convergence and nonlinear stability of the lagrange galerkin method for the navier stokes equations the lagrange galerkin method is a numerical technique for solving convection dominated diffusion problems based on combining a special discretisation of the lagrangian material derivative along particle trajectories with a galerkin finite element method we present optimal error estimates for the lagrange galerkin mixed finite element approximation of the navier stokes equations in a velocity pressure formulation the method is shown to be nonlinearly stable

the finite element method with nodes moving along the characteristics for convection diffusion equations we examine the convergence properties of the finite element method with nodes moving along the characteristics for one dimensional convection diffusion equations for linear elements we demonstrate optimal rates of convergence in thel 2 h 1 andl norms both linear and nonlinear problems are considered

a second order in time modified lagrange galerkin finite element method for the incompressible navier stokes equations we introduce a second order in time modified lagrange galerkin mlg method for the time dependent incompressible navier stokes equations the main ingredient of the new method is the scheme proposed to calculate in a more efficient manner the galerkin projection of the functions transported along the characteristic curves of the transport operator we present error estimates for velocity and pressure in the framework of mixed finite elements when either the mini element or the p2 p1 taylor hood element are used

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

discontinuous galerkin approximations of the stokes and navier stokes equations numerical schemes to compute approximate solutions of the evolutionary stokes and navier stokes equations are studied the schemes are discontinuous in time and conforming in space and of arbitrarily high order fully discrete error estimates are derived and dependence of the viscosity constant is carefully tracked it is shown that the errors are bounded by projection errors of the exact solution which exhibit optimal rates when the solutions are smooth

nonlinear oblique boundary value problems for nonlinear elliptic equations on considere le probleme aux valeurs limites a derivee oblique non lineaire pour des equations uniformement elliptiques quasi lineaires et totalement non lineaires d ordre 2 les operateurs elliptiques satisfont des conditions de structure naturelle on donne des theoremes d existence pour les solutions

characteristic stabilized finite element method for the transient navier stokes equations abstract based on the lowest equal order conforming finite element subspace xh mh i e p1 p1 or q1 q1 elements a characteristic stabilized finite element method for transient navier stokes problem is proposed the proposed method has a number of attractive computational properties parameter free avoiding higher order derivatives or edge based data structures and averting the difficulties caused by trilinear terms existence uniqueness and error estimates of the approximate solution are proved by applying the technique of characteristic finite element method finally a serious of numerical experiments are given to show that this method is highly efficient for transient navier stokes problem

a rotational velocity correction projection method for unsteady incompressible magnetohydrodynamics equations abstract this paper provides a rotational velocity correction projection method for the unsteady incompressible magnetohydrodynamics mhd equations in this method the electrical field variable e was eliminated in this way the gauss s law can be conserved the velocity variable u and the pressure field p are solved by a rotational velocity correction projection method the optimal error estimates of the l 2 norm will be given finally we shall present some numerical results to confirm our analysis it shows that our method is effective and can conserve the gauss s law very well

time stepping along characteristics with incomplete iteration for a galerkin approximation of miscible displacement in porous media miscible displacement of one incompressible fluid by another in a porous medium is modeled by a nonlinear coupled system of two partial differential equations the pressure equation is elliptic while the concentration equation is parabolic but normally convection dominated a sequential backward difference time stepping scheme is defined it approximates the pressure by a standard galerkin procedure and the concentration by a combination of a galerkin method and the method of characteristics optimal convergence rates in l 2 and h 1 are demonstrated for this scheme and for a modified version in which the algebraic equations at each time step are solved approximately by a limited number of preconditioned conjugate gradient iterations time stepping along the characteristics of the hyperbolic part of the concentration equation is shown to result in smaller time truncation errors than those of standard methods numerical results published elsewhere have confirmed that larger time steps are appropriate

defect correction finite element method for the stationary incompressible magnetohydrodynamics equation in this study we give the defect correction finite element method for the stationary incompressible mhd equation firstly the nonlinear mhd equation is solved with an artificial viscosity term then the numerical solution is improved on the same grid by a linearized defect correction technique then we give the numerical analysis including stability analysis and error analysis the numerical analysis proves that our method is stable and has an optimal convergence rate then we give some numerical results from the numerical results we can see that our method is efficient for solving the mhd equations

