an inf sup stable and residual free bubble element for the oseen equations we investigate the residual free bubble method for the linearized incompressible navier stokes equations starting with a nonconforming inf sup stable element pair for approximating the velocity and pressure we enrich the velocity space by discretely divergence free bubble functions to handle the influence of strong convection an important feature of the method is that the stabilization does not generate an additional coupling between the mass equation and the momentum equation as is the case for the streamline upwind petrov galerkin method applied to equal order interpolation furthermore the discrete solution is piecewise divergence free a property which is useful for the mass balance in transport equations coupled with the incompressible navier stokes equations

a hierarchical multiscale framework for problems with multiscale source terms this paper presents a hierarchical multiscale framework for problems that involve multiscale source terms an assumption on the additive decomposition of the source function results in consistent decoupling of the fully coupled system and constitutes the new method the structure of this decomposition is investigated and its mathematical implications are delineated this method results in variational embedding of fine scale information that is derived from the fine scale equations in the corresponding coarse scale equations it therefore provides a mathematically consistent way of bridging information between disparate spatial scales in the response function that are induced by multiscale forcing functions

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

a globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions we analyze a new galerkin finite element method for numerically solving a linear convection dominated convection diffusion problem in two dimensions the method is shown to be convergent uniformly in the perturbation parameter of order h1 2 in a global energy norm which is stronger than the l2 norm this order is optimal in this norm for our choice of trial functions

nearly h1 optimal finite element methods abstract we examine the problem of finding the h 1 projection onto a finite element space of an unknown field satisfying a specified boundary value problem solving the projection problem typically requires knowing the exact solution we circumvent this issue and obtain a petrov galerkin formulation which achieves h 1 optimality requiring weighting functions to be defined locally on the element level permits only approximate h 1 optimality in multi dimensional configurations we investigate the relation between our formulation and other stabilized fem formulations we show in particular that our formulation leads to a derivation of the supg method in special cases the present formulation reduces to that of residual free bubbles finally we present guidelines for obtaining the petrov weight functions and include a numerical example for the helmholtz equation

convergence analysis of the residual free bubble method for bilinear elements we present an a priori error analysis for the residual free bubble rfb method applied to diffusion convection equations with a dominating and antisymmetric convection operator special attention is paid to the case of piecewise bilinears in mathbb r 2 and the theoretical comparison with the streamline diffusion finite element sdfe equivalently streamline upwind petrov galerkin supg method

residual free bubbles for advection diffusion problems the general error analysis we develop the general a priori error analysis of residual free bubble finite element approximations to linear elliptic convection dominated diffusion problems subject to homogeneous dirichlet boundary condition optimal order error bounds are derived in various norms using piecewise polynomial finite elements of degree greater than or equal to 1

variational multiscale element free galerkin method for 2d burgers equation a new numerical method which is based on the coupling between variational multiscale method and meshfree methods is developed for 2d burgers equation with various values of re the proposed method takes full advantage of meshfree methods therefore no mesh generation and mesh recreation are involved meanwhile compared with the variational multiscale finite element method a strong assumption that is the fine scale vanishes identically over the element boundaries although non zero within the elements is not needed subsequently two problems which have an available analytical solution of their own are solved to analyze the convergence behaviour of the proposed method finally a 2d burgers equation having large re is solved and the results have also been compared with the ones computed by two other methods the numerical results show that the proposed method can indeed obtain accurate numerical results for 2d burgers equation having large re which does not refer to the choice of a proper stabilization parameter

a multiscale stabilized finite element method for the advection diffusion equation this paper presents a multiscale method that yields a stabilized finite element formulation for the advection diffusion equation the multiscale method arises from a decomposition of the scalar field into coarse resolved scale and fine unresolved scale the resulting stabilized formulation possesses superior properties like that of the supg and the gls methods a significant feature of the present method is that the definition of the stabilization term appears naturally and therefore the formulation is free of any user designed or user defined parameters another important ingredient is that since the method is residual based it satisfies consistency ab initio based on the proposed formulation a family of 2 d elements comprising 3 and 6 node triangles and 4 and 9 node quadrilaterals has been developed numerical results show the good performance of the method on uniform skewed as well as composite meshes and confirm convergence at optimal rates 2004 elsevier b v all rights reserved

