su n tensor product multiplicities and virtual berenstein zelevinsky triangles information on su n tensor product multiplicities is neatly encoded in berenstein zelevinsky triangles here we study a generalization of these triangles by allowing negative as well as non negative integer entries for a fixed triple product of weights these generalized berenstein zelevinsky triangles span a lattice in which one may move by adding integer linear combinations of so called virtual triangles inequalities satisfied by the coefficients of the virtual triangles describe a polytope the tensor product multiplicities may be computed as the number of integer points in this convex polytope as our main result we present an explicit formula for this discretised volume as a multiple sum as an application we also address the problem of determining when a tensor product multiplicity is non vanishing the solution is represented by a set of inequalities in the dynkin labels we also allude to the question of when a tensor product multiplicity is greater than a given non negative integer

on maps preserving operators of local spectral radius zero abstract let l x be the algebra of all bounded linear operators on a complex banach space x we describe surjective linear maps on l x that satisfy r t x 0 r t x 0 for every x x and t l x we also describe surjective linear maps on l x that satisfy r t x 0 r t x 0 for every x x and t l x furthermore we characterize maps not necessarily linear nor surjective on l x which satisfy r t s x 0 if and only if r t s x 0 for every x x and t s l x

surjective linear maps preserving local spectra let b x be the algebra of all bounded linear operators on an infinite dimensional complex banach space x we characterize linear surjective and continuous maps on b x preserving different local spectral quantities at a nonzero fixed vector

stability of the supg finite element method for transient advection diffusion problems implicit time integration coupled with supg discretization in space leads to additional terms that provide consistency and improve the phase accuracy for convection dominated flows recently it has been suggested that for small courant numbers these terms may dominate the streamline diffusion term ostensibly causing destabilization of the supg method while consistent with a straightforward finite element stability analysis this contention is not supported by computational experiments and contradicts earlier von neumann stability analyses of the semidiscrete supg equations this prompts us to re examine finite element stability of the fully discrete supg equations a careful analysis of the additional terms reveals that regardless of the time step size they are always dominated by the consistent mass matrix consequently supg cannot be destabilized for small courant numbers numerical results that illustrate our conclusions are reported

an a posteriori error estimator for an unsteady advectiondiffusionreaction problem in this work we introduce an a posteriori error estimator of the residual type for the unsteady advectiondiffusionreaction problem for the discretization in time we use an implicit euler scheme and a continuous piecewise linear triangular finite elements for the space together with a stabilized scheme we prove that the approximation error is bounded by above and below by the error estimator using that an adaptive algorithm is proposed analyzed and tested numerically to prove the efficiency of our approach

a numerical study of a posteriori error estimators for convection diffusion equations this paper presents a numerical study of a posteriori error estimators for convection diausion equations the study involves the gradient indicator an a posteriori error estimator which is based on gradient recovery three residual based error estimators for diaerent norms and two error estimators which are defined by solutions of local neumann problems they are compared with respect to the reliable estimation of the global error and with respect to the accuracy of the computed solutions on adaptively refined grids the numerical study shows for both criteria of comparison that none of the considered error estimators works satisfactorily in all tests 2000 elsevier science s a all rights reserved msc 65n50 65n30

robust a posteriori error estimates for nonstationary convection diffusion equations we consider discretizations of convection dominated nonstationary convection diffusion equations by a stable theta schemes in time and conforming finite elements in space on locally refined isotropic meshes for these discretizations we derive a residual a posteriori error estimator the estimator yields upper bounds on the error which are global in space and time and lower bounds that are global in space and local in time the error estimates are fully robust in the sense that the ratio between upper and lower bounds is uniformly bounded in time does not depend on any step size in space or time nor on any relation between these both and is uniformly bounded with respect to the size of the convection moreover the estimates are uniform with respect to the size of the zero order reaction term and also hold for the limit case of vanishing reaction

a posteriori error estimates for the crank nicolson method for parabolic equations we derive optimal order a posteriori error estimates for time discretizations by both the crank nicolson and the crank nicolson galerkin methods for linear and nonlinear parabolic equations we examine both smooth and rough initial data our basic tool for deriving a posteriori estimates are second order crank nicolson reconstructions of the piecewise linear approximate solutions these functions satisfy two fundamental properties i they are explicitly computable and thus their difference to the numerical solution is controlled a posteriori and ii they lead to optimal order residuals as well as to appropriate pointwise representations of the error equation of the same form as the underlying evolution equation the resulting estimators are shown to be of optimal order by deriving upper and lower bounds for them depending only on the discretization parameters and the data of our problem as a consequence we provide alternative proofs for known a priori rates of convergence for the crank nicolson method

