discontinuous galerkin finite element methods for second order hyperbolic problems abstract we prove a priori and a posteriori error estimates for a finite element method for linear second order hyperbolic equations based on a space time finite element discretization with the basis functions being continuous in space and discontinuous in time

superconvergence of solution derivatives for the shortley weller difference approximation of poisson s equation part i smoothness problems the finite difference method fdm using the shortley weller approximation can be viewed as a special kind of the finite element methods fems using the piecewise bilinear and linear functions and involving some integration approximation when u c3 s i e u c3 0 s and f c2 s the superconvergence rate o h2 of solution derivatives in discrete h1 norms by the fdm is derived for rectangular difference grids where h is the maximal mesh length of difference grids used and the difference grids are not confined to be quasiuniform comparisons are made on the analysis by the maximum principle and the fem analysis conversions between the fdm and the linear and bilinear fems are discussed and numerical experiments are provided to support superconvergence analysis made

translation representations for automorphic solutions of the wave equation in non euclidean spaces i this paper deals with the spectral theory of the laplace beltrami operator acting on automorphic functions in n dimensional hyperbolic space hn we study discrete subgroups which have a fundamental polyhedron f with a finite number of sides and infinite volume concerning these we have shown previously that the spectrum of contains at most a finite number of point eigenvalues in 1 2 n 1 2 0 and none less than 1 2 n 1 2 here we prove that the spectrum of is absolutely continuous and of infinite multiplicity in 1 2 n 1 2 our approach uses the non euclidean wave equation introduced by faddeev and pavlov energy ef is defined as ut ut u lu where the bracket is the l2 scalar product over a fundamental polyhedron with respect to the invariant volume of the hyperbolic metric energy is conserved under the group of operator u t relating initial data to data at time t we construct two isometric representations of the space of automorphic data by l2 r n which transmute the action of u t into translation these representations are given explicitly in terms of integrals of the data over horospheres in part ii we shall show the completeness of these representations utt lu 0 l 1 2 n 1 2

time discontinuous stabilized space time finite elements for timoshenko beams stabilized space time e nite element methods for the transient computational analysis of the elastodynamics of timoshenko beams have been developed the underlying time discontinuous galerkin formulation is implicit unconditionally stable higher order accurate and robust the employed interpolations are continuous in space and time inside time slabs but discontinuous in time between adjacent time slabs to suppress spurious numerical oscillations near discontinuities or high gradients galerkin least squares stabilization has been applied further improvement of the numerical representation of the transient elastic wave propagation phenomena has been achieved by specially designed galerkin gradient least squares operators because of the reduced phase and amplitude errors of these e nite elements much coarser spatial meshes may be used without loss of accuracy the resulting reduction of the number of unknowns leads to a signie cant decrease of computer time furthermore a globaladaptivetime stepping strategy has beenemployed basedon thetemporal jump residual of thetimee niteelement method as error estimator numerical examples of transientelastic wave propagation in timoshenko beams involving a wide spectrum of wave numbers and frequencies demonstrate the good performance of the developed methods

global superconvergence of finite element methods for biharmonic equations and blending surfaces abstract in this paper new numerical algorithms of finite element methods fem are reported for both biharmonic equations and 3d blending surfaces to achieve the global superconvergence o h3 o h4 in h2 norms this is significant compared with the optimal convergence o h2 the algorithms are simple because only an a posteriori interpolant solution is needed such a global convergence method was originated by lin and his colleagues in 1 3 for only the clamped boundary conditions recently we extended the global superconvergence to other boundary conditions such as the simple support condition the periodic boundary and the natural boundary condition moreover we apply in 4 6 this global superconvergence to the fem using the penalty techniques for biharmonic equations and blending problems also to reach o h3 and o h4 for quasiuniform and uniform ij respectively currently we develop in 7 8 and in this paper the fem using the penalty plus hybrid techniques to reduce the condition number down to o h 4 o h 5 of the associated matrix while retaining superconvergence o h3 o h4 since instability is severe for biharmonic equations any reduction of the condition number is crucial by the new algorithms in this paper not only can a great deal of cpu time be saved but also the complicated biharmonic equations and blending surfaces may be solved in double precision numerical experiments are carried out to support the theoretical conclusions