a relationship between stabilized finite element methods and the galerkin method with bubble functions abstract a relation between stabilized finite element methods and the galerkin method employing interpolations with bubble functions is established for the advective diffusive model and for the linearized compressible navier stokes equations the bubble functions are shown to help in stabilizing the advective operator without recourse to upwinding or any other numerical strategy in particular for the advective diffusive model the galerkin method employing piecewise linears with bubble functions is shown to be equivalent to the streamline upwind petrov galerkin supg method in the diffusive limit

quasi optimality of the supg method for the one dimensional advection diffusion problem in this paper we propose quasi optimal error estimates in various norms for the streamline upwind petrov galerkin supg method applied to the linear one dimensional advection diffusion problem we follow the classical argument due to babuska and brezzi therefore the goal of this work is the proof of the inf sup and of the continuity conditions for the bilinear stabilized variational form with respect to suitable norms these norms are suggested by our previous work g sangalli technical report 1221 ian cnr pavia italy 2001 numer math to appear in which we analyze the continuous multidimensional advection diffusion operator we obtain these results by means of function space interpolation

a bubble stabilized finite element method for dirichlet constraints on embedded interfaces we examine a bubble stabilized finite element method for enforcing dirichlet constraints on embedded interfaces by embedded we refer to problems of general interest wherein the geometry of the interface is assumed independent of some underlying bulk mesh as such the robust imposition of dirichlet constraints using a lagrange multiplier field is not trivial to focus issues we consider a simple one sided problem that is representative of a wide class of evolving interface problems the bulk field is decomposed into coarse and fine scales giving rise to coarse scale and fine scale one sided sub problems the fine scale solution is approximated with bubble functions permitting static condensation and giving rise to a stabilized form bearing strong analogy with a classical method importantly the method is simple to implement readily extends to multiple dimensions obviates the need to specify any free stabilization parameters and can lead to a symmetric positive definite system of equations the performance of the method is then examined through several numerical examples the accuracy of the lagrange multiplier is compared to results obtained using a local version of the domain integral method the variational multiscale approach proposed herein is shown to both stabilize the lagrange multiplier and improve the accuracy of the post processed fluxes copyright 2006 john wiley sons ltd

a petrov galerkin enriched method a mass conservative finite element method for the darcy equation starting from the non stable p1 p0 discretization we build enhanced methods for the darcy equation which are stable and locally mass conservative the methods are derived in a petrov galerkin framework where both velocity and pressure trial spaces are enriched with multiscale functions these functions solve local problems correcting the residuals of the strong equations in each element and interior edge which leads to a velocity space enhanced with functions belonging to the lowest order raviart thomas space several numerical tests validate the methods

on the stabilization parameter in the subgrid scale approximation of scalar convection diffusion reaction equations on distorted meshes abstract in this paper we revisit the definition of the stabilization parameter in the finite element approximation of the convection diffusion reaction equation the starting point is the decomposition of the unknown into its finite element component and a subgrid scale that needs to be approximated in order to incorporate the distortion of the mesh into this approximation we transform the equation for the subgrid scale within each element to the shape regular reference domain the expression for the subgrid scale arises from an approximate fourier analysis and the identification of the wave number direction where instabilities are most likely to occur the final outcome is an expression for the stabilization parameter that accounts for anisotropy and the dominance of either convection or reaction terms in the equation

endomorphisms of c algebras cross products and duality for compact groups on definit une nouvelle sorte de produit croise d une c algebre par des semigroupes d endomorphismes avec des proprietes reliees a la symetrie de permutation et a l existence de conjugues un groupe compact est alors defini intrinsequement par son action duale sur le produit croise cette construction donne une caracterisation des duaux de groupes compacts abstraits qui est hors d atteinte de la theorie de dualite de taunaka kein et est independante de cette theorie ceci resout le probleme de demontrer l existence d un groupe de jauge global compact en physique des particules etant seulement donnees les observables locales

on a multi scale element free galerkin method for the stokes problem abstract in this paper a multi scale element free galerkin method is presented for the stokes problem the new method is based on the hughes variational multi scale formulation and arises from a decomposition of the velocity field into coarse resolved scales and fine unresolved scales in this method an unresolved model is obtained in which unresolved scales are incorporated analytically through the bubble functions modeling of the unresolved scales corrects the lack of the stability of the standard element free galerkin formulation and the resulting stabilized formulation possesses superior properties like that of the streamline upwind petrov galerkin supg method and the galerkin least squares gls method the method allows equal order basis for pressure and velocity because it violates the celebrated babuska brezzi condition a significant feature of the present method is that the structure of the stabilization tensor appears naturally via the solution of the fine scale problem numerical results for example problems confirm that this method has some excellent properties such as better stability and accuracy

