two level method based on finite element and crank nicolson extrapolation for the time dependent navier stokes equations a fully discrete two level finite element method the two level method is presented for solving the two dimensional time dependent navier stokes problem the method requires a crank nicolson extrapolation solution u h tau 0 p h tau 0 on a spatial time coarse grid j h tau 0 and a backward euler solution u h tau p h tau on a space time fine grid j h tau the error estimates of optimal order of the discrete solution for the two level method are derived compared with the standard crank nicolson extrapolation method the one level method based on a space time fine grid j h tau the two level method is of the error estimates of the same order as the one level method in the h1 norm for velocity and the l2 norm for pressure however the two level method involves much less work than the one level method

finite element error estimates for 3d exterior incompressible flow with nonzero velocity at infinity we consider a stationary incompressible navier stokes flow in a 3d exterior domain with nonzero velocity at infinity in order to approximate this flow we use the stabilized p1 p1 finite element method proposed by rebollo numer math 79 283 319 1998 following an approach by guirguis and gunzburger model math anal numer 21 445 464 1987 we apply this method to the navier stokes system with oseen term in a truncated exterior domain under a pointwise boundary condition on the artificial boundary this leads to a discrete problem whose solution approximates the exterior flow as is shown by error estimates

extended deligne lusztig varieties for general and special linear groups abstract we give a generalisation of deligne lusztig varieties for general and special linear groups over finite quotients of the ring of integers in a non archimedean local field previously a generalisation was given by lusztig by attaching certain varieties to unramified maximal tori inside borel subgroups in this paper we associate a family of so called extended deligne lusztig varieties to all tamely ramified maximal tori of the group moreover we analyse the structure of various generalised deligne lusztig varieties and show that the unramified varieties including a certain natural generalisation do not produce all the irreducible representations in general on the other hand we prove results which together with some computations of lusztig show that for sl 2 f q 2 with odd q the extended deligne lusztig varieties do indeed afford all the irreducible representations

regular characters of groups of type an over discrete valuation rings let o be a complete discrete valuation ring with finite residue field k of odd characteristic let g be a general or special linear group or a unitary group defined over o and let g denote its lie algebra for every positive integer p let k l be the l th principal congruence subgroup of g o a continuous irreducible representation of g o is called regular of level p if it is trivial on kl 1 and its restriction to kl kl 1 similar or equal to g k consists of characters with g k over bar stabiliser of minimal dimension in this paper we construct the regular characters of g o compute their degrees and show that the latter satisfy ennola duality we give explicit uniform formulae for the regular part of the representation zeta functions of these groups c 2017 published by elsevier inc

on energy inequality smoothness and large time behavior in l 2 for weak solutions of the navier stokes equations in exterior domains soit un domaine exterieur de r n n 3 4 a frontiere lisse sur le cylindre d espace temps 0 on considere le probleme de navier stokes non stationnaire u t u u u p f dans 0 u 0 dans 0 u 0 sur 0 u x 0 a x avec des donnees prescrites f et a

a simplified two level method for the steady navier stokes equations abstract a simplified two level method for solving the steady two dimensional incompressible navier stokes equations is investigated in this article the simplified two level method involves solving one small nonlinear navier stokes problem on the coarse mesh and one linear stokes problem hence linear with positive definite symmetric part on the fine mesh the uniform optimal error estimates with respect to large for the standard galerkin method and the two level method are proven finally some numerically tests to confirm the theoretical results of the simplified two level method and the standard galerkin method are provided

a conforming finite element method for the time dependent navier stokes equations we study a spatial discretization of the stokes problem in a domain of mathbb r 2 or mathbb r 3 by a finite element method the unknowns being the velocity and the pressure optimal error

