approximation order of bivariate spline interpolation for arbitrary smoothness by using the algorithm of nurnberger and riessinger 1995 we construct hermite interpolation sets for spaces of bivariate splines sqr 1 of arbitrary smoothness defined on the uniform type triangulations it is shown that our hermite interpolation method yields optimal approximation order for q 3 5r 1 in order to prove this we use the concept of weak interpolation and arguments of birkhoff interpolation

on smooth multivariate spline functions in this paper the dimensions of bivariate spline spaces with simple cross cut grid partitions are determined and expressions of their basis functions are given consequently the closures of these spaces over all partitions of the same type can be determined a somewhat more detailed study on bivariate splines with rectangular grid partitions is included the results in this paper can be applied to problems on interpolation and approximation by bivariate spline functions

construction of local c 1 quartic spline elements for optimal order approximation this paper is concerned with a study of approximation order and construction of locally supported elements for the space s 4 1 of c 1 pp piecewise polynomial functions on an arbitrary triangulation of a connected polygonal domain in r 2 it is well known that even when a is a three directional mesh 1 the order of approximation of s 4 1 1 is only 4 not 5 the objective of this paper is two fold i a local clough tocher refinement procedure of an arbitrary triangulation a is introduced so as to yield the optimal fifth order of approximation where locality means that only a few isolated triangles need refinement and ii locally supported hermite elements are constructed to achieve the optimal order of approximation

the bezout number for linear piecewise algebraic curves a piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function this paper discusses the bezout number the maximum number of intersections between two linear piecewise algebraic curves whose intersections are finite on regular triangulations we give an upper bound of the bezout number for linear piecewise algebraic curves bn 1 0 1 0 d on the triangulation with an odd interior vertex for the triangulations which satisfy a vertex coloring condition we compute the exact value of the bezout number bn 1 0 1 0 d

a three level average implicit finite difference scheme to solve equation obtained by coupling the rosenau kdv equation and the rosenau rlw equation in the present work a mathematical model to obtain the solution of the nonlinear wave by coupling the rosenau kdv equation and the rosenau rlw equation is proposed the solution properties are also derived a numerical tool is applied to the model by using a three level average implicit finite difference technique the fundamental conservative properties of the equation are preserved by the presented numerical scheme and the existence and uniqueness of the numerical solution are proved moreover the convergence and stability of the numerical solution are also shown the new method give second order accurate in time and space thus the presented results can be constructed to demonstrate the viability of the new model

scattered data interpolation using c 2 supersplines of degree six we show how c2 supersplines of degree six can be used to interpolate hermite data at the vertices of a quadrangulation we also present error bounds that show that our method has full approximation order seven and compare its efficiency with other c2interpolation methods in the literature

on the dimension of bivariate spline spaces of smoothnessr and degreed 3r 1 we consider the well known spaces of bivariate piecewise polynomials of degreed defined over arbitrary triangulations of a polygonal domain and possessingr continuous derivatives globally to date dimension formulae for such spaces have been established only whend 3r 2 except for the special case wherer 1 andd 4 in this paper we establish dimension formulae for allr 1 andd 3r 1 for almost all triangulations

a deterministic model for the distribution of the stopping time in a stochastic equation and its numerical solution in this work we consider a stochastic differential equation that generalizes the well known paris equation from fracture of materials the model describes the propagation of cracks on solids and it includes a deterministic summand and a stochastic component in terms of a brownian motion the use of ito s stochastic integral gives an equivalent stochastic integral equation that is further generalized here we note that the probability distribution of the stopping time of the general model satisfies a deterministic diffusion advection partial differential equation for which the solution is known only in a reduced number of particular cases motivated by these analytical results we develop a fast finite difference method to approximate the distribution of the stopping time the method is an explicit exponential like technique that preserves the main features of a probability distribution namely the non negativity the boundedness from above by 1 as well as the spatial monotonicity moreover the method is a monotone technique that is also capable of preserving the temporal monotonicity of the approximations these properties of the proposed methodology are thoroughly established in the present manuscript a continuity condition of the numerical solutions in terms of the initial conditions and the temporal computational parameter is established also together with a limiting property of the methodology when the free parameter tends to infinity for comparison purposes we are providing an implicit and stable discretization of the mathematical model which has a second order of convergence but for which conditions that guarantee the positivity the boundedness and the monotonicity of approximations are not available the numerical simulations obtained with implementations of our techniques show that the explicit method is an efficient scheme that preserves the characteristics of interest non negativity boundedness from above by 1 and monotonicity and that the numerical approximations are in good agreement with the known exact solutions

