysis for the new elements are presented comprehensive numerical experiments are also conducted to confirm the theoretical analysis and to reveal the super convergence for some of these new elements

on the locking of the finite element method in thermoelasticity in this article we discuss the solution u to the stationary problem of thermoelasticity and the approximation un of u obtained with a standard galerkin method in the incompressible limit when the lame coefficient lambda becomes infinite un is subject to a divergence constraint div un pnt where pn is a given approximation operator acting on the temperature field t this mechanism provokes a locking phenomenon similar to the one observed in elasticity and studied by babuska and suri in numer math 62 1992 pp 439 464 the purpose of this work is to give a sufficient and necessary condition on the discrete spaces vn used in the galerkin formulation to avoid locking and to get an estimation of the error u un 1 which is optimal and independent of lambda this criterion will be used to construct two practical examples of methods free from locking for the usual thermoelasticty problem where the temperature field is the solution of a poisson equation the first example is based on the h version of the finite element method the second one on the p version

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

a block preconditioner for high order mixed finite element approximations to the navier stokes equations an iterative solver for the linear system arising from a discretization by high order mixed finite elements for the linearized steady state navier stokes problem is presented two mixed finite elements are considered mathbf p k 1 p k and mathbf p k 2 p k the iterative solver is based on the gmres method with right preconditioning where the preconditioner has a block upper triangular form computational results show that the convergence rate seems to be independent of the grid size parameter h and the polynomial degree k but is slightly affected by the increasing of the reynolds number

subdivision stabilised immersed b spline finite elements for moving boundary flows an immersed finite element method is presented to compute flows with complex moving boundaries on a fixed cartesian grid the viscous incompressible fluid flow equations are discretized with b spline basis functions the two scale relation for b splines is used to implement an elegant and efficient technique to satisfy the lbb condition on non grid aligned fluid domains and at moving boundaries the boundary conditions are enforced with a consistent penalty method as originally proposed by nitsche in addition a special extrapolation technique is employed to prevent the loss of numerical stability in presence of arbitrarily small cut cells the versatility and accuracy of the proposed approach is demonstrated by means of convergence studies and comparisons with previous experimental and computational investigations

max norm estimates for stokes and navier stokes approximations in convex polyhedra we prove error estimates in the maximum norm namely in w 1 infty omega 3 times l infty omega w1 3 l for the stokes and navier stokes equations in convex three dimensional domains omega with simplicial boundaries we modify the weighted l 2 l2 estimates for regularized green functions used earlier by us which impose restrictions on the domain beyond convexity the new ingredient is a holder regularity estimate proved recently by v maz ya and j rossmann for the stokes system on polyhedra we also extend the error analysis to w 1 r omega 3 times l r omega w1 r 3 lr for 1

first layer formulas for characters of sl n c some problems concerning the decomposition of certain characters of sl n c are studied from a combinatorial point of view the specific characters considered include those of the exterior and symmetric algebras of the adjoint representation and the euler characteristic of hanlon s so called macdonald complex a general recursion is given for computing the irreducible decomposition of these characters the recursion is explicitly solved for the first layer representations which are the irreducible representations corresponding to partitions of n in the case of the exterior algebra this settles a conjecture of gupta and hanlon a further application of the recursion is used to give a family of formal laurent series identities that generalize the equal parameter q dyson theorem

a unified framework for a posteriori error estimation for the stokes problem in this paper a unified framework for a posteriori error estimation for the stokes problem is developed it is based on h 1 0 omega d conforming velocity reconstruction and underline boldsymbol h mathrm div omega conforming locally conservative flux stress reconstruction it gives guaranteed fully computable global upper bounds as well as local lower bounds on the energy error in order to apply this framework to a given numerical method two simple conditions need to be checked we show how to do this for various conforming and conforming stabilized finite element methods the discontinuous galerkin method the crouzeix raviart nonconforming finite element method the mixed finite element method and a general class of finite volume methods the tools developed and used include a new simple equilibration on dual meshes and the solution of local poisson type neumann problems by the mixed finite element method numerical experiments illustrate the theoretical developments

a posteriori error estimations for mixed finite element approximations to the navier stokes equations based on newton type linearization abstract in this paper we derive a posteriori error estimates for inf sup stable mixed finite element approximations to the evolutionary navier stokes equations we reduce the problem of getting a posteriori error estimations of a non linear evolutionary problem to that of getting a posteriori estimations of a linear steady problem the main idea is based on the fact that the solutions of some newton type linearized navier stokes equations around the plain galerkin approximations approach the solution of the original navier stokes equations with a bigger rate of convergence than the plain galerkin method as a consequence the difference between the galerkin approximations and the solution of the linearized problem is an estimator of the galerkin error moreover since the galerkin approximation of the evolutionary navier stokes equations is also the galerkin approximation of the linearized equations any a posteriori estimator of the error in the newton type linearized navier stokes equations is also an a posteriori error estimator of the full navier stokes equations

