polynomials for infinite domain spectral elements in this note it is found that laguerre type elements may be used in conjunction with legendre type elements to resolve more accurately regions of outflow and infinite boundaries

error analysis of time delay estimation using a finite integration time correlator a methodology is presented for analyzing the error associated with time delay estimation using a finite integration time correlator processing waveforms received at two separate sensors the type of signal considered is a sinusoid whose amplitude is randomly modulated the signals are assumed to be imbedded in additive gaussian noise before they are correlated the received waveforms are converted to a lower center frequency by mixers whose local oscillators are assumed to contain phase noise by direct calculation in the time domain the variance of the error in the time delay estimate is shown to be a function of integration time signal to noise ratios signal and noise bandwidths and phase noise variance the phase noise is shown to limit the accuracy of the time delay estimate however without phase noise using the methodology the accuracy is shown to approach that obtained by the maximum likelihood estimator

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

galerkin analysis for schrodinger equation by wavelets we consider the perturbed schrodinger equation which is an elliptic operator with unbounded coefficients we use wavelets adapted to the schrodinger operator to deal with problems on the unbounded domain the wavelets are constructed from hermite functions which characterizes the space generated by the schrodinger operator we show that the galerkin matrix can be pre conditioned by a diagonal matrix so that its condition number is uniformly bounded moreover we introduce a periodic pseudo differential operator and show that its discrete galerkin matrix under periodic wavelet system is equal to the galerkin matrix for the equation with unbounded coefficients under the hermite system the convergence is proved in the l2 topology

finite calculus formulation for incompressible solids using linear triangles and tetrahedra summary many finite elements exhibit the so called volumetric locking in the analysis of incompressible or quasi incompressible problems in this paper a new approach is taken to overcome this undesirable effect the starting point is a new setting of the governing differential equations using a finite calculus fic formulation the basis of the fic method is the satisfaction of the standard equations for balance of momentum equilibrium of forces and mass conservation in a domain of finite size and retaining higher order terms in the taylor expansions used to express the different terms of the differential equations over the balance domain the modified differential equations contain additional terms which introduce the necessary stability in the equations to overcome the volumetric locking problem the fic approach has been successfully used for deriving stabilized finite element and meshless methods for a wide range of advective diffusive and fluid flow problems the same ideas are applied in this paper to derive a stabilized formulation for static and dynamic finite element analysis of incompressible solids using linear triangles and tetrahedra examples of application of the new stabilized formulation to linear static problems as well as to the semi implicit and explicit 2d and 3d non linear transient dynamic analysis of an impact problem and a bulk forming process are presented copyright 2004 john wiley sons ltd

hermite spectral methods with a time dependent scaling for parabolic equations in unbounded domains hermite spectral methods are investigated for linear diffusion equations and nonlinear convection diffusion equations in unbounded domains when the solution domain is unbounded the diffusion operator no longer has a compact resolvent which makes the hermite spectral methods unstable to overcome this difficulty a time dependent scaling factor is employed in the hermite expansions which yields a positive bilinear form as a consequence stability and spectral convergence can be established for this approach the present method plays a similar role in the stability of the similarity transformation technique proposed by funaro and kavian math comp 57 1991 pp 597 619 however since coordinate transformations are not required the present approach is more efficient and is easier to implement in fact with the time dependent scaling the resulting discretization system is of the same form as that associated with the classical straightforward but unstable hermite spectral method numerical experiments are carried out to support the theoretical stability and convergence results

jacobi approximations in certain hilbert spaces and their applications to singular differential equations jacobi approximations in certain hilbert spaces are investigated several weighted inverse inequalities and poincare inequalities are obtained some approximation results are given singular differential equations are approximated by using jacobi polynomials this method keeps the spectral accuracy some linear problems and a nonlinear logistic equation are considered the stabilities and the convergences of proposed schemes are proved strictly the main idea and techniques used in this paper are also applicable to other singular problems in multiple dimensional spaces