a small time solutions for the korteweg de vries equation using spline approximation in this paper the one dimensional nonlinear korteweg de vries kdv equation is numerically solved using second order spline approximation the test problems concerning the propagation of a solution and two solution interaction are used to validate the proposal scheme and it is found to be both accurate and efficient at small times also it is shown that the second order spline approximation may be used effectively at small times when the exact solution of the kdv equation is not known

swapping edges of arbitrary triangulations to achieve the optimal order of approximation in the representation of scattered data by smooth pp piecewise polynomial functions perhaps the most important problem is to find an optimal triangulation of the given sample sites called vertices of course the notion of optimality depends on the desirable properties in the approximation or modeling problems in this paper we are concerned with optimal approximation order with respect to the given order r of smoothness and degree k of the polynomial pieces of the smooth pp functions we will only consider c1 pp approximation with r 1 and k 4 the main result in this paper is an efficient method for triangulating any finitely many arbitrarily scattered sample sites such that these sample sites are the only vertices of the triangulation and that for any discrete data given at these sample sites there is a c1 piecewise quartic polynomial on this triangulation that interpolates the given data with the fifth order of approximation

nonlocality distillation and postquantum theories with trivial communication complexity we first present a protocol for deterministically distilling nonlocality building upon a recent result of forster et al phys rev lett 102 120401 2009 our protocol which is optimal for two copy distillation works efficiently for a specific class of postquantum nonlocal boxes which we term correlated nonlocal boxes in the asymptotic limit all correlated nonlocal boxes are distilled to the maximally nonlocal box of popescu and rohrlich then taking advantage of a result of brassard et al phys rev lett 96 250401 2006 we show that all correlated nonlocal boxes make communication complexity trivial and therefore appear very unlikely to exist in nature astonishingly some of these nonlocal boxes are arbitrarily close to the set of classical correlations this result therefore gives new insight to the problem of why quantum nonlocality is limited

an explicit unconditionally stable variable time step method for one dimensional stefan problems modelling of heat conduction processes with phase changes benefits from the application of variable time step methods when the behaviour of the moving boundary is not known a priori due to convergence and stability constraints only implicit difference equations have been used with these methods implicit methods show a significant loss of accuracy and exhibit convergence difficulties when used for relatively slow or rapid moving boundary problems

approximation by cubic c 1 splines on arbitrary triangulations in this paper we present an efficient representation for bivariate piecewise cubicc 1 splines on arbitrary triangulations a numerical method is discussed for computing the dimension of the spaces 3 1 of these splines we consider subspaces ofs 3 1 satisfying certain boundary conditions some applications are given where piecewise cubicc 1 functions are used to solve interpolation problems and least squares approximation problems

interpolation by bivariate linear splines we give a characterization of lagrange interpolation sets for the spaces of continuous bivariate linear splines on regular triangulations the characterization is based on a complete description of the zero sets of such splines

