a new a posteriori error estimate for the morley element in this paper we present a posteriori error analysis for the nonconforming morley element of the fourth order elliptic equation we propose a new residual based a posteriori error estimator and prove its reliability and efficiency these results refine those of beirao da veiga et al numer math 106 165 179 2007 by dropping two edge jump terms in both the energy norm of the error and the estimator and those of wang and zhang local a priori and a posteriori error estimates of finite elements for biharmonic equation research report 13 2006 by showing the efficiency in the sense of verfurth a review of a posteriori error estimation and adaptive mesh refinement techniques wiley teubner new york 1996 moreover the normal component in the estimators of beirao da veiga et al numer math 106 165 179 2007 and wang and zhang local a priori and a posteriori error estimates of finite elements for biharmonic equation research report 13 2006 is dropped and therefore only the tangential component of the stress on each edge comes into the estimator in addition we generalize these results to three dimensional case

on the development of a high order compact scheme for exhibiting the switching and dissipative solution natures in the camassa holm equation in this paper a two step iterative solution algorithm for solving the camassa holm equation which involves only the first order derivative term is presented in each set of the u p and u m differential equations one is governed by the inviscid nonlinear convection reaction equation for the time evolving fluid velocity component along the horizontal direction the other equation is known as the inhomogeneous helmholtz equation the resulting reduction of differential order facilitates us to develop the flux discretization scheme in a stencil with comparatively fewer points for accurately predicting the unidirectional propagation of the shallow water wave the modified equation analysis for eliminating several leading discretization error terms and the fourier analysis for minimizing a particular type of wave like error are employed in this study the fifth order spatially accurate combined compact upwind scheme is developed in a three point stencil for approximating the first order derivative term for the purpose of retaining a long term accurate hamiltonian and multi symplectic geometric structures in camassa holm equation the time integrator or time stepping scheme chosen in this study should conserve symplecticity another main emphasis of conducting the present calculation of camassa holm equation is to shed light on the conservation of hamiltonians up to the time before wave breaking we also intended to elucidate the switching scenario by virtue of the peakon peakon interaction problem and the dissipative scenario after the time of head on collision in the peakon antipeakon interaction problem

sov approach for integrable quantum models associated with general representations on spin 1 2 chains of the 8 vertex reflection algebra the analysis of the transfer matrices associated with the most general representations of the 8 vertex reflection algebra on spin 1 2 chains is here implemented by introducing a quantum separation of variables sov method which generalizes to these integrable quantum models the method first introduced by sklyanin for representations reproducing in their homogeneous limits the open xyz spin 1 2 quantum chains with the most general integrable boundary conditions we explicitly construct representations of the 8 vertex reflection algebras for which the transfer matrix spectral problem is separated then in these sov representations we get the complete characterization of the transfer matrix spectrum eigenvalues and eigenstates and its non degeneracy moreover we present the first fundamental step toward the characterization of the dynamics of these models by deriving determinant formulae for the matrix elements of the identity on separated states which particularly apply to transfer matrix eigenstates a comparison of our analysis of the 8 vertex reflection algebra with that of niccoli g 2012 j stat mech p10025 faldella s et al 2014 j stat mech p01011 for the 6 vertex leads to an interesting remark in that there is a profound similarity in both the characterization of the spectral problems and the scalar products which exists for these two different realizations of the reflection algebra once they are described by the sov method as will be shown in a future publication this remarkable similarity will be the basis of a simultaneous determination of the form factors of local operators of integrable quantum models associated with general reflection algebra representations of both 8 vertex and 6 vertex type

sixth order compact scheme combined with multigrid method and extrapolation technique for 2d poisson equation we develop a sixth order finite difference discretization strategy to solve the two dimensional poisson equation which is based on the fourth order compact discretization multigrid method richardson extrapolation technique and an operator based interpolation scheme we use multigrid v cycle procedure to build our multiscale multigrid algorithm which is similar to the full multigrid method fmg the multigrid computation yields fourth order accurate solution on both the fine grid and the coarse grid a sixth order accurate coarse grid solution is computed by using the richardson extrapolation technique then we apply our operator based interpolation scheme to compute sixth order accurate solution on the fine grid numerical experiments are conducted to show the solution accuracy and the computational efficiency of our new method compared to sun zhang s sixth order richardson extrapolation compact rec discretization strategy using alternating direction implicit adi method and the standard fourth order compact difference foc scheme using a multigrid method

a compact finite difference scheme on a non equidistant mesh abstract a fourth order compact finite difference scheme is presented for second order partial differential equations the scheme is derived for a one dimensional non equidistant mesh which makes it particularly useful in problems with sharp boundary layers the inclusion of general boundary conditions does not reduce the order of the scheme for the boundary points the scheme is tested on a representative model equation we shortly discuss its stability in the simplest time dependent heat flow problem and its use in more dimensions

a robust prolongation operator for non nested finite element methods the paper is to present a general framework for designing a prolongation operator for general nonnested finite element methods the main idea is to take the inclusion interpolation in the unrefined region that consists of all unrefined elements and the canonical interpolation of the quasi interpolation in the quasi strict refined region that consists of children elements which are not neighbors of the unrefined region and define a careful mixture in the small layer between these two regions this leads to a robust prolongation operator that has both local projection and uniform boundedness properties for most of nonnested finite element methods in literature

new a posteriori error estimate and quasi optimal convergence of the adaptive nonconforming wilson element in this paper we establish the quasi optimal convergence of the adaptive nonconforming wilson element on the rectangular mesh the main ingredients are a new a posteriori error estimator and a crucial observation that there is some special orthogonality between the conforming part and the nonconforming part in the energy inner product which helps us to show the quasi orthogonality and the discrete reliability finally we integrate these components in a usual way to achieve the quasi optimal convergence

a posteriori error estimates for nonconforming finite element methods computable a posteriori error bounds for a large class of nonconforming finite element methods are provided for a model poisson problem in two and three space dimensions besides a refined residual based a posteriori error estimate an averaging estimator is established and an l 2 estimate is included the a posteriori error estimates are reliable and efficient the proof of reliability relies on a helmholtz decomposition

quantum kz equation with q 1 and correlation functions of the xxz model in the gapless regime an integral solution to the quantum knizhnik zamolodchikov qkz equation with q 1 is presented upon specialization it leads to a conjectural formula for correlation functions of the xxz model in the gapless regime the validity of this conjecture is verified in special cases including the nearest neighbour correlator with an arbitrary coupling constant and general correlators in the xxx and xy limits

