isogeometric analysis using t splines we explore t splines a generalization of nurbs enabling local refinement as a basis for isogeometric analysis we review t splines as a surface design methodology and then develop it for engineering analysis applications we test t splines on some elementary two dimensional and three dimensional fluid and structural analysis problems and attain good results in all cases we summarize the current status of t splines their limitations and future possibilities

non oscillatory and non diffusive solution of convection problems by the iteratively reweighted least squares finite element method in this paper we first introduce the least squares finite element method lsfem for two dimensional steady state pure convection problems with smooth solutions and compare the lsfem with other finite element methods we prove that the lsfem has the same stability estimate as the original equation i e the lsfem has better control of the streamline derivative than the streamline upwinding petrov galerkin method numerical convergence rates are given to show that the lsfem is almost optimal then we use this lsfem as a framework to develop an iteratively reweighted least squares finite element method irlsfem to obtain non oscillatory and non diffusive solutions for problems with contact discontinuities this new method produces a highly accurate numerical solution that has a sharp discontinuity in one element a number of examples solved by using triangular and bilinear elements are presented to show that the method can convect contact discontinuities without error

enforcement of constraints and maximum principles in the variational multiscale method abstract we present a new theoretical framework for the enforcement of constraints in variational multiscale vms analysis the theory is first presented in an abstract operator format and subsequently specialized for the steady advection diffusion equation the approach borrows heavily from results in constrained and convex optimization an exact expression for the fine scales is derived in terms of variational derivatives of the constraints lagrange multipliers and a fine scale green s function the methodology described enables the development of numerical methods which satisfy predefined attributes a practical and effective procedure for solving the steady advection diffusion equation is presented based on a vms inspired stabilized method weakly enforced dirichlet boundary conditions and enforcement of a maximum principle and conservation constraint

a finite element technique for solving first order pdes in lp an approximation technique for solving first order pdes in l p omega 1 le p infty is proposed the method is a generalization of the least squares method to non hilbertian settings a priori and a posteriori error estimates are proven numerical tests in l 1 omega show that this type of technique can handle discontinuities without resorting to limiting procedures

representation theory of a hecke algebra of g r p n abstract in our previous paper 1 we introduced a notion of a hecke algebra for a series of complex reflection groups z r z k n which is the group of n by n permutation matrices with entries running through rth roots of unity the ordinary representation of the algebra is also studied in it the results are direct generalizations of hoefsmit s 4 for hecke algebras of classical weyl groups in this paper we establish a similar theory for the remaining series of complex reflection groups g r p n p r p 1 is the case we previously considered in 1 the construction of the hecke algebra was soon generalized to the case p 2 by broue and malle 2 motivated by a much deeper background in 2 they conjectured that these hecke algebras over a ring of l adic integers are candidates for endomorphism rings of certain deligne lusztig modules over the ring which arise in modular representation theory related to algebraic groups they also defined hecke algebras of complex reflection groups for not all but various kinds of exceptional and two dimensional cases in the same context the first section starts with the definition of the hecke algebra and then we prove that they actually have desirable properties the second section is for explicit construction of irreducible representations we call them the representations of semi normal form the author deeply thanks professor broue and professor malle since he was able to learn much from them and they hinted him to the problem

the p and hp versions of the finite element method for problems with boundary layers we study the uniform approximation of boundary layer functions exp x d for x 0 1 d 0 1 by the p and hp versions of the finite element method for the p version with fixed mesh we prove super exponential convergence in the range p 1 2 e 2d we also establish for this version an overall convergence rate of o p 1 ln p in the energy norm error which is uniform in d and show that this rate is sharp up to the ln p term when robust estimates uniform in d 0 1 are considered for the p version with variable mesh i e the hp version we show that exponential convergence uniform in d 0 1 is achieved by taking the first element at the boundary layer to be of size o pd numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates even when few degrees of freedom are used and when d is as small as e g 10 8 they also illustrate the superiority of the hp approach over other methods including a low order h version with optimal exponential mesh refinement the estimates established in this paper are also applicable in the context of corresponding spectral element methods