a structure preserving bhattacharya method for nonlinear parabolic equations with fractional diffusion and advection abstract in this work we investigate a parabolic equation with nonlinear reaction and fractional diffusion and advection terms of the riesz type the model under investigation is a fractional generalization of the well known burgers fisher and burgers huxley models from population and fluid dynamics which are equations that admit positive bounded and monotone solutions some of them being traveling waves a variable step bhattacharya type finite difference scheme based on fractional centered differences is proposed to approximate the solutions of the parabolic partial differential equation the method is an explicit technique which under suitable parameter conditions is capable of preserving the positivity the boundedness and the monotonicity of the approximations moreover the method preserves the constant solutions of the fractional partial differential equation under investigation the properties of consistency stability and convergence of the technique are established thoroughly in this manuscript along with some a priori bounds for the numerical solutions some illustrative simulations and numerical comparisons against other numerical techniques are carried out in order to show that the method preserves these features of the approximations

a numerical method for oxygen diffusion and absorption in a sike cell oxygen diffusion in a sike cell with simultaneous absorption is an important problem and has a wide range of medical applications the problem mathematically formulated through two different stages the present paper concerns mainly on the second stage in which the injected oxygen through the cell starts absorbing the basic idea of the proposed numerical method is to form a double linear system of equations each of dimension 4x4 the first system is formed through applying first second third and fourth moments and using assumed profile for the concentration containing four unknowns functions of time numerical solution through a proposed scheme leads to the unknown functions the second system is formed through applying the boundary conditions given in addition to another assumed condition that is the concentration at x 0 is unknown function of time the results of the first system becomes an entry data for the second one leading to the concentration at the fixed surface x 0 the results obtained by the present method were compared with two different methods and the results gave a good agreement

higher order boundary element methods for transient diffusion problems part ii singular flux formulation despite the significant number of publications on boundary element methods bem for time dependent problems of heat diffusion there still remain issues that need to be addressed most importantly accuracy of the numerical modelling although very precise for steady state problems the common boundary element methods applied to transient problems do not yield highly accurate numerical solutions this paper investigates the reasons that prohibit achievement of a high level of accuracy for transient heat diffusion problems with continuous temperature and bounded heat flux solutions in order to greatly enhance the commonly used boundary element formulations we propose higher order time interpolation functions including quadratic and quartic approximations we show that the use of higher order time functions greatly reduces the numerical error concentrated in the corner regions and results in very good uniformity of the flux and temperature distributions along the boundaries for problems where uniform distributions are expected in order to highlight the importance of proper resolution both in time and space for the transient problems we consider one and two dimensional formulations in this paper high order boundary elements using quartic shape functions as well as high order bi quartic volume cells are used to attain mesh independent numerical solutions we consider four transient heat diffusion problems that possess exact solutions to investigate the convergence rate and accuracy of the higher order boundary element formulations a very high level of accuracy is possible for both one and two dimensional formulations additionally we show that the accuracy of a commercially available finite element code is far less than that of the boundary element methods for a given spatial and temporal discretization copyright 2002 john wiley sons ltd

a modified bhattacharya exponential method to approximate positive and bounded solutions of the burgers fisher equation in this work we investigate numerically the classical burgers fisher equation using a modified bhattacharya method the partial differential equation under investigation possesses nonnegative and bounded solutions and under some suitable parameter conditions these solutions are traveling waves it is well known that the use of the bhattacharya approach leads to the design of numerical techniques that are sensitive to zero solutions however in this manuscript we provide a correction of that technique in order to approximate solutions of the burgers fisher equation that are bounded in 0 1 the proposed methodology is explicit and we establish thoroughly the capability of the technique to preserve the non negativity the boundedness and the monotonicity of the numerical approximations as well as the constant solutions of the continuous model the new class of methods introduced in this work considers the presence of a free parameter and we show that this family tends to an explicit and standard discretization of the burgers fisher equation when the free parameter tends to infinity some simulations illustrate the main features of the method