a type of adaptive c0 non conforming finite element method for the helmholtz transmission eigenvalue problem abstract the transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering in this paper for the helmholtz transmission eigenvalue problem in r d d 2 3 we discuss the a posteriori error estimates and adaptive algorithm of a type of c 0 non conforming finite elements including the morley zienkiewicz element and the modified zienkiewicz element we give the a posteriori error estimators for approximate eigenpair and prove their reliability and efficiency for eigenfunctions we also analyze their reliability for eigenvalues numerical results not only in r 2 but also in r 3 indicate that the a posteriori error estimators are sharp and the adaptive algorithm is efficient for computing real and complex transmission eigenvalues as expected

long time asymptotic solution structure of camassa holm equation subject to an initial condition with non zero reflection coefficient of the scattering data in this article we numerically revisit the long time solution behavior of the camassa holm equation ut uxxt 2ux 3uux 2uxuxx uuxxx the finite difference solution of this integrable equation is sought subject to the newly derived initial condition with delta function potential our underlying strategy of deriving a numerical phase accurate finite difference scheme in time domain is to reduce the numerical dispersion error through minimization of the derived discrepancy between the numerical and exact modified wavenumbers additionally to achieve the goal of conserving hamiltonians in the completely integrable equation of current interest a symplecticity preserving time stepping scheme is developed based on the solutions computed from the temporally symplecticity preserving and the spatially wavenumber preserving schemes the long time asymptotic ch solution characters can be accurately depicted in distinct regions of the space time domain featuring with their own quantitatively very different solution behaviors we also aim to numerically confirm that in the two transition zones their long time asymptotics can indeed be described in terms of the theoretically derived painleve transcendents another attempt of this study is to numerically exhibit a close connection between the presently predicted finite difference solution and the solution of the painleve ordinary differential equation of type ii in two different transition zones

derivative optimized empirical mode decomposition for the hilbert huang transform in the empirical mode decomposition emd for the hilbert huang transform hht a nonlinear and non stationary signal is adaptively decomposed by an hht into a series of intrinsic mode functions imfs with the lowest one as the trend at each step of the emd the low frequency component is mainly determined by the average of the upper envelope consisting of local maxima and the lower envelope consisting of local minima the high frequency component is the deviation of the signal relative to the low frequency component the fact that no local maximum and minimum can be determined at the two end points leads to detrend uncertainty and in turn causes uncertainty in the hht to reduce such uncertainty hermitian polynomials are used to obtain the upper and lower envelopes with the first derivatives at the two end points q l q r as parameters which are optimally determined on the base of minimum temporal variability of the low frequency component in the each step of the decomposition this well posed mathematical system is called the derivative optimized emd demd with the demd the end effect and detrend uncertainty are drastically reduced and scales are separated naturally without any a priori subjective selection criterion

the adaptive ciarlet raviart mixed method for biharmonic problems with simply supported boundary condition abstract in this paper we study the adaptive fashion of the ciarlet raviart mixed method for biharmonic equation eigenvalue problem with simply supported boundary condition in r d we propose an a posteriori error indicator of the ciarlet raviart approximate solution for the biharmonic equation and an a posteriori error indicator of the ciarlet raviart approximate eigenfuction and prove the reliability and efficiency of the indicators we also give an a posteriori error indicator for the approximate eigenvalue and prove its reliability we design an adaptive ciarlet raviart mixed method with piecewise polynomials of degree less than or equal to m and numerical experiments show that numerical eigenvalues obtained by the method can achieve the optimal convergence order o d o f 2 m d d 2 m 2 3 d 3 m 3

solution of the bethe ansatz equations with complex roots for finite size the spin s1 isotropic and anisotropic chains the bethe ansatz equations for spin s s or 1 integrable vertex models and magnetic chains where the ground state is formed by complex roots are investigated for finite size n it is shown that the finite size corrections to the imaginary parts of the roots bethe strings for n 1 are given by alpha m n sigma eta where eta is the real part of the roots sigma eta is the density of the real parts and m is the index of a root within a string the constants alpha m are determined by a set of algebraic equations and are given explicitly by alpha m 1 pi ln cos 1 2 pi s m 1 s 1 cos 1 2 pi s m s 1 for the best known s 1 case alpha 0 ln 2 2 pi these results are found through a generalisation of the euler maclaurin formula including nonanalytic contributions in n 1 which turn out to be essential in the solution of the present problem

essentially compact schemes for unsteady viscous incompressible flows a new fourth order accurate finite difference scheme for the computation of unsteady viscous incompressible flows is introduced the scheme is based on the vorticity stream function formulation it is essentially compact and has the nice features of a compact scheme with regard to the treatment of boundary conditions it is also very efficient at every time step or runge kutta stage only two poisson like equations have to be solved the poisson like equations are amenable to standard fast poisson solvers usually designed for second order schemes detailed comparison with the second order scheme shows the clear superiority of this new fourth order scheme in resolving both the boundary layers and the gross features of the flow this efficient fourth order scheme also made it possible to compute the driven cavity flow at reynolds number 106on a 10242grid at a reasonable cost fourth order convergence is proved under mild regularity requirements this is the first such result to our knowledge

error reduction of the adaptive conforming and nonconforming finite element methods with red green refinement in this paper we analyze the convergence of the adaptive conforming and nonconforming p 1 finite element methods with red green refinement based on standard dorfler marking strategy since the mesh after refining is not nested into the one before the usual galerkin orthogonality or quasi orthogonality for newest vertex bisection does not hold for this case to overcome such a difficulty we develop some new quasi orthogonality instead under certain condition on the initial mesh condition a consequently we show convergence of the adaptive methods by establishing the reduction of some total errors to weaken the condition on the initial mesh we propose a modified red green refinement and prove the convergence of the associated adaptive methods under a much weaker condition on the initial mesh condition b furthermore we also develop an initial mesh generator which guarantee that all the interior triangles are equilateral triangles satisfy condition a and the other triangles containing at least one vertex on the boundary satisfy condition b