characters of finite reductive lie algebras abstract for any finite group of lie type g q deligne and lusztig p deligne g lusztig representations of reductive groups over finite fields ann of math 2 103 1976 103 161 defined a family of virtual q l characters r t g of g q such that any irreducible character of g q is an irreducible constituent of at least one of the r t g in this paper we study analogues of this result for characters of the finite reductive lie algebra g q where g lie g motivated by the results of e letellier fourier transforms of invariant functions on finite reductive lie algebras lecture notes in math vol 1859 springer verlag 2005 and g lusztig representations of reductive groups over finite rings represent theory 8 2004 1 14 we define two families r t g and r t g of virtual q l characters of g q we prove that they coincide when is in general position and that they differ in general we verify that any character of g q appears in some r t g we conjecture that this is also true if r t g is replaced by r t g

a robust solver for a second order mixed finite element method for the cahn hilliard equation abstract we develop a robust solver for a second order mixed finite element splitting scheme for the cahn hilliard equation this work is an extension of our previous work in which we developed a robust solver for a first order mixed finite element splitting scheme for the cahn hilliard equation the key ingredient of the solver is a preconditioned minimal residual algorithm with a multigrid preconditioner whose performance is independent of the spatial mesh size and the time step size for a given interfacial width parameter the dependence on the interfacial width parameter is also mild

a structure preserving fem for the uniaxially constrained mathbf q tensor model of nematic liquid crystals we consider the one constant landau de gennes model for nematic liquid crystals the order parameter is a traceless tensor field mathbf q which is constrained to be uniaxial mathbf q s mathbf n otimes mathbf n d 1 mathbf i where mathbf n is a director field s in mathbb r is the degree of orientation and d ge 2 is the dimension building on similarities with the one constant ericksen energy we propose a structure preserving finite element method for the computation of equilibrium configurations we prove stability and consistency of the method without regularization and gamma convergence of the discrete energies towards the continuous one as the mesh size goes to zero we design an alternating direction gradient flow algorithm for the solution of the discrete problems and we show that such a scheme decreases the energy monotonically finally we illustrate the method s capabilities by presenting some numerical simulations in two and three dimensions including non orientable line fields

stabilized finite element method based on the crank nicolson extrapolation scheme for the time dependent navier stokes equations this paper provides an error analysis for the crank nicolson extrapolation scheme of time discretization applied to the spatially discrete stabilized finite element approximation of the two dimensional time dependent navier stokes problem where the finite element space pair x h m h for the approximation u n h p n h of the velocity u and the pressure p is constructed by the low order finite element the q 1 p 0 quadrilateral element or the p 1 p 0 triangle element with mesh size h error estimates of the numerical solution u n h p n h to the exact solution u t n p t n with t n 0 t are derived

numerical analysis of a modified finite element nonlinear galerkin method a fully discrete modified finite element nonlinear galerkin method is presented for the two dimensional equation of navier stokes type the spatial discretization is based on two finite element spaces xh and xh defined on a coarse grid with grid size h and a fine grid with grid size h h respectively the time discretization is based on the euler explicit scheme with respect to the nonlinear term we analyze the stability and convergence rate of the method comparing with the standard finite element galerkin method and the nonlinear galerkin method this method can admit a larger time step under the same convergence rate of same order hence this method can save a large amount of computational time finally we provide some numerical tests on this method the standard finite element galerkin method and the nonlinear galerkin method which are in a good agreement with the theoretical analysis

stability and convergence of iterative methods related to viscosities for the 2d 3d steady navier stokes equations abstract some finite element iterative methods related to viscosities are designed to solve numerically the steady 2d 3d navier stokes equations the two level finite element iterative methods are designed to solve numerically the steady 2d 3d navier stokes equations for a large viscosity such that a strong uniqueness condition holds the two level finite element iterative methods consist of using the stokes newton and oseen iterations of m times on a coarse mesh with mesh size h and computing the stokes newton and oseen correction of one time on a fine grid with mesh size h h moreover the one level oseen finite element iterative method based on a fine mesh with a small mesh size is designed to solve numerically the steady 2d 3d navier stokes equations for small viscosity such that a weak uniqueness condition holds the uniform stability and convergence of these methods with respect to and mesh sizes h and h and iterative times m are provided