a class of discontinuous petrov galerkin methods part iv the optimal test norm and time harmonic wave propagation in 1d the phase error or the pollution effect in the finite element solution of wave propagation problems is a well known phenomenon that must be confronted when solving problems in the high frequency range this paper presents a new method with no phase errors for one dimensional 1d time harmonic wave propagation problems using new ideas that hold promise for the multidimensional case the method is constructed within the framework of the discontinuous petrov galerkin dpg method with optimal test functions we have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm in this paper we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm this is answered in the specific case of the helmholtz equation with l 2 norm as the energy norm we obtain uniform stability with respect to the wave number we illustrate the method with a number of 1d numerical experiments using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally a 1d theoretical stability analysis is also developed

adaptive streamline diffusion finite element methods for stationary convection diffusion problems adaptive finite element methods for stationary convectiondiffusion problems are designed and analyzed the underlying discretization scheme is the shock capturing streamline diffusion method the adaptive algorithms proposed are based on a posteriori error estimates for this method leading to reliable methods in the sense that the desired error control is guaranteed a priori error estimates are used to show that the algorithms are efficient in a certain sense

proof of the modular branching rule for cyclotomic hecke algebras abstract we prove the modular branching rule of the cyclotomic hecke algebras which has remained open

modular branching rules and the mullineux map for hecke algebras of type a in a series of papers 21 22 23 kleshchev has recently proved a branching rule for modular representations of the symmetric group s r on r symbols fix a field f of arbitrary characteristic p then kleshchev s branching rule gives a precise description of the socle of the restriction of an irreducible fs r module to fs r 1 in 24 kleshchev applies this branching rule to give a purely combinatorial description of the mullineux map which describes the irreducible fs r module obtained by tensoring an arbitrary irreducible fs r module with the 1 dimensional sign representation in 13 see also 3 ford and kleshchev show with some technical combinatorics that kleshchev s description of the mullineux map is equivalent to a quite different algorithm conjectured by mullineux 27 in 1979 thus proving the so called mullineux conjecture the main purpose of this paper is to prove the quantum analogues of these results replacing the group algebra fs r with the corresponding hecke algebra h r over f at an arbitrary root of unity q f our main result is the hecke algebra analogue of kleshchev s modular branching rule it turns out that the branching rule for h r only depends on the integer where is the smallest natural number such that 1 q q 1 0 in f if q 1 is precisely the characteristic p of f and our branching rule specializes to the classical case considered by kleshchev as a consequence of this branching rule combining kleshchev s argument in 24 with the known block structure of h r from 9 we are able to deduce the quantum analogue of kleshchev s description of the corresponding mullineux map for h r the proof by ford and kleshchev in 13 that this is equivalent to mullineux s original algorithm does not depend on p being prime so our result is equivalent by 13 to a quantum version of the mullineux conjecture replacing p by we remark that there is now a quite different proof that kleshchev s algorithm gives the mullineux map using crystal bases this was first observed for fields of characteristic 0 in 25 modulo a conjecture which has now been proved 17 2 relating crystal bases for a certain affine quantum group to decomposition matrices of h r over fields of characteristic 0 the result in arbitrary characteristic can be deduced from the characteristic 0 case by an

decomposable specht modules for the iwahori hecke algebra hf 1 sn abstract let s denote the specht module defined by dipper and james for the iwahori hecke algebra h n of the symmetric group s n when e 2 we determine the decomposability of all specht modules corresponding to hook partitions a 1 b we do so by utilising the brundan kleshchev isomorphism between h and a khovanov lauda rouquier algebra and working with the relevant klr algebra using the set up of kleshchev mathas ram when n is even we easily arrive at the conclusion that s is indecomposable when n is odd we find an endomorphism of s and use it to obtain a generalised eigenspace decomposition of s