hybrid laplace transform finite difference method for transient heat conduction problems the new method involving the combined use of the laplace transform and the finite difference method is applicable to the problem of time dependent heat flow systems the present method removes the time derivatives from the governing differential equation using the laplace transform and then solves the associated equation with the finite difference method the transformed temperature is inverted numerically by the method of honig and hirdes to obtain the result in the physical quantities the present results are compared in tables with exact solutions and those obtained from the combined use of the laplace transform and the finite element method it is found that the present method is reliable and efficient

a small time solutions for the korteweg de vries equation in this paper a heat balance integral hbi method is applied to the one dimensional non linear korteweg de vries kdv equation prescribed by appropriate homogenous boundary conditions and a set of initial conditions to obtain its approximate analytical solutions at small times it is shown that the hbi solutions obtained by the method may be used effectively at small times when the exact solution of the kdv equation is not known

an explicit conditionally stable finite difference equation for heat conduction problems this paper reports a new explicit and conditionally stable finite difference equation for heat conduction it predicts results with an accuracy comparable with or better than that obtainable by other methods stability of operation can be extended to any desired degree by subdividing the basic time step and increasing the number of nodes some existing difference equations are special cases of the new equation reported in this paper the new solution has been tested by calculating the response of a slab to transient and progressive waves whose analytical solutions are known

developments in bivariate spline interpolation the aim of this survey is to describe developments in the field of interpolation by bivariate splines we summarize results on the dimension and the approximation order of bivariate spline spaces and describe interpolation methods for these spaces moreover numerical examples are given

a new numerical algorithm for 2d moving boundary problems using a boundary element method phase change problems are of practical importance and can be found in a wide range of engineering applications in the present paper two proposed numerical algorithms are developed the first one is general for phase change problems while the second one is for ablation problems the boundary elements method is used as a mathematical tool in conjunction with the proposed algorithms two test examples were solved and the results agree with the physics of the problems

a hybrid numerical method for the kdv equation by finite difference and sinc collocation method abstract in this paper we propose a hybrid numerical method for the kdv equation more precisely we discretize the temporal derivative of kdv equation by a weighted scheme and treat the implicitly nonlinear term with the combination of finite difference and sinc collocation method the stability analysis is presented and numerical experiments illustrate the efficiency and stability of the proposed hybrid method the motions and the interactions of solitary waves relying on particular initial boundary conditions are also simulated

petrov galerkin methods for nonlinear dispersive waves abstract petrov galerkin methods based on piecewise linear interpolants for the korteweg de vries and related equations are studied both accuracy and stability are analysed for the linearised case it is shown that in the nonlinear case the order of accuracy of the standard galerkin procedure is reduced and an alternative technique is therefore proposed which retains the fourth order accuracy in space this is found to perform well when compared with finite difference or other finite element schemes

stability of optimal order approximation by bivariate splines over arbitrary triangulations let a be a triangulation of some polygonal domain in r2 and skr a the space of all bivariate cr piecewise polynomials of total degree 3r 2 that can be used to provide the highest order of approximation with the property that the approximation constant of this order is independent of the geometry of a with the exception of the smallest angle in the partition this result is obtained by means of a careful choice of locally supported basis functions which however require a very technical proof to justify their stability in optimal order approximation a new formulation of smoothness conditions for piecewise polynomials in terms of their b net representations is derived for this purpose

nonlocality is transitive we show a transitivity property of nonlocal correlations there exist tripartite nonsignaling correlations of which the bipartite marginals between a and b as well as b and c are nonlocal and any tripartite nonsignaling system between a b and c consistent with them must be such that the bipartite marginal between a and c is also nonlocal this property represents a step towards ruling out certain alternative models for the explanation of quantum correlations such as hidden communication at finite speed whereas it is not possible to rule out this model experimentally it is the goal of our approach to demonstrate this explanation to be logically inconsistent either the communication cannot remain hidden or its speed has to be infinite the existence of a three party system that is pairwise nonlocal is of independent interest in the light of the monogamy property of nonlocality