a coupled ericksen allen cahn model for liquid crystal droplets abstract we present a model and discretization that couples the ericksen model of liquid crystals with variable degree of orientation to the allen cahn equations with a mass constraint the coupled system models liquid crystal droplets with anisotropic surface tension effects due to the liquid crystal molecular alignment the total energy consists of the ericksen energy phase field allen cahn energy and a weak anchoring energy that couples the liquid crystal to the diffuse interface we describe our discretization of the total energy along with a method to compute minimizers via a discrete gradient flow algorithm which has a strictly monotone energy decreasing property numerical experiments are given in three dimensions that illustrate a wide variety of droplet shapes that result from their interaction with defects

finite element approximation for the viscoelastic fluid motion problem in this article the variational formulation of the two dimensional viscoelastic fluid motion problem and its finite element approximation are considered an local error estimate for the velocity with h1 norm and the pressure with l2 norm is obtained and a uniform error estimate for the velocity and pressure with the above norms is provided if the given data satisfies the uniqueness condition

coupling boundary integral and finite element methods for the oseen coupled problem abstract in this paper we represent an oseen coupled problem and related numerical method for solving the nonstationary navier stokes problem in an unbounded domain the oseen coupled problem consists of a coupling between the navier stokes problem in an inner region and oseen problem in an outer region the related numerical method consists of coupling the boundary integral and the finite element method to solve the coupled problem the variational formulation of the coupled problem and its well posedness are obtained the optimal error estimates between the numerical solution of the oseen coupled problem and the exact solution of the navier stokes problem are provided

on the approximation of the unsteady navier stokes equations by finite element projection methods this paper provides an analysis of a fractional step projection method to compute incompressible viscous flows by means of finite element approximations the analysis is based on the idea that the appropriate functional setting for projection methods must accommodate two different spaces for representing the velocity fields calculated respectively in the viscous and the incompressible half steps of the method such a theoretical distinction leads to a finite element projection method with a poisson equation for the incremental pressure unknown and to a very practical implementation of the method with only the intermediate velocity appearing in the numerical algorithm error estimates in finite time are given an extension of the method to a problem with unconventional boundary conditions is also considered to illustrate the flexibility of the proposed method

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

flags and orbits of connected reductive groups over local rings we prove that generic higher deligne lusztig representations over truncated formal power series are non nilpotent when the parameters are non trivial on the biggest reduction kernel of the centre we also establish a relation between the orbits of higher deligne lusztig representations of mathrm sl n and of mathrm gl n then we introduce a combinatorial analogue of deligne lusztig construction for general and special linear groups over local rings this construction generalises the higher deligne lusztig representations and affords all the nilpotent orbit representations and for mathrm gl n it also affords all the regular orbit representations as well as the invariant characters of the lie algebra

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

calculation of incompressible viscous flows by an unconditionally stable projection fem this paper investigates the numerical performance of a finite element implementation of a new incremental fractional step method to compute steady and unsteady incompressible viscous flows under general boundary conditions and using unstructured meshes a variational framework is adopted which accommodates two different spaces for representing and approximating the velocity fields calculated respectively in the viscous and inviscid phases of the method but which leads to a very simple numerical scheme in terms of only one discrete velocity field an unconditionally stable semi implicit approximation of the nonlinear term is used to eliminate any time step restriction as far as the numerical stability is concerned numerical results for five test problems in two dimensions are reported to illustrate the flexibility of the proposed method

convergence analysis of a finite element projection lagrange galerkin method for the incompressible navier stokes equations this paper provides a convergence analysis of a fractional step method to compute incompressible viscous flows by means of finite element approximations in the proposed algorithm the convection the diffusion and the incompressibility are treated in three different substeps the convection is treated first by means of a lagrange galerkin technique whereas the diffusion and the incompressibility are treated separately in two subsequent substeps by means of a projection method it is shown that provided the time step delta t is of cal o h d 4 where h is the meshsize and d is the space dimension 2 leq d leq 3 the proposed method yields for finite time t an error of cal o h l 1 delta t in the l2 norm for the velocity and an error of cal o h l delta t in the h1 norm or the l2 norm for the pressure where l is the polynomial degree of the approximate velocity

felicity a matlab c toolbox for developing finite element methods and simulation modeling this paper describes a matlab c finite element toolbox called felicity for simulating various types of systems of partial differential equations e g coupled elliptic parabolic problems usin