stabilized formulation for the advection diffusion absorption equation using finite calculus and linear finite elements abstract a stabilized finite element method fem for the steady state advection diffusion absorption equation is presented the stabilized formulation is based on the modified governing differential equations derived via the finite calculus fic method the basis of the method is detailed for the 1d problem it is shown that the stabilization terms act as a non linear additional diffusion governed by a single stabilization parameter a critical constant value of this parameter ensuring a stabilized solution using two node linear elements is given more accurate numerical results can be obtained by using a simple expression of the non linear stabilization parameter depending on the signs of the numerical solution and of its derivatives a straightforward two steps algorithm yielding a stable and accurate solution for all the range of physical parameters and boundary conditions is described the extension to the multi dimensional case is briefly described numerical results for 1d and 2d problems are presented showing the efficiency and accuracy of the new stabilized formulation

on the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations an analytical formula expressing the ultraspherical coefficients of an expansion for an infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is stated in a more compact form and proved in a simpler way than the formula suggested by phillips and karageorghis 27 1990 823 a new formula expressing explicitly the integrals of ultraspherical polynomials of any degree that has been integrated an arbitrary number of times of ultraspherical polynomials is given the tensor product of ultraspherical polynomials is used to approximate a function of more than one variable formulae expressing the coefficients of differentiated expansions of double and triple ultraspherical polynomials in terms of the original expansion are stated and proved some applications of how to use ultraspherical polynomials for solving ordinary and partial differential equations are described

a new generalized laguerre spectral approximation and its applications a new family of generalized laguerre polynomials is introduced various orthogonal projections are investigated some approximation results are established as an example of their important applications the mixed spherical harmonic generalized laguerre approximation is developed a mixed spectral scheme is proposed for a three dimensional model problem its convergence is proved numerical results demonstrate the high accuracy of this new spectral method

time delay estimation by generalized cross correlation methods the problem of estimating time delay by cross correlation methods is reexamined for the whole class of stationary signals expressions are derived for the estimation mean square error mse by the cross correlation method and are shown to be identical to previously published results for gaussian signals the generalized cross correlation method is also analyzed and the optimal weight function for this method is derived it is shown to be identical to that derived for gaussian signals by the maximum likelihood method for the cross correlation method a simplified mse expression is derived which is to be used instead of a previously published result

gegenbauer approximation in certain hilbert spaces and its applications to singular differential equations gegenbauer approximation in certain hilbert spaces is investigated several weighted inverse inequalities and poincare inequalities are obtained some approximation results are given singular differential equations are approximated by using gegenbauer polynomials this method keeps the spectral accuracy the main idea and techniques used in this paper are also applicable to some singular problems in multiple dimensional spaces

rational legendre pseudospectral approach for solving nonlinear differential equations of lane emden type lane emden equation is a nonlinear singular equation in the astrophysics that corresponds to the polytropic models in this paper a pseudospectral technique is proposed to solve the lane emden type equations on a semi infinite domain the method is based on rational legendre functions and gauss radau integration the method reduces solving the nonlinear ordinary differential equation to solve a system of nonlinear algebraic equations the comparison of the results with the other numerical methods shows the efficiency and accuracy of this method

a complete orthonormal system of functions in l 2 infty infty space a new complete orthonormal system of functions in the l 2 infty infty space is introduced the system consists of two sequences composed of odd and even functions respectively unlike hermite and laguerre sets of functions which behave exponentially at infinity the new system exhibits asymptoticbehavior x 1 for the odd sequence and x 2 for the even one formulae representing products derivatives etc in series in the system are developed a nonlinear differential equation with a requirement for summability of the square of its solutions instead of boundary conditions is solved the example displays most of the features of the new method important for essentially nonlinear problems