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

analysis of mixed finite elements methods for the stokes problem a unified approach we develop a method for the analysis of mixed finite element methods for the stokes problem in the velocity pressure formulation a technical macroelement condition which is sufficient for the classical babuska brezzi inequality to be valid is introduced using this condition we are able to verify the stability and optimal order of convergence of several known mixed finite element methods

a galerkin characteristic unified finite element method for moving thermal fronts in porous media abstract we investigate the performance of a unified finite element method for the numerical solution of moving fronts in porous media under non isothermal flow conditions the governing equations consist of coupling the darcy equation for the pressure to two convection diffusion reaction equations for the temperature and depth of conversion the aim is to develop a non oscillatory unified galerkin characteristic method for efficient simulation of moving fronts in porous media the method is based on combining the modified method of characteristics with a galerkin finite element discretization of the governing equations the main feature of the proposed unified finite element method is that the same finite element space is used for all solutions to the problem including the pressure velocity temperature and concentration analysis of convergence and stability is also presented in this study and error estimates in the l 2 norm are established for the numerical solutions in addition due to the lagrangian treatment of convection terms the standard courant friedrichs lewy condition is relaxed and the time truncation errors are reduced in the diffusion reaction part we verify the method for the benchmark problem of moving fronts around an array of cylinders the numerical results obtained demonstrate the ability of the proposed method to capture the main flow features

a second order characteristics finite element scheme for natural convection problems in this paper a second order characteristics finite element scheme is applied to the numerical solution of natural convection problems firstly after recalling the mathematical model a second order time discretization of the material time derivative is introduced next fully discretized schemes are proposed by using finite element methods numerical results for the two dimensional problem of buoyancy driven flow in a square cavity with differentially heated side walls are given and compared with a reference solution

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

on quotients of hom functors and representations of finite general linear groups i abstract this is a second article on quotients of hom functors and their applications to the representation theory of finite general linear groups in a nondescribing characteristic after some general results on quotients of hom functors and their connection to the harish chandra theory these constructions are used to obtain a full classification of the l modular irreducible representations of gl n q for some prime power q which is not divisible by the prime l and to explain some facts on their harish chandra series and decomposition numbers

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

convergence analysis of three finite element iterative methods for the 2d 3d stationary incompressible magnetohydrodynamics abstract three finite element iterative methods are designed and analyzed for solving 2d 3d stationary incompressible magnetohydrodynamics mhd by a new technique strong uniqueness conditions for both stokes type iterative method iterative method i and newton iterative method iterative method ii are obtained which are weaker than the ones reported in open literature stability and optimal convergence rates for the above two methods are derived where the iterative method ii has an exponential convergent part with respect to iterative step m moreover oseen type iterative method iterative method iii is unconditionally stable and convergent under a uniqueness condition finally performance of the three proposed methods is investigated by numerical experiments

uniform estimates for characteristics mixed finite method for transient advection dominated diffusion problems in two dimensional space we prove uniform optimal order error estimates for the characteristics mixed finite element for advection diffusion equations in the proof the interpolation operators are used instead of the mixed elliptic projections the generic constants of the estimates do not explicitly depend on e but depend linearly on sobolev norms of the true solution we prove uniform error estimates for the characteristics mixed finite element method for two dimensional transient convection dominated diffusion equations in the sense that the generic constants in the error estimates depend linearly on certain sobolev norms of the true solution but not on the scaling diffusion parameter e numerical experiments are presented to confirm our theoretical findings

unconditional stability and error estimates of the modified characteristics fems for the time dependent incompressible mhd equations abstract this paper focuses on the unconditional stability and convergence of characteristics type methods for the time dependent incompressible mhd equations for this purpose we introduce a new characteristics time discrete system the optimal error estimates in l 2 and h 1 norms for the typical modified characteristics finite element method unconditionally can be deduced while the whole previous works require certain time step restrictions some numerical experiments document performance of the characteristics type methods for the time dependent incompressible mhd equations

a finite element semi lagrangian method with l2 interpolation summary high order accurate methods for convection dominated problems have the potential to reduce the computational effort required for a given order of solution accuracy the state of the art in this field is more advanced for eulerian methods than for semi lagrangian slag methods in this paper we introduce a new slag method that is based on combining the modified method of characteristics with a high order interpolating procedure the method employs the finite element method on triangular meshes for the spatial discretization an l2 interpolation procedure is developed by tracking the feet of the characteristic lines from the integration nodes numerical results are illustrated for a linear advection diffusion equation with known analytical solution and for the viscous burgers equation the computed results support our expectations for a robust and highly accurate finite element slag method copyright 2012 john wiley sons ltd