applications of the pseudo residual free bubbles to the stabilization of convection diffusion reaction problems it is known that the enrichment of the polynomial finite element space of degree 1 by bubble functions results in a stabilized scheme of the supg type for the convection diffusion reaction problems in particular the residual free bubbles rfb can assure stabilized methods but they are usually difficult to compute unless the configuration is simple therefore it is important to devise numerical algorithms that provide cheap approximations to the rfb functions contributing a good stabilizing effect to the numerical method overall here we propose a stabilization technique based on the rfb method and particularly designed to treat the most interesting case of small diffusion we replace the rfb functions by their cheap yet efficient approximations which retain the same qualitative behavior the approximate bubbles are computed on a suitable sub grid the choice of whose nodes are critical and determined by minimizing the residual of a local problem with respect to l 1 norm the resulting numerical method has similar stability features with the rfb method for the whole range of problem parameters this fact is also confirmed by numerical experiments we also note that the location of the sub grid nodes suggested by the strategy herein coincides with the one in brezzi et al math models methods appl sci 13 445 461 2003

a novel least squares finite element method enriched with residual free bubbles for solving convection dominated problems in this paper we devise a novel least squares finite element method lsfem for solving scalar convection dominated convection diffusion problems first we convert a second order convection diffusion problem into a first order system formulation by introducing the gradient mathbf p kappa nabla u as an additional variable the lsfem using continuous piecewise linear elements enriched with residual free bubbles for both variables u and mathbf p is applied to solve the first order mixed problem the residual free bubble functions are assumed to strongly satisfy the associated homogeneous second order convection diffusion equations in the interior of each element up to the contribution of the linear part and vanish on the element boundary to implement this two level least squares approach a stabilized method of galerkin least squares type is used to approximate the residual free bubble functions this enriched lsfem not only inherits the advantages of the primitive lsfem such as the resulting linear system being symmetric and positive definite but also exhibits the characteristics of the residual free bubble method without involving stability parameters several numerical experiments are given to demonstrate the effectiveness of the proposed enriched lsfem the accuracy and computational cost of this enriched lsfem are also compared with those of the primitive lsfem we find that for a small diffusivity kappa the enriched lsfem is much better able to capture the nature of layer structure in the solution than the primitive lsfem even if the primitive lsfem uses a very fine mesh or higher order elements in other words the enriched lsfem provides a significant improvement with a lower computational cost over the primitive lsfem for solving convection dominated problems

finite element methods for convection diffusion problems using exponential splines on triangles abstract a new family of petrov galerkin finite element methods on triangular grids is constructed for singularly perturbed elliptic problems in two dimensions it uses divergence free trial functions that form a natural generalization of one dimensional exponential trial functions this family includes an improved version of the divergence free finite element method used in the pltmg code numerical results show that the new method is able to compute strikingly accurate solutions on coarse meshes an analysis of the use of slotboom variables shows that they are theoretically unsatisfactory and explains why certain petrov galerkin methods lose stability when generalized from one to two dimensions

streamline design of stability parameters for advection diffusion problems the dependence of the computation of advective diffusive transport phenomena on the orientation of the mesh with respect to the flow direction is analyzed poor performance of the classical galerkin finite element method in the convection dominated regime is alleviated by stabilization we propose definitions of the stability parameter that rationally incorporate the flow direction numerical tests compare the performance of the proposed methods with established techniques

further considerations on residual free bubbles for advective diffusive equations abstract we further consider the galerkin method for advective diffusive equations in two dimensions the finite dimensional space employed is of piecewise polynomials enriched with residual free bubbles rfb we show that in general this method does not coincide with the supg method unless the piecewise polynomials are spanned by linear functions furthermore a simple stability analysis argument displays the effect of the rfb on the reduced space of piecewise polynomials which in some situations is not equivalent to streamline diffusion for bilinears

bundles of c categories we introduce the notions of multiplier c category and continuous bundle of c categories as the categorical analogues of the corresponding c algebraic notions every symmetric tensor c category with conjugates is a continuous bundle of c categories with base space the spectrum of the c algebra associated with the identity object we classify tensor c categories with fibre the dual of a compact lie group in terms of suitable principal bundles this also provides a classification for certain c algebra bundles with fibres fixed point algebras of od

