postprocessing the galerkin method the finite element case a postprocessing technique developed earlier for spectral methods is extended here to galerkin finite element methods for dissipative evolution partial differential equations the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed this technique increases the convergence rate of the finite element method to which it is applied and this is done at almost no additional computational cost the numerical experiments presented here show that the resulting postprocessed method is computationally more efficient than the method to which it is applied say quadratic finite elements as well as standard methods of similar order of convergence as the postprocessed one say cubic finite elements the error analysis of the new method is performed in l2 and in l infty norms

on the global convergence of a class of functional differential equations with applications in neural network theory abstract we study a system of retarded functional differential equations which generalise both the hopfield neural network model as well as hybrid network models of the cellular neural network type our main results give sufficient conditions for the global asymptotic stability of such systems and are milder than previously known conditions for the hybrid models when specialised to neural networks our models allow us to consider several different types of activation functions including piecewise linear sigmoids and unbounded activations as well as the usualc smooth sigmoids these issues are vital in the applications we also study neural network models with nonconstant delaysr t

superconvergence in finite element methods and meshes that are locally symmetric with respect to a point consider a second order elliptic boundary value problem in any number of space dimensions with locally smooth coefficients and solution consider also its numerical approximation by standard conforming finite element methods with for example fixed degree piecewise polynomials on a quasi uniform mesh family the h method it will be shown that if the finite element function spaces are locally symmetric about a point x 0 with respect to the antipodal map x to x 0 x x 0 then superconvergence ensues at xo under mild conditions on what happens outside a neighborhood of x 0 for piecewise polynomials of even degree superconvergence occurs in function values for piecewise polynomials of odd degree it occurs in derivatives

least squares solutions as solutions of a perturbation form of the galerkin methods interior pointwise error estimates and pollution effect first order div least squares ls finite element methods for second order elliptic partial differential equations are considered the main idea of this paper is a consideration of the ls solution u h for the primary function u as the solution of a perturbation form of the galerkin methods consequently the results concerning the perturbation form of the galerkin methods can be applied for the ls methods here we use the pointwise error estimates for the perturbation form provided by schatz a h schatz pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates siam j numer anal 38 2000 1269 1293 to obtain the pointwise estimates for the ls solution the estimates consist of the local truncation error and pollution terms we provide estimates for pollution terms our process for arriving at the estimates involves obtaining the norm of a linear functional which accounts for the perturbation our estimates are valid for both smooth and nonsmooth problems for the smooth problems the pollution is weaker than the local truncation error for the nonsmooth problems we provide a criterion for determining the strength of the pollution in particular we identify superconvergent points for nonconvex polygonal domains

superconvergence analysis of finite element method for the time dependent schrodinger equation in this paper we consider the two dimensional time dependent schrodinger equation firstly we use the rectangular lagrange type finite element of order p to get a semi discrete scheme of the equation and discuss the superconvergence error estimate in the h 1 norm secondly we use the crank nicolson method in time to get a fully discrete scheme of the equation and the superconvergence estimate in the h 1 norm can be obtained in this scheme finally a numerical example with the order p 1 is provided to verify our theoretical results

stability analyses of cellular neural networks with continuous time delay the stability of cellular neural networks cnns with continuous time delay is investigated in the present paper a main theorem together with a few corollaries related to the existence and boundedness of the solution for this kind of cnns as well as its global asymptotic stability is derived some numerical simulation examples aimed at justifying the theoretical results are also given

stability analysis of a single neuron model with delay in this paper we study the asymptotic behavior and numerical approximation of the single neuron model equation x t dx t af x t bf x t i t 0 1 where d 0 and f x 0 5 x 1 x 1 we obtain new sufficient conditions for global asymptotic stability of constant equilibriums of 1 give several numerical examples to illustrate our results and formulate conjectures on the asymptotic behavior of the solutions based on our numerical experiments

two approximation techniques for functional differential equations abstract some authors proposed to approximate the solutions of delay differential equations by ordinary differential equations to eliminate in part the difficulties presented by time delays this technique termed the chain method and its modified version are discussed in this paper for general nonlinear time dependent delay differential equations the modified version is based on some heuristic approximation in biological modelling and the difference in properties of the original and the new modified approximation techniques is demonstrated on a delay differential equation of an epidemic model

a numerical study of some hessian recovery techniques on isotropic and anisotropic meshes spaces of continuous piecewise linear finite elements are considered to solve a poisson problem and several numerical methods are investigated to recover second derivatives numerical results on 2d and 3d isotropic and anisotropic meshes indicate that the quality of the results is strongly linked to the mesh topology and that no convergence can be insured in general

superconvergence of h div finite element approximations for the stokes problem by l2 projection methods a general superconvergence result of h div finite element approximations for the stokes equations is established by using l 2 projection method regularity assumptions for the stokes problem with regular partitions is required numerical experiments are given to verify the theoretical results

