optimal bounds for a lagrange interpolation inequality for piecewise linear continuous finite elements in two space dimensions optimal bounds for a lagrange interpolation inequality for piecewise linear continuous finite elements in two space dimensions

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

goal oriented error control in adaptive mixed fem for signorini s problem abstract this paper outlines goal oriented finite element error control for signorini s problem the discretization is based on a mixed formulation proposed by hlavacek et al which is extended to higher order polynomials a duality argument based on a variational inequality is applied which allows for the estimates in h as well as hp adaptivity numerical results confirm the applicability of the theoretical findings

a posteriori error analysis for stabilised finite element approximations of transport problems we develop the a posteriori error analysis of stabilised finite element approximations to linear transport problems via duality arguments two alternative dual problems are considered one is based on the formal adjoint of the hyperbolic differential operator the other on the transposition of the bilinear form for the stabilised finite element method we show both analytically and through numerical experiments that the second approach is superior in the sense that it leads to sharper a posteriori error bounds and more economical adaptively refined meshes

free vibration analysis of a uniform multi span beam carrying multiple spring mass systems abstract the literature regarding the free vibration analysis of single span beams carrying a number of spring mass systems is plenty but that of multi span beams carrying multiple spring mass systems is fewer thus this paper aims at determining the exact solutions for the natural frequencies and mode shapes of a uniform multi span beam carrying multiple spring mass systems firstly the coefficient matrices for an intermediate pinned support an intermediate spring mass system left end support and right end support of a uniform beam are derived next the numerical assembly technique for the conventional finite element method is used to establish the overall coefficient matrix for the whole vibrating system finally equating the last overall coefficient matrix to zero one determines the natural frequencies of the vibrating system and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes in this paper the natural frequencies and associated mode shapes of the vibrating system are obtained directly from the differential equation of motion of the continuous beam and no other assumptions are made thus the last solutions are the exact ones the effects of attached spring mass systems on the free vibration characteristics of the 1 4 span beams are studied

a time space flux corrected transport finite element formulation for solving multi dimensional advection diffusion reaction equations abstract we present a time space flux corrected transport fct finite element formulation for the multi dimensional time dependent advection diffusion reaction equation monotonic solutions can be achieved with the presented method while large time steps with courant number c r 1 are used numerical verification as well as a grid convergence analysis are carried out for 1d and 2d benchmark problems a 1d burgers equation with various reynolds numbers is solved with the time space fct method in addition a 2d transport problem with and without a nonlinear reaction inside of a cavity is considered finally the newly developed time space fct formulation is applied for modeling biofilm growth problems based on a continuum mathematical model it turns out that the time space fct method helps to reduce numerical dissipation and guarantees comparable numerical dispersion of the solution at the same time comparing to numerical solutions obtained by a time space finite incremental calculus fic method

on a fem method for a linearized version of the oldroyd s problem abstract we present a study of a finite element method fem for the approximation of a linearized version of the oldroyd s problem the method is based on a work in zamm 65 1985 449 concerning the continuous problem on a splitting of the momentum equation and on a modification of the streamline upwind petrov galerkin method in comput meth appl mech engrg 45 1984 285 it is also formally connected to the modified elastic viscous split stress method in j non newtonian fluid mech 60 1995 27 this method allows us to improve some numerical results expected for more standard methods in particular we obtain convergence results in the case of a null newtonian fraction of the viscosity

the exact solutions for the natural frequencies and mode shapes of non uniform beams with multiple spring mass systems abstract for a beam carrying n spring mass systems if the left side and right side of each attaching point and each end of the beam are regarded as nodes then considering the compatibility of deformations and the equilibrium of forces between the two adjacent beam segments at each attaching point and incorporating with the equation of motion for each spring mass system simultaneous equations may be obtained for the th attaching point where the unknowns for the simultaneous equations are composed of the integration constants for the eigenfunctions of the th and 1 th beam segments and the associated modal displacements of the th sprung mass it is evident that if these unknowns are considered as the nodal displacements then the coefficient matrix of the simultaneous equations will be equivalent to the element stiffness matrix for the th attaching point associated with the th and 1 th beam segments in view of the last fact one may use the numerical assembly method nam for the conventional finite element method to obtain the overall simultaneous equations for the overall n attaching points associated with the overall n 1 beam segments by taking into account the boundary conditions of the whole beam the solutions for the coefficient determinant of the overall simultaneous equations to be equal to zero will give the exact natural frequencies of the constrained beam carrying multiple n spring mass systems and the substitution of each corresponding values of the integration constants into the associated eigenfunctions for each attaching point will determine the corresponding mode shapes since no discretization on the continuous beam was made in the present approach nam the natural frequencies and the corresponding mode shapes obtained are the exact ones