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

variational multiscale element free galerkin method for the water wave problems a new numerical method which is based on the coupling between variational multiscale method and meshfree methods is developed for the water wave problems in which the free surface capturing technique is used to capture the position of the free surface the proposed method takes full advantage of meshfree methods therefore no mesh generation and mesh reconstruction are involved meanwhile due to that the proposed method belongs to meshfree methods thus it is suitable for the highly deformed free surface flow problems finally two water wave problems are solved and the results have also been analyzed the numerical results show that the proposed method can indeed obtain accurate numerical results for the water wave problems which does not refer to the choice of a proper stabilization parameter

on efficient least squares finite element methods for convection dominated problems abstract this paper focuses on the least squares finite element method and its three variants for obtaining efficient numerical solutions to convection dominated convection diffusion problems the coercivity estimates for the corresponding homogeneous least squares energy functionals are derived and based on which error estimates are established one of the common advantages of these least squares methods is that the resulting linear system is symmetric and positive definite numerical experiments that demonstrate the theoretical analysis of the developed methods are presented it was observed that the primitive least squares method performs poorly for convection dominated problems while the stabilized streamline diffusion and negatively stabilized streamline diffusion least squares methods perform considerably better for interior layer problems and the negatively stabilized streamline diffusion least squares method is able to better capture the boundary layer behavior when compared with other least squares methods but all the least squares methods do not give reasonable results for problem possessing both interior and boundary layer structures in the solution

a variational multiscale method based on bubble functions for convection dominated convection diffusion equation this work presents a variational multiscale method based on polynomial bubble functions as subgrid scale and a numerical implementation based on two local gauss integrations this method can be implemented easily and efficiently for the convection dominated problem static condensation of the bubbles suggests the stability of the method and we establish its global convergence representative numerical tests are presented

the nearly optimal petrov galerkin method for convection diffusion problems abstract the nearly optimal petrov galerkin nopg method is employed to improve finite element computation of convection dominated transport phenomena the design of the nopg method for convection diffusion is based on consideration of the advective limit nonetheless the resulting method is applicable to the entire admissible range of problem parameters an investigation of the stability properties of this method leads to a coercivity inequality the convergence features of the nopg method for convection diffusion are studied in an error analysis that is based on the stability estimates the proposed method compares favorably to the performance of an established technique on several numerical tests

on a two level finite element method for the incompressible navier stokes equations we consider the galerkin finite element method for the incompressible navier stokes equations in two dimensions where the finite dimensional space s employed consist of piecewise polynomials enriched with residual free bubble rfb functions to find the bubble part of the solution a two level finite element method tlfem is described and its application to the navier stokes equation is displayed numerical solutions employing the tlfem are presented for three benchmark problems we compare the numerical solutions using the tlfem with the numerical solutions using a stabilized method

finite element method solution of electrically driven magnetohydrodynamic flow the magnetohydrodynamic mhd flow in a rectangular duct is investigated for the case when the flow is driven by the current produced by electrodes placed one in each of the walls of the duct where the applied magnetic field is perpendicular the flow is steady laminar and the fluid is incompressible viscous and electrically conducting a stabilized finite element with the residual free bubble rfb functions is used for solving the governing equations the finite element method employing the rfb functions is capable of resolving high gradients near the layer regions without refining the mesh thus it is possible to obtain solutions consistent with the physical configuration of the problem even for high values of the hartmann number before employing the bubble functions in the global problem we have to find them inside each element by means of a local problem this is achieved by approximating the bubble functions by a nonstandard finite element method based on the local problem equivelocity and current lines are drawn to show the well known behaviours of the mhd flow those are the boundary layer formation close to the insulated walls for increasing values of the hartmann number and the layers emanating from the endpoints of the electrodes the changes in direction and intensity with respect to the values of wall inductance are also depicted in terms of level curves for both the velocity and the induced magnetic field

a comparison of uniformly convergent difference schemes for two dimensional convection diffusion problems galerkin and petrov galerkin finite element methods are used to obtain new finite difference schemes for the solution of linear two dimensional convection diffusion problems numerical estimates are made of the rates of convergence of these schemes uniformly with respect to the perturbation parameter and these uniform rates are shown to compare favourably with those of established methods