compact structure preserving approach to solitary wave in shallow water modeled by the rosenau rlw equation abstract a mass preserving scheme a nonlinear algorithm based on modification of a finite difference method to the rosenau rlw equation is proposed subject to homogeneous boundary conditions the key feature of the method for improving the accuracy of approximate solutions is to develop a compact higher order scheme together with an iterative algorithm for solving the nonlinear implicit scheme the derivatives for space discretization are approximated by using the algorithm dealing with a five point stencil in addition a three level average difference technique is used to perform time discretization the conservation of mass and both the existence and uniqueness of the numerical solution are proved the stability and convergence of the numerical solution with order o 4 2 h 2 h 4 are also confirmed for efficiency analysis numerical results show that the computational efficiency of the compact scheme is much higher than that of non compact schemes moreover long time behavior is also used to validate the capability of the present method

multivariate spline spaces in principle all of the properties and applications of multivariate spline functions can be derived from the basic theories in chapter 1 the expressions of the multivariate spline functions in 2 of chapter 1 are inconvenient to be applied directly because the expressions also rely on the solutions of the global conformality conditions so as to determine all the smooth cofactors

upper and lower bounds on the dimension of multivariate spline spaces we give upper and lower bounds on the dimensions of multivariate spline spaces defined on triangu lations in mathbb r k the bounds are optimal in a certain sense

an explicit basis for c 1 quartic by various bivariate splines we establish the dimension of the space of c 1 bivariate piecewise quartic polynomials defined on a triangulation of a connected polygonal domain our approach is to construct a minimal determining set and an associated explicit basis for the space for general triangulations the minimal determining set must be defined globally

static and dynamic deflections of plates of arbitrary geometry by a new finite difference approach finite difference solutions for the static and dynamic displacements of a plate undergoing vibration are presented the approach presented differs from the conventional methods in which the derivatives are expressed by their difference equivalents here the difference equations are obtained as solutions to the fourth order biharmonic equation a single space varying drive number is found which varies from node to node and characterizes the true mode shape of the plate at a node the technique presented can be applied to finite elements of triangular rectangular or quadrilateral geometry without any restriction the marcus difference solution is seen to be a particular case of the general difference solution derived in this paper to give static deflection the analysis is confined to small deflections of plate but the technique can also be applied to appropriate differential equations for large deflections the method has applications in the determination of the displacements of structures having arbitrary boundary conditions evaluation of complicated modal response associated with the traditional approach is thus avoided

new hybrid laplace transform finite element method for three dimensional transient heat conduction problem the paper presents results obtained by the implementation of a new hybrid laplace transform finite element method developed by the authors the present method removes the time derivatives from the governing differential equation using the laplace transform and then solves the associated equation with the finite element method previously reported hybrid laplace transform finite element methods1 have been confined to one nodal solution at a time when applied to many nodes it takes an excessive amount of computer time by using a similarity transform method on the matrix of the complex number coefficients this restriction is removed and the reported new method provides a more useful tool for the solution of linear transient problems test examples are used to show that the basic accuracy is comparable to that obtainable by analytical finite difference and finite element methods

on nonlocality as a resource theory and nonlocality measures with the advent of device independent quantum information processing nonlocality is nowadays regarded as a resource that can be used for various tasks using the analogy of entanglement theory we approach nonlocality from this perspective in order to do so we analyze in full detail the operations that can be implemented in this scenario and under which nonlocality cannot increase this provides a theoretical basis from which to study how to order and quantify nonlocal behavior finally we review several nonlocality measures and discuss their validity from this point of view this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

quadrilateral finite elements of fvs type and class c rho let cal p be some partition of a planar polygonal domain omega into quadrilaterals given a smooth function u we construct piecewise polynomial functions upsilon in c rho omega of degree n 3 rho for rho odd and n 3 rho 1 for rho even on a subtriangulation tau 4 of cal p the latter is obtained by drawing diagonals in each q in cal p and upsilon vert q is a composite quadrilateral finite element generalizing the classical c 1 cubic fraeijs de veubeke and sander or fvs quadrilateral the function upsilon interpolates the derivatives of u up to order rho rho 2 at the vertices of cal p polynomial degrees obtained in this way are minimal in the family of interpolation schemes based on finite elements