the finite difference streamline diffusion methods for sobolev equations with convection dominated term in this paper we give two finite difference streamline diffusion fdsd schemes for solving linear sobolev equations with convection dominated term stability and optimal error estimates for them are derived in suitable norms

on inf sup stabilized finite element methods for transient problems we consider the behavior of inf sup stabilization in the context of transient problems with multiple time scales our motivation for studying this setting is provided by reacting flows problems for which small time steps are necessary in the integration process we show that for algorithms defined through a process wherein spatial and temporal discretizations are separated the coupling of implicit time integration with spatial inf sup stabilization may lead to anomalous pressure behavior including the onset of spurious oscillations for very small time steps effectively this coupling introduces a stability criterion resulting in a dependence between the spatial grid size and the time step we illustrate our theoretical results by numerical examples that demonstrate the stability criterion 2004 elsevier b v all rights reserved

a stabilized mixed finite element method for darcy flow we develop new stabilized mixed finite element methods for darcy flow stability and an a priori error estimate in the stability norm are established a wide variety of convergent finite elements present themselves unlike the classical galerkin formulation which requires highly specialized elements an interesting feature of the formulation is that there are no mesh dependent parameters numerical tests confirm the theoretical results 2002 elsevier science b v all rights reserved

subgrid stabilized projection method for 2d unsteady flows at high reynolds numbers a subgrid stabilization technique is developed for solving the two dimensional incompressible navier stokes equations at high reynolds numbers the time marching algorithm is based on a well established fractional step pressure correction projection method the advection diffusion step is enriched by an implicit subgrid stabilizing term and by an explicit dissipative shock capturing term the former is calculated by means of a hierarchical finite element setting the latter is included to avoid gibbs phenomenon in the boundary layer convergence tests on prototypical two dimensional examples are reported and the method is used to simulate the viscous incompressible flows around the airfoil naca0012 at zero incidence and reynolds numbers ranging from 105 to 106

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

on discontinuous galerkin methods discontinuous galerkin methods have received considerable attention in recent years for problems in which advection and diffusion terms are present several alternatives for treating the diffusion and advective fluxes have been introduced this report summarizes some of the methods that have been proposed several numerical examples are included in the paper these present discontinuous galerkin solutions of one dimensional problems with a scalar variable results are presented for diffusion reaction problems and advection diffusion problems we discuss the performance of various formulations with respect to accuracy as well as stability of the method copyright 2003 john wiley sons ltd

convergence analyses of galerkin least squares methods for symmetric advective diffusive forms of the stokes and incompressible navier stokes equations abstract symmetric advective diffusive forms of the stokes and incompressible navier stokes equations are presented the galerkin least squares method for advective diffusive equations is used for both systems and is related to other stabilized methods previously studied the presentation reveals that the convergence analysis for advective diffusive equations as applied before to a linearized form of the compressible navier stokes equations carries over in a straightforward manner to the stokes problem and to a linearized form of the incompressible navier stokes equations

multiscale finite element modeling of diffusion reaction equation using bubble functions with bilinear and triangular elements a multiscale finite element method based on the bubble functions is used to generate stable and accurate solutions for the diffusion reaction equation for both dissipation and production cases this equation shows multiscale behaviour when the process becomes reaction dominated to validate the approach the numerical results obtained for a benchmark problem are compared with the analytical solution further numerical experiments are carried out to investigate the performance of the developed technique over a range damkohler number the numerical experiments are performed using bilinear and triangular elements

explicit finite element methods for symmetric hyperbolic equations a family of explicit space time finite element methods for the initial boundary value problem for linear symmetric hyperbolic systems of equations is described and analyzed the method generalizes the discontinuous galerkin method and as is typical for this method obtains error estimates of order o h n 1 2 for approximations by polynomials of degree le n

mackey constraints for james s compactness theorem and risk measures abstract let a be a closed convex bounded and non weakly compact subset of a banach space e with 0 a let us fix a convex and weakly compact subset w of e a functional z 0 e with z 0 a 0 and 0 then there is a linear form x 0 b p w z 0 i e x 0 e and sup w w x 0 w z 0 w which does not attain its infimum on a and such that x 0 a 0 as a consequence we show that a fatou coherent monetary utility function u 1 ess inf has the lebesgue property if and only if the convolution u 1 u 2 is fatou for every fatou coherent monetary utility function u 2