a two level enriched finite element method for a mixed problem the simplest pair of spaces is made inf sup stable for the mixed form of the darcy equation the key ingredient is to enhance the finite element spaces inside a petrov galerkin framework with functions satisfying element wise local darcy problems with right hand sides depending on the residuals over elements and edges the enriched method is symmetric locally mass conservative and keeps the degrees of freedom of the original interpolation spaces first we assume local enrichments exactly computed and we prove uniqueness and optimal error estimates in natural norms then a low cost two level finite element method is proposed to effectively obtain enhancing basis functions the approach lays on a two scale numerical analysis and shows that well posedness and optimality is kept despite the second level numerical approximation several numerical experiments validate the theoretical results and compares favourably in some cases our results with the classical raviart thomas element

a robust a posteriori estimator for the residual free bubbles method applied to advection diffusion problems we develop the a posteriori error analysis for the rfb method applied to the linear advection diffusion problem the numerical error measured in suitable norms is estimated in terms of the numerical residual the robustness is investiged in the sense that we prove uniform equivalence between a norm of the numerical residual and a particular norm of the error

multiscale stabilized solutions to one dimensional systems of conservation laws abstract we present a variational multiscale formulation for the numerical solution of one dimensional systems of conservation laws the key idea of the proposed formulation originally presented by hughes comput methods appl mech engrg 127 1995 387 401 is a multiple scale decomposition into resolved grid scales and unresolved subgrid scales incorporating the effect of the subgrid scales onto the coarse scale problem results in a finite element method with enhanced stability properties capable of accurately representing the sharp features of the solution in the formulation developed herein the multiscale split is invoked prior to any linearization of the equations special attention is given to the choice of the matrix of stabilizing coefficients and the discontinuity capturing diffusion the methodology is applied to the one dimensional simulation of three phase flow in porous media and the shallow water equations these numerical simulations clearly show the potential and applicability of the formulation for solving highly nonlinear nearly hyperbolic systems on very coarse grids application of the numerical formulation to multidimensional problems is presented in a forthcoming paper

morita homotopy theory of c categories abstract in this article we establish the foundations of the morita homotopy theory of c categories concretely we construct a cofibrantly generated simplicial symmetric monoidal quillen model structure denoted by m mor on the category c 1 cat of small unital c categories the weak equivalences are the morita equivalences and the cofibrations are the functors which are injective on objects as an application we obtain an elegant description of brown green rieffel s picard group in the associated homotopy category ho m mor we then prove that ho m mor is semi additive by group completing the induced abelian monoid structure at each hom set we obtain an additive category ho m mor 1 and a composite functor c 1 cat ho m mor ho m mor 1 which is characterized by two simple properties inversion of morita equivalences and preservation of all finite products finally we prove that the classical grothendieck group functor becomes co represented in ho m mor 1 by the tensor unit object

finite difference approximations of multidimensional unsteady convection diffusion reaction equations in this paper the numerical approximation of unsteady convection diffusion reaction equations with finite difference method on a special grid is studied in the convection or reaction dominated regime we extend the method 19 which was designed for multidimensional steady convection diffusion reaction equations to unsteady problems we investigate two possible different ways of combining the discretization in time and in space where the sequence of the discretizations is interchanged discretization in time is performed by using crank nicolson and backward euler finite difference schemes while for the space discretization we consider the method 19 numerical tests are presented to show good performance of the method

two level finite element method and its application to the helmholtz equation a two level finite element method is introduced and its application to the helmholtz equation is considered the method retains the desirable features of the galerkin method enriched with residual free bubbles while it is not limited to discretizations using elements with simple geometry the method can be applied to other equations and to irregular shaped domains

mass lumping emanating from residual free bubbles a nodally exact scheme is derived for a model equation in 1d involving zeroth and second order terms the method is derived using residual free bubbles in conjunction with the galerkin approximation it is shown that this approach leads to the mass lumping scheme for sufficiently small mesh sizes

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

a nonlinear subgrid method for advection diffusion problems abstract this work presents a general framework for approximating advection diffusion equations based on principles of scale separation a two level decomposition of the discrete approximation space is performed and the local problem is modified to capture both local and nonlocal discontinuities the new feature is the local control resulting from decomposing the velocity field into the resolved and unresolved scales and requiring the satisfaction of the discrete model problem at the element level for a minimum kinetic energy associated to the unresolved scales this procedure leads to a nonlinear subgrid model that acts only on the unresolved scales but does not require any tuned up parameter it can be considered a self adaptive method such that the amount of the subgrid viscosity is automatically introduced according to the residual of the resolved scale at element level