a space time finite element method for the wave equation a numerical method is introduced which uses finite elements in time and space simultaneously to solve the wave equation an error analysis for this scheme is presented and the results of several computations this scheme is accurate for time slabs of arbitrary thickness without introducing extra least squares or stabilization terms

spurious resonances in numerical time integration methods for linear dynamics this paper deals with the accuracy of time integration methods for linear dynamics when applied near the resonance condition an approach for the analysis is considered which allows spurious resonance conditions to be detected the analysis of newmark methods shows the existence of such conditions which can adversely affect the quality of numerical computations as an alternative a higher order algorithm which can be viewed as a generalization of the trapezoidal rule is investigated the analysis reveals that the spurious disturbance near the resonance condition is greatly reduced the reported numerical tests confirm the theoretical predictions and demonstrate that high quality simulations can be obtained by means of higher order algorithms

η superconvergence of finite element approximations in the interior of general meshes of triangles abstract in this paper we introduce a new definition of superconvergence tne superconvergence which generalizes the classical idea of superconvergence to general meshes we show that this new definition can be employed to determine the regions of least error in any element in the interior of any grid by using a computer based approach we present numerical results for the standard displacement finite element method for the scalar equation of orthotropic heat conduction for meshes of conforming triangles of degree p 1 p 5 and elements in the interior of the mesh the results demonstrate that unlike classical superconvergence superconvergence is applicable to the complex grids which are employed in practical engineering computations

use of post processing to increase the order of accuracy of the trapezoidal rule at time integration of linear elastodynamics problems in this paper we suggest a very simple and effective post processing procedure to increase the order of accuracy in time for numerical results obtained by the trapezoidal rule we first derive a new exact closed form a priori error estimator for time integration of linear elastodynamics equations by the trapezoidal rule with non uniform time increments based on this error estimator we suggest a new post processing procedure containing additional time integration of elastodynamics equations by the trapezoidal rule with few time increments that systematically improves the order of accuracy of numerical results with the increase in the number of additional time increments used for post processing for example the use of just one additional time increment for post processing after time integration with any number of uniform time increments renders the order of accuracy of numerical results equal to 10 3 numerical examples of the application of the new techniques to a system with a single degree of freedom and to a multi degree system confirm the corresponding increase in the order of convergence of numerical results after post processing because the same trapezoidal rule is used for basic computations and post processing the new technique retains all of the properties of the trapezoidal rule requires no writing of a new computer program for its implementation and can be easily used with any existing commercial and research codes for elastodynamics

high order hierarchical a and l stable integration methods numerical integration of stiff first order systems of differential equations is considered it is shown how a c continuous polynomial time discretization in conjunction with a weighted residual method can be used to derive methods corresponding to the diagonal and first subdiagonal pade approximants in this manner a stable schemes of order 2k and l stable schemes of order 2k l are obtained at least for k 4 the methods are hierarchical in the sense that a scheme with a given accuracy embraces the equations of all lower order methods that have the same stability type

a consistent approach for mixed stress finite element formulations in linear elastodynamics abstract in this paper a mixed stress formulation for linear elastodynamic analysis based on a modified hellinger reissner functional and a consistent approach for selecting finite element approximations are presented the key idea in this new approach is to choose stress approximations by taking into account for suitable modes to equilibrate inertia forces feasibility and effectiveness of the proposed approach are numerically verified through three benchmark tests