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

superconvergence analysis of second and third order rectangular edge elements with applications to maxwell s equations abstract superconvergence for the second and third order edge elements is investigated on nonuniform rectangular meshes first we develop the explicit expression for the nedelec interpolation based on the hierarchical basis functions then we prove that the pointwise interpolation error estimates are one order higher at element gauss points than the standard analysis can provide using the superconvergence at gauss points we establish the discrete l 2 norm superconvergence for the solution of maxwell s equations solved by both the second and third order rectangular edge elements numerical results justifying our theoretical analysis are presented

convergence rates of some iterative methods for nonsymmetric algebraic riccati equations arising in transport theory we determine and compare the convergence rates of various fixed point iterations for finding the minimal positive solution of a class of nonsymmetric algebraic riccati equations arising in transport theory

the local superconvergence of the trilinear element for the three dimensional poisson problem abstract in this paper we shall combine the finite element theory of green s function presented in this paper the extrapolation technique and the local symmetric technique to investigate the local superconvergence of the trilinear element for the three dimensional poisson equation

fast gaussian elimination with partial pivoting for matrices with displacement structure fast o n 2 implementation of gaussian elimination with partial pivoting is designed for matrices possessing cauchy like displacement structure we show how toeplitz like toeplitz plus hankel like and vandermonde like matrices can be transformed into cauchy like matrices by using discrete fourier cosine or sine transform matrices in particular this allows us to propose a new fast o n 2 toeplitz solver gko which incorporates partial pivoting a large set of numerical examples showed that gko demonstrated stable numerical behavior and can be recommended for solving linear systems especially with nonsymmetric indefinite and ill conditioned positive definite toeplitz matrices it is also useful for block toeplitz and mosaic toeplitz toeplitz block matrices the algorithms proposed in this paper suggest an attractive alternative to look ahead approaches where one has to jump over ill conditioned leading submatrices which in the worst case requires o n 3 operations

superconvergence of h div finite element approximations for the stokes problem by local l 2 projection methods in this paper a general superconvergence result of h div finite element approximations for the stokes equations is established by using the local l 2 projection method local regularity assumptions for the stokes problem on regular partitions are required in order to derive superconvergence conclusions several numerical experiments are given to verify the theoretical results

global dynamics of hopfield neural networks involving variable delays abstract in this article a model describing dynamics of hopfield neural networks involving variable delays is considered existence and uniqueness of the equilibrium point under fairly general and easily verifiable conditions are also established further we derive sufficient criteria of global asymptotic stability gas of the equilibrium point

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

a new approach to richardson extrapolation in the finite element method for second order elliptic problems this paper presents a nonstandard local approach to richardson extrapolation when it is used to increase the accuracy of the standard finite element approximation of solutions of second order elliptic boundary value problems in mathbb r n n ge 2 the main feature of the approach is that it does not rely on a traditional asymptotic error expansion but rather depends on a more easily proved weaker a priori estimate derived in 19 called an asymptotic error expansion inequality in order to use this inequality to verify that the richardson procedure works at a point we require a local condition which links the different subspaces used for extrapolation roughly speaking this condition says that the subspaces are similar about a point i e any one of them can be made to locally coincide with another by a simple scaling of the independent variable about that point examples of finite element subspaces that occur in practice and satisfy this condition are given

superconvergence of quadratic finite volume method on triangular meshes abstract this paper is concerned with the superconvergence properties of the quadratic finite volume method fvm on triangular meshes for elliptic equations we proved the 3rd order superconvergence rate of the gradient approximation u h u i 1 o h 3 and the 4th order superconvergence rate of the function value approximation u h u i 0 o h 4 for the quadratic fvm on triangular meshes here u h is the fvm solution and u i is the piecewise quadratic lagrange interpolation of the exact solution it should be pointed out that the superconvergence phenomena of fvm strongly depends on the construction of dual mesh specially for quadratic fvms the scheme presented in this paper is the unique scheme which holds superconvergence