a finite difference scheme for multidimensional convection diffusion reaction equations abstract in this paper a finite difference scheme is proposed for multidimensional convection diffusion reaction equations particularly designed to treat the most interesting case of small diffusion it is based closely on the work endur and nesliturk 2011 application of the method to multidimensional convection diffusion reaction equation is based on a simple splitting of the convection diffusion reaction equation and then joining their approximations obtained with endur and nesliturk 2011 the method adapts very well to all regimes with continuous transitions from one regime to another numerical tests show good performance of the method and superiority with respect to well known stabilized finite element methods

bubble based stabilization for the helmholtz equation a comparison of two bubble enriched methods derived by different considerations indicates that the methods are identical in some cases thus series representations of auxiliary functions derived independently for the two methods turn out to be equivalent prior to truncation three such series for time harmonic acoustics are considered dispersion analysis points to the more efficient series representation and provides guidelines for the number of terms to be retained numerical tests confirm the validity of these practical guidelines copyright 2006 john wiley sons ltd

c categories groupoid actions equivariant kk theory and the baum connes conjecture in this article we give a characterisation of the baum connes assembly map with coecien ts the technical tools needed are the k theory of c categories and equivariant kk theory in the world of groupoids

a three level finite element method for the instationary incompressible navier stokes equations we present a two and a three level finite element method for the numerical simulation of incompressible flow governed by the navier stokes equations within the theoretical framework of the variational multiscale method and exploiting the concept of residual free bubbles we propose a separation of three different scale groups namely large resolved scales small resolved scales and unresolved scales the resolution of the large and small scales takes place on levels 1 and 2 with the aid of diverse approaches the dynamic calculation of a subgrid viscosity representing the effect of the unresolved scales constitutes level 3 of our algorithm the proposed algorithm is tested for various laminar flow situations and compared to the results obtained via an unusual stabilized finite element method it is supposed to be the basic scheme also for turbulent flow in a subsequent study

finite difference approximations of multidimensional convection diffusion reaction problems with small diffusion on a special grid a numerical scheme for the convection diffusion reaction cdr problems is studied herein we propose a finite difference method on a special grid for solving cdr problems particularly designed to treat the most interesting case of small diffusion we use the subgrid nodes in the link cutting bubble lcb strategy 5 to construct a numerical algorithm that can easily be extended to the higher dimensions the method adapts very well to all regimes with continuous transitions from one regime to another we also compare the performance of the present method with the streamline upwind petrov galerkin supg and the residual free bubbles rfb methods on several benchmark problems the numerical experiments confirm the good performance of the proposed method

bubble functions prompt unusual stabilized finite element methods a second order linear scalar differential equation including a zero th order term is approximated using first the standard galerkin method enriched with bubble functions static condensation of the bubbles suggest an unusual stabilized finite element method for which we establish a convergence study and obtain successful numerical simulations the method is generalized to allow for a convection operator in the equation this work may be employed as a starting point for simulation of nonlinear equations governing turbulence phenomena flows with chemical reactions and other important problems

a priori error analysis of residual free bubbles for advection diffusion problems we develop an a priori error analysis of a finite element approximation to the elliptic advection diffusion equation eps delta u conv cdot nabla u f subject to a homogeneous dirichlet boundary condition based on the use of residual free bubble functions an optimal order error bound is derived in the so called stability norm biggl eps nabla v 2 l 2 omega sum t h t conv cdot nabla v 2 l 2 t biggr 1 2 where ht denotes the diameter of element t in the subdivision of the computational domain

bubble and multiscale stabilization of bilinear finite element methods for transient advection diffusion equations on rectangular grids in this paper a petrov galerkin type stabilization for a time dependent advection diffusion equation is considered we first enrich the bilinear test space with bubble functions and the bilinear trial space with a special combination of bubble and multiscale functions for the steady state advection diffusion equation it is known that solving the residual equation obtained by the bubble elimination procedure is as difficult as solving the steady case of the original problem which makes the enriched methods quite costly for two dimensional problems in this study we instead utilize their cheap and efficient approximations in each rectangular element then we suggest a recipe for a proper adaptation of these functions combined with the generalized euler time integration to the unsteady problem some numerical experiments for typical model problems are presented to illustrate the accuracy of our method

streamline upwind petrov galerkin method supg vs residual free bubbles rfb abstract the purpose of this paper is to present the residual free bubbles rfb method and to show its relationship with the streamline upwind petrov galerkin method supg the paper is organized as follows after the introduction we show how the failure of classical finite element methods for convection dominated convection diffusion problem can be explained by energy considerations then we describe the residual free bubbles ideas and in the last section we show the equivalence of rfb and supg in a simple case