piecewise algebraic curve a piecewise algebraic curve is defined by a bivariate spline function using the techniques of the b net form of bivariate splines function discriminant sequence of polynomial cf yang lu et al sci china ser e 39 6 1996 628 and yang lu et al nonlinear algebraic equation system and automated theorem proving shanghai scientific and technological education publishing house shanghai 1996 and the number of sign changes in the sequence of coefficients of the highest degree terms of sturm sequence we determine the number of real intersection points of two piecewise algebraic curves whose common points are finite a lower bound of the number of real intersection points is given in terms of the method of rotation degree of vector field

a conservative fourth order stable finite difference scheme for the generalized rosenau kdv equation in both 1d and 2d abstract in the present work a conservative fourth order stable finite difference scheme is proposed to solve the generalized rosenau kdv equation in both 1d and 2d the existence uniqueness and mass and energy conservations of the numerical solution are proved using the discrete energy method the new scheme is convergent with o 2 h 4 and unconditionally stable numerical experiments are carried out to show that the scheme is efficient and reliable

on c 2 quintic spline functions over triangulations of powell sabin s type abstract given a triangulation of a polygonal domain we find a refinemet of of by choosing t in a neighborhood of the center of the inscribed circle of each triangle of t connecting t to the vertices of the triangle t and connecting t to t if t shares an interior edge with t or to the midpoint e of any boundary edge e of t the resulting triangulation is a triangulation of powell sabin s type we investigate a c 2 quintic spline space s 5 2 whose elements are c 3 only at t s we give a dimension formula for this spline space show how to construct a locally supported basis display an interpolation scheme and prove that this spline space has the full approximation order

structure and dimension of multivariate spline space of lower degree on arbitrary triangulation in this paper we discuss the structure of multivariate spline spaces on arbitrary triangulation by using the methods and results of smoothing cofactor and generator basis of modules on the base of analyzing the algebraic and geometric results about singularity of s21 ms we build the structure of triangulation and give some useful geometric conditions such that s 1 space is singular and we obtain an algebraic condition which is necessary and sufficient for the singularity of s 1 spaces as well as their dimension formulae moreover the structure matrix of spline spaces over any given partition is defined which has been used to discuss the structure of s31 and s21 spaces over arbitrary triangulation and to prove the nonsingularity of s31 spaces this partially settles a conjecture on the singularity of spline spaces in wang et al multivariate spline and its applications kluwer press dordrecht 2002 academic press beijing 1994 in chinese meanwhile the dimension formulae of s31 s21 spaces and the dimension formulae of s 1 1 spaces are also given in this paper

integrable discretization of time and its application on the fourier pseudospectral method to the korteweg de vries equation this paper presents a new integrable discretization of the korteweg de vries kdv equation different from other discrete analogues we discretize the variable time and obtain an integrable differential difference system this system has the original kdv equation as a standard limit when the step size tends to zero the main idea is based on hirota s bilinear method and backlund transformation and can be applied to other integrable systems by applying the fourier pseudospectral method to the space variable we derive a new numerical scheme for the kdv equation numerical results are found to agree with the exact solution and the first five conservation quantities are preserved quite well

interpolation by c 1 splines of degree q 4 on triangulations let be an arbitrary regular triangulation of a simply connected compact polygonal domain ir and let s1q denote the space of bivariate polynomial splines of degree q and smoothness 1 with respect to we develop an algorithm for constructing point sets admissible for lagrange interpolation by s1q if q 4 in the case q