accuracy of one step integration schemes for damped forced linear structural dynamics summary this paper proposes an energy based measure for the evaluation of the local truncation error of two level one step integration schemes the measure applies to multiple degree of freedom systems and does not necessarily require modal reduction to a scalar model it naturally handles the structural damping and external forcing terms that are generally and mistakenly neglected in error analyses and it segregates the error associated with the free and forced response components of the problem to illustrate the approach two examples associated with the application of the trapezoidal scheme and of a high order scheme proposed in the literature are analyzed the latter reveals the shortcomings of the standard approach that is based on the undamped unforced linear oscillator and therefore highlights the need for the proposed framework indeed the scheme order of accuracy is below expectation when structural damping or external forcing is considered in the numerically dissipative setting developments on the basis of the time discontinuous galerkin tdg method are then proposed to recover the scheme high order accuracy additionally they show the similarity that exists between schemes related to the tdg method and the ones obtained by integration by parts of the equation of motion copyright 2014 john wiley sons ltd

a discontinuous galerkin finite element method for dynamic and wave propagation problems in non linear solids and saturated porous media a time discontinuous galerkin finite element method dgfem for dynamics and wave propagation in non linear solids and saturated porous media is presented the main distinct characteristic of the proposed dgfem is that the specific p3 p1 interpolation approximation which uses piecewise cubic hermite s polynomial and linear interpolations for both displacements and velocities in the time domain is particularly proposed consequently continuity of the displacement vector at each discrete time instant is exactly ensured whereas discontinuity of the velocity vector at the discrete time levels still remains the computational cost is then obviously saved particularly in the materially non linear problems as compared with that required for the existing dgfem both the implicit and explicit algorithms are developed to solve the derived formulations for linear and materially non linear problems numerical results illustrate good performance of the present method in eliminating spurious numerical oscillations and in providing much more accurate solutions over the traditional galerkin finite element method using the newmark algorithm in the time domain copyright 2003 john wiley sons ltd

structural dynamic analysis by a time discontinuous galerkin finite element method this paper studies a time discontinuous galerkin finite element method for structural dynamic problems by which both displacements and velocities are approximated as piecewise linear functions in the time domain and may be discontinuous at the discrete time levels a new iterative solution algorithm which involves only one factorization for each fixed time step size and a few iterations at each step is presented for solving the resulted system of coupled equations by using the jumps of the displacements and the velocities in the total energy norm as error indicators an adaptive time stepping procedure for selecting the proper time step size is described numerical examples including both single dof and multi dof problems are used to illustrate the performance of these algorithms comparisons with the exact results and or the results by the newmark integration scheme are given it is shown that the time discontinuous galerkin finite element method discussed in this study possesses good accuracy third order and stability properties its numerical implementation is not difficult and the higher computational cost needed in each time step is compensated by use of a larger time step size

explicit predictor multicorrector time discontinuous galerkin methods for non linear dynamics abstract explicit predictor multicorrector time discontinuous galerkin tdg methods developed for linear structural dynamics are formulated and implemented in a form suitable for arbitrary non linear analysis of structural dynamics problems the formulation is intended to inherit the accuracy properties of the exact parent implicit tdg methods to this end suitable predictors and correctors are designed to achieve third order accuracy large stability limits and controllable numerical dissipation by means of an algorithmic parameter as the study of a general non linear case is rather complex the analysis of the convergence properties of the resulting algorithms are restricted to conservative duffing oscillators for which closed form solutions are available it is shown that the main properties of the underlying parent scheme can be retained finally results of representative numerical simulations relevant to duffing oscillators and to a stiff spring pendulum discretized with finite elements illustrate the performance of the numerical schemes and confirm the analytical estimates

an efficient time discontinuous galerkin procedure for non linear structural dynamics this paper presents an effective time discontinuous galerkin procedure for non linear dynamics the procedure is based on a cheap iterative algorithm which is rather different from previous efforts in fact the algorithm is designed so that the corrected solutions inherit the desired stability and dissipative properties and the iterations serve only to improve accuracy some numerical tests illustrate the good performance of the present procedure which appears to be competitive with the available implementations of time discontinuous galerkin methods