the space continuous discontinuous galerkin method abstract the space discontinuous finite element method for a scalar diffusion convection reaction problem is introduced and analyzed in the solution method the boundary conditions and the continuity of the solution at the element interfaces are forced in a weak sense the convergence rate of the numerical solution is shown however to be the same as in the usual approach based on a continuous approximation finally the benefits of the method with discontinuous continuous approximation in h adaptation of the finite element space are illustrated with the aid of numerical calculations

a streamline upwind full galerkin method for space time convection dominated transport problems an original space time finite element approach for the solution of the diffusion convection equation is proposed in this paper a slight manipulation of the differential equation suggests that transient transport problems may in fact be seen as steady state space time transport problems accurately and easily soluble by the standard galerkin technique however concerning convective transport involving sharp fronts or coarse discretization it is shown that implementation of dissipation along space time trajectories significantly improves the solutions classical comparative test problems are run to establish the performances of this method and to show the limits of the more sophisticated petrov and taylor galerkin schemes evocation of a possible space time anisotropy generated by usual finite difference time stepping procedures as well as comparative analysis of amplification matrices help to understand the accuracy and the robustness of the proposed approach

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

anisotropic mesh refinement in stabilized galerkin methods the numerical solution of a convection diffusion reaction model problem is considered in two and three dimensions a stabilized finite element method of galerkin least square type accomodates diffusion dominated as well as convection and or reaction dominated situations the resolution of boundary layers occuring in the singularly perturbed case is achieved using anisotropic mesh refinement in boundary layer regions in this paper the standard analysis of the stabilized galerkin method on isotropic meshes is extended to more general meshes with boundary layer refinement simplicial lagrangian elements of arbitrary order are used

an adaptive finite element method with crosswind diffusion for low mach steady laminar combustion we investigate adaptive finite element methods for low mach steady laminar combustion the finite element discretization of the flame equations involves least squares control of streamline derivatives and pressure velocity coupling as well as a new shock capturing term based on nonlinear crosswind diffusion yielding a suitable discrete maximum principle for the discrete solution a posteriori error estimates derived from the dual weighted residual method are used to refine the mesh adaptively numerical results are presented for a bunsen flame with simple chemistry on locally refined as well as fully unstructured delaunay meshes solution quality is evaluated in terms of overall flame characteristics including length lift off and width as well as undershoots in species and temperature profiles

boundaries of a convex set and interpolation sets the structure of the boundary of a convex set c is a field of intensive research in functional analysis the classical boundary is the set ext c of the extreme points of c at least when some compactness is assumed and c is recovered from ext c by means of the integral representation theory in this paper a more general notion of boundary definition 1 i is considered such a boundary needs not contain or even meet the extreme points and thus the classical tools are not available however through r c james s technique and a remarquable result of s simons a convex set can often be recovered in a strong sense from its boundary sect 1 tight connections are established between this notion lacunarity in harmonic analysis sect 2 banach spaces containing or not n sect 3 compactness and duality theory sect 4 finally sect 5 is devoted to miscellaneous results and questions in the spirit of the work and to examples showing the necessity of the assumptions we made as a general remark we did not systematically try to state the results in their most general form as far as it was not necessary to introduce new ideas for doing so this is specially true for the results of sect 2 where we worked in the duality 2 of the little fourier analysis although the techniques we used remain valid within the general frame of locally compact abelian groups

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

a new projection based stabilized method for steady convection dominated convection diffusion equations abstract we propose a new projection based stabilized method for the steady convection dominated convection diffusion equations this method directly increases the l 2 stability instead of the h 1 stability which the variational multiscale vms methods do numerical analysis and numerical experiments illustrate and confirm that our new method has effective numerical performances on convection dominated convection diffusion problems

specifying nodes at multiple locations for any normal mode of a linear elastic structure abstract spring mass oscillators have long been used as vibration absorbers to quench excess vibration in structural systems in this paper sprung masses are used as a passive means of inducing multiple nodes for any normal mode of an arbitrarily supported linear elastic structure when the parameters of the elastically mounted masses are properly chosen their attachment locations can be made to coincide exactly with the nodes of the structure thereby allowing nodes to be imposed at multiple locations anywhere along the combined assembly moreover when the desired node locations are closely spaced it is possible to specify a region of nearly zero amplitudes for a particular normal mode a procedure to guide the proper selection of the spring mass parameters in order to induce multiple nodes is outlined in detail and numerical experiments are performed to verify the utility of the proposed scheme of imposing nodes at multiple locations

edge stabilization for galerkin approximations of convection diffusion reaction problems n this paper we recall a stabilization technique for finite element methods for convection diffusion reaction equations originally proposed by j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 207 216 lecture notes in phys 58 springer berlin 1976 mr0440955 55 13823 the method uses least square stabilization of the gradient jumps across element boundaries we prove that the method is stable in the hyperbolic limit and prove optimal a priori error estimates we address the question of monotonicity of discrete solutions and present some numerical examples illustrating the theoretical results