derivation of stabilized equations for numerical solution of advective diffusive transport and fluid flow problems abstract the concept of the so called artificial or balancing diffusion used to stabilize the numerical solution of advective diffusive transport and fluid flow problems is revised in this paper it is shown that the standard forms of the balancing diffusion terms usually chosen in a heuristic manner can be naturally found by introducing higher order approximations in the derivation of the governing differential equations via standard conservation or equilibrium principles this allows us to reinterpret many stabilization algorithms and concepts used in every day practice by numerical analysts and also provides an expression for computing the stabilization parameter

analysis of the dpg method for the poisson equation we give an error analysis of the recently developed dpg method applied to solve the poisson equation and a convection diffusion problem we prove that the method is quasioptimal error estimates in terms of both the mesh size h and the polynomial degree p for various element shapes can be derived from our results results of extensive numerical experiments are also presented

the discontinuous petrov galerkin method for elliptic problems abstract we propose a novel discontinuous mixed finite element formulation for the solution of second order elliptic problems fully discontinuous piecewise polynomial finite element spaces are used for the trial and test functions the discontinuous nature of the test functions at the element interfaces allows to introduce new boundary unknowns that on the one hand enforce the weak continuity of the trial functions and on the other avoid the need to define a priori algorithmic fluxes as in standard discontinuous galerkin methods static condensation is performed at the element level leading to a solution procedure based on the sole interface unknowns the resulting family of discontinuous dual primal mixed dpm finite element methods is presented in the one and two dimensional cases in the one dimensional case we show the equivalence of the method with implicit runge kutta schemes of the collocation type exhibiting optimal behavior and we develop the stability analysis numerical experiments in one and two dimensions demonstrate the order accuracy of the new method confirming the results of the analysis

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

robust dpg method for convection dominated diffusion problems we propose and analyze a discontinuous petrov galerkin dpg method for convection dominated diffusion problems that provides robust l 2 error estimates for the field variables which are quasi optimal in the energy norm a key feature of the method is to construct test functions defined by a variational formulation with bilinear form test norm specifically designed for the goal of robustness a main theoretical ingredient is a stability analysis of the adjoint problem numerical experiments underline our theoretical results and in particular confirm robustness of the dpg method for well chosen test norms

a new family of stable elements for nearly incompressible elasticity based on a mixed petrov galerkin finite element formulation adding to the classical hellinger reissner formulation another residual form of the equilibrium equation a new petrov galerkin finite element method is derived it fits within the framework of a mixed finite element method and is proved to be stable for rather general combinations of stress and displacement interpolations including equal order discontinuous stress and continuous displacement interpolations which are unstable within the galerkin approach error estimates are presented using the babuska brezzi theory and numerical results confirm these estimates as well as the good accuracy and stability of the method

submodule structure of some specht modules in this paper we show how homomorphisms between certain permutation modules for the symmetric group s may be used to give information about the submodule structure of specht modules for hook and two rowed diagrams we continue the investigation of the properties of the hook module at characteristic 2 which was begun in 3 4 here we give precise conditions for the module to be completely reducible and to have a unique minimal submodule we also show that the two rowed specht module has a unique minimal submodule over a field of any characteristic let k be any field and x x i xn a set of independent commutative indeterminates we may define the action of s on any polynomal in the elements of x by u xi xu for each u in s we denote by scnprvr the specht module corresponding to a young diagram with n r nodes in the first row and r nodes in the second and by s n rv1 the module for a diagram with n r nodes in the top row and r rows of length 1 then s n rlr and s n r are generated over ks by t and u respectively where

analysis of a multiscale discontinuous galerkin method for convection diffusion problems we study a multiscale discontinuous galerkin method introduced in t j r hughes g scovazzi p bochev and a buffa comput meth appl mech engrg 195 2006 pp 2761 2787 that reduces the computational complexity of the discontinuous galerkin method seemingly without adversely affecting the quality of results for a stabilized variant we are able to obtain the same error estimates for the convection diffusion equation as for the usual discontinuous galerkin method we assess the stability of the unstabilized case numerically and find that the inf sup constant is positive bounded uniformly away from zero and very similar to that for the usual discontinuous galerkin method