a simple subgrid scale stabilized method for the advection diffusion reaction equation abstract a subgrid scale method has been developed for the advection diffusion reaction equation with a simple intrinsic time scale parameter the proposed formulation combines very good accuracy and stability in all the physical regimes of the equation that is in the exponential regime for both negative and positive source terms and in the propagation regime numerical examples confirm the robustness and accuracy of the method

a combined fic tdg finite element approach for the numerical solution of coupled advection diffusion reaction equations with application to a bioregulatory model for bone fracture healing summary numerical schemes for the approximative solution of advection diffusion reaction equations are often flawed because of spurious oscillations caused by steep gradients or dominant advection or reaction in addition for strong coupled nonlinear processes which may be described by a set of hyperbolic pdes established time stepping schemes lack either accuracy or stability to provide a reliable solution in this contribution an advanced numerical scheme for this class of problems is suggested by combining sophisticated stabilization techniques namely the finite calculus fic fem scheme introduced by onate et al with time discontinuous galerkin tdg methods whereas the former one provides a stabilization technique for the numerical treatment of steep gradients for advection dominated problems the latter ensures reliable solutions with regard to the temporal evolution a brief theoretical outline on the superior behavior of both approaches will be presented and underlined with related computational tests the performance of the suggested fic tdg finite element approach will be discussed exemplarily on a bioregulatory model for bone fracture healing proposed by geris et al which consists of at least 12 coupled hyperbolic evolution equations copyright 2012 john wiley sons ltd

a generalized galerkin method for steady convection diffusion problems with application to quadratic shape function elements abstract a generalization of the standard galerkin finite element method is considered to enable it to deal successfully with steady convection diffusion problems the proposed method employs a generalized governing equation which is obtained by subtracting from the original differential equation the scalar product of its gradient by a vector of free parameters associated with each of the coordinate directions the generalized equation is successively discretized by the standard bubnov galerkin finite element method the effectiveness of the method is illustrated for the case of quadratic local interpolations in one and two space dimensions

combined finite element and spectral approximation of the navier stokes equations we present a method for the numerical approximation of navier stokes equations with one direction of periodicity in this direction a fourier pseudospectral method is used in the two others a standard f e m is applied we prove optimal rate of convergence where the two parameters of discretization intervene independently on presente une methode d approximation numerique des equations de navier stokes possedant une direction de periodicite dans cette direction une methode pseudospectrale basee sur des developpements en serie de fourier est utilisee dans les deux autres on applique une methode d elements finis standard on montre que la convergence est optimale et que les deux parametres de discretisation peuvent etre choisis de facon independante

a high resolution petrov galerkin method for the 1d convection diffusion reaction problem abstract we present the design of a high resolution petrov galerkin hrpg method using linear finite elements for the problem defined by the residual r t u x k 2 x 2 s f where k s 0 the structure of the method in 1d is identical to the consistent approximate upwind petrov galerkin cau pg method a c galeao e g dutra do carmo a consistent approximate upwind petrov galerkin method for the convection dominated problems comput methods appl mech engrg 68 1988 83 95 except for the definitions of the stabilization parameters such a structure may also be attained via the finite calculus fic procedure e onate derivation of stabilized equations for numerical solution of advective diffusive transport and fluid flow problems comput methods appl mech engrg 151 1998 233 265 e onate j miquel g hauke stabilized formulation for the advection diffusion absorption equation using finite calculus and linear finite elements comput methods appl mech engrg 195 2006 3926 3946 by an appropriate definition of the characteristic length the prefix high resolution is used here in the sense popularized by harten i e second order accuracy for smooth regular regimes and good shock capturing in nonregular regimes the design procedure embarks on the problem of circumventing the gibbs phenomenon observed in l 2 projections next we study the conditions on the stabilization parameters to circumvent the global oscillations due to the convective term a conjuncture of the two results is made to deal with the problem at hand that is usually plagued by gibbs global and dispersive oscillations in the numerical solution it is shown that the method indeed reproduces stabilized high resolution numerical solutions for a wide range of values of u k s and f finally some remarks are made on the extension of the hrpg method to multidimensions