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

improved eigenvalues for combined dynamical systems using a modified finite element discretization scheme new approaches are presented to discretize an arbitrarily supported linear structure carrying various lumped attachments specifically the exact eigendata i e the exact natural frequencies and mode shapes of the linear structure without the lumped attachments are first used to modify its finite element mass and stiffness matrix so that the eigensolutions of the discretized system coincide with the exact modes of vibration this is achieved by identifying a set of minimum changes in the finite element system matrices and enforcing certain constraint conditions once the updated matrices for the linear structure are found the finite element assembling technique is then used to include the lumped attachments by adding their parameters to the appropriate elements in the modified mass and stiffness matrices numerical experiments show that for the same number of elements the proposed scheme returns higher natural frequencies that are substantially more accurate than those given by the finite element model alternatively the proposed discretization scheme allows one to efficiently and accurately determine the higher natural frequencies of a combined system without increasing the number of elements in the finite element model

variational multiscale analysis the fine scale green s function projection optimization localization and stabilized methods we derive an explicit formula for the fine scale green s function arising in variational multiscale analysis the formula is expressed in terms of the classical green s function and a projector which defines the decomposition of the solution into coarse and fine scales the theory is presented in an abstract operator format and subsequently specialized for the advection diffusion equation it is shown that different projectors lead to fine scale green s functions with very different properties for example in the advection dominated case the projector induced by the h 1 0 seminorm produces a fine scale green s function which is highly attenuated and localized these are very desirable properties in a multiscale method and ones that are not shared by the l 2 projector by design the coarse scale solution attains optimality in the norm associated with the projector this property combined with a localized fine scale green s function indicates the possibility of effective methods with local character for dominantly hyperbolic problems the constructs lead to a new class of stabilized methods and the relationship between h 1 0 optimality and the streamline upwind petrov galerkin supg method is described

dynamic analysis of a multi span uniform beam carrying a number of various concentrated elements abstract this paper employs the numerical assembly method nam to determine the exact frequency response amplitudes of a multiple span beam carrying a number of various concentrated elements and subjected to a harmonic force and the exact natural frequencies and mode shapes of the beam for the case of zero harmonic force first the coefficient matrices for the intermediate concentrated elements pinned support applied force left end support and right end support of a beam are derived next the overall coefficient matrix for the whole vibrating system is obtained using the numerical assembly technique of the conventional finite element method fem finally the exact dynamic response amplitude of the forced vibrating system corresponding to each specified exciting frequency of the harmonic force is determined by solving the simultaneous equations associated with the last overall coefficient matrix the graph of dynamic response amplitudes versus various exciting frequencies gives the frequency response curve for any point of a multiple span beam carrying a number of various concentrated elements for the case of zero harmonic force the above mentioned simultaneous equations reduce to an eigenvalue problem so that natural frequencies and mode shapes of the beam can also be obtained

frequency response of a combined structure using a modified finite element method a modified finite element method is proposed to analyze the frequency response of an arbitrarily supported linear structure carrying various lumped attachments this is accomplished by updating the finite element mass and stiffness matrices of the linear structure without any lumped attachments using its exact natural frequencies and mode shapes so that the eigensolutions of the updated linear structure coincide with the exact eigendata once the updated matrices for the linear structure are found the finite element assembling technique is then exploited to include the lumped attachments by adding their parameters to the appropriate elements in the modified mass and stiffness matrices after assembling the global mass and stiffness matrices the frequency response of the combined system can be easily obtained numerical experiments show that using only a few finite elements the proposed scheme returns natural frequencies and frequency responses that are nearly identical to those obtained by using a finite element model with a very fine mesh the new method is easy to apply and efficient to use and it can be extended to determine the frequency response of any combined linear structure over any specified range of excitation frequencies

one sided james compactness theorem abstract we present some extensions of classical results that involve elements of the dual of banach spaces such as bishop phelp s theorem and james compactness theorem but restricting ourselves to sets of functionals determined by geometrical properties the main result which answers a question posed by f delbaen is the following let e be a banach space such that b e is convex block compact let a and b be bounded closed and convex sets with distance d a b 0 if every x e with sup x b inf x a attains its infimum on a and its supremum on b then a and b are both weakly compact we obtain new characterizations of weakly compact sets and reflexive spaces as well as a result concerning a variational problem in dual banach spaces