a second order in time local projection stabilized lagrange galerkin method for navier stokes equations at high reynolds numbers we present a stabilized backward difference formula of order 2 lagrange galerkin method to integrate the incompressible navier stokes equations at high reynolds numbers re the stabilization of the conventional lagrange galerkin method is done via a local projection technique for inf sup stable finite elements we prove that for a finite time t the a priori error estimate for velocity in a mesh dependent norm is o h m t 2 whereas the error for pressure in the l 2 l 2 d norm is o h m t 2 with error constants that are independent of the re 1 here m denotes the degree of the polynomials of the velocity finite element space the size of the stabilization parameters is calculated from the velocity error estimate in a way that the error is optimal when the solution is sufficiently smooth numerical examples at high reynolds numbers show the robustness of our method

existence of capillary gravity water waves with piecewise constant vorticity abstract in this paper we construct small amplitude periodic capillary gravity water waves with a piecewise constant vorticity distribution they describe water waves traveling on superposed linearly sheared currents that have different vorticities this is achieved by associating to the height function formulation of the water wave problem a diffraction problem where we impose suitable transmission conditions on each line where the vorticity function has a jump the solutions of the diffraction problem found by using local bifurcation theory are the desired solutions of the hydrodynamical problem

a parallel adaptive barotropic model of the atmosphere the parallel adaptive model plasma has been developed for modeling a barotropic atmosphere this model adapts the computational grid at every time step according to a physical error indicator thus compared to uniform grid experiments the number of grid points is reduced significantly at the same time the error increases only slightly when comparing with uniform grid solutions for the discretization of the underlying spherical shallow water equations a lagrange galerkin method is used the unstructured triangular grid is maintained by the grid generator amatos and the large linear systems are solved by the parallel solver interface fossi experimental convergence is shown by means of steady state and unsteady analytical solutions plasma yields satisfactory results for quasi standard experiments that is the rossby haurwitz wave and zonal flows over an isolated mountain

a new error analysis of characteristics mixed fems for miscible displacement in porous media the method of characteristics type is especially effective for convection dominated diffusion problems due to the nature of characteristic temporal discretization the method allows one to use a large time step in many practical computations while all previous theoretical analyses always required certain restrictions on the time stepsize here we present a new analysis to establish unconditionally optimal error estimates for a modified method of characteristics with a mixed finite element approximation to the miscible displacement problem in mathbb r d d 2 3 for this purpose we introduce a new characteristic time discrete system we prove that the l 2 error bound the characteristic time discrete systemof the fully discrete method of characteristics to the time discrete system is tau independent and the numerical solution is bounded in w 1 infty norm unconditionally with the boundedness optimal error estimates are established in a traditional manner numerical results confirm our th

evolution galerkin methods and their supraconvergence evolution galerkin methods for partial differential equations of the form u t l u 0 are characterised by i the use of some form of approximation to the corresponding evolution operator e t and ii projection onto an approximation space s h to obtain u n in this paper we concentrate on characteristic galerkin and lagrange galerkin methods to derive basic error estimates for multidimensional convection problems methods covered include those using recovery techniques to improve accuracy many schemes exhibit a supraconvergence phenomenon and a general technique for its analysis is given together with a number of particular examples

a modified lagrange galerkin method for a fluid rigid system with discontinuous density in this paper we propose a new characteristics method for the discretization of the two dimensional fluid rigid body problem in the case where the densities of the fluid and the solid are different the method is based on a global weak formulation involving only terms defined on the whole fluid rigid domain to take into account the material derivative we construct a special characteristic function which maps the approximate rigid body at the k 1 th discrete time level into the approximate rigid body at k th time convergence results are proved for both semi discrete and fully discrete schemes