reliability and efficiency of an anisotropic zienkiewicz zhu error estimator abstract in this paper we study the efficiency and the reliability of an anisotropic a posteriori error estimator in the case of the poisson problem supplied with mixed boundary conditions the error estimator may be classified as a residual based one but its novelty is twofold firstly it employs anisotropic estimates of the interpolation error for linear triangular finite elements and secondly it makes use of the zienkiewicz zhu recovery procedure to approximate the gradient of the exact solution finally we describe the adaptive procedure used to obtain a numerical solution satisfying a given accuracy and we include some numerical test cases to assess the robustness of the proposed numerical algorithm

superconvergence results on mildly structured triangulations abstract many results on superconvergence for recovered gradients of piecewise linear galerkin approximations on triangular mesh partitions to the weak solutions of elliptic boundary value problems in two dimensions have been proved in recent years these were obtained first for regular fully structured partitions in which the mid points of the diagonals of all quadrilaterals formed by pairs of adjacent elements are coincident this condition was then relaxed to allow for strongly regular meshes in which the distance between the above mid points is o h2 h being the mesh size parameter in this paper these conditions are weakend still further to the case of globally mildly structured meshes where the mid point distance is o h1 0

time continuous galerkin methods for linear heat conduction problems abstract a petrov galerkin formulation of the semidiscrete linear transient heat conduction problem is given from which a unified set of a stable higher order time integration methods is derived accuracy and stability properties can be controlled by an algorithmic parameter introduced in the definition of the test functions in this manner a stable algorithms of order 2p and l stable algorithms of order 2p 1 are derived from a unique formulation in addition algorithms with intermediate accuracy and stability properties are made available the effects of varying the algorithmic parameter are investigated numerical tests are included to support the theoretical analysis

time discontinuous galerkin methods with energy decaying correction for non linear elastodynamics in this paper a new time discontinuous galerkin tdg formulation for non linear elastodynamics is presented the new formulation embeds an energy correction which ensures truly energy decaying thus allowing to achieve unconditional stability that as shown in the paper is not guaranteed by the classical tdg formulation the resulting method is simple and easily implementable into existing finite element codes moreover it inherits the desirable higher order accuracy and high frequency dissipation properties of the classical formulation numerical results illustrate the very good performance of the proposed formulation copyright 2010 john wiley sons ltd

time discontinuous material point method for transient problems abstract this paper presents a time discontinuous material point method tdmpm for transient problems such as the wave propagation and impact responses in solids by dividing the continuous time domain into discrete time intervals the weak form of the tdmpm is established by considering the discrete grid based governing equations constraint and discontinuity conditions the displacement and velocity fields in a time interval are interpolated with the piecewise cubic and linear functions respectively by substituting the assumed displacement and velocity fields into the weak form a novel computational framework for the grid displacements and velocities at the discrete time instants is constructed in the new formulations the displacement field at each time instant remains to be continuous whereas the velocity field at the time instant becomes discontinuous these unique features ensure the tdmpm could properly capture the discontinuous characteristics and control the spurious numerical oscillations two numerical examples under the impact loading are used to verify the proposed method two representative impact problems are then presented for further verification and demonstration besides the corresponding contact algorithm adopted in the tdmpm is shown to be capable of capturing the correct contact behavior with higher fidelity and less computational cost than the mpm the presented results illustrate that the tdmpm could successfully control the spurious numerical oscillations associated with transient simulations

the asymptotic distribution of lattice points in euclidean and non euclidean spaces the counting numbers for discrete subgroups of motions in euclidean and non euclidean spaces are obtained using the wave equation as the principal tool in dimensions 2 and 3 the error estimates are close to the best known