combined compact difference scheme for the time fractional convection diffusion equation with variable coefficients combined compact finite difference scheme for solving the time fractional convection diffusion reaction equation is given the primary unknown and its derivative are solved simultaneously the coefficients of the fractional partial differential equation are variable both dirichlet and robin boundary conditions are discussed and the algorithm gives high order convergence results fourth order combined compact finite difference scheme is given for solving the time fractional convection diffusion reaction equation with variable coefficients we introduce the flux as a new variable and transform the original equation into a system of two equations compact difference is used as a high order approximation for spatial derivatives of integer order in the coupled partial differential equations the caputo fractional derivative is discretized by a high order approximation both dirichlet and robin boundary conditions are discussed convergence analysis is given for the problem of integer order with constant coefficients under some assumption numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

spectral compact difference hybrid computation of passive scalar in isotropic turbulence an accurate and efficient hybrid numerical method is developed for direct numerical simulation of passive scalar in homogeneous turbulence for the schmidt number 1 and 50 the hybrid method uses the standard fourier spectral method for the incompressible navier stokes equation and the combined compact difference scheme for the passive scalar transport equation accuracy of the method is carefully examined by comparing with the full spectral method regarding the spectra probability density function field structure of the passive scalar and is found to be very satisfactory the computational time for the hybrid method is decreased by 26 for the schmidt number 1 when compared to the full spectral method and by 77 for the schmidt number 50 when the number of grid points for the velocity field is reduced under the scale separation

error estimators for nonconforming finite element approximations of the stokes problem in this paper we define and analyze a posteriori error estimators for nonconforming approximations of the stokes equations we prove that these estimators are equivalent to an appropriate norm of the error for the case of piecewise linear elements we define two estimators both of them are easy to compute but the second is simpler because it can be computed using only the right hand side and the approximate velocity we show how the first estimator can be generalized to higher order elements finally we present several numerical examples in which one of our estimators is used for adaptive refinement

adaptive finite element methods with convergence rates adaptive finite element methods for numerically solving elliptic equations are used often in practice only recently 12 17 have these methods been shown to converge however this convergence analysis says nothing about the rates of convergence of these methods and therefore does in principle not guarantee yet any numerical advantages of adaptive strategies versus non adaptive strategies the present paper modifies the adaptive method of morin nochetto and siebert 17 for solving the laplace equation with piecewise linear elements on domains in 2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm which is equivalent to the h1 norm namely it is shown that whenever s 0 and the solution u is such that for each n 1 it can be approximated to accuracy o n s in the energy norm by a continuous piecewise linear function on a triangulation with n cells using complete knowledge of u then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process moreover the number of arithmetic computations in the proposed method is also of order o n for each n 1 the construction and analysis of this adaptive method relies on the theory of nonlinear approximation

a unifying theory of a posteriori error control for nonconforming finite element methods residual based a posteriori error estimates were derived within one unifying framework for lowest order conforming nonconforming and mixed finite element schemes in carstensen numer math 100 617 637 2005 therein the key assumption is that the conforming first order finite element space v h c annulates the linear and bounded residual l written v h c subseteq rm ker ell that excludes particular nonconforming finite element methods ncfems on parallelograms in that v h c not subset rm ker ell the present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates also for mortar and discontinuous galerkin methods the key assumption is the existence of some bounded linear operator pi v h c rightarrow v h nc with some elementary properties it is conjectured that the more general hypothesis h1 h3 can be established for all known ncfems applications on various nonstandard finite element schemes for the laplace stokes and navier lame equations illustrate the presented unifying theory of a posteriori error control for ncfem

a unifying theory of a posteriori finite element error control residual based a posteriori error estimates are derived within a unified setting for lowest order conforming nonconforming and mixed finite element schemes the various residuals are identified for all techniques and problems as the operator norm l of a linear functional of the form in the variable of a sobolev space v the main assumption is that the first order finite element space is included in the kernel ker l of l as a consequence any residual estimator that is a computable bound of l can be used within the proposed frame without further analysis for nonconforming or mixed fe schemes applications are given for the laplace stokes and navier lame equations

three point combined compact difference schemes for time fractional advection diffusion equations with smooth solutions a ccd method with the l1 formula is proposed for solving time fractional advection diffusion equations the method is globally at least fifth order accurate in space and 2 th order accurate in time the unconditional stability is proved by means of fourier analysis numerical experiments are conducted to demonstrate the efficiency the present method preserves the higher order accuracy for advection dominated problems with the smooth solutions in this paper we study the numerical solution for the time fractional advection diffusion equation tfade a high order accurate three point combined compact difference scheme with the l1 formula is proposed to solve a class of tfades the method is globally 2 th order accurate in time and at least fifth order accurate in space for the constant coefficient tfades subject to periodic boundary conditions where is the order of time fractional derivative in the governing equation the unconditional stability and the high order convergence are proved using the fourier analysis method the proposed method can be extended to treat the variable coefficient tfades subject to other local boundary conditions several numerical examples are computed to validate the numerical accuracy effectiveness and robustness of the present method for tfades with smooth solutions and different boundary conditions moreover the proposed algorithm preserves the higher order accuracy for advection dominated problems with the smooth solutions

the stability of numerical boundary treatments for compact high order finite difference schemes the stability characteristics of various compact fourth and sixth order spatial operators are assessed with the theory of gustafsson kreiss and sundstrom g k s for the semidiscrete initial boundary value problem these results are generalized to the fully discrete case with a recently developed theory of kreiss and wu in all cases favorable comparisons are obtained between g k s theory eigenvalue determination and numerical simulation the conventional definition of stability then is sharpened to include only those spatial discretizations that are asymptotically stable bounded left half plane eigenvalues many of the higher order schemes that are g k s stable are not asymptotically stable a series of compact fourth and sixth order schemes that are both asymptotically and g k s stable for the scalar case are then developed

a conservative phase field method for solving incompressible two phase flows in this paper a conservative phase field method based on the work of sun and beckermann y sun c beckermann sharp interface tracking using the phase field equation j comput phys 220 2007 626 653 for solving the two and three dimensional two phase incompressible navier stokes equations is proposed the present method can preserve the total mass as the cahn hilliard equation but the calculation and implementation are much simpler than that the dispersion relation preserving schemes are utilized for the advection terms while the helmholtz smoother is applied to compute the surface tension force term to verify the proposed method several benchmarks are examined and shown to have good agreements with previous results it also shows that the satisfactions of mass conservations are guaranteed