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

a term by term stabilization algorithm for finite element solution of incompressible flow problems this paper introduces a stabilization technique for finite element numerical solution of 2d and 3d incompressible flow problems it may be applied to stabilize the discretization of the pressure gradient and also of any individual operator term such as the convection curl or divergence operators with specific levels of numerical diffusion for each one of them its computational complexity is reduced with respect to usual residual based stabilization techniques we consider piecewise affine finite elements for which we obtain optimal error bounds for steady navier stokes and also for generalized stokes equations including convection we include some numerical experiment in well known 2d test cases that show its good performances

towards a unified finite element method for the stokes equations we present an algorithm for solving the stokes system based on the scott vogelius technique together with the iterated penalty method we show that by projecting the scott vogelius pressure onto

generalized postprocessed approximations to the navier stokes equations based on two grids abstract in this paper we generalize previous postprocessed approximations to the navier stokes equations the idea of postprocessing in the context of mixed finite element approximations is the following once a standard galerkin mixed finite element approximation is computed using a mesh we will call coarse of size h at a fixed time t we postprocess the approximation using a finer mesh of size h h by solving a steady problem with data based on the galerkin approximation in the literature one can find different ways to postprocess the galerkin approximation depending on the problem that is solved over the fine mesh at the final time t basically one can solve a stokes problem an oseen type problem or a newton type problem in the present paper we present a method based on several parameters that generalize the above postprocessing techniques depending on the values chosen for the different parameters one can recover one of the old known postprocessing procedures but also produce a new different method we get error bounds for the generalized method valid for any of the values of the different parameters in all the cases the postprocessed method has a rate of convergence one unit bigger than the rate of convergence of the plain galerkin method in terms of the coarse mesh h and optimal in terms of the fine mesh h the computational added cost of postprocessing is however negligible for the error analysis we do not assume nonlocal compatibility conditions for the true solutions of the navier stokes equations some numerical experiments show the performance of the method for different values of the parameters

the importance of the selberg integral it has been remarked that a fair measure of the impact of atle selberg s work is the number of mathematical terms that bear his name one of these is the selberg integral an n dimensional generalization of the euler beta integral we trace its sudden rise to prominence initiated by a question to selberg from enrico bombieri more than thirty years after its initial publication in quick succession the selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the macdonald conjectures it further initiated the study of q analogues which in turn enriched the macdonald conjectures we review these developments and proceed to exhibit the sustained prominence of the selberg integral as evidenced by its central role in random matrix theory calogero sutherland quantum many body systems knizhnik zamolodchikov equations and multivariable orthogonal polynomial theory

the dual brunn minkowski theory for bounded borel sets dual affine quermassintegrals and inequalities abstract this paper develops a significant extension of e lutwak s dual brunn minkowski theory originally applicable only to star shaped sets to the class of bounded borel sets the focus is on expressions and inequalities involving chord power integrals random simplex integrals and dual affine quermassintegrals new inequalities obtained include those of isoperimetric and brunn minkowski type a new generalization of the well known busemann intersection inequality is also proved particular attention is given to precise equality conditions which require results stating that a bounded borel set almost all of whose sections of a fixed dimension are essentially convex is itself essentially convex

a simple construction of a fortin operator for the two dimensional taylor hood element abstract a fortin operator is constructed to verify the discrete inf sup condition for the lowest order taylor hood element and its variant in two dimensions the approach is closely related to the recent work by mardal et al 2013 that is based on the isomorphism of the tangential edge bubble function space to a subspace of the lowest order edge element space a more precise characterization of this subspace and a numerical quadrature are introduced to simplify the analysis and remove the mesh restriction the constructed fortin operator is stable in both h 1 and l 2 norm for general shape regular triangulations

a consistent and stabilized continuous discontinuous galerkin method for fourth order incompressible flow problems this paper presents a new consistent and stabilized finite element formulation for fourth order incompressible flow problems the formulation is based on the c 0 interior penalty method the galerkin least square gls scheme which assures that the formulation is weakly coercive for spaces that fail to satisfy the inf sup condition and considers discontinuous pressure interpolations a stability analysis through a lemma establishes that the proposed formulation satisfies the inf sup condition thus confirming the robustness of the method this lemma indicates that at the element level there exists an optimal or quasi optimal gls stability parameter that depends on the polynomial degree used to interpolate the velocity and pressure fields the geometry of the finite element and the fluid viscosity term numerical experiments are carried out to illustrate the ability of the formulation to deal with arbitrary interpolations for velocity and pressure and to stabilize large pressure gradients