quasi linear variable exponent boundary value problems with wentzell robin and wentzell boundary conditions let p c0 1 be such that 1 p p 1 p p let rn rn be a bounded w1 p w1 p extension domain and let be an upper d ahlfors measure supported on with d n p n d n p n we investigate the solvability of a class of quasi linear boundary value problems involving the p p laplace and p p laplace beltrami operators and either classical wentzell robin boundary conditions p ud u p 2 u dhn 1 u p 2ud 0on or general fully wentzell type boundary conditions of the form p ud p ud u p 2 u dhn 1 u p 2ud 0on where l d l d is such that infx x 0 for some constant 0 0 0 0 and u dhn 1 denotes the generalized p p normal derivative on in the interpretative sense we prove that the realization of the p p laplace operator with both of the above boundary conditions generate nonlinear ultracontractive submarkovian c0c0 semigroups on l2 dx l2 d l2 dx l2 d and hence their associated first order cauchy problems are both well posed on lq dx lq d lq dx lq d for all measurable function q with 1 q q 1 q q in addition we investigate the associated quasi linear elliptic problem with general wentzell boundary conditions and obtain existence uniqueness and global regularity of weak solutions to this equation

convergence of a finite element ale method for the stokes equations in a domain depending on time we consider the approximation of the unsteady stokes equations in a time dependent domain when the motion of the domain is given more precisely we apply the finite element method to an arbitrary lagrangian eulerian ale formulation of the system our main results state the convergence of the solutions of the semi discretized with respect to the space variable and of the fully discrete problems towards the solutions of the stokes system

evolution galerkin methods for hyperbolic systems in two space dimensions the subject of the paper is the analysis of three new evolution galerkin schemes for a system of hyperbolic equations and particularly for the wave equation system the aim is to construct methods which take into account all of the infinitely many directions of propagation of bicharacteristics the main idea of the evolution galerkin methods is the following the initial function is evolved using the characteristic cone and then projected onto a finite element space a numerical comparison is given of the new methods with already existing methods both those based on the use of bicharacteristics as well as commonly used finite difference and finite volume methods we discuss the stability properties of the schemes and derive error estimates

a semi lagrangian high order method for navier stokes equations we present a semi lagrangian method for advection diffusion and incompressible navier stokes equations the focus is on constructing stable schemes of second order temporal accuracy as this is a crucial element for the successful application of semi lagrangian methods to turbulence simulations we implement the method in the context of unstructured spectral hp element discretization which allows for efficient search interpolation procedures as well as for illumination of the nonmonotonic behavior of the temporal advection error of the form o tk xp 1 t we present numerical results that validate this error estimate for the advection diffusion equation and we document that such estimate is also valid for the navier stokes equations at moderate or high reynolds number two and three dimensional laminar and transitional flow simulations suggest that semi lagrangian schemes are more efficient than their eulerian counterparts for high order discretizations on nonuniform grids

numerical method of characteristics for one dimensional blood flow mathematical modeling at the level of the full cardiovascular system requires the numerical approximation of solutions to a one dimensional nonlinear hyperbolic system describing flow in a single vessel this model is often simulated by computationally intensive methods like finite elements and discontinuous galerkin while some recent applications require more efficient approaches e g for real time clinical decision support phenomena occurring over multiple cardiac cycles iterative solutions to optimization inverse problems and uncertainty quantification further the high speed of pressure waves in blood vessels greatly restricts the time step needed for stability in explicit schemes we address both cost and stability by presenting an efficient and unconditionally stable method for approximating solutions to diagonal nonlinear hyperbolic systems theoretical analysis of the algorithm is given along with a comparison of our method to a discontinuous galerkin implementation lastly we demonstrate the utility of the proposed method by implementing it on small and large arterial networks of vessels whose elastic and geometrical parameters are physiologically relevant

arbitrary lagrangian eulerian method for navier stokes equations with moving boundaries this article deals with the numerical approximation of navier stokes equations in a domain with moving boundaries using the arbitrary lagrangian eulerian ale method we transform the problem from a moving domain to a fixed reference domain through an artificial domain velocity then we apply the characteristic method to solve the navier stokes equation in the fixed domain suitable boundary conditions are used for the interfaces three examples are provided to show the efficiency of the present method

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

a spectral lagrange galerkin method for convection dominated diffusion problems abstract the spectral lagrange galerkin method is a numerical technique for time dependent convection diffusion problems based on combining a lagrangian formulation of the equations with the spectral method the resulting scheme can be shown to be unconditionally stable for linear advection diffusion equations and optimal convergence rates may be obtained both the spectral accuracy and the efficiency of the scheme are examined theoretically and by various numerical experiments and extensions of the method to give high accuracy in time are presented

a finite element semi lagrangian explicit runge kutta chebyshev method for convection dominated reaction diffusion problems explicit runge kutta chebyshev methods have proved to be efficient for reaction diffusion problems of moderate stiffness in this paper we extend such an efficiency to convection dominated reaction diffusion problems by giving a formulation of these methods in a semi lagrangian framework using c0 finite elements of degree m 2 as the space discretization method we also study the convergence in the l2 norm of the methods proposed in this paper