quasi optimal convergence rate for an adaptive hybridizable c 0 discontinuous galerkin method for kirchhoff plates in this paper we present an adaptive hybridizable c 0 discontinuous galerkin hcdg method for kirchhoff plates a reliable and efficient a posteriori error estimator is produced for this hcdg method quasi orthogonality and discrete reliability are established with the help of a postprocessed bending moment and the discrete helmholtz decomposition based on these the contraction property between two consecutive loops and complexity of the adaptive hcdg method are studied thoroughly the key points in our analysis are a postprocessed normal normal continuous bending moment from the hcdg method solution and a lifting of jump residuals from inter element boundaries to element interiors

a new error analysis of nonconforming bergan s energy orthogonal element for the extended fisher kolmogorov equation abstract a nonconforming finite element method fem for short is proposed and analyzed for the fourth order extended fisher kolmogorov efk for short equation by employing the bergan s energy orthogonal plate element because the shape function and its first derivatives of this element are discontinuous at the element s vertices which is quite different from the conventional finite elements used in the existing literature a series of novel approaches including some a priori bounds and interpolation function splitting are developed to present a new error analysis for deriving optimal estimates of order o h in energy norm for both the semi discrete and backward euler fully discrete schemes at last numerical experiments are also provided to verify the theoretical analysis

highly accurate compact mixed methods for two point boundary value problems finite difference method and finite element method are widely used partly for their simplicity though these methods can obtain first order or second order of accuracy in this paper we give two highly accurate fourth order and sixth order of accuracy respectively while still quite simple schemes for two point boundary value problems we call them compact mixed methods since they can obtain numerical solutions for unknown function and its first derivative simultaneously to yield best numerical solutions we combine our compact mixed schemes with a direct solver and an iterative solver then we employ fourier method to analyze differencing errors shows that our compact mixed formulae are closer to the true wavenumber numerical experiments show compact mixed schemes are very efficient and highly accurate techniques

efficient and accurate numerical simulation of acoustic wave propagation in a 2d heterogeneous media abstract in this paper a compact fourth order finite difference scheme is derived to solve the 2d acoustic wave equation in heterogenous media the pade approximation is used to obtain fourth order accuracy in both temporal and spatial dimensions and the alternating direction implicit adi technique is used to reduce the computational cost due to the non constant wave velocity the conventional adi method is hard to implement as the algebraic manipulation cannot be used here a novel numerical strategy is proposed in this work so that the compact scheme still maintains fourth order accuracy in time and space the fourth order convergence order was firstly proved by theoretical error analysis then was confirmed by numerical examples it was shown that the proposed method is conditionally stable with a courant friedrichs lewy cfl condition that is comparable to other existing finite difference schemes several numerical examples were solved to demonstrate the efficiency and accuracy of the new algorithm

finite size effects and 2 string deviations in the spin 1 xxz chains in an earlier work suzuki 2004 j phys a math gen 37 11957 proposed a set of nonlinear integral equations nlie to describe the excited state spectrum of the integrable spin 1 xxz chain in its repulsive regime in this paper we extend his equations for the attractive regime of the model and calculate analytically the conformal spectrum of the spin chain we also discuss the typical root configurations of the thermodynamic limit as well as the 2 string deviations of certain excited states of the model special objects appearing in the nlie are also treated with special care

accuracy study of the ido scheme by fourier analysis the numerical accuracy of the interpolated differential operator ido scheme is studied with fourier analysis for the solutions of partial differential equations pdes advection diffusion and poisson equations the ido scheme solves governing equations not only for physical variable but also for first order spatial derivative spatial discretizations are based on hermite interpolation functions with both of them in the fourier analysis for the ido scheme the fourier coefficients of the physical variable and the first order derivative are coupled by the equations derived from the governing equations the analysis shows the ido scheme resolves all the wavenumbers with higher accuracy than the fourth order finite difference fd and compact difference cd schemes for advection equation in particular for high wavenumbers the accuracy is superior to that of the sixth order combined compact difference ccd scheme the diffusion and poisson equations are also more accurately solved in comparison with the fd and cd schemes these results show that the ido scheme guarantees highly resolved solutions for all the terms of fluid flow equations

a differentially interpolated direct forcing immersed boundary method for predicting incompressible navier stokes equations in time varying complex geometries a dispersion relation preserving dual compact scheme developed in cartesian grids is applied together with the immersed boundary method to solve the flow equations in irregular and time varying domains the artificial momentum forcing term applied at certain points in cells containing fluid and solid allows an imposition of velocity condition to account for the motion of solid body we develop in this study a differential based interpolation scheme which can be easily extended to three dimensional simulation the results simulated from the proposed immersed boundary method agree well with other numerical and experimental results for the chosen benchmark problems the accuracy and fidelity of the ib flow solver developed to predict flows with irregular boundaries are therefore demonstrated

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

on solutions of the camassa holm equation the techniques that have been developed for the application of the inverse scattering transform method to the solution of the camassa holm equation principally by constantin are implemented we use this approach first to represent the known solitary wave solution in a simple parametric form and then second to obtain the general two and three soliton solutions these latter two solutions require rather extensive use of mathematical packages mathematica and maple in order to complete the construction of solutions a number of examples are presented the phase shifts evident after an interaction are found and the special limit that recovers the peakon solutions is discussed

finite temperature correlation functions from discrete functional equations we present a new approach to the static finite temperature correlation functions of the heisenberg chain based on functional equations an inhomogeneous generalization of the n site density operator is considered the lattice path integral formulation with a finite but arbitrary trotter number allows us to derive a set of discrete functional equations with respect to the spectral parameters we show that these equations yield a unique characterization of the density operator our functional equations are a discrete version of the reduced q knizhnik zamolodchikov equations which played a central role in the study of the zero temperature case as a natural result and independent of the arguments given by jimbo et al 2009 j phys a math theor 42 304018 we prove that the inhomogeneous finite temperature correlation functions have the same remarkable structure as for zero temperature they are a sum of products of nearest neighbour correlators

exactly solvable models of interacting spin s particles in one dimension we consider the eigenspectrum solution of a many body problem of interacting spin s particles that can be solvable within the generalized bethe ansatz method we assume that the interactions are encoded in terms of an arbitrary u 1 invariant factorizable s matrix the exact solution of the spin part is based on a unified formulation of the quantum inverse scattering method for an arbitrary 2s 1 dimensional monodromy matrix the respective eigenstates are shown to be given in terms of 2s creation fields by a general new recurrence relation this allows us to derive the spectrum and the respective bethe ansatz equations