derivative superconvergent points in finite element solutions of harmonic functions a theoretical justification finite element derivative superconvergent points for harmonic functions under local rectangular mesh are investigated all superconvergent points for the finite element space of any order that is contained in the tensor product space and contains the intermediate family can be predicted in the case of the serendipity family results are given for finite element spaces of order below 6 the results justify the computer findings of babugka et al

natural superconvergence points in three dimensional finite elements a systematic and analytic process is conducted to identify natural superconvergence points of high degree polynomial c 0 finite elements in a three dimensional setting this identification is based upon explicitly constructing an orthogonal decomposition of local finite element spaces derivative and function value superconvergence points are investigated for both the poisson and the laplace equations superconvergence results are reported for hexahedral pentahedral and tetrahedral elements up to certain degrees

numerical approximation of the solutions of delay differential equations on an infinite interval using piecewise constant arguments the idea of approximating the solutions of delay differential equations by equations with piecewise constant arguments epca has been suggested by gyijri l who proved convergence of the method for linear and nonlinear delay equations on compact intervals and under certain conditions also on the half line the approximating equations with piecewise constant argument can in turn be solved by use of difference equations the latter then also provide an approximation of the original delay equation the theory of epca was initiated and studied by cooke and wiener in 2 and 3 in this paper we begin to investigate the question of whether in this method of approximation the asymptotic dynamics of the delay differential equations are preserved a number of authors have dealt with the problem of showing that discretization of ordinary differential equations does not significantly alter the basic qualitative features readers may refer to the papers of kloeden and lorenz 4 and beyn 5 for some of this work for delay differential equations questions of this sort have been studied by cryer 6 barwell 7 zennaro 8 and later authors the paper of strehmel et al 9 contains an up to date list of references in our method of approximation the delay equation is first replaced by an epca and then by a difference equation and our objective is to relate the qualitative dynamics of these three equations we obtain results for autonomous and nonautonomous equations with one or many delays for autonomous equations the resulting difference algorithm is just a simple euler scheme but for nonautonomous equations it includes other possibilities in order to make the method and the nature of results as clear as possible we begin with a particular test equation k t pz t 7 p 0 7 0

the method of lines for the solutions of some nonlinear partial differential equations abstract in this paper we deal with the method of lines in the case of certain first order nonlinear partial differential equations as well as equations of the parabolic type a new aspect compared to earlier investigations is that the system we consider may contain partial and ordinary differential equations alike and in addition the r h s of the equations may involve some nonlinear operators the domain of definition of the solutions is bounded by time dependent functions our general results are demonstrated on a simple system which describes a biological model

improving the rate of convergence of high order finite elements on polygons and domains with cusps let u and uv v be the solution and respectively the discrete solution of the non homogeneous dirichlet problem u f on u 0 for any m and any bounded polygonal domain we provide a construction of a new sequence of finite dimensional subspaces vn such that where f hm 1 is arbitrary and c is a constant that depends only on and not on n we do not assume u hm 1 the existence of such a sequence of subspaces was first proved in a ground breaking paper by babuska 8 our method is different it is based on the homogeneity properties of sobolev spaces with weights and the well posedness of non homogeneous dirichlet problem in suitable sobolev spaces with weights for which we provide a new proof and which is a substitute of the usual shift theorems for boundary value problems in domains with smooth boundary our results extended right away to domains whose boundaries have conical points we also indicate some of the changes necessary to deal with domains with cusps our numerical computation are in agreement with our theoretical results

an estimate for the three dimensional discrete green s function and applications in this paper we first introduce definitions of the regularized green s function and the discrete green s function in three dimensions with complicated arguments estimates for these two functions are then derived finally we give an application of the estimate for the discrete green s function to the high accuracy analysis of the three dimensional block finite element approximation

finite element superconvergence on shishkin mesh for 2 d convection diffusion problems in this work the bilinear finite element method on a shishkin mesh for convection diffusion problems is analyzed in the two dimensional setting a superconvergence rate o n 2 ln2 n en 1 5lnn in a discrete e weighted energy norm is established under certain regularity assumptions this convergence rate is uniformly valid with respect to the singular perturbation parameter e numerical tests indicate that the rate o n 2ln2 n is sharp for the boundary layer terms as a by product an e uniform convergence of the same order is obtained for the l2 norm furthermore under the same regularity assumption an e uniform convergence of order n 3 2ln5 2n en 1ln1 2n in the l norm is proved for some mesh points in the boundary layer region