computing beltrami fields for solving the nonlinear equations governing force free fields an iterative methodology based on the splitting of the problem is described on the basis of this splitting three families of subproblems have to be solved numerically the first problem consists to find a potential field a mixed hybrid method is used to solve it the second problem which is a curl div system is solved by means of a mixed method the last problem is a transport equation which is approximated using a streamline diffusion technique numerical three dimensional experiments and results are given to illustrate the efficiency of the method

reproducing kernel hierarchical partition of unity part ii applications in this part of the work the meshless hierarchical partition of unity proposed in 1 referred here as part i is used as a multiple scale basis in numerical computations to solve practical problems the applications discussed in the present work fall into two categories 1 a wavelet adaptivity refinement procedure and 2 a so called wavelet petrov galerkin procedure in the applications of wavelet adaptivity the hierarchical reproducing kernels are used as a multiple scale basis to compute the numerical solutions of the helmholtz equation a model equation of wave propagation problems and to simulate shear band formation in an elasto viscoplastic material a problem dictated by the presence of the high gradient deformation in both numerical experiments numerical solutions with high resolution are obtained by inserting the wavelet like basis into the primary interpolation function basis a process that may be viewed as a spectral p type refinement by using the interpolant that has synchronized convergence property as a weighting function a wavelet petrov galerkin procedure is proposed to stabilize computations of some pathological problems in numerical computations such as advection diffusion problems and stokes flow problem it offers an alternative procedure in stablized methods and also provides some insight or new interpretation of the method detailed analysis has been carried out on the stability and convergence of the wavelet petrov galerkin method copyright 1999 john wiley sons ltd

a coercive james s weak compactness theorem and nonlinear variational problems abstract we discuss a perturbed version of james s sup theorem for weak compactness that not only properly generalizes that classical statement but also some recent extensions of this central result the sublevel sets of an extended real valued and coercive function whose subdifferential is surjective are relatively weakly compact furthermore we apply it to generalize and unify some facts in mathematical finance and to prove that the unique possible framework in the development of an existence theory for a wide class of nonlinear variational problems is the reflexive one

a space time finite element method for structural acoustics in infinite domains part 1 formulation stability and convergence abstract a space time finite element method for solution of the exterior structural acoustics problem involving the interaction of vibrating elastic structures submerged in an infinite acoustic fluid is formulated in particular time discontinuous galerkin and galerkin least squares gls variational formulations for coupled structural acoustics in unbounded domains are developed and analyzed for stability and convergence the formulation employs a finite computational fluid domain surrounding the structure and incorporates time dependent non reflecting boundary conditions on the fluid truncation boundary energy estimates are obtained which allow us to prove the unconditional stability of the method for the coupled fluid structure problem with absorbing boundaries the methods developed are especially useful for the application of adaptive solution strategies for transient acoustics in which unstructured space time meshes are used to track waves propagating along space time characteristics an important feature of the space time formulation is the incorporation of temporal jump operators which allow for finite element interpolations that are discontinaous in time for additional stability least squares operators based on local residuals of the structural acoustics equations including the non reflecting boundary conditions are incorporated the energy decay estimates and high order accuracy predicted by our a priori error estimates are demonstrated numerically in a simple canonical example

boundaries of asplund spaces we study the relationship between the classical combinatorial inequalities of simons and the more recent i property of fonf and lindenstrauss we obtain a characterization of strong boundaries for asplund spaces using the new concept of finitely self predictable set strong properties for w k analytic boundaries are established as well as a sup lim sup theorem for baire maps

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

nonlocal convection diffusion problems and finite element approximations abstract nonlocal models are becoming increasingly popular in numerical simulations of important application problems ranging from anomalous diffusion in heterogeneous environment to crack and fracture in composites in this paper we propose and compare some nonlocal models for convection diffusion problems and study their finite element approximations by piecewise constant and piecewise bilinear basis functions our focus is on the convection dominated case through analysis and numerical experiments we demonstrate the advantage of the upwind models which are more consistent with the physical nature and allow more stable and robust numerical approximations on the other hand the central model may produce unphysical oscillations

decoupled algorithm for solving phan thien tanner viscoelastic fluid by finite element method in this paper we present a decoupled algorithm for the viscoelastic fluid flow obeying the phan thien tanner ptt differential model the method consists to solve alternatively a stokes like problem by weighted least squares wls finite element method and the constitutive equation by streamline upwind petrov galerkin supg method a priori error estimate for the wls supg finite element method is derived the existence and uniqueness of the approximate solution are obtained using fixed point techniques we show that the mapping of the decoupled algorithm is locally contracting the planar channel flow problem is used to illustrate our theoretical results we encounter no limit of the weissenberg number for reasonable values of e a dimensionless material parameter in our calculations