a new error analysis for discontinuous finite element methods for linear elliptic problems the standard a priori error analysis of discontinuous galerkin methods requires additional regularity on the solution of the elliptic boundary value problem in order to justify the galerkin orthogonality and to handle the normal derivative on element interfaces that appear in the discrete energy norm in this paper a new error analysis of discontinuous galerkin methods is developed using only the h k weak formulation of a boundary value problem of order 2k this is accomplished by replacing the galerkin orthogonality with estimates borrowed from a posteriori error analysis and by using a discrete energy norm that is well defined for functions in h k

a quadrilateral morley element for biharmonic equations in this paper we propose a morley type finite element for quadrilateral meshes to solve biharmonic problems for each quadrilateral q the finite element space is defined by the span of p 2 q plus two functions in p 3 q each of the cubic polynomials is the product of a pair of equations of opposite edges and the equation of the bimedian between them the degrees of freedom consist of the values at vertices and integrals of normal derivatives over edges optimal orders of convergence are proved both in discrete h 2 and h 1 seminorms several numerical tests confirm the convergence analysis

a posteriori error estimates for nonconforming finite element methods for fourth order problems on rectangles the a posteriori error analysis of conforming finite element discretisations of the biharmonic problem for plates is well established but nonconforming discretisations are more easy to implement in practice the a posteriori error analysis for the morley plate element appears very particular because two edge contributions from an integration by parts vanish simultaneously this crucial property is lacking for popular rectangular nonconforming finite element schemes like the nonconforming rectangular morley finite element the incomplete biquadratic finite element and the adini finite element this paper introduces a novel methodology and utilises some conforming discrete space on macro elements to prove reliability and efficiency of an explicit residual based a posteriori error estimator an application to the morley triangular finite element shows the surprising result that all averaging techniques yield reliable error bounds numerical experiments confirm the reliability and efficiency for the established a posteriori error control on uniform and graded tensor product meshes

vorticity boundary condition and related issues for finite difference schemes this paper discusses three basic issues related to the design of finite difference schemes forunsteadyviscous incompressible flows using vorticity formulations the boundary condition for vorticity an efficient time stepping procedure and the relation between these schemes and the ones based on velocity pressure formulation we show that many of the newly developed global vorticity boundary conditions can actually be written as some local formulas derived earlier we also show that if we couple a standard centered difference scheme with third or fourth order explicit runge kutta methods the resulting schemes havenocell reynolds number constraints for high reynolds number flows these schemes are stable under the cfl condition given by the convective terms finally we show that the classical mac scheme is the same as thom s formula coupled with second order centered differences in the interior in the sense that one can define discrete vorticity in a natural way for the mac scheme and get the same values as the ones computed from thom s formula we use this to derive an efficient fourth order runge kutta time discretization for the mac scheme from the one for thom s formula we present numerical results for driven cavity flow at high reynolds number 105

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

the compact and crank nicolson adi schemes for two dimensional semilinear multidelay parabolic equations this paper deals with numerical solutions of initial boundary value problems of the two dimensional semilinear multidelay parabolic equations two types of alternating direction implicit adi schemes are suggested the unique solvability convergence and unconditional stability of the schemes are analyzed and hence the corresponding criteria are established especially by using the discrete energy method it is proven that the compact adi scheme can attain second order accuracy in time and fourth order accuracy in space and the crank nicolson adi scheme has second order accuracy in both time and space numerical experiments are performed to verify the efficiency and accuracy of the both schemes

optimal adaptive nonconforming fem for the stokes problem this paper presents an optimal nonconforming adaptive finite element algorithm and proves its quasi optimal complexity for the stokes equations with respect to natural approximation classes the proof does not explicitly involve the pressure variable and follows from a novel discrete helmholtz decomposition of deviatoric functions

robust a posteriori error estimation for the nonconforming fortin soulie finite element approximation we obtain a computable a posteriori error bound on the broken energy norm of the error in the fortin soulie finite element approximation of a linear second order elliptic problem with variable permeability this bound is shown to be efficient in the sense that it also provides a lower bound for the broken energy norm of the error up to a constant and higher order data oscillation terms the estimator is completely free of unknown constants and provides a guaranteed numerical bound on the error

an unconditionally stable linearized ccdadi method for generalized nonlinear schrdinger equations with variable coefficients in two and three dimensions in this paper we propose a three level linearly implicit combined compact difference method ccd together with alternating direction implicit method adi for solving the generalized nonlinear schrdinger equation nlse with variable coefficients in two and three dimensions the method is sixth order accurate in space variable and second order accurate in time variable fourier analysis shows that the method is unconditionally stable comparing to the nonlinear ccdpradi scheme for solving the 2d cubic nlse with constant coefficients li etal 2015 current method is a linear scheme which generally requires much less computational cost moreover current method can handle 3d problems with variable coefficients naturally finally numerical results for both 2d and 3d cases are presented to illustrate the advantages of the proposed method

a three point combined compact difference scheme a new three point combined compact difference ccd scheme is developed for numerical models the major features of the ccd scheme are three point implicit sixth order accuracy and inclusion of boundary values due to its combination of the first and second derivatives the ccd scheme becomes more compact and more accurate than normal compact difference schemes the efficient twin tridiagonal for calculating derivatives and triple tridiagonal for solving partial difference equation with the ccd scheme methods are also presented besides the ccd scheme has sixth order accuracy at periodic boundaries and fifth order accuracy at nonperiodic boundaries the possibility of extending to a three point eighth order scheme is also included