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

a modified characteristic finite element method for a fully nonlinear formulation of the semigeostrophic flow equations this paper develops a fully discrete modified characteristic finite element method for a coupled system consisting of the fully nonlinear monge ampere equation and a transport equation the system is the eulerian formulation in the dual space for b j hoskins semigeostrophic flow equations which are widely used in meteorology to model frontogenesis to overcome the difficulty caused by the strong nonlinearity we first formulate at the differential level a vanishing moment approximation of the semigeostrophic flow equations a methodology recently proposed by the authors x feng and m neilan j sci comput 38 2009 pp 74 98 which involves approximating the fully nonlinear monge ampere equation by a family of fourth order quasilinear equations we then construct a fully discrete modified characteristic finite element method for the regularized problem it is shown that under certain mesh constraint the proposed numerical method converges with an optimal order rate of convergence in particular the obtained error bounds show explicit dependence on the regularization parameter varepsilon numerical tests are also presented to validate the theoretical results and to gauge the efficiency of the proposed fully discrete modified characteristic finite element method

a characteristic finite element algorithm for the 3 d navier stokes equations using unstructured grids abstract this paper discusses a finite element algorithm for solution of the 3 d navier stokes equations that is based on the method of characteristics the algorithm which is based on a time slitting of the convective terms from the generalized stokes problem comprises a point wise characteristic integration of the convective terms and uzawa integration of the generalized stokes problem the most costly part of the characteristic integration scheme is the search for an element containing the foot of a given characteristic the present study uses a very efficient algorithm for element searching which is particularly useful for complex geometries the implementation of the algorithm is discussed in the paper and the resulting code s performance is compared to that of a code based on a more conventional fourth order runge kutta convection integration a variety of flow problems are tested including an analytical solution of the navier stokes equations flow in a lid driven cavity flow over a backward facing step and flow in an end to side anastomoses in all cases the proposed algorithm demonstrated performance superior to that of a conventional explicit treatment of the convective terms

a unipotent support for irreducible representations 1 1 let g f be the group of fq rational points of a connected reductive algebraic group g defined over a finite field f throughout this paper we shall assume that the characteristic of the ground field is large the following is part of problem ii in l9 let p be an irreducible complex character of g f show that there is a unique umpotent class c in g which has the property that cga c cfqi p g 0 and has maximal dimension among classes with this property one of the results of this paper is a solution of this problem we also prove the following refinement if g e gf is such that tr g p 0 then the unipotent part of g lies in c or in a conjugacy class of dimension dim c we call c the unipotent support of p another map from the set of irreducible representations of gf to unipotent classes in g has been defined in l8 13 43 this was the composition of two maps one from irreducible representations of gf to a set of parameters in the langlands dual of g and one from this set of parameters to unipotent classes in g another conjectural definition of this map which did not pass through the langlands dual has been proposed by kawanaka kl k2 k3 in terms of generalized gelfand graev representations in analogy with real lie groups he uses the name wave front set for the unipotent class attached to a representation one of the results of this paper is a proof see section 11 of kawanaka s conjecture that his wave front set is well defined and that it coincides with the one defined in l8 this has been known earlier from the work of kawanaka in certain special cases we also show that the unipotent support of an irreducible representation coincides with the wave front set of the dual representation

on the existence uniqueness and finite element approximation of solutions of the equations of stationary incompressible magnetohydrodynamics the authors consider the equations of stationary incompressible magneto hydrodynamics posed in a bounded domain in three dimensions and treat the full coupled system of equations with inhomogeneous boundary conditions under certain conditions on the data they show that the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed they discuss a finite element discretization of the equations and prove an optimal estimate for the error of the approximate solution

linear second order elliptic equations with venttsel boundary conditions we prove the existence of solutions of quasilinear second order elliptic equations with quasilinear venttsel boundary conditions in both classical and weak sense under natural structure conditions