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

multigrid methods for saddle point problems optimality systems abstract we develop multigrid methods for an elliptic distributed optimal control problem on convex domains that are robust with respect to a regularization parameter we prove the uniform convergence of the w cycle algorithm and demonstrate the performance of v cycle and w cycle algorithms through numerical experiments

long time unconditional stability of a two level hybrid method for nonstationary incompressible navier stokes equations abstract this paper presents a two level maccormack rapid solver method for solving a 2d time dependent navier stokes problem that models an incompressible fluid flow a decoupling approach based on the coupling term via spatial extrapolation is proposed for devising decoupled marching algorithms for the nonstationary incompressible model in this hybrid method an explicit maccormack scheme provides a h 1 optimal velocity approximation and a l 2 optimal pressure approximation by solving a nonlinear navier stokes problem on a coarse grid with mesh size h while an implicit crank nicolson algorithm consists in dealing with the fully discrete linear generalized stokes problem on a fine grid with mesh size h h the theoretical result suggests that our method is unconditionally stable over long time intervals while the numerical evidences both confirm the theoretic analysis and show that the algorithm is convergent the tests also suggest that our method is both cheaper and faster than the two level finite element galerkin scheme

representations of automorphism groups of finite o modules of rank two abstract let o be a complete discrete valuation domain with finite residue field in this paper we describe the irreducible representations of the groups aut m for any finite o module m of rank two the main emphasis is on the interaction between the different groups and their representations an induction scheme is developed in order to study the whole family of these groups coherently the results obtained depend on the ring o in a very weak manner mainly through the degree of the residue field in particular a uniform description of the irreducible representations of gl 2 o p l is obtained where p is the maximal ideal of o

convergence and stability of finite element nonlinear galerkin method for the navier stokes equations in this article we present a new fully discrete finite element nonlinear galerkin method which are well suited to the long time integration of the navier stokes equations spatial discretization is based on two grid finite element technique time discretization is based on euler explicit scheme with variable time step size moreover we analyse the boundedness convergence and stability condition of the finite element nonlinear galerkin method our discussion shows that the time step constraints of the method depend only on the coarse grid parameter h and the time step constraints of the finite element galerkin method depend on the fine grid parameter h h under the same convergence accuracy

numerics and hydrodynamic stability toward error control in computational fluid dynamics we critically review the available error analysis in computational fluid dynamics cfd and come to the conclusion that the existing error estimates are meaningless in most cases of interest we propose a new approach to error analysis in cfd aiming at reliable and efficient adaptive quantitative error control this is based on a precise analysis of hydrodynamic stability coupled with galerkin orthogonality we prove a priori and a posteriori type error estimates in a model case for pipe flow formulate corresponding adaptive algorithms and discuss the potential of this approach for adaptive error control in cfd

global ln solution and its decay property for the navier stokes equations in half space r n soit r n x 1 x 2 x n 1 x n x n 0 n 2 dans q r n 0 on considere les equations de navier stokes u t u u u 0 dans q divu 0 dans q u x u n 0 0 u 0 a on demontre l existence et l unicite d une l n solution globale forte pour des donnees initiales petites

approximating solutions of linear elliptic pde s on a smooth manifold using local kernel abstract a mesh free numerical method for solving linear elliptic pde s using the local kernel theory that was developed for manifold learning is proposed in particular this novel approach exploits the local kernel theory which allows one to approximate the kolmogorov operator associated with ito diffusion processes on compact riemannian manifolds without boundary or with neumann boundary conditions using an integral operator theoretical justification for the convergence of this numerical technique is provided under the standard conditions for the existence of the weak solutions of the pdes numerical results on various instructive examples ranging from pde s defined on flat and non flat manifolds with known and unknown embedding functions show accurate approximation with error on the order of the kernel bandwidth parameter