a new compact difference scheme for second derivative in non uniform grid expressed in self adjoint form a single parameter family of self adjoint compact difference sacd schemes is developed for discretizing the laplacian operator in self adjoint form developed implicit scheme is formally second order accurate with respect to truncation error with a free parameter a which helps control the numerical properties in the spectral plane the sacd scheme is analyzed in the spectral plane for its resolution properties for both periodic and non periodic problems using the matrix spectral analysis t k sengupta g ganeriwal s de analysis of central and upwind schemes j comput phys 192 2 2003 677 694 the major objective here is to identify the advantages of the new scheme over the traditional explicit second order cd2 scheme in discretizing the laplacian operator in self adjoint form this appears in navier stokes equation and in other transport equations and boundary value problems bvp expressed in orthogonal co ordinate systems either in physical or in transformed plane we also compare the developed method with the higher order compact schemes for non uniform grids to demonstrate the accuracy of sacd scheme we have tested it for i bi directional wave propagation problem given by the second order wave equation and ii an elliptic bvp as in the stommel ocean model for the stream function these examples help infer the properties of sacd scheme when solving different types of partial differential equations most importantly the effects of grid stretching and choice of value of the free parameter a are investigated here we also compare the present implicit compact method with explicit compact method known as the higher order compact hoc method also the practical applications of the sacd scheme are explored by solving the navier stokes equation for the cases of a a flow inside a lid driven cavity and b the receptivity and instability of an external adverse pressure gradient flow over a flat plate in the former unsteadiness of the flow is captured and in the latter the receptivity of the flow is studied in causing flow instability by triggering tollmien schlichting waves the new scheme shows a marked improvement over the explicit scheme for low reynolds number steady flow in lid driven cavity whereas for the flow in the same geometry at higher reynolds numbers efficacy of the scheme is established by showing the formation of a triangular vortex and secondary vortical structures presented scheme is perfectly capable of expressing the diffusion operator accurately as shown via the capturing of instability waves inside the shear layer

fast computation in adaptive tree approximation adaptive methods of approximation arise in many settings including numerical methods for pdes and image processing they can usually be described by a tree which records the adaptive decisions this paper is concerned with the fast computation of near optimal trees based on n adaptive decisions the best tree based on n adaptive decisions could be found by examining all such possibilities however this is exponential in n and could be numerically prohibitive the main result of this paper is to show that it is possible to find near optimal trees using computations linear in n

convergence and optimality of adaptive mixed finite element methods the convergence and optimality of adaptive mixed finite element methods for the poisson equation are established in this paper the main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality a quasi orthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result this discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation

convergence and optimality of the adaptive morley element method this paper is devoted to the convergence and optimality analysis of the adaptive morley element method for the fourth order elliptic problem a new technique is developed to establish a quasi orthogonality which is crucial for the convergence analysis of the adaptive nonconforming method by introducing a new parameter dependent error estimator and further establishing a discrete reliability property sharp convergence and optimality estimates are then fully proved for the fourth order elliptic problem

framework for the a posteriori error analysis of nonconforming finite elements this paper establishes a unified framework for the a posteriori error analysis of a large class of nonconforming finite element methods the theory assures reliability and efficiency of explicit residual error estimates up to data oscillations under the conditions h1 h2 and applies to several nonconforming finite elements the crouzeix raviart triangle element the han parallelogram element the nonconforming rotated nr parallelogram element of rannacher and turek the constrained nr parallelogram element of hu and shi the p 1 element on parallelograms due to park and sheen and the dssy parallelogram element the theory is extended to include 1 irregular meshes with at most one hanging node per edge

conservative form of interpolated differential operator scheme for compressible and incompressible fluid dynamics the proposed scheme which is a conservative form of the interpolated differential operator scheme ido cf can provide high accurate solutions for both compressible and incompressible fluid equations spatial discretizations with fourth order accuracy are derived from interpolation functions locally constructed by both cell integrated values and point values these values are coupled and time integrated by solving fluid equations in the flux forms for the cell integrated values and in the derivative forms for the point values the ido cf scheme exactly conserves mass momentum and energy retaining the high resolution more than the non conservative form of the ido scheme a direct numerical simulation of turbulence is carried out with comparable accuracy to that of spectral methods benchmark tests of riemann problems and lid driven cavity flows show that the ido cf scheme is immensely promising in compressible and incompressible fluid dynamics studies

the completion of locally refined simplicial partitions created by bisection recently in found comput math 7 2 2007 245 269 we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations which is essentially the method from sinum 38 2000 466 488 by morin nochetto and siebert converges with the optimal rate the number of triangles n in the output partition of such a method is generally larger than the number m of triangles that in all intermediate partitions have been marked for bisection because additional bisections are needed to retain conforming meshes a key ingredient to our proof was a result from numer math 97 2004 219 268 by binev dahmen and devore saying that n n 0 cm for some absolute constant c where n 0 is the number of triangles from the initial partition that have never been bisected in this paper we extend this result to bisection algorithms of n simplices with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions

the multiple soliton solution of the camassa holm equation this paper refines johnso s implementation of constantin s method for solving the camassa holm equation for a multiple soliton solution an analytical formula for the q y and an explicit relation between x and y are found an algorithm of solving for u y is presented how to introduce time variable t into the solution is also clearly explained

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

correlation functions of the higher spin xxx chains using the algebraic bethe ansatz we consider the correlation functions of the integrable higher spin chains we apply a method recently developed for the spin heisenberg chain based on the solution of the quantum inverse problem we construct a representation for the correlation functions on a finite chain for arbitrary spin then we show how the string solutions of the bethe equations can be considered in the framework of this approach in the thermodynamic limit finally a multiple integral representation for the spin 1 zero temperature correlation functions is obtained in the thermodynamic limit

high order compact adi methods for parabolic equations in this paper we develop a sixth order compact scheme coupled with alternating direction implicit adi methods and apply it to parabolic equations in both 2 d and 3 d unconditional stability is proved for linear diffusion problems with periodic boundary conditions numerical examples supporting our theoretical analysis are provided

new error estimates of the morley element for the plate bending problems in this paper we derive some new error estimates of a nonconforming morley element for the plate bending problems our aim is fourfold first it shows that the consistency error of the morley element can be controlled by its corresponding approximation error up to an arbitrarily high order term especially which can be of arbitrary order provided that f on the right hand side is piecewise smooth enough thus the total energy norm is dominated alone by the approximation error second as a byproduct we derive the error estimate under the regularity assumption u h 2 s with s 0 1 which fills the gap in the a priori error estimate of the morley element with low regularity for s 0 1 2 third based on this new error bound a robust convergence is proved even if the solution is exactly in h 2 finally by a slight modification of the variational formulation a new error estimate is also derived for the case f h 1 the key role is played by some quasi interpolation operators and the weak continuous properties of morley finite element space the main ideas and results presented here are motivated by mao and shi 2010 and can be viewed as its extension to fourth order problems