discontinuous petrov galerkin method with optimal test functions for thin body problems in solid mechanics we study the applicability of the discontinuous petrov galerkin dpg variational framework for thin body problems in structural mechanics our numerical approach is based on discontinuous piecewise polynomial finite element spaces for the trial functions and approximate local computation of the corresponding optimal test functions in the timoshenko beam problem the proposed method is shown to provide the best approximation in an energy type norm which is equivalent to the l2 norm for all the unknowns uniformly with respect to the thickness parameter the same formulation remains valid also for the asymptotic euler bernoulli solution as another one dimensional model problem we consider the modelling of the so called basic edge effect in shell deformations in particular we derive a special norm for the test space which leads to a robust method in terms of the shell thickness finally we demonstrate how a posteriori error estimator arising directly from the discontinuous variational framework can be utilized to generate an optimal hp mesh for resolving the boundary layer

comparison of some finite element methods for solving the diffusion convection reaction equation abstract in this paper we describe several finite element methods for solving the diffusion convection reaction equation none of them is new although the presentation is non standard in an effort to emphasize the similarities and differences between them in particular it is shown that the classical supg method is very similar to an explicit version of the characteristic galerkin method whereas the taylor galerkin method has a stabilization effect similar to a sub grid scale model which is in turn related to the introduction of bubble functions

an adaptive characteristic petrov galerkin finite element method for convection dominated linear and nonlinear parabolic problems in one space variable a new adaptive finite element method for convection dominated problems is presented a special feature of the method is that it is based on a petrov galerkin scheme for spatial approximation at a typical time step which employs test functions chosen so that the approximate solution coincides with the exact solution at the nodes of finite element grid this procedure makes possible the derivation of truly local a posteriori error estimates and a very effective solver numerical examples are discussed which illustrate the efficiency and effectively of the method

a class of discontinuous petrov galerkin methods part i the transport equation considering a simple model transport problem we present a new finite element method while the new method fits in the class of discontinuous galerkin dg methods it differs from standard dg and streamline diffusion methods in that it uses a space of discontinuous trial functions tailored for stability the new method unlike the older approaches yields optimal estimates for the primal variable in both the element size h and polynomial degree p and outperforms the standard upwind dg method

revisiting stabilized finite element methods for the advective diffusive equation we give a brief overview of stabilized finite element methods and illustrate the developments applied to the advection diffusion equation this article presents a concise perspective of the developments emanated from the works started in the 1980s through today 2005 elsevier b v all rights reserved pacs 47 11 j

the irreducible representations of the alternating group which remain irreducible in characteristic p let p be an odd prime and an the alternating group of degree n we determine which ordinary irreducible representations of u n remain irreducible in characteristic p verifying the author s conjecture from 2010 given the preparatory work done in 2010 our task is to determine which self conjugate partitions label specht modules for the symmetric group in characteristic p having exactly two composition factors this is accomplished through the use of the robinson brundan kleshchev i restriction functors together with known results on decomposition numbers for the symmetric group and additional results on the mullineux map and homomorphisms between specht modules

adaptive petrov galerkin methods for first order transport equations we propose stable variational formulations for certain linear unsymmetric operators with first order transport equations in bounded domains serving as the primary focus of this paper the central objective is to develop for such classes adaptive solution concepts with provable error reduction to adaptively resolve anisotropic solution features such as propagating singularities the presently proposed variational formulations allow in particular the employment of trial spaces spanned by directional representation systems since such systems typically given as frames are known to be stable only in l 2 special emphasis is placed on l 2 stable formulations the proposed stability concept is based on perturbations of certain ideal test spaces in petrov galerkin formulations see also l f demkowicz and j gopalakrishnan comput methods appl mech engrg 199 2010 pp 1558 1572 l demkowicz and j gopalakrishnan numer methods partial differential equations 27 2011 pp 70 105 j