a finite element method for time dependent convection diffusion equations we present a finite element method for time dependent convection diffusion equations the method is explicit and is applicable with piecewise polynomials of degree n 2 in the limit of zero diffusion it reduces to a recently analyzed finite element method for hyperbolic equations near optimal error estimates are derived numerical results are given

robust preconditioning estimates for convection dominated elliptic problems via a streamline poincare friedrichs inequality this paper is devoted to the streamline diffusion finite element method combined with equivalent preconditioning for solving convection dominated elliptic problems the preconditioner is obtained from the streamline diffusion inner product it is proved that the obtained convergence is robust i e bounded independently of the perturbation parameter varepsilon for proper convection vector fields the key to the estimates is an improved streamline poincare friedrichs inequality

a vertex based scheme on polyhedral meshes for advection reaction equations with sub mesh stabilization we devise and analyze vertex based schemes on polyhedral meshes to approximate advection reaction equations error estimates of order o h3 2 are established in the discrete inf sup stability norm which includes the mesh dependent weighted advective derivative the two key ingredients are a local polyhedral reconstruction map leaving affine polynomials invariant and a local design of stabilization whereby gradient jumps are only penalized across some subfaces in the interior of each mesh cell numerical results are presented on three dimensional polyhedral meshes

a new spectral element method for the reliable computation of viscoelastic flow abstract a new stabilised spectral element method has been developed by the authors for the accurate integration of the mixed elliptic hyperbolic system of partial differential equations governing certain viscoelastic flows the method is illustrated by solving the benchmark problem of the flow of an oldroyd b and a ptt fluid past a cylinder in a channel results are presented to demonstrate the advantages of the proposed method over traditional galerkin type procedures in terms of accuracy and stability

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

computational schemes for non linear elasto dynamics this paper deals with the development of computational schemes for the dynamic analysis of non linear elastic systems the focus of the investigation is on the derivation of unconditionally stable time integration schemes presenting high frequency numerical dissipation for these types of problem at first schemes based on galerkin and time discontinuous galerkin approximations applied to the equations of motion written in the symmetric hyperbolic form are proposed though useful these schemes require casting the equations of motion in the symmetric hyperbolic form which is not always possible furthermore this approaches to unacceptably high computational costs next unconditionally stable schemes are proposed that do not rely on the symmetric hyperbolic form both energy preserving and energy decaying schemes are derived numerical examples are presented to demonstrate the accuracy and efficiency of the proposed schemes copyright 1999 john wiley sons ltd

convergence of a residual based artificial viscosity finite element method we present a residual based artificial viscosity finite element method to solve conservation laws the galerkin approximation is stabilized by only residual based artificial viscosity without any least squares supg or streamline diffusion terms we prove convergence of the method applied to a scalar conservation law in two space dimensions toward an unique entropy solution for implicit time stepping schemes

iterative local solvers for distributed krylov schwarz method applied to convection diffusion problems abstract nowadays supercomputers can be used to solve large scale problems that come from simulation of industrial or research problems however those machines are usually inacessible to most industries and university laboratories around the world in this work we present an iterative solver a krylov schwarz method ksm to be used in a collection of workstations under pvm the subdomain problems are solved by using many methods in order to show how the choice of the local solvers affects the overall performance of the distributed ksm

optimal h p finite element methods abstract a theory of optimal petrov galerkin h p version finite element approximations is presented the optimal scheme is denned relative to a fine mesh solution space and relative to an arbitrary symmetric bilinear form the optimal method leads to a symmetric positive definite stiffness matrix which is independent of the coefficients of the given problem exhibits extra superconvergence properties and has a relative error that can be calculated exactly at each point in the problem domain various generalizations are also discussed including the connection of these methods with certain preconditioning schemes

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

non weak strong solutions in gas dynamics a c11 p version stlsfef in lagrangian frame of reference using ρ u p primitive variables in this paper we present computations of the non weak solutions of class c11 of transient navier stokes equations for compressible flow in lagrangian frame of reference using space time least squares finite element formulation with primitive variables u p for high speed compressible flows the solutions reported here possess the same orders of continuity as the governing differential equations the role of diffusion i e viscosity physical or artificial and thermal conductivity on shock structure is demonstrated compression of air in a rigid cylinder by a rigid massless and frictionless piston is used as a model problem true time evolutions of class c11 are reported beginning with the first time step until steady shock conditions are achieved comparisons with analytical solutions are presented when possible copyright 2001 john wiley sons ltd

adaptive finite element methods for conservation laws based on a posteriori error estimates we prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension and design corresponding adaptive methods the proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem together with the galerkin orthogonality of the finite element method the strong stability estimate uses the entropy condition for the system in an essential way 1995 john wiley sons inc

stability analysis of the forward euler scheme for the convection diffusion equation using the supg formulation in space the stability and accuracy of the forward euler scheme for the semidiscrete problem arising from the space discretization of the convection diffusion equation using the supg formulation are analysed both linear and quadratic finite elements are considered the stability limits are derived for the one dimensional problem and a heuristic criterion is proposed for the multidimensional equation some numerical experiments are conducted in order to assess the performance of different finite elements