an analytic treatment of finite size corrections in the spin 1 antiferromagnetic xxz chain it is known that the standard method for calculating finite size corrections in bethe ansatz solvable systems is not applicable to the takhtajan babujian model and its anisotropic xxz generalisation the authors develop a new analytic method explicitly avoiding root densities and associated problems nonlinear integral equations are derived whose solutions yield the correct central charges c 1 and c 3 2 for the spin 1 2 and spin 1 xxz chains respectively in the spin 1 case the authors obtain as a by product the finite size deviation of the bethe ansatz roots from the 2 string formation

an antiperiodic dynamical six vertex model i complete spectrum by sov matrix elements of the identity on separate states and connections to the periodic eight vertex model the spin 1 2 highest weight representations of the dynamical six vertex and the standard eight vertex yang baxter algebra on a finite chain are considered in this paper in particular the integrable quantum models associated with the corresponding transfer matrices under antiperiodic boundary conditions for the dynamical six vertex case and periodic boundary conditions for the eightvertexcaseareanalyzedhere fortheantiperiodicdynamicalsix vertextransfer matrix defined on chains with an odd number of sites we adapt sklyanin s quantum separation of variable sov method and explicitly construct the sov representations from the original space of the representations in this way we provide the complete characterization of the eigenvalues and the eigenstates proving also the simplicity of its spectrum moreover we characterize the matrix elements of the identity on separated states of this model by determinant formulae the matrices entering these determinants have elements given by sums over the sov spectrum of the product of the coefficients of the separate states this sov analysis is done without any need to be reduced to the case of the so called elliptic roots of unit and the results derived here define the required setup to extend to the dynamical six vertex model the approach recently developed by the author and collaborators to compute the form factors of the local operators in the sov framework for the periodic eight vertex transfer matrix we prove that its eigenvalues have to satisfy a fixed system of equations in the case of a chain with an odd number of sites this system of equations is the same entering in the sov characterization of the antiperiodic dynamical six vertex transfer matrix spectrum this implies that the set of the periodic eight vertex eigenvalues is contained in the set of the antiperiodic dynamical six vertex eigenvalues a criterion is introduced to find simultaneous eigenvalues of these two transfer matrices and associate with any of such eigenvalues one nonzero eigenstate of the periodic eight vertex transfer matrix by using the sov results moreover a preliminary discussion

compact θ method for the generalized delay diffusion equation abstract the generalized diffusion equation with a delay has inherent complex nature because its analytical solutions are hard to obtain therefore one has to seek numerical methods especially the high order accurate ones for their approximate solutions in this paper we have established the results of the numerical asymptotic stability of the compact method for the generalized delay diffusion equation it shows that the compact method is asymptotically stable if and only if k r t h 2 10 cos h 12 1 cos h 1 2 for 0 1 2 and is unconditionally asymptotically stable for 1 2 1 respectively the convergent results in the maximum norm are studied according to the consistency analysis and lax theorem in the end a series of numerical tests on stability and convergence are carried out to support our theoretical results

a compact high order alternating direction implicit method for three dimensional acoustic wave equation with variable coefficient abstract efficient and accurate numerical simulation of seismic wave propagation is important in various geophysical applications such as seismic full waveform inversion fwi problem however due to the large size of the physical domain and requirement on low numerical dispersion many existing numerical methods are inefficient for numerical modelling of seismic wave propagation in an heterogeneous media despite the great efforts that have been devoted during the past decades it still remains a challenging task in the development of efficient and accurate finite difference method for multi dimensional acoustic wave equation with variable velocity in this paper we proposed a pade approximation based finite difference scheme for solving the acoustic wave equation in three dimensional heterogeneous media the new method is obtained by combining the pade approximation and a novel algebraic manipulation the efficiency of the new algorithm is further improved through the alternative directional implicit adi method the stability of the new algorithm has been theoretically proved by energy method the new method is conditionally stable with a better courant friedrichs lewy condition cfl condition which has been verified numerically extensive numerical examples have been solved which demonstrated that the new method is accurate efficient and stable

a three point sixth order nonuniform combined compact difference scheme a three point nonuniform combined compact difference nccd scheme with sixth order accuracy is proposed for numerical models the nccd scheme is a generalization of the previously proposed combined compact difference ccd scheme with a global hermitan polynomial spline and has major improved features such as error and computational cpu time reduction for nonperiodic boundaries additional sixth or fifth order nonuniform boundary conditions are proposed the nccd scheme with either sixth or fifth order additional boundary conditions can increase the accuracy and decrease the cpu time about 1 2 orders of magnitude compared to the ccd scheme

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

form factors and complete spectrum of xxx antiperiodic higher spin chains by quantum separation of variables the antiperiodic transfer matrices associated to higher spin representations of the rational 6 vertex yang baxter algebra are analyzed by generalizing the approach introduced recently in the framework of sklyanin s quantum separation of variables sov for cyclic representations spin 1 2 highest weight representations and also for spin 1 2 representations of the 6 vertex reflection algebra such sov approach allow us to derive exactly results which represent complicate tasks for more traditional methods based on bethe ansatz and baxter q operator in particular we both prove the completeness of the sov characterization of the transfer matrix spectrum and its simplicity then the derived characterization of local operators by sklyanin s quantum separate variables and the expression of the scalar products of separate states by determinant formulae allow us to compute the form factors of the local spin operators by one determinant formulae similar to those of the scalar products

global spectral analysis for convection diffusion reaction equation in one and two dimensions effects of numerical anti diffusion and dispersion abstract convection diffusion reaction cdr equation plays a central role in many disciplines of engineering science and finances as a consequence importance of analysis of numerical methods for the accurate solution of cdr equation has motivated the present research we have used the global spectral analysis to characterize all the three important physical processes in terms of the non dimensional numerical parameters namely the non dimensional wavenumber kh courant friedrich lewy cfl number n c the peclet number pe and the damkohler number da for the purpose of illustration we have focused on two space time discretization schemes known for accuracy and robustness the basic properties relate to numerical issues arising for the numerical amplification factor numerical diffusion coefficient numerical phase speed and numerical group velocity with the help of model one dimensional 1d and two dimensional 2d cdr equations we have reported the numerical property charts for the cases i when all the processes of convection diffusion and reaction are of same order with critical numerical behaviour enforcing low values of da for the 1d cdr equation studied here ii the 2d cdr equation considered is diffusion reaction dominated and as a consequence this enforces da to be larger we have thoroughly analyzed these cases to identify the essential roles of anti diffusion on the critical n c and pe values which in turn decides admissible space and time steps to be used with the discretization schemes the property charts have been used to calibrate the analysis with two model equations one of which has an exact solution for a 1d cdr equation and the second case for the 2d cdr equation has numerical solution available in the literature these cases help to identify the importance of such analysis in explaining the utility of the choice one can exercise in fixing the numerical parameters this also identifies and explains some hitherto unknown numerical problems for cdr equation and their alleviation techniques