a high order characteristics method for the incompressible navier stokes equations abstract we analyze a high order characteristics method for the navier stokes equations we focus on the cases of the first second and third order in time schemes with finite element spatial discretization a numerical comparison between the first and second order schemes is done for steady or transient states flows

unconditional stability and error estimates of modified characteristics fems for the navier stokes equations the paper is concerned with the unconditional stability and convergence of characteristics type methods for the time dependent navier stokes equations we present optimal error estimates in l 2 l2 and h 1 h1 norms for a typical modified characteristics finite element method unconditionally while all previous works require certain time step restrictions the analysis is based on an iterated characteristic time discrete system with which the error function is split into a temporal error and a spatial error with a rigorous analysis to the characteristic time discrete system we prove that the difference between the numerical solution and the solution of the time discrete system is tau independent where tau denotes the time stepsize thus numerical solution in w 1 infty w1 is bounded and optimal error estimates can be obtained in a traditional way numerical results confirm our analysis and show clearly the unconditional stability and convergence of the modified characteristics finite element method for the time dependent navier stokes equations the approach used in this paper can be easily extended to many other characteristics based methods

on the relationship between semi lagrangian and lagrange galerkin schemes following a previous result stating their equivalence under constant advection speed semi lagrangian and lagrange galerkin schemes are compared in this paper in the situation of variable coefficient advection equations once known that semi lagrangian schemes can be proved to be equivalent to area weighted lagrange galerkin schemes via a suitable definition of the basis functions we will further prove that area weighted lagrange galerkin schemes represent a small more precisely an o delta t perturbation of exact lagrange galerkin schemes this equivalence implies a general result of stability for semi lagrangian schemes

modified lagrange galerkin methods of first and second order in time for convection diffusion problems we introduce modified lagrange galerkin mlg methods of order one and two with respect to time to integrate convection diffusion equations as numerical tests show the new methods are more efficient but maintaining the same order of convergence than the conventional lagrange galerkin lg methods when they are used with either p 1 or p 2 finite elements the error analysis reveals that 1 when the problem is diffusion dominated the convergence of the modified lg methods is of the form o h m 1 h 2 t q q 1 or 2 and m being the degree of the polynomials of the finite elements 2 when the problem is convection dominated and the time step t is large enough the convergence is of the form o frac h m 1 delta t h 2 delta t q 3 as in case 2 but with t small then the order of convergence is now o h m h 2 t q 4 when the problem is convection dominated the convergence is uniform with respect to the diffusion parameter x t so that when 0 and the forcing term is also equal to zero the error tends to that of the pure convection problem our error analysis shows that the conventional lg methods exhibit the same error behavior as the mlg methods but without the term h 2 numerical experiments support these theoretical results

unconditional convergence and optimal error estimates of a galerkin mixed fem for incompressible miscible flow in porous media in this paper we study the unconditional convergence and error estimates of a galerkin mixed fem with the linearized semi implicit euler scheme for the equations of incompressible miscible flow in porous media we prove that the optimal l 2 error estimates hold without any time step convergence conditions while all previous works require certain time step restrictions our theoretical results provide a new understanding on commonly used linearized schemes the proof is based on a splitting of the error into two parts the error from the time discretization of the pdes and the error from the finite element discretization of corresponding time discrete pdes the approach used in this paper can be applied to more general nonlinear parabolic systems and many other linearized semi implicit time discretizations

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

modular irreducibility of cuspidal unipotent characters we prove a long standing conjecture of geck which predicts that cuspidal unipotent characters remain irreducible after ell reduction to this end we construct a progenerator for the category of representations of a finite reductive group coming from generalised gelfand graev representations this is achieved by showing that cuspidal representations appear in the head of generalised gelfand graev representations attached to cuspidal unipotent classes as defined and studied in geck and malle j lond math soc 2 53 63 78 1996

analysis of coupling iterations based on the finite element method for stationary magnetohydrodynamics on a general domain this paper considers finite element approximation and three coupled type iterations of stationary magnetohydrodynamics mhd equations on a general lipschitz domain an additional lagrange multiplier r is introduced related to divergence free of magnetic field b and the b is analyzed in h curl space which is proposed in schotzau 2004 in finite element discretization the hydrodynamic unknowns are approximated by stable finite element pairs and the magnetic unknown is discretized by curl conforming nedelec element spaces the well posedness of this formula and the optimal error estimate are provided based on this for numerical implementation of this scheme we propose and discuss three coupled type iterative methods which are stable and convergent under different conditions specifically iteration i is stable and convergent under strong condition iteration ii is stable and convergent under weaker condition iteration iii is unconditionally stable and convergent under the weakest condition finally some numerical tests confirm our theoretical analysis