on the stabilization parameter in the subgrid scale approximation of scalar convection diffusion reaction equations on distorted meshes abstract in this paper we revisit the definition of the stabilization parameter in the finite element approximation of the convection diffusion reaction equation the starting point is the decomposition of the unknown into its finite element component and a subgrid scale that needs to be approximated in order to incorporate the distortion of the mesh into this approximation we transform the equation for the subgrid scale within each element to the shape regular reference domain the expression for the subgrid scale arises from an approximate fourier analysis and the identification of the wave number direction where instabilities are most likely to occur the final outcome is an expression for the stabilization parameter that accounts for anisotropy and the dominance of either convection or reaction terms in the equation

a positivity preserving variational method for multi dimensional convectiondiffusionreaction equation a new positivity preserving variational ppv procedure is proposed to solve the convectiondiffusionreaction cdr equation through the generalization of stabilized finite element methods the present variational procedure offers minimal phase and amplitude errors for different regimes associated with convection diffusion and reaction effects by means of fourier analysis we first review the shortcomings of the galerkin least squares gls and the subgrid scale sgs methods during the change in sign of the reaction coefficient that motivates us for the present linear stabilization as a combined gls sgs methodology discrete upwind operator with a solution dependent nonlinear term is then introduced in high gradient regions which enables the positivity preserving property in the variational formulation direct extension to multi dimensions is carried out by considering the principle streamline and crosswind directions the efficacy of the method is demonstrated by the systematic accuracy and stability analyses in one and two dimensions results show the reduction of oscillations in the solution in one and two dimensional cases and a remarkable reduction in the phase error is observed for the cases with negative reaction coefficient the proposed formulation provides a superior solution in the reaction dominated as well as the convection dominated regimes due to the minimization of spurious oscillations and accurate capturing of the high gradient regions

a fast local maximum likelihood estimator for time delay estimation a fast algorithm for the local maximum likelihood determination of the difference of arrival time of a common signal at two spatially separated sensors with uncorrelated noise is given the fast algorithm consists of locally maximizing the cross correlation function from the two wide band signals by using newton s method for finding the root of an equation the probability density function of one iteration of newton s method is explicitly computed in terms of exponential and error functions using a theorem by rice on the probability density of local maxima of gaussian processes the probability density of the local maxima of the cross correlator is obtained these results are new when both the signal and the noises have flat power spectral densities the mean square error mse of two iterates of newton s method is practically equal to the mse computed from the probability density of the local maxima of the cross correlator via rice s theorem the above holds if the starting point used in newton s method is within a quarter signal resolution binwidth from the true delay and the signal to noise ratio snr at the cross correlator output is 15 db or higher the mse of the local maximum estimator obtained from rice s theorem is almost equal to the cramer rao bound even for low snr i e 5 db

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

streamline design of stability parameters for advection diffusion problems the dependence of the computation of advective diffusive transport phenomena on the orientation of the mesh with respect to the flow direction is analyzed poor performance of the classical galerkin finite element method in the convection dominated regime is alleviated by stabilization we propose definitions of the stability parameter that rationally incorporate the flow direction numerical tests compare the performance of the proposed methods with established techniques

finite element stabilization parameters computed from element matrices and vectors abstract we propose new ways of computing the stabilization parameters used in the stabilized finite element methods such as the streamline upwind petrov galerkin supg and pressure stabilizing petrov galerkin pspg formulations the parameters are computed based on the element level matrices and vectors which automatically take into account the local length scales advection field and the reynolds number we describe how we compute these parameters in the context of first a time dependent advection diffusion equation and then the navier stokes equations of unsteady incompressible flows

derivation of stabilized equations for numerical solution of advective diffusive transport and fluid flow problems abstract the concept of the so called artificial or balancing diffusion used to stabilize the numerical solution of advective diffusive transport and fluid flow problems is revised in this paper it is shown that the standard forms of the balancing diffusion terms usually chosen in a heuristic manner can be naturally found by introducing higher order approximations in the derivation of the governing differential equations via standard conservation or equilibrium principles this allows us to reinterpret many stabilization algorithms and concepts used in every day practice by numerical analysts and also provides an expression for computing the stabilization parameter