hp fem for reaction diffusion equations i robust exponential convergence a singularly perturbed reaction diffusion equation in two dimensions is considered we assume analyticity of the input data i e the boundary of the domain is an analytic curve and the right hand side is analytic we show that the hp version of the finite element method leads to robust exponential convergence provided that one layer of needle elements of width o p eps is inserted near the domain boundary that is the rate of convergence is o mbox rm exp b p and independent of the perturbation parameter eps additionally we show that the use of numerical quadrature for the evaluation of the stiffness matrix and the load vector retains the exponential rate of convergence in particular the spectral element method based on the use of a gauss lobatto quadrature rule with p 1 x p 1 points yields robust exponential convergence

stable adaptive control for not necessarily stable discrete linear plants with bounded non linear inputs abstract this paper deals with the design of stable adaptive control schemes for discrete plants under non linear bounded inputs two algorithms are studied one of them being a generalization of the d step ahead standard algorithm while the second one involves the use of a time decreasing adaptation gain the class of inputs for which the algorithms perform in a stable fashion include saturations relays and saturated dead zones the plant stability assumption is not requested to prove standard non asymptotic stability results the mechanism to achieve scheme s stability is the use of an extra weighted additive term in the adaptation error which takes into account the output deviation with respect to that associated with linear inputs

schubert calculus and torsion explosion the author observes that certain numbers occurring in schubert calculus for sl also occur as entries in intersection forms controlling decompositions of soergel bimodules in higher rank these numbers grow exponentially this observation gives many counter examples to the expected bounds in lusztig s conjecture on the characters of simple rational modules for sl over fields of positive characteristic the examples also give counter examples to the james conjecture on decomposition numbers for symmetric groups

specht modules labelled by hook bipartitions i abstract brundan kleshchev and wang equip the specht modules s over the cyclotomic khovanov lauda rouquier algebra h n with a homogeneous z graded basis in this paper we begin the study of graded specht modules labelled by hook bipartitions n m 1 m in level 2 of h n which are precisely the hecke algebras of type b with quantum characteristic at least three we give an explicit description of the action of the khovanov lauda rouquier algebra generators 1 n 1 on the basis elements of s n m 1 m introducing certain specht module homomorphisms we construct irreducible submodules of these specht modules and thereby completely determine the composition series of specht modules labelled by hook bipartitions

modeling subgrid viscosity for advection diffusion problems we analyse the effect of the subgrid viscosity on a finite element discretisation with piecewise linear elements of a linear advection diffusion scalar equation we point out the importance of a proper tune up of the viscosity coefficient and we propose a heuristic method for obtaining reasonable values for it the extension to more general problems is then hinted in the last section

multiscale stabilization for convection diffusion equations with heterogeneous velocity and diffusion coefficients abstract we present a new stabilization technique for multiscale convection diffusion problems stabilization for these problems has been a challenging task especially for the case with high peclet numbers our method is based on a constraint energy minimization idea and the discontinuous petrov galerkin formulation in particular the test functions are constructed by minimizing an appropriate energy subject to certain orthogonality conditions and are related to the trial space the resulting test functions have a localization property and can therefore be computed locally we will prove the stability and present several numerical results our numerical results confirm that our test space gives a good stability in the sense that the solution error is close to the best approximation error

on locking and robustness in the finite element method a numerical scheme for the approximation of a parameter dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value a robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter precise mathematical definitions for these terms are developed their quantitative characterization is given and some general theorems involving locking and robustness are proven a model problem involving heat transfer is analyzed in detail using this mathematical framework and various related computational results are described applications to some different problems involving locking are presented

automatically stable discontinuous petrov galerkin methods for stationary transport problems quasi optimal test space norm we investigate the application of the discontinuous petrov galerkin dpg finite element framework to stationary convection diffusion problems in particular we demonstrate how the quasi optimal test space norm improves the robustness of the dpg method with respect to vanishing diffusion we numerically compare coarse mesh accuracy of the approximation when using the quasi optimal norm the standard norm and the weighted norm our results show that the quasi optimal norm leads to more accurate results on three benchmark problems in two spatial dimensions we address the problems associated to the resolution of the optimal test functions with respect to the quasi optimal norm by studying their convergence numerically in order to facilitate understanding of the method we also include a detailed explanation of the methodology from the algorithmic point of view

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100