superconvergence of discontinuous galerkin method for scalar nonlinear hyperbolic equations in this paper we study the superconvergence behavior of the semi discrete discontinuous galerkin dg method for scalar nonlinear hyperbolic equations in one spatial dimension superconvergence results for problems with fixed and alternating wind directions are established on the one hand we prove that if the wind direction is fixed i e the derivative of the flux function is bounded away from zero both the cell average error and numerical flux error at cell interfaces converge at a rate of 2k 1 when upwind fluxes and piecewise polynomials of degree k are used moreover we also prove that the function value approximation of the dg solution is superconvergent at interior right radau points and the derivative value approximation is superconvergent at interior left radau points with an order of k 2 and k 1 respectively as a byproduct we show a k 2 th order superconvergence of the dg solution towards the gauss radau projection of the exact solution on the other hand superconvergence resu

uniform approximation of a nonlinear delay equation on infinite intervals approximation of delay differential equations by sequences of ordinary differential equations has received considerable attention in the literature see e g 9 1 21 and the references therein a somewhat different approximation methodology motivated by the conceptual connection between models of biological compartmental systems with and without pipes described by delay and nondelay equations respectively was proposed in 4 and 5 to preserve a conservation of material type condition under the approximation a nice feature of the latter approach is that the steady state solutions for the original and approximated systems are identical this property of the approximation method leads naturally to the following question see also s is the convergence of the approximate solutions uniform on the infinite interval 0 a in this paper under mild smoothness conditions on the solution of the original problem we provide an affirmative answer to the above question for a class of scalar nonlinear delay equations this class includes equations used in population dynamics and epidemics 3 and compartmental systems with pipes 6 71

fast iterative schemes for nonsymmetric algebraic riccati equations arising from transport theory for a class of nonsymmetric algebraic riccati equations arising from transport theory lu siam j matrix anal appl 26 2005 pp 679 685 reformulated them into a couple of fixed point equations in a vector form and constructed an iterative method for finding their minimal positive solutions in this paper based on this fixed point equation we propose several fast and effective iterative methods for solving the above mentioned special type of nonsymmetric algebraic riccati equations these iterative methods can be categorized into the class of nonlinear block relaxation iterations and can also be considered as accelerated variants of the known fixed point iteration according to the principle of using the currently available information as promptly as possible the monotone convergence theorems about these new iterative methods are established and the sharper bounds about the solutions of the nonsymmetric algebraic riccati equations are derived numerical implementations show that the new iterative methods are feasible and effective solvers for the nonsymmetric algebraic riccati equations arising from transport theory

asymptotic constancy and periodicity for a single neuron model with delay abstract we study the single neuron model equation x t d x t a f x t b f x t 1 i t 0 where d 0 b 0 and f x 1 2 x 1 x 1 we examine the asymptotic behaviour of the solutions and confirm that although most of them are convergent there exists a periodic solution if d a b i

superconvergence of numerical gradient for weak galerkin finite element methods on nonuniform cartesian partitions in three dimensions abstract a superconvergence error estimate for the gradient approximation of the second order elliptic problem in three dimensions is analyzed by using weak galerkin finite element scheme on the uniform and non uniform cubic partitions due to the loss of the symmetric property from two dimensions to three dimensions this superconvergence result in three dimensions is not a trivial extension of the recent superconvergence result in two dimensions li et al 0000 from rectangular partitions to cubic partitions the error estimate for the numerical gradient in the l 2 norm arrives at a superconvergence order of o h r 1 5 r 2 when the lowest order weak galerkin finite elements consisting of piecewise linear polynomials in the interior of the elements and piecewise constants on the faces of the elements are employed a series of numerical experiments are illustrated to confirm the established superconvergence theory in three dimensions

ultraconvergence of zz patch recovery at mesh symmetry points the ultraconvergence property of the zienkiewicz zhu gradient patch recovery technique based on local discrete least squares fitting is established for a large class of even order finite elements the result is valid at all rectangular mesh symmetry points different smoothing strategies are discussed and numerical examples are demonstrated

ultraconvergence of high order fems for elliptic problems with variable coefficients in this paper we investigate local ultraconvergence properties of the high order finite element method fem for second order elliptic problems with variable coefficients under suitable regularity and mesh conditions we show that at an interior vertex which is away from the boundary with a fixed distance the gradient of the post precessed kth k ge 2 k 2 order finite element solution converges to the gradient of the exact solution with order mathcal o h k 2 mathrm ln h 3 o hk 2 lnh 3 the proof of this ultraconvergence property depends on a new interpolating operator some new estimates for the discrete green s function a symmetry theory derived in 26 and the richardson extrapolation technique in 20 numerical experiments are performed to demonstrate our theoretical findings

derivative superconvergent points in finite element solutions of poisson s equation for the serendipity and intermediate families a theoretical justification finite element derivative superconvergent points for the poisson equation under local rectangular mesh in the two dimensional case and local brick mesh in the three dimensional situation are investigated all superconvergent points for the finite element space of any order that is contained in the tensor product space and contains the intermediate family can be predicted in case of the serendipity family the results are given for finite element spaces of order below 7 any finite element space that contains the complete polynomial space will have at least all superconvergent points of the related serendipity family

existence of algebraic matrix riccati equations arising in transport theory we consider the existence of positive solutions of a certain class of algebraic matrix riccati equations with two parameters c 0 c 1 and 0 1 here c denotes the fraction of scattering per collision and is an angular shift equations of this class are induced via invariant imbedding and the shifted gauss legendre quadrature formula from a simple transport model by establishing the existence of positive solutions of such equations the problem of the convergence of some iterative schemes for solving them can be completely solved

local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients abstract assume that u x satisfies the problem l u x x i a i j u x j f x x u x 0 x in this article using interpolation postprocessing technique we will investigate the local ultraconvergence of the primal variable and the derivative of finite element approximation of u x using piecewise polynomials of degrees bi k k 3 over a rectangular partition assume that k 3 is odd and x 0 is an interior vertex satisfying x 0 c using the new interpolation postprocessing formula presented in this study we show that the primal variable and the derivative of the post processed finite element solution using piecewise of degrees bi k k 3 at x 0 converge to the primal variable and the derivative of the exact solution with order o h k 3 ln h under suitable regularity and mesh conditions respectively finally we use numerical experiments to illustrate our theoretical findings

is 2k conjecture valid for finite volume methods this paper is concerned with superconvergence properties of a class of finite volume methods of arbitrary order over rectangular meshes our main result is to prove the 2k conjecture at each vertex of the underlying rectangular mesh the bi k degree finite volume solution approximates the exact solution with an order o h 2k where h is the mesh size as byproducts superconvergence properties for finite volume discretization errors at lobatto and gauss points are also obtained all theoretical findings are confirmed by numerical experiments

finite element differential forms on cubical meshes we develop a family of finite element spaces of differential forms defined on cubical meshes in any number of dimensions the family contains elements of all polynomial degrees and all form degrees in two dimensions these include the serendipity finite elements and the rectangular bdm elements in three dimensions they include a recent generalization of the serendipity spaces and new h curl and h div finite element spaces spaces in the family can be combined to give finite element subcomplexes of the de rham complex which satisfy the basic hypotheses of the finite element exterior calculus and hence can be used for stable discretization of a variety of problems the construction and properties of the spaces are established in a uniform manner using finite element exterior calculus

stability in cellular neural networks with a piecewise constant argument in this paper by using the concept of differential equations with piecewise constant arguments of generalized type 1 4 a model of cellular neural networks cnns 5 6 is developed the lyapunov razumikhin technique is applied to find sufficient conditions for the uniform asymptotic stability of equilibria global exponential stability is investigated by means of lyapunov functions an example with numerical simulations is worked out to illustrate the results

a structure preserving doubling algorithm for nonsymmetric algebraic riccati equation in this paper we propose a structure preserving doubling algorithm sda for the computation of the minimal nonnegative solution to the nonsymmetric algebraic riccati equation nare based on the techniques developed for the symmetric cases this method allows the simultaneous approximation to the minimal nonnegative solutions of the nare and its dual equation requiring only the solutions to two linear systems and several matrix multiplications per iteration similar to newton s method and the fixed point iteration methods for solving nares we also establish global convergence for sda under suitable conditions using only elementary matrix theory we show that sequences of matrices generated by sda are monotonically increasing and quadratically convergent to the minimal nonnegative solutions of the nare and its dual equation numerical experiments show that the sda algorithm is feasible and effective and outperforms newton s iteration and the fixed point iteration methods

global stability for cellular neural networks with time delay a sufficient condition related to the existence of a unique equilibrium point and its global asymptotic stability for cellular network networks with delay dcnns is derived it is shown that the condition relies on the feedback matrices and is independent of the delay parameter furthermore this condition is less restrictive than that given in the literature

complete stability in multistable delayed neural networks we investigate the complete stability for multistable delayed neural networks a new formulation modified from the previous studies on multistable networks is developed to derive componentwise dynamical property an iteration argument is then constructed to conclude that every solution of the network converges to a single equilibrium as time tends to infinity the existence of 3n equilibria and 2n positively invariant sets for the n neuron system remains valid under the new formulation the theory is demonstrated by a numerical illustration

a modified structure preserving doubling algorithm for nonsymmetric algebraic riccati equations from transport theory we consider the nonsymmetric algebraic riccati equation arising in transport theory where the nxn coefficient matrices a b c and e involved in the equation are rank structured after a balancing strategy the matrix email protected t is the minimal positive solution of the dual algebraic riccati equation we can simplify the structure preserving doubling algorithm sda to this special equation and give a modified sda which has less computational cost at each iteration step also we use numerical experiments to show the effectiveness of our new methods

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates this part contains new interior pointwise error estimates for the finite element method for second order elliptic problems in mathbb r n global estimates were considered in part i in the sense to be discussed below these sharpen known interior quasi optimal l infty and w 1 infty estimates in that they indicate a more local dependence of the error at a point on the derivatives of the solution near the point the higher the order of the finite element the more local the behavior of the finite element approximation as a consequence of these estimates new types of local error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in mathbb r n and have applications to superconvergence extrapolation and a posteriori estimates for both smooth and nonsmooth problems

the optimal convergence of finite element methods for the three dimensional second order elliptic boundary value problem with singularity abstract let x 0 assume that g x 0 x satisfies the elliptic boundary value problem l g x 0 x x j a i j x g x 0 x x i x x 0 in g x 0 x 0 on where 3 is a convex polyhedral domain in this article we propose specially graded meshes t h furthermore we obtain an l 2 error estimate of order h k 1 ln h 3 2 for approximations with a p k finite element method

an interior estimate of superconvergence for finite element solutions for second order elliptic problems on quasi uniform meshes by local projections this paper establishes some superconvergence estimates for finite element solutions of second order elliptic problems by a projection method depending only on local properties of the domain and the finite element solution the projection method is a postprocessing procedure that constructs a new approximation by using the method of least squares in particular some local superconvergence estimates in the l2 and l infty norms are derived for the local projections of the galerkin finite element solution the results have two prominent features first they are established for any quasi uniform meshes which are of practicalimportance in scientific computation second they are derived on the basis of local properties of the domain and the solution for the second order elliptic problem therefore the result of this paper can be employed to provide useful a posteriori error estimators in practical computing

averaging for superconvergence verification and application of 2d edge elements to maxwell s equations in metamaterials abstract in this paper a simple averaging technique for edge elements is justified to improve the standard convergence rate hence achieving superconvergence when they are used to solve the maxwell s equations for simplicity here we focus on the lowest order triangular edge element which is widely used in practice though there exists no natural superconvergence points for the numerical solution obtained on such an edge element superconvergence can be obtained after a simple average of solutions over the neighboring elements a comprehensive analysis for the lowest order triangular edge element is carried out and one order higher convergence rate than the standard interpolation error estimate is proved for the averaged solution at midpoints of those interior edges of parallelograms formed by two triangles i e superconvergence happens at the parallelogram centers we also provide detailed analysis to explain why several cases do not have superconvergence extensive numerical results consistent with our analysis are presented

high accuracy analysis of two nonconforming plate elements we prove the superconvergence of morley element and the incomplete biquadratic nonconforming element for the plate bending problem under uniform rectangular meshes we obtain a superconvergence property at the symmetric points of the elements and a global superconvergent result by a proper postprocessing method

some error estimates for the lumped mass finite element method for a parabolic problem we study the spatially semidiscrete lumped mass method for the model homogeneous heat equation with homogeneous dirichlet boundary conditions improving earlier results we show that known optimal order smooth initial data error estimates for the standard galerkin method carry over to the lumped mass method whereas nonsmooth initial data estimates require special assumptions on the triangulation we also discuss the application to time discretization by the backward euler and crank nicolson methods

superconvergence of the velocity along the gauss lines in mixed finite element methods superconvergence for the velocity along the gauss lines is derived for the mixed finite element method for the second order elliptic equation in rectangular domains these estimates are based on the rectangular finite elements of raviart thomas type

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

stability in asymmetric hopfield nets with transmission delays sufficient conditions are derived for the delay independent stability of the equilibria in hopfield s graded response networks of the type dxi t dt bixi t nj 1aijfj jxj t ij fi t i 1 2 n when the external inputs fi are held temporally uniform a generalization to continuously distributed delays is briefly indicated several illustrative examples are numerically simulated and the results of simulations are graphically displayed