the non locality of n noisy popescu rohrlich boxes we quantify the amount of non locality contained in n noisy versions of the so called popescu rohrlich boxes prbs i e bipartite systems violating the chsh bell inequality maximally following the approach by elitzur popescu and rohrlich we measure the amount of non locality of a system by representing it as a convex combination of a local behaviour with maximal possible weight and a non signalling system we show that the local part of n systems each of which approximates a prb with probability 1 e is of order en 2 in the isotropic and equal to 3e n in the maximally biased case

hybrid laplace transform weighting function scheme for transient multidimensional conservation equations abstract a hybrid laplace transform weighting function scheme is developed for solving time dependent multidimensional conservation equations the new method removes the time derivatives from the governing differential equations using the laplace transform and solves the associated equation with the weighting function scheme the similarity transform method is used to treat the complex coefficient system of the equations which allows the simplest form of complex number functions to be obtained and then to use the partial fractions method or a numerical method to invert the laplace transform and transform the functions to the physical plane three different examples have been analyzed by the present method the present method solutions are compared in tables with the exact solutions and those obtained by the other numerical methods it is found that the present method is a reliable and efficient numerical tool

bivariate spline interpolation at grid points we describe algorithms for constructing point sets at which interpolation by spaces of bivariate splines of arbitrary degree and smoothness is possible the splines are defined on rectangular partitions adding one or two diagonals to each rectangle the interpolation sets are selected in such a way that the grid points of the partition are contained in these sets and no large linear systems have to be solved our method is to generate a net of line segments and to choose point sets in these segments which satisfy the schoenberg whitney condition for certain univariate spline spaces such that a principle of degree reduction can be applied in order to include the grid points in the interpolation sets we give a sufficient schoenberg whitney type condition for interpolation by bivariate splines supported in certain cones this approach is completely different from the known interpolation methods for bivariate splines of degree at most three our method is illustrated by some numerical examples

distilling non locality two parts of an entangled quantum state can have a correlation in their joint behavior under measurements that is unexplainable by shared classical information such correlations are called non local and have proven to be an interesting resource for information processing since non local correlations are more useful if they are stronger it is natural to ask whether weak non locality can be amplified we give an affirmative answer by presenting the first protocol for distilling non locality in the framework of generalized non signaling theories our protocol works for both quantum and non quantum correlations this shows that in many contexts the extent to which a single instance of a correlation can violate a chsh inequality is not a good measure for the usefulness of non locality a more meaningful measure follows from our results

approximation order from certain spaces of smooth bivariate splines on a three direction mesh determination de l ordre exact d approximation de certains espaces de fonctions splines regulieres a deux variables

multivariate spline and algebraic geometry abstract the purpose of this survey is to emphasize the special relationship between multivariate spline and algebraic geometry we will not only point out the algebraic geometric method of multivariate spline but also the algebraic geometric background and essence of multivariate spline especially we have made an introduction to the so called piecewise algebraic curve piecewise algebraic variety and some of their properties

macro elements and stable local bases for splines on clough tocher triangulations macro elements of arbitrary smoothness are constructed on clough tocher triangle splits these elements can be used for solving boundary value problems or for interpolation of hermite data and are shown to be optimal with respect to spline degree we conjecture they are also optimal with respect to the number of degrees of freedom the construction provides local bases for certain superspline spaces defined over clough tocher refinements of arbitrary triangulations these bases are shown to be stable as a function of the smallest angle in the triangulation which in turn implies that the associated spline spaces have optimal order approximation power

on super splines and finite elements super splines are spaces of multivariate splines consisting of piecewise polynomials defined on triangulations where the degree of smoothness enforced at the vertices is larger than the degree of smoothness enforced across the edges dimension formulae for super spline spaces are established and the construction of explicit bases of locally supported splines is presented in addition it is shown that many classical finite element spaces can be interpreted as super spline spaces providing a link between spline theory and the highly applicable finite element theory

exponential finite difference method applied to korteweg de vries equation for small times the korteweg de vries equation is numerically solved by using the exponential finite difference technique the accuracy of computed solutions is examined by comparison with other numerical and analytical solutions using two examples the close results agreement between the current results and the exact solutions confirms that the proposed finite difference procedure is an effective technique for the solution of the korteweg de vries equation at the small times

the comparative performance of some finite difference equations for transient heat conduction the established finite difference equations to estimate one dimensional transient heat flow in solids are the classical form explicit and the crank nicolson and pure implicit forms both implicit they are all based on finite difference approximation to the fourier continuity equation to these are now added three more explicit forms exponential linear exponential inverse cosine and polynomial which are based on exact solutions to the fourier equation the performance of each of the six equations is tested against the exact results of a well known step excitations problem the groeber model the tests consist of examining i finite difference errors arising from a single implementation of each form at different stages in the transient cooling process ii the errors that accumulate during part or all of the cooling process both as regards any bias that is introduced and also a measure of variance and iii the run times in executing the various forms the nondimensional time step r was treated as the independent variable and can be made arbitrarily large by use of a simple time division procedure otherwise r for use with the classical form it is shown that having regard to both error and run time the polynomial form appears to be the most efficient estimator

fully discrete galerkin methods for the korteweg de vries equation abstract fully discrete schemes for the numerical simulation of solutions of the periodic initial value problem for the korteweg de vries equation are introduced implemented and tested of special interest are stable schemes featuring rates of convergence of order higher than two in both the spatial and temporal variable a careful analysis of the relative and absolute efficiency of these schemes is carried out and one of the schemes is applied to demonstrate that solutions of certain generalized korteweg de vries equations apparently may develop singularities in finite time

a comparative investigation on recurrence formulae in finite difference methods to solve the transient heat conduction equation the pade approximation is introduced into the finite difference method but if the time step is chosen too large relatively to the element size the euler method pade 0 1 approximation and the crank nicolson solution pade 1 1 approximation lead to significant oscillations in contrast the full implicit scheme pade 1 0 approximation does not show this oscillatory behaviour but is more inaccurate compared to these time stepping algorithms the presented weighted time step method is seen to offer definite advantages

on the approximation power of splines on triangulated quadrangulations we study the approximation properties of the bivariate spline spaces srt cal s r 3r hbox diamondsuit kern 0 78em raise0 10786ex hbox of smoothness r and degree 3r defined on triangulations lai hbox diamondsuit kern 0 78em raise0 10786ex hbox which are obtained from arbitrary nondegenerate convex quadrangulations by adding the diagonals of each quadrilateral

hybrid method for transient response of circular pins the combined application of the laplace transform and the finite element method is used to analyse the transient response of circular pins the present method removes the time dependence terms from the governing differential equations and boundary conditions using the laplace transform and then the eigenfunction expansion method is applied to reduce the two dimensional boundary value problem to that of one dimension accordingly the final transformed equation can easily be solved by the finite element method the transformed temperature is inverted to the physical quantity numerically the present results agree well with analytical solutions in addition it is seen that the results of axisymmetric transient heat conduction problems with the central node at r 0 can accurately be obtained using the present method

instability in the dimension of spaces of bivariate piecewise polynomials of degree 2 r and smoothness order r the spaces of bivariate splines on a particular triangulation are considered the dimension of the space of splines with degree 2 and with smoothness order 1 on this triangulation has long been known to depend on the geometry of the triangulation this result is extended to the space of splines of degree 2r and smoothness order r the dimension of the spline space is shown to have a similar dependence on geometry for all values of r geqq 1

c1 quintic interpolation over triangles two explicit representations two explicit representation of a c1 quintic interpolant over triangles are given these representations are derived by generalization of coons methods and bernstein bezier methods respectively

monotonicity preserving surface interpolation c 1 monotone quadratic splines are analyzed this motivates an algorithm for interpolating monotone data given on a rectangular grid with a c 1 monotone quadratic spline surface error estimates an operation count and numerical examples are given