optimized prefactored compact schemes the numerical simulation of aeroacoustic phenomena requires high order accurate numerical schemes with low dispersion and dissipation errors in this paper we describe a strategy for developing high order accurate prefactored compact schemes requiring very small stencil support these schemes require fewer boundary stencils and offer simpler boundary condition implementation than existing compact schemes the prefactorization strategy splits the central implicit schemes into forward and backward biased operators using fourier analysis we show it is possible to select the coefficients of the biased operators such that their dispersion characteristics match those of the original central compact scheme and their numerical wavenumbers have equal and opposite imaginary components this ensures that when the forward and backward stencils are added the original central compact scheme is recovered to extend the resolution characteristic of the schemes an optimization strategy is employed in which formal order of accuracy is sacrificed in preference to enhanced resolution characteristics across the range of wavenumbers realizable on a given mesh the resulting optimized schemes yield improved dispersion characteristics compared to the standard sixth and eighth order compact schemes making them more suitable for high resolution numerical simulations in gas dynamics and computational aeroacoustics the efficiency accuracy and convergence characteristics of the new optimized prefactored compact schemes are demonstrated by their application to several test problems

a discrete helmholtz decomposition with morley finite element functions and the optimality of adaptive finite element schemes the discrete reliability of a finite element method is a key ingredient to prove optimal convergence of an adaptive mesh refinement strategy and requires the interchange of a coarse triangulation and some arbitrary refinement of it one approach for this is the careful design of an intermediate triangulation with one level refinements and with the remaining difficulty to design some interpolation operator which maps a possibly nonconforming approximation into the finite element space based on the finer triangulation this paper enfolds the second possibility of some novel discrete helmholtz decomposition for the nonconforming morley finite element method this guarantees the optimality of a standard adaptive mesh refining algorithm for the biharmonic equation numerical examples illustrate the crucial dependence of the bulk parameter and the surprisingly short pre asymptotic range of the adaptive morley finite element method

multiple coarse grid acceleration for multiscale multigrid computation the multiscale multigrid method uses an iterative refinement procedure with the richardson extrapolation technique to obtain a higher order solution the computational cost for the iterative refinement procedure may be significant for some ill conditioned coefficient matrices in this paper we proposed an alternative strategy using multiple coarse grids to eliminate the iterative refinement procedure and thus accelerate the multiscale multigrid computation numerical investigations show that our multiple coarse grid computing strategy is more efficient and scalable than the iterative refinement procedure the multiscale multigrid method with multiple coarse grid strategy is used to solve two dimensional 2d poisson equation and convection diffusion equation but the idea can be used to solve other partial differential equations

numerical modelling of linear and nonlinear diffusion equations by compact finite difference method in this work accurate solutions to linear and nonlinear diffusion equations were introduced a combination of a sixth order compact finite difference scheme in space and a low storage third order total variation diminishing runge kutta scheme in time have been used for treatment of these equations the computed results with the use of this technique have been compared with the exact solution to show the accuracy of it here the approximate solution to the diffusion equations has been obtained easily and elegantly with neither transforming nor linearizing the equation the present method is seen to be a very good alternative method to some existing techniques for realistic problems

numerical solutions to a two component camassa holm equation abstract in the present paper a three step iterative algorithm for solving a two component camassa holm 2ch equation is presented in the first step the time dependent equation for the horizontal fluid velocity with nonlinear convection is solved then an inhomogeneous helmholtz equation is solved finally the equation for modeling the transport of density is solved in the third step the differential order of 2ch equation has been reduced in order to facilitate numerical scheme development in a comparatively smaller grid stencil in this study a fifth order spatially accurate upwinding combined compact difference scheme uccd5 which differs from that in sheu et al 2011 is developed in a four point grid stencil for approximating the first order derivative term for the purpose of retaining long time hamiltonians in the 2ch equation the time integrator or time stepping scheme chosen is symplectic various numerical experiments such as the single peakon peakon antipeakon interaction and dam break problems are conducted to illustrate the effectiveness of the proposed numerical method it is shown that both the hamiltonians and casimir functions are conserved well for all problems

a high order finite difference method with richardsonextrapolation for 3d convection diffusion equation in this paper we extend the sun and zhang s 24 work on high order finite difference method which is based on the richardson extrapolation technique and an operator interpolation scheme for the one and two dimensional steady convection diffusion equations to the three dimensional case firstly we employ a fourth order compact difference scheme to get the fourth order accurate solution on the fine and the coarse grids then we use the richardson extrapolation technique by combining the two approximate solutions to get a sixth order accurate solution on coarse grid finally we apply an operator interpolation scheme to achieve the sixth order accurate solution on the fine grid during this process we use alternating direction implicit adi method to solve the resulting linear systems numerical experiments are conducted to verify the accuracy and effectiveness of the present method

an inexpensive method for the evaluation of the solution of the lowest order raviart thomas mixed method for the model problem operatorname div a nabla u f we show how the approximate solution produced by the raviart thomas lecture notes in mathematics 606 springer verlag berlin 1977 mixed method of lowest degree can be obtained from the solution of the p 1 nonconforming galerkin method modified in a virtually cost free manner

a posteriori error estimates for the morley plate bending element a local a posteriori error indicator for the well known morley element for the kirchhoff plate bending problem is presented the error indicator is proven to be both reliable and efficient the technique applied is general and it is shown to have also other applications

convergence analysis of an adaptive nonconforming finite element method an adaptive nonconforming finite element method is developed and analyzed that provides an error reduction due to the refinement process and thus guarantees convergence of the nonconforming finite element approximations the analysis is carried out for the lowest order crouzeix raviart elements and leads to the linear convergence of an appropriate adaptive nonconforming finite element algorithm with respect to the number of refinement levels important tools in the convergence proof are a discrete local efficiency and a quasi orthogonality property the proof does neither require regularity of the solution nor uses duality arguments as a consequence on the data control no particular mesh design has to be monitored