effective condition number for finite difference method for solving the linear algebraic equations ax b with the symmetric and positive definite matrix a from elliptic equations the traditional condition number in the 2 norm is defined by cond l n where l and n are the maximal and minimal eigenvalues of the matrix a respectively the condition number is used to provide the bounds of the relative errors from the perturbation of both a and b such a cond can only be reached by the worst situation of all rounding errors and all b for the given b the true relative errors may be smaller or even much smaller than the cond which is called the effective condition number in chan and foulser effectively well conditioned linear systems siam j sci statist comput 9 1988 963 969 and christiansen and hansen the effective condition number applied to error analysis of certain boundary collocation methods j comput appl math 54 1 1994 15 36 in this paper we propose the new computational formulas for effective condition number cond eff and define the new simplified effective condition number cond e for the latter we only need the eigenvector corresponding to the minimal eigenvalue of a which can be easily obtained by the inverse power method in this paper we also apply the effective condition number for the finite difference method for poisson s equation the difference grids are not supposed to be quasiuniform under a non orthogonality assumption the effective condition number is proven to be o 1 for the homogeneous boundary conditions such a result is extraordinary compared with the traditional cond o hmin 2 where hmin is the minimal meshspacing of the difference grids used for the non homogeneous neumann and dirichlet boundary conditions the effective condition number is proven to be o h 1 2 and o h 1 2hmin 1 respectively where h is the maximal meshspacing of the difference grids numerical experiments are carried out to verify the analysis made

new error estimates of bi cubic hermite finite element methods for biharmonic equations for the global superconvergence over the entire solution domain originated by lin and his colleagues 7 8 this paper gives a framework of new estimates and new proofs for the basic estimates for bounds of u ui xx xxds u ui xy xyds and u ui xx yyds which reveal more intrinsic characteristics and easier understanding and better readable suppose that the solution is smooth enough and the solution domain can be split into quasiuniform rectangular elements ij with the maximal boundary length h the study of 7 8 dealt with only the clamped boundary condition for biharmonic equations to obtain the global superconvergence o h4 in h2 norms under uniform rectangles ij for the solution u h6 this paper is devoted to other kinds of important boundary conditions such as the simple support condition the natural boundary condition and their mixed types where the different boundary conditions are subject to different edges of new error estimates are derived theoretically and verified numerically to reach the global superconvergence o h3 5 and o h4 for different boundary conditions on different edges of under uniform ij note that the new superconvergence estimates in this paper are essential in practical applications because different boundary conditions are needed in 3d blending surfaces 5 and in the combined methods for singularity problems 6

discontinuous galerkin finite element in time for solving periodic differential equations the dynamics of rotating systems can be modeled by ordinary differential equations with periodic time varying coefficients and forcing discontinuous galerkin dg methods which are known to be higher order accurate a stable and locally conservative are used to solve such equations in this work dg methods are shown to find the steady state periodic solution of a constant coefficient structural dynamics problem with periodic forcing and b helicopter rotor rigid blade flapping equation which is a parametric ordinary differential equation with periodic time varying coefficients and periodic forcing convergence of the numerical solution obtained is studied for both the above equations while h and p are refined separately it is found that the dg method works well for these problems and allows the development of an element level error estimate which can be used to monitor convergence and hence develop adaptive schemes it is clear from the results that even though higher order polynomials result in much faster convergence with reduced computational effort it also results in deterioration of the numerical performance for p 12 two adaptive strategies based on the element level displacement jump norm i e h and hp refinement are proposed and demonstrated on problems with higher harmonic forcing primary controls and parametric variations

the norsett time integration methodology for finite element transient analysis this paper presents a set of methods for time integration of problems arising from finite element semidiscretizations the purpose is to obtain computationally efficient methods which possess higher order accuracy and controllable dissipation in the spurious high modes the methods are developed and analysed by a general collocation methodology which leads to the class of norsett approximants an algorithmic parameter is used to achieve an effective control over numerical dissipation moreover a simple and efficient implementation scheme is presented at each time step algorithms based on p order collocation polynomials require the solution of p sets of linear algebraic equations with the same coefficient matrix in this way a single factorization is needed and no transformations are required to recover the approximate solution at the end or within the time interval to demonstrate the performance of the proposed algorithms a wide experimental evaluation is carried out on typical test problems in finite element transient analysis