stability of spherically symmetric subsonic flows and transonic shocks under multidimensional perturbations abstract we develop a method that works in general product riemannian manifold to decompose the three dimensional steady full compressible euler system which is of elliptic hyperbolic composite mixed type for subsonic flows the method is applied to show stability of spherically symmetric subsonic flows and transonic shocks in space r 3 under multidimensional perturbations of boundary conditions

numerical analysis of a characteristic stabilized finite element method for the time dependent navier stokes equations with nonlinear slip boundary conditions abstract based on a characteristic method this work is concerned with a finite element approximation to the time dependent navier stokes equations with nonlinear slip boundary conditions since this slip boundary condition of friction type contains a subdifferential property its continuous variational problem is formulated as an inequality which can turn into an equality problem by using a powerful regularized method then a fully discrete characteristic scheme under the stabilized lower order finite element pairs is proposed for the equality problem optimal error estimates for velocity and pressure are derived under the corresponding l 2 h 1 norms finally a smooth problem test is reported to demonstrate the theoretically predicted convergence order and the expected slip phenomena and the simulation of a bifurcated blood flow model is displayed to illustrate the efficiency of the proposed method

a mixed finite element method with exactly divergence free velocities for incompressible magnetohydrodynamics abstract we introduce and analyze a mixed finite element method for the numerical discretization of a stationary incompressible magnetohydrodynamics problem in two and three dimensions the velocity field is discretized using divergence conforming brezzi douglas marini bdm elements and the magnetic field is approximated by curl conforming nedelec elements the h 1 continuity of the velocity field is enforced by a dg approach a central feature of the method is that it produces exactly divergence free velocity approximations and captures the strongest magnetic singularities we prove that the energy norm error is convergent in the mesh size in general lipschitz polyhedra under minimal regularity assumptions and derive nearly optimal a priori error estimates for the two dimensional case we present a comprehensive set of numerical experiments which indicate optimal convergence of the proposed method for two dimensional as well as three dimensional problems

generalized gelfand graev representations of exceptional simple algebraic groups over a finite field i introduction let ffi be an adjoint simple algebraic group of type e n 6 7 8 f 4 or g2 over an algebraically closed field k of characteristic p 0 we assume that 5 has a fixed fq rational structure for a finite subfield fq of k the purpose of this paper is to determine explicitly the values of the irreducible characters of g ffi fq at the unipotent elements when p is good 38 i 4 3 in this part i we shall do this only for large p and the full result will be proved in the forthcoming part ii after submitting the first version of the present paper the author was informed by lusztig that he solved analogous problems for classical groups using his theory of character sheaves and that he can also recover the result of this paper mentioned above our method depends upon 1 a parametrization of the irreducible characters of g and the determination of their multiplicities in deligne lusztig virtual representations lusztig 25 2 the computation of the green functions of g in large characteristic springer 36 7 1 shoji 33 beynon and spaltenstein 3 4 3 the determination of the values of generalized gelfand graev characters 20 of g in good characteristic see section 3 by 1 and 2 the uniform parts of the irreducible characters are already known explicitly in large characteristic this implies that in large characteristic our problem for groups of type e 6 o r e 7 has already been solved it should be mentioned here that the whole character table of the g2 group is also known by chang and ree 8 hence in the g2 case this paper gives just another approach to a part of the result of chang and ree if g is of type g2 f4 or e 8 the restriction of an irreducible character r of g to the set of unipotent elements can be written as a sum of its uniform part with a function of the form

pseudospectral method for the good boussinesq equation we prove the nonlinear stability and convergence of a fully discrete pseudospectral scheme for the good boussinesq equation utt uxxxx uxx u2 xx numerical comparisons with finite difference schemes are also reported

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

the neumann problem for equations of monge ampere type we prove the existence of a classical solution to a neumann type problem for nonlinear elliptic equations of monge ampere type the methods depend upon the establishment of a priori derivative estimates up to order two and yield a sharp result for the equation of prescribed gauss curvature we also present applications to asymptotic problems which approach the non unique case and maximum principles for linear equations of aleksandrov bakelman type