the finite element method for mhd flow at high hartmann numbers abstract a stabilized finite element method using the residual free bubble functions rfb is proposed for solving the governing equations of steady magnetohydrodynamic duct flow a distinguished feature of the rfb method is the resolving capability of high gradients near the layer regions without refining mesh we show that the rfb method is stable by proving that the numerical method is coercive even not only at low values but also at moderate and high values of the hartmann number numerical results confirming theoretical findings are presented for several configurations of interest the approximate solution obtained by the rfb method is also compared with the analytical solution of shercliff s problem

k homology of c categories and symmetric spectra representing k homology we define k homology groups k for small c categories in terms of hilbert modules over the c category we also define a functor a f from the category of small c categories into the category of c algebras and show that there is a natural isomorphism k cal c cong k a f cal c in addition we give an easy construction of a functor mathbb k from the category of c algebras into the category of symmetric spectra which represents k homology i e we show that the functor mathbb k comes with a natural isomorphism k a cong pi mathbb k a for c algebras a it then follows that the composition mathbb k a f provides a functor that can be used in the davis luck approach for constructing the baum connes map

the discontinuous enrichment method for multiscale analysis computation naturally separates scales of a problem according to the mesh size a variety of improved numerical methods are described by multiscale considerations differing in the treatment of the unresolved fine scales the discontinuous enrichment method provides a unique multiscale approach to computation by employing fine scales that contain solutions of the homogeneous partial differential equation in a discontinuous framework the method thus combines relative ease of implementation with improved numerical performance these properties are demonstrated for both multiscale wave and transport problems pointing to the potential of considerable savings in computational resources

virtual bubbles and galerkin least squares type methods ga l s abstract the equivalence between stabilized finite element methods or galerkin least squares tyoe methods ga l s and the standard galerkin method with bubble functions is established in an abstract framework the results are applicable to various finite element spaces including high order elements and applications to the advective diffusive model and to the stokes problem are presented illustrating the potential of the abstract theory introduced here

on the choice of stabilizing sub grid for convection diffusion problem on rectangular grids a stabilizing sub grid which consists of a single additional node in each rectangular element is analyzed for solving the convection diffusion problem especially in the case of small diffusion we provide a simple recipe for spotting the location of the additional node that contributes a very good stabilizing effect to the overall numerical method we further study convergence properties of the method under consideration and prove that the standard galerkin finite element solution on augmented grid produces a discrete solution that satisfies the same type of a priori error estimates that are typically obtained with the supg method some numerical experiments that confirm the theoretical findings are also presented

on the stability of residual free bubbles for convection diffusion problemsand their approximation by a two level finite element method we consider the galerkin finite element method for partial diffferential equations in two dimensions where the finite dimensional space used consists of piecewise isoparametric polynomials enriched with bubble functions writing l for the differential operator we show that for elliptic convection diffusion problems the component of the bubble enrichment that stabilizes the method is essentially equivalent to a petrov galerkin method with an l spline exponentially fitted trial space and piecewise polynomial test space the remaining component of the bubble influences the accuracy of the method a stability inequality recently obtained by brezzi franca and russo for a limiting case of bubbles applied to convection diffusion problems is shown to be slightly weaker than the standard stability inequality that is obtained for the sdfem supg method thereby demonstrating that the bubble approach is in general slightly less stable than the streamline diffusion method when the trial functions are piecewise linear we show that residual free bubbles are as stable as sdfem supg and we extend this stability inequality to include positive mesh peclet numbers in the convection dominated regime approximate computations of the residual free bubbles are performed using a two level finite element method

global and local error analysis for the residual free bubbles method applied to advection dominated problems we prove the stability and a priori global and local error analysis for the residual free bubbles rfb finite element method applied to advection dominated advection diffusion problems

on the choice of a stabilizing subgrid for convection diffusion problems abstract supg and residual free bubbles are closely related methods that have been used with success to stabilize a certain number of problems including advection dominated flows in recent times a slightly different idea has been proposed to choose a suitable subgrid in each element and then solving standard galerkin on the augmented grid for this however the correct location of the subgrid node s plays a crucial role here for the model problem of linear advection diffusion equations we propose a simple criterion to choose a single internal node such that the corresponding plain galerkin scheme on the augmented grid provides the same a priori error estimates that are typically obtained with supg or rfb methods