extended framework of hamilton s principle for continuum dynamics abstract hamilton s principle is the variational principle for dynamical systems and it has been widely used in mathematical physics and engineering however it has a critical weakness termed end point constraints which means that in the weak form we cannot use the given initial conditions properly by utilizing a mixed formulation and sequentially assigning initial conditions this paper presents a novel extended framework of hamilton s principle for continuum dynamics to resolve such weakness the primary applications lie in an elastic and a j 2 viscoplastic continuum dynamics the framework is simple and initiates the development of a space time finite element method with the proper use of initial conditions non iterative numerical algorithms for both elasticity and j 2 viscoplasticity are presented

translation representations for automorphic solutions of the wave equation in non euclidean spaces the case of finite volume soit un sous groupe discret d automorphismes de h n a polyedre fondamental de volume fini de nombre de cotes fini et a n cusps soit l operateur de laplace beltrami agissant sur des fonctions automorphes par rapport a on demontre que a un spectre absolument continu de multiplicite uniforme n sur n 1 2 2 plus un spectre standard discret

penalty combinations of the ritz galerkin and finite difference methods for singularity problems abstract penalty combination of the ritz galerkin and finite difference methods is presented for solving elliptic boundary value problems with singularities the superconvergence rate o h2 of solution derivatives by the combination can be achieved while using quasiuniform rectangular difference grids where h is the maximal mesh length of difference grids used in the finite difference method and 0 is an arbitrarily small number it is due to its simplicity that the penalty combination of the ritz galerkin and finite difference methods is highly recommended for solving the complicated problems with multiple singularities and multiple interfaces

solution of linear elastodynamics problems with space time finite elements on structured and unstructured meshes abstract a new implicit approach for the solution of linear elastodynamics problems on structured and unstructured meshes with space time finite elements is suggested new weak formulations are based on time continuous tcg and discontinuous tdg galerkin methods it was shown that for structured meshes the new tcg method has a higher order of accuracy by a factor of two than that of the standard tcg method at the same number of degrees of freedom the accuracy of the new tcg method is one order higher than the accuracy of the standard tdg method the introduction of new weighting functions for the weak formulations of the tcg and tdg methods allows control of the high frequency behavior without degradation of accuracy of the methods a variable number of elements in the time direction as well as a variable order of polynomial time approximations of finite elements for time slabs allows the implementation of a coupled space time adaptive procedure using structured or unstructured meshes numerical examples for 1 d problems demonstrate the effectiveness of the technique

a space time discontinuous galerkin method for the solution of the wave equation in the time domain in recent years the focus of research in the field of computational acoustics has shifted to the medium frequency regime and multiscale wave propagation this has led to the development of new concepts including the discontinuous enrichment method its basic principle is the incorporation of features of the governing partial differential equation in the approximation in this contribution this concept is adapted for the simulation of transient problems governed by the wave equation we present a space time discontinuous galerkin method with lagrange multipliers where the shape approximation in space and time is based on solutions of the homogeneous wave equation the use of hierarchical wave like basis functions is enabled by means of a variational formulation that allows for discontinuities in both the spatial and the temporal discretizations numerical examples in one space dimension demonstrate the outstanding performance of the proposed method compared with conventional space time finite element methods copyright 2008 john wiley sons ltd

a particular integral bem time discontinuous fem methodology for solving 2 d elastodynamic problems abstract this study proposes a time discontinuous galerkin finite element method fem for solving second order ordinary differential equations in the time domain the equations are formulated using a particular integral boundary element method bem in the space domain for elastodynamic problems the particular integral bem technique depends only on elastostatic displacement and traction fundamental solutions without resorting to commonly used complex fundamental solutions for elastodynamic problems based on the time discontinuous galerkin fem the unknown displacements and velocities are approximated as piecewise linear functions in the time domain and are permitted to be discontinuous at the discrete time levels this leads to stable and third order accurate solution algorithms for ordinary differential equations numerical results using the time discontinuous galerkin fem are compared with results using a conventional finite difference method the houbolt method both methods are employed for a particular integral bem analysis in elastodynamics this comparison reveals that the time discontinuous galerkin fem is more stable and more accurate than the traditional finite difference methods