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

on stability accuracy and fast solvers for finite element approximations of the axisymmetric stokes problem by hood taylor elements we provide a proof of both the stability and the approximation property for the finite element approximations of the axisymmetric stokes problem by continuous piecewise polynomials of degree kappa 1 for the velocity and continuous piecewise polynomials of degree kappa for the pressure with any kappa geq1 new techniques are designed so that in this perspective by a simple transformation the existing theory developed in three dimensional cartesian coordinates can be effectively exploited in fact this perspective provides a new way of developing theories for the axisymmetric stokes problems and it can be applied potentially to other problems as well a simple illustration is provided for the application in the development and analysis of fast solvers for the resulting discrete saddle point problems sample numerical experiments have been presented as well to confirm the theoretical results

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

the busemann petty problem for arbitrary measures the busemann petty problem asks whether symmetric convex bodies in n with smaller n 1 dimensional volume of central hyperplane sections necessarily have smaller n dimensional volume the answer to this problem is affirmative for n 4 and negative for n 5 in this paper we generalize the busemann petty problem to essentially arbitrary measure in place of the volume we also present applications of the latter result by proving several inequalities concerning the measure of sections of convex symmetric bodies in n

new anisotropic a priori error estimates we prove a priori anisotropic estimates for the l 2 and h 1 interpolation error on linear finite elements the full information about the mapping from a reference element is employed to separate the contribution to the elemental error coming from different directions this new h 1 error estimate does not require the maximal angle condition the analysis has been carried out for the 2d case but may be extended to three dimensions numerical experiments have been carried out to test our theoretical results

computing electromagnetic eigenmodes with continuous galerkin approximations costabel and dauge proposed a variational setting to solve numerically the time harmonic maxwell equations in 3d polyhedral geometries with a continuous approximation of the electromagnetic field in order to remove spurious eigenmodes three computational strategies are then possible the original method which requires a parameterization of the variational formulation the second method which is based on an a posteriori filtering of the computed eigenmodes and the third method which uses a mixed variational setting so that all spurious modes are removed a priori in this paper we discuss the relative merits of the approaches which are illustrated by a series of 3d numerical examples 2008 elsevier b v all rights reserved

divergence free finite elements on tetrahedral grids for k 6 it was shown two decades ago that the p k p k 1 mixed element on triangular grids approximating the velocity by the continuous p k piecewise polynomials and the pressure by the discontinuous p k 1 piecewise polynomials is stable for all k 4 provided the grids are free of a nearly singular vertex the problem with the method in 3d was posted then and remains open the problem is solved partially in this work it is shown that the p k p k 1 element is stable and of optimal order in approximation on a family of uniform tetrahedral grids for all k 6 the analysis is to be generalized to non uniform grids when we can deal with the complicity of 3d geometry for the divergence free elements the finite element spaces for the pressure can be avoided in computation if a classic iterated penalty method is applied the finite element solutions for the pressure are computed as byproducts from the iterate solutions for the velocity numerical tests are provided

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

a stable and scalable hybrid solver for rate type non newtonian fluid models we present and analyze hybrid discretization schemes for rate type non newtonian fluids models the method employs higher order conforming approximations for velocity and pressure of the stokes equation and lower order approximations such as piecewise linear or piecewise constant for conformation tensor to reduce the accuracy gap the constitutive equation is discretized on a refined mesh which is obtained by subdividing the mesh for the velocity and pressure the temporal discretization is made by the standard semi lagrangian scheme the fully discrete nonlinear system is shown to be solved iteratively by applying the three steps 1 locating the characteristic feet of fluid particles 2 solving the constitutive equation and 3 solving the momentum and continuity equations stokes type equation to achieve a scalability of the solution process we employ an auxiliary space preconditioning method for the solution to the conforming finite element methods for the stokes equation this method is basically two grid method in which lower order finite element spaces are employed as auxiliary spaces it is shown to not only lead to the mesh independent convergence but also improve robustness and scalability stability analysis shows that if t o h d where d is the dimension of domain then the scheme admits a globally unique solution a number of full 3d test cases are provided to demonstrate the advantages of the proposed numerical techniques in relation to efficiency robustness and weak scalability

flag numbers and floating bodies abstract we investigate weighted floating bodies of polytopes we show that the weighted volume depends on the complete flags of the polytope this connection is obtained by introducing flag simplices which translate between the metric and combinatorial structure our results are applied in spherical and hyperbolic space this leads to new asymptotic results for polytopes in these spaces we also provide explicit examples of spherical and hyperbolic convex bodies whose floating bodies behave completely different from any convex body in euclidean space