unlocking with residual free bubbles residual free bubbles are derived for the timoshenko beam problem eliminating these bubbles the resulting formulation is form identical in using the following tricks to the standard variational formulation i one point reduced integration on the shear energy term ii replace its coefficient 1 2 by 1 2 hk212 in each element iii modify consistently the right hand side this final formulation is legally obtained in that the galerkin method enriched with residual free bubbles is developed using full integration throughout furthermore this method is nodally exact by construction

a bubble stabilized least squares finite element method for steady mhd duct flow problems at high hartmann numbers in this paper we devise a stabilized least squares finite element method using the residual free bubbles for solving the governing equations of steady magnetohydrodynamic duct flow we convert the original system of second order partial differential equations into a first order system formulation by introducing two additional variables then the least squares finite element method using c 0 linear elements enriched with the residual free bubble functions for all unknowns is applied to obtain approximations to the first order system the most advantageous features of this approach are that the resulting linear system is symmetric and positive definite and it is capable of resolving high gradients near the layer regions without refining the mesh thus this approach is possible to obtain approximations consistent with the physical configuration of the problem even for high values of the hartmann number before incoorperating the bubble functions into the global problem we apply the galerkin least squares method to approximate the bubble functions that are exact solutions of the corresponding local problems on elements therefore we indeed introduce a two level finite element method consisting of a mesh for discretization and a submesh for approximating the computations of the residual free bubble functions numerical results confirming theoretical findings are presented for several examples including the shercliff problem

the discontinuous enrichment method we propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it the enrichment contains free space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field continuity of the enrichment across element interfaces is enforced weakly by lagrange multipliers in this manner we expect to attain high coarse mesh accuracy without significant degradation of conditioning assuring good performance of the computation at any mesh resolution examples of application to acoustics and advection diffusion are presented

stabilized finite element approximation of transient incompressible flows using orthogonal subscales abstract in this paper we present a stabilized finite element method to solve the transient navier stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales the latter are approximately accounted for so as to end up with a stable finite element problem which in particular allows to deal with convection dominated flows and the use of equal velocity pressure interpolations three main issues are addressed the first is a method to estimate the behavior of the stabilization parameters based on a fourier analysis of the problem for the subscales secondly the way to deal with transient problems discretized using a finite difference scheme is discussed finally the treatment of the nonlinear term is also analyzed a very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space in the transient case this simplifies considerably the numerical scheme

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 new duality theory for compact groups the tannaka krein duality theory characterizes the category g of finite dimensional continuous unitary representations of a compact group as a subcategory of the category of hilbert spaces we prove a more powerful result characterizing g as an abstract category every strict symmetric monoidalc category with conjugates which has subobjects and direct sums and for which thec algebra of endomorphisms of the monoidal unit reduces to the complex numbers is isomorphic to a category g for a compact groupg unique up to isomorphism

bundles of c categories ii c dynamical systems and dixmier douady invariants we introduce a cohomological invariant arising from a class in nonabelian cohomology this invariant generalises the dixmier douady class and encodes the obstruction to a c algebra bundle being the fixed point algebra of a gauge action as an application the duality breaking for group bundles vs tensor c categories with nonsimple unit is discussed in the setting of nistor troitsky gauge equivariant k theory there is a map assigning a nonabelian gerbe to a tensor category and triviality of the gerbe is equivalent to the existence of a dual group bundle at the c algebraic level this corresponds to studying c algebra bundles with fibre a fixed point algebra of the cuntz algebra and in this case our invariant describes the obstruction to finding an embedding into the cuntz pimsner algebra of a vector bundle

on generalized torsion tensor fields and the reduced fiber bundle the reduction of a connection form from a fiber bundle p to a subbundle q is examined in detail defining generalized torsion forms we show how the usual maurer cartan structural equations have to be modified examples and applications to classical general relativity and gauge theories are outlined

a positive and bounded finite element approximation of the generalized burgers huxley equation abstract we present a finite element scheme capable of preserving the nonnegative and bounded solutions of the generalized burgers huxley equation proofs of existence and uniqueness of a solution to the continuous problem together with some results concerning the boundedness and the nonnegativity of the solution are given under appropriate conditions on the mesh and the initial and boundary data boundedness and nonnegativity of the finite element approximation are established an a priori error estimate for the approximation is also derived numerical experiments are presented which support the derived theoretical results