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

the nearly optimal petrov galerkin method for convection diffusion problems abstract the nearly optimal petrov galerkin nopg method is employed to improve finite element computation of convection dominated transport phenomena the design of the nopg method for convection diffusion is based on consideration of the advective limit nonetheless the resulting method is applicable to the entire admissible range of problem parameters an investigation of the stability properties of this method leads to a coercivity inequality the convergence features of the nopg method for convection diffusion are studied in an error analysis that is based on the stability estimates the proposed method compares favorably to the performance of an established technique on several numerical tests

error analysis and bounds in time delay estimation we have analyzed the error propagation in the classical delay estimation based on the identification of the maximum of the cross correlation between the input signals the resulting error propagation model has the form of a linear combination of the errors additional to the signal samples these errors are weighted by the derivative of the input signals at the sampling instants based on the error propagation model we have further found an upper bound to the square of the estimation error the use of the model is demonstrated by two simple examples

mixed laguerre legendre pseudospectral method for incompressible fluid flow in an infinite strip in this paper we investigate the mixed laguerre legendre interpolation approximation and its application some approximation results are established a mixed laguerre legendre pseudospectral scheme is constructed for incompressible fluid flow in an infinite strip its stability and convergence are proved numerical results show the efficiency of this new approach

the eigenvalues of hermite and rational spectral differentiation matrices we derive expressions for the eigenvalues of spectral differentiation matrices for unbounded domains in particular we consider galerkin and collocation methods based on hermite functions as well as rational functions a fourier series combined with a cotangent mapping we show that i first derivative matrices have purely imaginary eigenvalues and second derivative matrices have real and negative eigenvalues ii for the hermite method the eigenvalues are determined by the roots of the hermite polynomials and for the rational method they are determined by the laguerre polynomials and iii the hermite method has attractive stability properties in the sense of small condition numbers and spectral radii

the superconvergent patch recovery spr and adaptive finite element refinement abstract the error estimation processes which have been developed by the authors depend on the accuracy with which stresses gradients can be recovered the recovery methods available currently such as the l 2 projection and averaging are inaccurate and an improved process is needed this paper presents a very much improved recovery process yielding superconvergent values throughout the domain details of this method named the superconvergent patch recovery are presented with test results showing its efficiency examples of adaptive h based refinement are given showing high local and overall efficiency indices for error estimation

pseudospectral method for non isotropic heat transfer using mixed hermite legendre interpolation abstract in this paper we develop a new mixed pseudospectral method for heat transfer by using generalized hermite functions and legendre polynomials in unbounded domains fundamental to spectral methods for various unbounded domain problems we establish some basic results on the mixed generalized hermite legendre interpolation as an example a new mixed generalized hermite legendre pseudospectral scheme is provided for non isotropic heat transfer its convergence is proved numerical results demonstrate the spectral accuracy of this approach

on the coefficients of differentiated expansions and derivatives of jacobi polynomials a formula expressing explicitly the derivatives of jacobi polynomials of any degree and for any order in terms of the jacobi polynomials themselves is proved another explicit formula which expresses the jacobi expansion coefficients of a general order derivative of an infinitely differentiable function in terms of its original jacobi coefficients is also given the results for the special case of ultraspherical polynomials are considered the results for chebyshev polynomials of the first and second kinds and for legendre polynomials are also noted an application of how to use jacobi polynomials for solving ordinary and partial differential equations is described