hierarchical higher order dissipative methods for transient analysis this work focuses on devising an efficient hierarchy of higher order methods for linear transient analysis equipped with an effective dissipative action on the spurious high modes of the response the proposed strategy stems from the norsett idea and is based on a multi stage algorithm designed to hierarchically improve accuracy while retaining the desired dissipative behaviour computational efficiency is pursued by requiring that each stage should involve just one set of implicit equations of the size of the problem to be solved as standard time integration methods and in addition all the stages should share the same coefficient matrix this target is achieved by rationally formulating the methods based on the discontinuous collocation approach the resultant procedure is shown to be well suited for adaptive solution strategies in particular it embeds two natural tools to effectively control the error propagation one estimates the local error through the next stage solution which is one order more accurate the other through the solution discontinuity at the beginning of the current time step which is permitted by the present formulation the performance of the procedure and the quality of the two error estimators are experimentally verified on different classes of problems some typical numerical tests in transient heat conduction and elasto dynamics are presented copyright 2006 john wiley sons ltd

a superconvergent linear fe approximation for the solution of an elliptic system of pdes the aim of this work is to study a nonstandard piecewise linear finite element method for elliptic systems of partial differential equations this nonstandard method was considered by the authors for scalar elliptic equations and for a planar elasticity problem the method enables us to compute a superconvergent numerical approximation to the solution of the system of partial differential equations

formulation and analysis of variational methods for time integration of linear elastodynamics abstract a general framework for variational formulations of linear elastodynamics is introduced variational formulations which originate from the principle of minimum potential energy the hellinger reissner principle and the hu washizu principle are considered they are characterized by the weak enforcement of the velocity displacement relation and of the initial displacement and velocity special cases are obtained by a priori enforcement of these conditions the time integration algorithms obtained discretizing in the space and time domains are studied with reference to accuracy stability and high frequency behaviour it is shown that some of these algorithms are very competitive with respect to classical finite difference methods the only drawback of variational methods seems to be the large size of the set of simultaneous equations which they generate effective solution procedures aiming at minimizing both the storage and the computational effort will be discussed in a separate paper

a posteriori error estimators in the finite element method in the paper we develop a structured approach to thea posteriori estimation of the error in the approximation obtained via the finite element method this aids the classification of existing estimators as well as allowing new estimators to be proposed for new situations a class of abstract estimators for finite elements of orderp 1 in re n n 2 3 based on exploiting the superconvergence phenomenon are analyzed

on a global superconvergence of the gradient of linear triangular elements abstract we study a simple superconvergent scheme which recovers the gradient when solving a second order elliptic problem in the plane by the usual linear elements the recovered gradient globally approximates the true gradient even by one order of accuracy higher in the l 2 norm than the piecewise constant gradient of the ritz galerkin solution a superconvergent approximation to the boundary flux is presented as well

scattering theory for twisted automorphic functions the purpose of this paper is to develop a scattering theory for twisted automorphic functions on the hyperbolic plane defined by a cofinite but not cocompact discrete group r with an irreducible unitary representation p and satisfying u yz p qy u z the lax phillips approach is used with the wave equation playing a central role incoming and outgoing subspaces are employed to obtain corresponding unitary translation representations r and r for the solution operator the scattering operator which maps r f into r f is unitary and commutes with translation the spectral representation of the scattering operator is a multiplicative operator which can be expressed in terms of the constant term of the eisenstein series when the dimension of p is one the elements of the scattering operator cannot vanish however when dim p 1 this is no longer the case

global superconvergence in combinations of ritz galerkin fem for singularity problems abstract this paper combines the piecewise bilinear elements with the singular functions to seek the corner singular solution of elliptic boundary value problems the global superconvergence rates o h 2 can be achieved by means of the techniques of lin and yan the construction and analysis of high efficient fem hobei university publishing hobei 1996 for different coupling strategies such as the nonconforming constraints the penalty integrals and the penalty plus hybrid integrals where 0 is an arbitrarily small number and h is the maximal boundary length of quasiuniform rectangles q ij used a little effort in computation is paid to conduct a posteriori interpolation of the numerical solutions u h only on the subregion used in finite element methods this paper also explores an equivalence of superconvergence between this paper and z c li internat j numer methods eng 39 1996 1839 1857 and j comput appl math 81 1997 1 17