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

dynamic characteristics of a beam and distributed spring mass system this paper studies dynamic characteristics of a beam with continuously distributed spring mass which may represent a structure occupied by a crowd of people dividing the coupled system into several segments and considering the distributed spring mass and the beam in each segment being uniform the equations of motion of the segment are established the transfer matrix method is applied to derive the eigenvalue equation of the coupled system it is interesting to note from the governing equations that the vibration mode shape of the uniformly distributed spring mass is proportional to that of the beam at the attached regions and can be discontinuous if the natural frequencies of the spring masses in two adjacent segments are different parametric studies demonstrate that the natural frequencies of the coupled system appear in groups in a group of frequencies all related modes have similar shapes the number of natural frequencies in each group depends on the number of segments having different natural frequencies with the increase of group order the largest natural frequency in a group monotonically approaches the natural frequency of corresponding order of the bare beam from the upper side whereas the others monotonically move towards those of the independent spring mass systems from the lower side numerical results show that the frequency coupling between the beam and the distributed spring mass mainly occurs in the low order of frequency groups especially in the first group in addition vibratory characteristics of the coupled system can be approximately represented by a series of discrete multi degrees of freedom system it also demonstrates that a beam on winkler elastic foundation and a beam with distributed solid mass are special cases of the proposed solution

free vibration of a uniform beam with multiple elastically mounted two degree of freedom systems abstract the free vibration of a beam with one or more elastically mounted two degree of freedom systems that translate and rotate is considered in this note the assumed modes method is applied to formulate the equations of motion and the natural frequencies of the system are found by solving for the roots of a given characteristic determinant if the number of attached spring mass systems is small one can exploit the sherman morrison woodbury determinant formula and reduce the characteristic determinant to one of smaller size which will be easier to code and more computationally efficient to solve

boundaries for banach spaces determine weak compactness a boundary for a real banach space is a subset of the dual unit sphere with the property that each element of the banach space attains its norm on an element of that subset trivially the pointwise convergence with respect to such a boundary is coarser than the weak topology on the banach space the boundary problem asks whether nevertheless both topologies have the same norm bounded compact sets

the exact solution for free vibration of uniform beams carrying multiple two degree of freedom spring mass systems abstract the literature regarding the exact solutions of natural frequencies and mode shapes of a uniform beam carrying multiple two degree of freedom 2 dof spring mass systems is rare thus this paper aims at studying this problem using the numerical assembly method nam first of all the equivalent springs for replacing the effect of a 2 dof spring mass system are determined next the coefficient matrix for a 2 dof spring mass system attached to the uniform beam is derived based on the compatibility of deformations and equilibrium of forces including moments the coefficient matrices for the left end and right end of the beam are also derived based on the various boundary conditions of the beam combining the coefficient matrices for all the 2 dof spring mass systems attached to the beam and the coefficient matrices for the boundary conditions of the beam one obtains the overall coefficient matrix of the constrained beam i e the beam carrying any number of 2 dof spring mass systems the product of the overall coefficient matrix and the vector for all the integration constants yields a set of simultaneous equations let the coefficient determinant of the last simultaneous equations equal to zero one obtains the frequency equation the roots of the last frequency equation denote the natural frequencies of the constrained beam substituting the roots of the frequency equation into the set of simultaneous equations one may determine the associated mode shapes of the constrained beam in this paper the exact solution refers to the one obtained from the continuous model instead of the discrete mode besides the accuracy of the analytical and numerical combined method ancm given by the existing literature is dependent on the total number of vibration modes considered but this is not true for the accuracy of the nam adopted here to confirm the reliability of the presented theory all the numerical results obtained from nam are compared with the corresponding ones obtained from the conventional finite element method fem and good agreement is achieved

the semi discrete streamline diffusion finite element method for time dependented convection diffusion problems abstract in this paper a semi discrete streamline diffusion finite element method sdfem for solving the time dependent convection dominated diffusion problem in one dimension is studied under the mild regularity assumptions the convergence stability results are derived finally some numerical experiments are given for a convection dominated problem with boundary layer at x 1 by using p1 element compared with the finite difference streamline diffusion method fdsdm the semi discrete sdfem is more stable and it can imitate the exact solution effectively