error estimation of hermite spectral method for nonlinear partial differential equations hermite approximation is investigated some inverse inequalities imbedding inequalities and approximation results are obtained a hermite spectral scheme is constructed for burgers equation the stability and convergence of the proposed scheme are proved strictly the techniques used in this paper are also applicable to other nonlinear problems in unbounded domains

a taylor galerkin method for convective transport problems a method is described to derive finite element schemes for the scalar convection equation in one or more space dimensions to produce accurate temporal differencing the method employs forward time taylor series expansions including time derivatives of second and third order which are evaluated from the governing partial differential equation this yields a generalized time discretized equation which is successively discretized in space by means of the standard bubnov galerkin finite element method the technique is illustrated first in one space dimension with linear elements and euler leap frog and crank nicolson time stepping several interesting relations with standard galerkin and recently developed petrov galerkin methods emerge and the new taylor galerkin schemes are found to exhibit particularly high phase accuracy with minimal numerical damping the method is successively extended to deal with variable coefficient problems and multi dimensional situations

modeling incompressible flows at low and high reynolds numbers via a finite calculus finite element approach we present a formulation for incompressible flows analysis using the finite element method fem the necessary stabilization for dealing with convective effects and the incompressibility condition is modeled via the finite calculus fic method the stabilization terms introduced by the fic formulation allow to solve a wide range of fluid flow problems for low and high reynolds numbers flows without the need for a turbulence model examples of application of the fic fem formulation to the analysis of 2d and 3d incompressible flows with moderate and large reynolds numbers are presented

a high resolution petrov galerkin method for the convection diffusion reaction problem part ii a multidimensional extension abstract a multidimensional extension of the hrpg method doi 10 1016 j cma 2009 10 009 using the lowest order block finite elements is presented first we design a nondimensional element number that quantifies the characteristic layers which are found only in higher dimensions this is done by matching the width of the characteristic layers to the width of the parabolic layers found for a fictitious 1d reaction diffusion problem the nondimensional element number is then defined using this fictitious reaction coefficient the diffusion coefficient and an appropriate element size next we introduce anisotropic element length vectors li and the stabilization parameters i i are calculated along these li except for the modification to include the new dimensionless number that quantifies the characteristic layers the definitions of i i are a direct extension of their counterparts in 1d using i i and li objective characteristic tensors associated with the hrpg method are defined the numerical artifacts across the characteristic layers are manifested as the gibbs phenomenon hence we treat them just like the artifacts formed across the parabolic layers in the reaction dominant case several 2d examples are presented that support the design objective stabilization with high resolution

a finite element method for fluid structure interaction with surface waves using a finite calculus formulation abstract a stabilized semi implicit fractional step finite element method fem for solving coupled fluid structure interaction problems involving free surface waves is presented the stabilized governing equations for the viscous incompressible fluid and the free surface are derived at a differential level via a finite calculus fic procedure a mesh updating technique based on solving a fictitious elastic problem on the moving mesh is described examples of the efficiency of the stabilized semi implicit algorithm for the analysis of fluid structure interaction problems in totally or partially submerged bodies is presented

spectral approximation of pattern forming nonlinear evolution equations with double well potentials of quadratic growth this paper is concerned with the analysis of a numerical algorithm for the approximate solution of a class of nonlinear evolution problems that arise as l 2 gradient flow for the modica mortola regularization of the functional v bv t d 1 1 e v 2 td v 1 2 k d k v k 2 here is the interfacial energy per unit length or unit area t d is the flat torus in d and is a nonnegative fourier multiplier that is continuous on d symmetric in the sense that for all d and that decays to zero at infinity such functionals feature in mathematical models of pattern formation in micromagnetics and models of diblock copolymers the resulting evolution equation is discretized by a fourier spectral method with respect to the spatial variables and a modified crank nicolson scheme in time optimal order a priori bounds are derived on the global error in the l 0 t l 2 t d norm