higher genus su n fusion multiplicities as polytope volumes we show how higher genus su n fusion multiplicities may be computed as the discretized volumes of certain polytopes the method is illustrated by explicit analyses of some su 3 and su 4 fusions but applies to all higher point and higher genus su n fusions it is based on an extension of the realm of berenstein zelevinsky triangles by including so called gluing and loop gluing diagrams the identification of the loop gluing diagrams is our main new result since they enable us to characterize higher genus fusions in terms of polytopes also the genus 2 0 point su 3 fusion multiplicity is found to be a simple binomial coefficient in the affine level

analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements this paper adresses the problem of determining upper and lower bounds for the effectivity index on the a posteriori estimate of the error in the finite element method these bounds are given explicitly for a certain concrete estimator for linear elements and unstructured triangular meshes they depend strongly on the geometry of the triangles and relatively weakly on the smoothness of the solution an example shows that the bounds are not over pessimistic in babuska plank and rodriguez 4 detailed numerical experimentation is given

consistent supg method for transient transport problems stability and convergence we consider the time space discretization of the transient advection equation discretization in space is performed by the streamline upwind petrov galerkin method and in time we use an a stable finite difference operator the formulation is strongly consistent in the sense that the time derivative is included in the stabilization term uniform stability of the general formulation is proved under a regularity condition on data or a moderate inverse cfl condition that allows for optimal choices of the discretization parameters both the backward euler method bdf1 the crank nicolson scheme and the second order backward differentiation formula bdf2 enter the framework and quasi optimal convergence is proved for these schemes

a posteriori error estimators for convection diffusion equations we derive a posteriori error estimators for convection diffusion equations with dominant convection the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local mesh peclet number the estimators are either based on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

element diameter free stability parameters for stabilized methods applied to fluids abstract stability parameters for stabilized methods in fluids are suggested the computation of the largest eigenvalue of a generalized eigenvalue problem replaces controversial definitions of element diameters and inverse estimate constants used heretofore to compute these stability parameters the design is employed in the advective diffusive model incompressible navier stokes equations and the stokes problem

on finite element methods for 3d time dependent convection diffusion reaction equations with small diffusion the paper studies finite element methods for the simulation of time dependent convection diffusion reaction equations with small diffusion the supg method a sold method and two types of fem fct methods the methods are assessed in particular with respect to the size of the spurious oscillations in the computed solutions at a 3d example with nonhomogeneous dirichlet boundary conditions and homogeneous neumann boundary conditions

an adaptive algorithm for the crank nicolson scheme applied to a time dependent convection diffusion problem an a posteriori upper bound is derived for the nonstationary convection diffusion problem using the crank nicolson scheme and continuous piecewise linear stabilized finite elements with large aspect ratio following lozinski et al 2009 13 a quadratic time reconstruction is used a space and time adaptive algorithm is developed to ensure the control of the relative error in the l 2 h 1 norm numerical experiments illustrating the efficiency of this approach are reported it is shown that the error indicator is of optimal order with respect to both the mesh size and the time step even in the convection dominated regime and in the presence of boundary layers

bounds of functional outputs for parabolic problems part i exact bounds of the discontinuous galerkin time discretization classical implicit residual type error estimators require using an underlying spatial finer mesh to compute bounds for some quantity of interest consequently the bounds obtained are only guaranteed asymptotically that is with respect to the reference solution computed with the fine mesh exact bounds that is bounds guaranteed with respect to the exact solution are needed to properly certify the accuracy of the results especially if the meshes are coarse the paper introduces a procedure to compute strict upper and lower bounds of the error in linear functional outputs of parabolic problems in this first part the bounds account for the error associated with the spatial discretization the error coming from the time marching scheme is therefore assumed to be negligible in front of the spatial error the time discretization is performed using the discontinuous galerkin method both for the primal and adjoint problems in the error estimation procedure equilibrated fluxes at interelement edges are calculated using hybridization techniques

finite element methods with symmetric stabilization for the transient convection diffusion reaction equation we consider implicit and semi implicit time stepping methods for finite element approximations of singularly perturbed parabolic problems or hyperbolic problems we are interested in problems where the advection dominates and stability is obtained using a symmetric weakly consistent stabilization operator in the finite element method several a stable time discretizations are analyzed and shown to lead to unconditionally stable and optimally convergent schemes in particular we show that the contribution from the stabilization leading to an extended matrix pattern may be extrapolated from previous time steps and hence handled explicitly without loss of stability and accuracy a fully explicit treatment of the stabilization term is obtained under a cfl condition

error analysis of the supg finite element discretization of evolutionary convection diffusion reaction equations conditions on the stabilization parameters are explored for different approaches in deriving error estimates for the streamline upwind petrov galerkin supg finite element stabilization of time dependent convection diffusion reaction equations exemplarily it is shown for the supg method combined with the backward euler scheme that standard energy arguments lead to estimates for stabilization parameters that depend on the length of the time step the stabilization vanishes in the time continuous limit however based on numerical experience this seems not to be the correct behavior for this reason the main focus of the paper consists in deriving estimates in which the stabilization parameters do not depend on the length of the time step it is shown that such estimates can be obtained in the case of time independent convection and reaction an error estimate for the time continuous case with the standard order of convergence is derived for stabilization parameters of the same form as they are optimal for the steady state problem analogous estimates are obtained for the fully discrete case using the backward euler method and the crank nicolson scheme numerical studies support the analytical results

a posteriori analysis of the finite element discretization of some parabolic equations nous considerons la discretisation d equations paraboliques soit lineaires soit semi lineaires par un schema d euler implicite en temps et par elements finis en espace l idee de cet article est de construire des indicateurs d erreur lies a l approximation en temps et en espace et de prouver leur equivalence avec l erreur dans le but de travailler avec des pas de temps adaptatifs et des maillages d elements finis adaptes a la solution

adaptive finite elements for a linear parabolic problem a posteriori error estimates for the heat equation in two space dimensions are presented a classical discretization is used euler backward in time and continuous piecewise linear triangular finite elements in space the error is bounded above and below by an explicit error estimator based on the residual numerical results are presented for uniform triangulations and constant time steps the quality of our error estimator is discussed an adaptive algorithm is then proposed successive delaunay triangulations are generated so that the estimated relative error is close to a preset tolerance again numerical results demonstrate the efficiency of our approach c 1998 elsevier science s a all rights reserved

an assessment of discretizations for convection dominated convection diffusion equations the performance of several numerical schemes for discretizing convection dominated convection diffusion equations will be investigated with respect to accuracy and efficiency accuracy is considered in measures which are of interest in applications the study includes an exponentially fitted finite volume scheme the streamline upwind petrov galerkin supg finite element method a spurious oscillations at layers diminishing sold finite element method a finite element method with continuous interior penalty cip stabilization a discontinuous galerkin dg finite element method and a total variation diminishing finite element method femtvd a detailed assessment of the schemes based on the hemker example will be presented

n point and higher genus osp 1 2 fusion we study affine osp 1 2 fusion the fusion in osp 1 2 conformal field theory for example higher point and higher genus fusion is discussed the fusion multiplicities are characterized as discretized volumes of certain convex polytopes and are written explicitly as multiple sums measuring those volumes we extend recent methods developed to treat affine su 2 fusion they are based on the concept of generalized berenstein zelevinsky triangles and virtual couplings higher point tensor products of finite dimensional irreducible osp 1 2 representations are also considered the associated multiplicities are computed and written as multiple sums

on an improved unusual stabilized finite element method for theadvective reactive diffusive equation an improved unusual finite element method is studied herein for a second order linear scalar differential equation including a zero order term the method consists in subtracting from the standard galerkin method a mesh dependent term suggested by static condensation of the bubbles based on this idea a new stabilized parameter is constructed which improves accuracy of the solution several numerical tests attest the efficiency of the present method and an error analysis is perfomed

a posteriori error estimation for hp version time stepping methods for parabolic partial differential equations the aim of this paper is to develop an hp version a posteriori error analysis for the time discretization of parabolic problems by the continuous galerkin cg and the discontinuous galerkin dg time stepping methods respectively the resulting error estimators are fully explicit with respect to the local time steps and approximation orders their performance within an hp adaptive refinement procedure is illustrated with a series of numerical experiments

the finite element method for parabolic equations in this first of two papers computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time independent space meshes the effectiveness of the error estimator is related to conditions on the solution regularity mesh family type and asymptotic range for the mesh size for clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used the results extend straight forwardly to systems of equations and higher order elements in one space dimension while the higher dimensional case requires additional considerations the theory presented here provides the basis for the analysis and adaptive construction of time dependent space meshes which is the subject of the second paper computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems

affine su 3 and su 4 fusion multiplicities as polytope volumes affine su 3 and su 4 fusion multiplicities are characterized as discretized volumes of certain convex polytopes the volumes are measured explicitly resulting in multiple sum formulae these are the first polytope volume formulae for higher rank fusion multiplicities the associated threshold levels are also discussed for any simple lie algebra we derive an upper bound on the threshold levels using a refined version of the gepner witten depth rule