numerical crosswind smear in the streamline diffusion method abstract the streamline diffusion method is a popular element method for approximating convection dominated transport equations the solution to such equations may develop boundary or internal layers in the crosswind direction orthogonal to the flow direction and this results in numerical crosswind layers it is known that the size of these layers is o h 1 2 or perhaps o h 3 4 h being the size of the mesh used in the numerical computation but in practice o h is observed if the gridlines of the method are aligned with the streamlines of the problem this connection is studied and an explanation is given

on a one sided james theorem abstract we provide a short proof of the following fact if x is a banach space a and b are bounded closed and convex sets with dist a b 0 and every x x with the property that sup x b inf x a attains its infimum on a and its supremum on b then both a and b are weakly compact

time finite element methods for structural dynamics summary time finite element methods are developed for the equations of structural dynamics the approach employs the time discontinuous galerkin method and incorporates stabilizing terms having least squares form these enable a general convergence theorem to be proved in a norm stronger than the energy norm results are presented from finite difference analyses of the time discontinuous galerkin and least squares methods with various temporal interpolations and commonly used finite difference methods for structural dynamics these results show that for particular interpolations the time finite element method exhibits improved accuracy and stability

a general approach to formulating the frequency equation for a beam carrying miscellaneous attachments beams carrying various lumped attachments have been studied extensively over the years in this paper a simple approach is proposed that can be used to readily determine the eigenvalues of an arbitrarily supported single span or multi span beam carrying any combination of lumped mass rotary inertia grounded translational or torsional spring grounded translational or torsional viscous damper an undamped or damped oscillator with or without a rigid body degree of freedom rather than solving a generalized eigenvalue problem to obtain the eigenvalues of the system a frequency equation is formulated instead whose solution can be easily solved either numerically or graphically the proposed scheme is easy to code and can be easily modified to accommodate beams with arbitrary supports and carrying any number of miscellaneous lumped attachments

a new approach for determining the natural frequencies and mode shapes of a uniform beam carrying any number of sprung masses abstract in theory one may obtain five equations from each attaching point of a spring mass system and two boundary equations from each end of the uniform beam hence for a uniform beam carrying n spring mass systems simultaneous equations of the form b c 0 will be obtained the solutions of b 0 where represents a determinant give the natural frequencies of the constrained beam and the substitution of each corresponding values of c j j 1 4 into the eigenfunction will define the associated mode shapes while the foregoing theory is simple the lengthy explicit mathematical expressions become impractical if the total number of spring mass systems is larger than two for this reason it was applied to do the free vibration analysis of a uniform beam carrying one spring mass system only in the existing literature the purpose of this paper is to present a numerical technique to apply the foregoing theory to obtain the exact solutions for the lowest several natural frequencies and mode shapes of a uniform beam carrying any number of spring mass systems with various boundary conditions to this end each integration constant c vi and each mode displacement z v v 1 n i 1 4 at the attaching point and the two ends of the beam are considered as nodal displacements of a finite beam element and are assigned an appropriate degree of freedom dof hence each associated coefficient matrix will be equivalent to the stiffness matrix of a beam element and the conventional numerical assembly technique for the finite element method fem may be used to determine the overall coefficient matrix b therefore the eigenvalue equation b c 0 is easily obtained

convergence of a fully discrete scheme for two dimensional neutron transport we prove an error estimate for a fully discrete method for the numerical solution of a two dimensional model problem in neutron transport theory based on using the discrete ordinates method for the

an upwind like discontinuous galerkin method for hyperbolic systems we investigate an upwind like dg method for solving first order hyperbolic problems written as the friedrichs systems under certain condition this dg scheme may be semi explicit such that the discrete equations can be solved layer by layer we give the stability analysis and error estimate of order k 1 2 in the dg norm in particular for some hyperbolic systems we show that the convergence rate is of order k 1 in the l 2 norm if the q k elements are used on rectangular meshes finally we provide some numerical experiments to illustrate the theoretical analysis

semidiscrete least squares methods for linear convection diffusion problem abstract in this paper some approximate methods for solving linear convection diffusion problems are presented the methods consist in discretizing with respect to time and solving the resulting convection dominated elliptic problem for fixed time by least squares finite element methods an analysis of least squares approximations is given including optimal order estimates for piecewise polynomial approximation spaces the model problem considered is the time dependent convection dominated linear convection diffusion equation numerical results for the burgers equation will also be presented

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis