numerical solutions of the modified burgers equation by petrov galerkin method abstract the modified burgers equation mbe is solved numerically by the petrov galerkin method using a linear hat function as the trial function and a cubic b spline function as the test function product approximation has been used in this method a linear stability analysis of the scheme shows it to be unconditionally stable the accuracy of the presented method is demonstrated by two test problems the numerical results are found in good agreement with the exact solutions

finite hilbert stability of bi canonical curves we prove that a generic canonically or bicanonically embedded smooth curve has semistable mth hilbert points for all m 2 we also prove that a generic bicanonically embedded smooth curve has stable mth hilbert points for all m 3 in the canonical case this is accomplished by proving finite hilbert semistability of special singular curves with mathbb g m action namely the canonically embedded balanced ribbon and the canonically embedded balanced doublea2k 1 curve in the bicanonical case we prove finite hilbert stability of special hyperelliptic curves namely wiman curves finally we give examples of canonically embedded smooth curves whose mth hilbert points are non semistable for low values of m but become semistable past a definite threshold

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

hamiltonian and linear space structure for damped oscillators i general theory the phase space of n damped linear oscillators is endowed with a bilinear map under which the evolution operator is symmetric this analogue of self adjointness allows properties familiar from conservative systems to be recovered e g eigenvectors are orthogonal under the bilinear map and obey sum rules initial value problems are readily solved and perturbation theory applies to the complex eigenvalues these concepts are conveniently represented in a biorthogonal basis

numerical approach for solving diffusion problems using cubic b spline collocation method abstract a numerical approach for modeling the removal of homogeneous solutes from the porous structure is presented pore diffusion equations controlling the overall process are expressed in terms of concentration of solute as a function of position in bed and washing period a computational scheme specifying knot vector s and control points de boor points to solve diffusion model using cubic b spline collocation method cscm is given the knot vector provides sufficient resolution and control points yield an accurate approximation of the desired curve convergence of numerical scheme is of o h 2 four problems have been worked out by the proposed scheme results are plotted graphically for fairly wide range of parameters a comparison with the previous techniques demonstrates the superiority of the proposed method on basis of computational time efficiency and accuracy

numerical solution of burgers equation by lattice boltzmann method in this paper a lattice boltzmann model for burgers equation is proposed through selecting equilibrium distribution function properly the model has been verified by three test examples the numerical results are found in good agreement with analytical solutions and numerical solutions reported in previous studies the advantages of the resulting method are algorithmic simplicity parallel computation easy handling of complicated boundary conditions so it is very easy to implement and deal with more complex problems

the canonical syzygy conjecture for ribbons green s canonical syzygy conjecture asserts a simple relationship between the clifford index of a smooth projective curve and the shape of the minimal free resolution of its homogeneous ideal in the canonical embedding we prove the analogue of this conjecture formulated by bayer and eisenbud for a class of non reduced curves called ribbons our proof uses the results of voisin and hirschowitz ramanan on green s conjecture for general smooth curves

two component eigenfunction expansion for open systems described by the wave equation i completeness of expansion the concept of eigenfunction expansions for the wave equation is generalized to open systems in which waves escape to the outside these non conservative systems are non hermitian in the usual sense it is shown that the natural framework is an eigenfunction expansion within a two component formalism that treats the wavefunction and its conjugate momentum together provided the system approaches spatial infinity rapidly without tails and possesses spatial discontinuities the expansion in terms of the eigenfunctions which are now quasinormal modes is shown to be valid

a moving petrov galerkin method for transport equations a time dependent transport equation is solved numerically by using a petrov galerkin technique on a time dependent grid a piecewise linear trial solution and piecewise hermite cubic test functions are used the same method is also applied to a time independent problem in which case it automatically generates a grid in a way which bears some resemblance to certain existing grid generation techniques several numerical examples are given

rank two acm bundles on the del pezzo threefold with picard number 3 abstract a del pezzo threefold f with maximal picard number is isomorphic to p 1 p 1 p 1 in the present paper we completely classify locally free sheaves e of rank 2 such that h i f e t 0 for i 1 2 and t z such a classification extends similar results proved by e arrondo and l costa regarding del pezzo threefolds with picard number 1

canonical double covers of minimal rational surfaces and the non existence of carpets his article delves into the relation between the deformation theory of finite morphisms to projective space and the existence of ropes embedded in projective space with certain invariants we focus on the case of canonical double covers x of a minimal rational surface y embedded in p n by a complete linear series and carpets on y canonically embedded in p n we prove that these canonical double covers always deform to double covers and that canonically embedded carpets on y do not exist this fact parallels the results known for hyperelliptic canonical morphisms of curves and canonical ribbons and the results for k3 double covers of surfaces of minimal degree and enriques surfaces and k3 carpets that canonical double covers of minimal rational surfaces should deform to double covers is not a priori obvious for the invariants of most of these surfaces lie on or above the castelnuovo line thus in principle deformations of such covers could have birational canonical maps in fact many canonical double covers of non minimal rational surfaces do deform to birational canonical morphisms we also map the region of the geography of surfaces of general type corresponding to the surfaces x and we compute the dimension of the irreducible moduli component containing x in certain cases we exhibit some interesting moduli components parameterizing surfaces s with the same invariants as x but with birational canonical map unlike x

two component eigenfunction expansion for open systems described by the wave equation ii linear space structure for a broad class of open systems described by the wave equation the eigenfunctions which are quasinormal modes provide a complete basis for simultaneously expanding outgoing wavefunctions in this paper the linear space structure associated with this expansion is developed under a modified inner product the time evolution operator is self adjoint even though energy is not conserved for the system alone thus the eigenfunctions are mutually orthogonal consequently the usual tools of eigenfunction expansions can be transcribed to these open systems as an example the time independent perturbation theory is developed in straightforward analogy with quantum mechanics giving the shift in both the real part and the imaginary part of the eigenvalues

hamiltonian and linear space structure for damped oscillators ii critical points the eigenvector expansion developed in the preceding paper for a system of damped linear oscillators is extended to critical points where eigenvectors merge and the time evolution operator h assumes a jordan block structure the representation of the bilinear map is obtained in this basis perturbations h around an mth order critical point generically lead to eigenvalue shifts 1 m dependent on only one matrix element with the m eigenvalues splitting in equiangular directions in the complex plane small denominators near criticality are shown to cancel

ribbons and their canonical embeddings we study double structures on the projective line and on certain other varieties with a view to having a nice family of degenerations of curves and k3 surfaces of given genus and clifford index our main interest is in the canonical embeddings of these objects with a view toward green s conjecture on the free resolutions of canonical curves we give the canonical embeddings explicitly and exhibit an approach to determining a minimal free resolution

biorthogonal multiwavelets on the interval for numerical solutions of burgers equation to numerically solve the burgers equation in this paper we propose a general method for constructing wavelet bases on the interval 0 1 derived from symmetric biorthogonal multiwavelets on the real line in particular we obtain wavelet bases with simple structures on the interval 0 1 from the hermite cubic splines in comparison with all other known constructed wavelets on the interval 0 1 our constructed wavelet bases on the interval 0 1 from the hermite cubic splines not only have good approximation and symmetry properties with extremely short supports but also employ a minimum number of boundary wavelets with a very simple structure these desirable properties make them to be of particular interest in numerical algorithms our constructed wavelet bases on the interval 0 1 are then used to solve the nonlinear burgers equation our method is based on the finite difference formula combined with the collocation method therefore our proposed numerical scheme in this paper is abbreviated as mfdcm mixed finite difference and collocation method some numerical examples are provided to demonstrate the validity and applicability of our proposed method which can be easily implemented to produce a desired accuracy

deformation of canonical morphisms and the moduli of surfaces of general type in this article we study the deformation of finite maps and show how to use this deformation theory to construct varieties with given invariants in a projective space among other things we prove a criterion that determines when a finite map can be deformed to a one to one map we use this criterion to construct new surfaces of general type with birational canonical map for different c 1 2 and the canonical map of the surfaces we construct is in fact a finite birational morphism our general results enable us to describe some new components of the moduli of surfaces of general type we also find infinitely many moduli spaces open image in new window having one component whose general point corresponds to a canonically embedded surface and another component whose general point corresponds to a surface whose canonical map is a degree 2 morphism

degenerations of k3 surfaces in projective space a k3 carpet s is a double structure on a rational normal scroll such that its dualizing sheaf is trivial and h1 os 0 in this note the authors show that every k3 carpet s can be smoothed i e there exists a flat family over a smooth curve with smooth generic fiber and with a special closed fiber isomorphic top s moreover they study the hilbert scheme of numerical k3 surfaces at the locus parametrizing k3 carpets characterizing those k3 carpets whose corresponding hilbert point is smooth the proof is based on the properties of the hyperelliptic linear systems on k3 surfaces

clifford indices of ribbons we present a theory of limit linear series for rational ribbonsthat is for schemes that are double structures on pi this allows us to define a linear series clifford index for ribbons our main theorem shows that this is the same as the clifford index of ribbons studied by eisenbud bayer in this same volume this allows us to prove that the clifford index is semicontinuous in degenerations from a smooth curve to a ribbon a result of fong 1993 then shows that ribbons may be deformed to smooth curves of the same clifford index thus the canonical curve conjecture of green 1984 would follow at least for a general smooth curve of each clifford index from the corresponding statement for ribbons

fourth order finite difference method for solving burgers equation abstract in this paper we present fourth order finite difference method for solving nonlinear one dimensional burgers equation this method is unconditionally stable the convergence analysis of the present method is studied and an upper bound for the error is derived numerical comparisons are made with most of the existing numerical methods for solving this equation

a framework for nonlinear shells based on generalized stress and strain measures a very general class of shell theories is established based on a unified variational principle and on expansions with respect to biorthogonal function systems the shell models derived are able to accommodate the flow theory of large strain thermoelastoplasticity including the effects due to unloading the specific case of a compressible neo hookean material with a von mises yield condition is discussed in detail

deformations of canonical double covers abstract in this paper we show that if x is a smooth variety of general type of dimension m 2 for which its canonical map induces a double cover onto y where y is the projective space a smooth quadric hypersurface or a smooth projective bundle over p 1 embedded by a complete linear series then the general deformation of the canonical morphism of x is again canonical and induces a double cover the second part of the article proves the non existence of canonical double structures on the rational varieties above mentioned our results have consequences for the moduli of varieties of general type of arbitrary dimension since they show that infinitely many moduli spaces of higher dimensional varieties of general type have an entire hyperelliptic component this is in sharp contrast with the case of curves or surfaces of lower kodaira dimension

exactly solvable path integral for open cavities in terms of quasinormal modes we evaluate the finite temperature euclidean phase space path integral for the generating functional of a scalar field inside a leaky cavity provided the source is confined to the cavity one can first of all integrate out the fields on the outside to obtain an effective action for the cavity alone subsequently one uses an expansion of the cavity field in terms of its quasinormal modes qnms the exact exponentially damped eigenstates of the classical evolution operator which previously have been shown to be complete for a large class of models dissipation causes the effective cavity action to be nondiagonal in the qnm basis the inversion of this action matrix inherent in the gaussian path integral to obtain the generating functional is therefore nontrivial but can be accomplished by invoking a novel qnm sum rule the results are consistent with those obtained previously using canonical quantization

linear systems of plane curves with base points of equal multiplicity in this article we address the problem of computing the dimenlsion of the space of plane curves of degree d with n general points of multiplicity m a conjecture of harbourne and hirschowitz implies that when d 3m the dimension is equal to the expected dimension given by the riemann roch theorem also systems for which the dimension is larger than expected should have a fixed part containing a multiple 1 curve we reformulate this conjecture by explicitly listing those systems which have unexpected dimension then we use a degeneration technique developed to show that the conjecture holds for all m 12

classification of quadruple galois canonical covers i in this article we classify quadruple galois canonical covers of smooth surfaces of minimal degree the classification shows that they are either non simple cyclic covers or bi double covers if they are bi double then they are all fiber products of double covers we construct examples to show that all the possibilities in the classification do exist there are implications of this classification that include the existence of families with unbounded geometric genus in sharp contrast with triple canonical covers and families with unbounded irregularity in sharp contrast with canonical covers of all other degrees together with the earlier known results on double and triple covers a pattern emerges that motivates some general questions on the existence of higher degree canonical covers some of which are answered in this article

a novel numerical scheme for solving burgers equation abstract in this paper we propose a novel numerical scheme for solving burgers equation the scheme is based on a cubic spline quasi interpolant and multi node higher order expansion which make the algorithm simple and easy to implement the numerical experiments show that the proposed method produces high accurate results

jordan blocks and generalized bi orthogonal bases realizations in open wave systems dissipation can sometimes be described by a non hermitian hamiltonian h whose left and right eigenvectors f j fj form a bi orthogonal basis bb for waves in a class of open systems this is known to lead to exact complete bb expansions if f j fj 0 for all j if not normalization seems impossible and many familiar formulae fail examples are given the problem is related to the merging of eigenmodes so that h can only be diagonalized to jordan blocks the resolution involves a generalized bb containing extra vectors whose dynamics are modified by polynomials in the time t the splitting of merged modes under a perturbation is also treated one thus obtains a non trivial extension of the bb formalism for open systems

numerical solutions of nonlinear burgers equation with modified cubic b splines collocation method abstract in this paper a numerical method is proposed to approximate the solution of the nonlinear burgers equation the method is based on collocation of modified cubic b splines over finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range we apply modified cubic b splines for spatial variable and derivatives which produce a system of first order ordinary differential equations we solve this system by using ssp rk43 or ssp rk54 scheme this method needs less storage space that causes less accumulation of numerical errors the numerical approximate solutions to the burgers equation have been computed without transforming the equation and without using the linearization illustrative eleven examples are included to demonstrate the validity and applicability of the technique easy and economical implementation is the strength of this method

gauss composition for p1 and the universal jacobian of the hurwitz space of double covers abstract in this paper we give an explicit description of the moduli space of line bundles on hyperelliptic curves including singular curves we study the universal jacobian j 2 g n of degree n line bundles over the hurwitz stack of double covers of p 1 by a curve of genus g our main results are the construction of a smooth irreducible universally closed but not separated moduli compactification j bd 2 g n of j 2 g n whose points we describe simply and explicitly as sections of certain vector bundles on p 1 a description of the global geometry and moduli properties of these stacks and a computation of the picard groups of j bd 2 g n and j 2 g n in the cases when n g is even an important ingredient of our work is the parametrization of line bundles on double covers by binary quadratic forms this parametrization generalizes the classical number theoretic correspondence between ideal classes of quadratic rings and integral binary quadratic forms which in particular gives the group law on integral binary quadratic forms first discovered by gauss

numerical solution of nonlinear burgers equation using high accuracy multi quadric quasi interpolation in this paper a numerical method is presented to approximate the solution to nonlinear burgers equation which is related to many scientific research topics a numerical scheme by using high accuracy multi quadric quasi interpolation is presented in which a kind of multi quadric quasi interpolant is used to approximate the derivatives of the solution in spatial domain and finite difference is used to approximate the derivatives of the solution in temporal domain the advantage of the scheme is that it is mesh free and in each time step only a multi quadric quasi interpolant is employed so that the algorithm is easy to implement the numerical results of this scheme are also shown and compared with other numerical schemes

appropriate stabilized galerkin approaches for solving two dimensional coupled burgers equations at high reynolds numbers abstract this paper aims to seek proper stabilized galerkin methods for solving the two dimensional coupled burgers equations at high reynolds numbers the stabilization techniques employed here include the streamline upwind petrov galerkin supg method the spurious oscillations at layers diminishing sold method and the characteristic galerkin cg method the first two methods are combined with the crank nicolson scheme for time discretization and the last one is applied in its semi implicit version different from most of the studies on the equations which are usually devoted to improving the accuracy of computed solution in the case of low reynolds numbers this paper mainly focuses on keeping the stability of the solution at high reynolds numbers which is significant in practical applications and also challenging in numerical computation we study two problems equipped with mixed boundary conditions and only dirichlet boundary conditions respectively numerical experiments reveal that the supg method is optimal for the former problem and the sold method is more appropriate for the latter one in addition the performances of these methods demonstrate the difference between the two problems which is seldom mentioned previously and might be helpful to other conventional methods intending to solve the problems at high reynolds numbers and last since sold methods have rarely been utilized to solve nonlinear unsteady problems before this study also indicates the potential of this class of methods to solve nonlinear unsteady convection dominated problems

waves in open systems via a biorthogonal basis dissipative quantum systems are sometimes phenomenologically described in terms of a non hermitian hamiltonian h with different left and right eigenvectors forming a biorthogonal basis it is shown that the dynamics of waves in open systems can be cast exactly into this form thus providing a well founded realization of the phenomenological description and at the same time placing these open systems into a well known framework the formalism leads to a generalization of norms and inner products for open systems which in contrast to earlier works is finite without the need for regularization the inner product allows transcription of much of the formalism for conservative systems including perturbation theory and second quantization

projective degenerations of k3 surfaces gaussian maps and fano threefolds in this article we exhibit certain projective degenerations of smoothk3 surfaces of degree 2g 2 in g whose picard group is generated by the hyperplane class to a union of two rational normal scrolls and also to a union of planes as a consequence we prove that the general hyperplane section of suchk3 surfaces has a corank one gaussian map ifg 11 org 13 we also prove that the general such hyperplane section lies on a uniquek3 surface up to projectivities finally we present a new approach to the classification of prime fano threefolds of index one which does not rely on the existence of a line

the generalized boundary element approach to burgers equation the generalized boundary element method is presented for the numerical solution of burgers equation the new method is based on the set of boundary integral equations derived for each subdomain by using the fundamental solution for the linearized differential operator of the equation the resulting system of quasi non linear equations is solved implicitly with use of a simple iterative procedure the adaptability and the accuracy of the proposed method are demonstrated by three examples and a comparison of the numerical results with the exact solution or other existing solutions is shown for the first example

a galerkin finite element approach to burgers equation a galerkin finite element method is presented for the numerical solution of burgers equation a linear recurrence relationship for the numerical solution of the resulting system of ordinary differential equations is found via a crank nicolson approach involving a product approximation it is shown that this method is capable of solving burgers equation accurately for a wide range of viscosity values the results show that the new method performs better than the most of the methods available in the literature

numerical algorithm for the determination of the potential of a conservative system from its normal mode spectra an efficient numerical algorithm is developed for determining the potential of a conservative system described by a klein gordon or schrodinger equation from its normal mode spectra the algorithm is based on first solving the linear problem of perturbative inversion by means of tikhonov regularization and then using a series of perturbative steps to approach the full potential by newton s method the algorithm is stable and convergence is rapid

susy transformations for quasinormal modes of open systems supersymmetry susy in quantum mechanics is extended from square integrable states to those satisfying the outgoing wave boundary condition in a klein gordon formulation this boundary condition allows both the usual normal modes and quasinormal modes with complex eigenvalues the simple generalization leads to three features the counting of eigenstates under susy becomes more systematic the linear space structure of outgoing waves nontrivially different from the usual hilbert space of square integrable states is preserved by susy and multiple states at the same frequency not allowed for normal modes are also preserved the existence or otherwise of susy partners is furthermore relevant to the question of inversion are open systems uniquely determined by their complex outgoing wave spectra

a class of high order compact difference schemes for solving the burgers equations abstract in this paper a class of high order compact difference method is introduced for solving the burgers equations firstly a linear high order compact difference scheme is proposed to solve the one dimensional burgers equation the scheme is fourth order accurate in space and second order accurate in time linear stability analysis is conducted to show the scheme is conditionally stable because only three grid points are involved in each time level thomas algorithm can be directly used to solve the tridiagonal linear system then this method is extended to solve the two dimensional and three dimensional coupled burgers equations numerical experiments are carried out to demonstrate the accuracy and dependability of the present method

syzygies of canonical curves and special linear series 0 introduction 105 1 line bundles on scrolls 109 2 scrolls and pencils 113 3 syzygies of a subvariety of a scroll 115 4 canonical curves 118 5 on the equation of rational surfaces 121 6 d gonal curves for d 5 126 7 proof of the main result 131 references 136

an exponentially fitted method for solving burgers equation in this paper an exponentially fitted method is used to numerically solve the one dimensional burgers equation the performance of the method is tested on the model involving moderately large reynolds numbers the obtained numerical results show that the method is efficient stable and reliable for solving burgers equation accurately even involving high reynolds numbers for which the exact solution fails copyright 2009 john wiley sons ltd

curves of degree two and ropes on a line their ideals and even liaison classes abstract in this paper we investigate degree two curves of arbitrary codimension this requires to study also ropes supported on a line after establishing several characterizations of ropes supported on a line we describe their homogeneous ideals and hartshorne rao modules in particular we characterize the arithmetically buchsbaum ropes then we describe the even gorenstein liaison classes of the non arithmetically buchsbaum non degenerate ropes generalizing a well know result of migliore on double lines of codimension two the result relies on the explicit description of certain arithmetically gorenstein curves with maximal tangent spaces at each point as a consequence we can decide if two curves of degree two belong to the same gorenstein liaison class finally we show as an application how ropes can be used in order to construct quasi extremal curves

a space time fully decoupled wavelet galerkin method for solving two dimensional burgers equations abstract a space time fully decoupled formulation for solving two dimensional burgers equations is proposed based on the coiflet type wavelet sampling approximation for a function defined on a bounded interval by applying a wavelet galerkin approach for spatial discretization nonlinear partial differential equations are first transformed into a system of ordinary differential equations in which all matrices are completely independent of time and never need to be updated in the time integration finally the mixed explicit implicit scheme is employed to solve the resulting semi discretization system by numerically studying three widely considered test problems results demonstrate that the proposed method has a much better accuracy and a faster convergence rate than many existing numerical methods most importantly the study also indicates that the present wavelet method is capable of solving the two dimensional burgers equation at high reynolds numbers

pareto optimal structures producing resonances of minimal decay under l1 type constraints abstract resonances optimization is studied under the constraint b 1 m on the nonnegative function b l 1 0 l representing the resonator structure the problem is to design for a given frequency r a structure that generates a resonance on the line i r with minimal possible decay rate im we generalize the problem replacing b by a nonnegative measure and show that optimal measures consist of finite number of point masses this yields non existence of optimizers for the problem over absolutely continuous measures we derive restrictions on optimal masses and their positions this reduces the original infinitely dimensional problem to optimization over four real parameters for low frequencies we explicitly find optimizers the technique is based on the two parameter perturbation method and the notion of local boundary point which is introduced as a generalization of local extrema to vector optimization problems special attention is paid to multiple and non differentiable resonances

an algorithm based on exponential modified cubic b spline differential quadrature method for nonlinear burgers equation in this paper the authors developed a new differential quadrature method exponential modified cubic b spline differential quadrature method expo mcb dqm by using exponential modified cubic b spline functions as test functions in the traditional differential quadrature method 32 the new method is tested on one and two dimensional nonlinear burgers equations to check the efficiency and accuracy of the proposed method five numerical problems have been considered the numerical results of the method are compared with some existing methods and found that the proposed numerical method produces more accurate results than existing methods stability analysis of the algorithm is also done by using matrix stability analysis method

dynamics in spectral solutions of burgers equation for low values of the viscosity coefficient burgers equation can develop sharp discontinuities which are difficult to simulate in a computer oscillations can occur by discretization through spectral collocation methods due to gibbs phenomena under a dynamic point of view these instabilities are related to bifurcations arising to the discretized equation for different values of discretized points herein a study is performed of the dynamics and bifurcations occurring in the spectral solutions of burgers equation with symmetry several phenomena are observed from limit cycles strange attractors to the presence of bistability with two periodic attractors with a periodic attractor and a strange attractor and with two strange attractors also other stable equilibrium points can occur diverse from the ones corresponding to the solution of burgers equation

a new low order least squares nonconforming characteristics mixed finite element method for burgers equation in this paper a new low order least squares nonconforming characteristics mixed finite element method mfem is considered for two dimensional burgers equation by use of two typical characters of the elements for approximating the velocity and flux variables a the consistency errors of the elements can be estimated as order o h 2 with respect to the mesh size h in the broken h 1 norm one order higher than their interpolation errors b the elements interpolation operators satisfy certain orthogonalities which lead to o h 2 order error estimates in l 2 norm and some superclose results the accuracy of corresponding variables in l 2 norm is improved by one order compared with other mfems in the existing literature it seems that the results provided herein have never been derived before