superconvergence of coupling techniques in combined methods for elliptic equations with singularities abstract several coupling techniques such as the nonconforming constraints penalty and hybrid integrals of the ritz galerkin and finite difference methods are presented for solving elliptic boundary value problems with singularities based on suitable norms involving discrete solutions at specific points superconvergence rates on solution derivatives are exploited by using five combinations e g the nonconforming combination the penalty combination combinations i and ii and symmetric combination for quasi uniform rectangular grids the superconvergence rates o h 2 of solution derivatives by all five combinations can be achieved where h is the maximal mesh length of difference grids used in the finite difference method and 0 is an arbitrarily small number superconvergence analysis in this paper lies in estimates on error bounds caused by the coupling techniques and their incorporation with finite difference methods therefore a similar analysis and conclusions may be extended to linear finite element methods using triangulation by referring to existing references moreover the five combinations having o h 2 of solution derivatives are well suited to solving engineering problems with multiple singularities and multiple interfaces

on some new recovery based a posteriori error estimators the paper is concerned with the formulation of recovery based a posteriori error estimators at first we analyze a variant of the well known zienkiewicz zhu method which is here formulated so as to be exact in one dimension for quadratic solutions on non uniform grids next we discuss two methods which operate directly on the solution rather than its gradient one is based on a solution enrichment using the zienkiewicz zhu recovered gradient while the other consists of a roughening of the solution followed by a zienkiewicz zhu like recovery the three new proposed methods are compared in terms of their effectivity indices and solution accuracy to the zienkiewicz zhu estimator and are applied to representative two and three dimensional problems

collocation methods with controllable dissipation for linear dynamics abstract this paper introduces a unified set of collocation methods for linear dynamics in which a parameter is used to control the position of the collocation points in this manner both a stable algorithms of order 2 p and l stable algorithms of order 2 p 1 are derived from collocation polynomials of degree p in addition algorithms with intermediate accuracy and stability properties are made available the effects of varying the algorithmic parameter are investigated with particular reference to the numerical dissipation of spurious high frequency modes numerical tests are reported which support the theoretical analysis and demonstrate the performance of the proposed algorithms

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

multiscale methods for mechanical science of complex materials bridging from quantum to stochastic multiresolution continuum recent development in the multiscale method based on the bridging scale concept is presented with an emphasis on complex material systems the bridging scale method bsm was originally proposed by wagner and liu j comput phys 2003 190 249 274 as an effective way of treating the interface in coupled atomistic continuum simulation since its publication the bsm has become a very useful paradigm that has been applied to solve a host of problems in mechanical sciences of complex material systems in this paper we present a review on the recent developments we first describe the application of bsm to the coupled atomistic continuum simulation of dynamic fracture the important extensions within the framework of space time method and multiscale non equilibrium molecular dynamics are then presented we then focus on the multiresolution continuum theory that inherits the bsm concept in the analysis of heterogeneous material structures recent work of incorporating statistical factors into this model based on the concurrent nested homogenization of randomness of the material structures is highlighted finally we present the use of the bridging scale concept in resolving the electron mechanical coupling mechanism the robustness of the bsm is demonstrated through many benchmark problems and application examples copyright 2010 john wiley sons ltd

superconvergence phenomenon in the finite element method arising from averaging gradients we study a superconvergence phenomenon which can be obtained when solving a 2nd order elliptic problem by the usual linear elements the averaged gradient is a piecewise linear continuous vector field the value of which at any nodal point is an average of gradients of linear elements on triangles incident with this nodal point the convergence rate of the averaged gradient to an exact gradient in thel 2 norm can locally be higher even by one than that of the original piecewise constant discrete gradient

superconvergence phenomena in the finite element method abstract we give a brief survey of superconvergence phenomena in finding a numerical solution of differential equations by finite elements several new results and open problems are introduced