on stabilized finite element formulations for incompressible advective diffusive transport and fluid flow problems abstract a new approach is presented to obtain stabilized finite element formulations such as streamline upwind petrov galerkin supg and galerkin least squares gls the procedure consists in modifying the equations to be solved and then obtaining the variational equations by the standard galerkin method the new formulation generates additional terms involving boundary integrals to standard stabilization techniques these terms compensate for the lack of consistency of the traditional supg and gls methods for which stabilization terms are added only on the element interiors while jumps of the residual across element faces are neglected a physical interpretation is provided of how the modified equations are obtained it is shown how stabilized formulations such as streamline upwind su and supg are recovered as special cases stabilization terms defined on the element interiors are always accompanied by additional boundary integrals the presence of the boundary integrals is shown to improve the numerical prediction for various viscous and nearly inviscid flows

on the construction of recurrence relations for the expansion and connection coefficients in series of jacobi polynomials formulae expressing explicitly the jacobi coefficients of a general order derivative integral of an infinitely differentiable function in terms of its original expansion coefficients and formulae for the derivatives integrals of jacobi polynomials in terms of jacobi polynomials themselves are stated a formula for the jacobi coefficients of the moments of one single jacobi polynomial of certain degree is proved another formula for the jacobi coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its original expanded coefficients is also given a simple approach in order to construct and solve recursively for the connection coefficients between jacobi jacobi polynomials is described explicit formulae for these coefficients between ultraspherical and jacobi polynomials are deduced of which the chebyshev polynomials of the first and second kinds and legendre polynomials are important special cases two analytical formulae for the connection coefficients between laguerre jacobi and hermite jacobi are developed

pseudospectral method for fisher equation in a disk abstract in this paper we develop a mixed jacobi fourier pseudospectral method for solving the fisher equation in a disc fisher s equation always plays a large role in many fields such as logistic population growth models tissue engineering nuclear reactions and neurophysiology etc it is very important to study how to solve these equations numerically in this work we employ the generalized jacobi approximation to simulate the singularity of solutions at the regional center some mixed jacobi fourier interpolation approximation results are established which play important roles in numerical simulation of various problems defined in a disc as an application the mixed jacobi fourier pseudospectral scheme is provided for the fisher equation in a disk the generalized stability and convergence of the proposed scheme are proved some numerical results are presented to demonstrate the efficiency of this new algorithm

mixed legendre hermite spectral method for heat transfer in an infinite plate nonisotropic heat transfer in an infinite plate is considered a weak formulation is derived which is appropriate for its numerical simulation the mixed legendre hermite spectral method is proposed for this problem its convergence is proved numerical results show the efficiency of this new approach

a nonlinear subgrid method for advection diffusion problems abstract this work presents a general framework for approximating advection diffusion equations based on principles of scale separation a two level decomposition of the discrete approximation space is performed and the local problem is modified to capture both local and nonlocal discontinuities the new feature is the local control resulting from decomposing the velocity field into the resolved and unresolved scales and requiring the satisfaction of the discrete model problem at the element level for a minimum kinetic energy associated to the unresolved scales this procedure leads to a nonlinear subgrid model that acts only on the unresolved scales but does not require any tuned up parameter it can be considered a self adaptive method such that the amount of the subgrid viscosity is automatically introduced according to the residual of the resolved scale at element level

on the gibbs phenomenon iv recovering exponential accuracy in a subinterval from a gegenbauer partial sum of a piecewise analytic function we continue the investigation of overcoming gibbs phenomenon i e obtaining exponential accuracy at all points including at the discontinuities themselves from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function we show that if we are given the first n expansion coefficients of an l 2 function f x in terms of either the trigonometrical polynomials or the chebyshev or legendre polynomials we can construct an exponentially convergent approximation to the point values of f x in any sub interval in which it is analytic

the optimal convergence of the h p version of the finite element method with quasi uniform meshes in the framework of the jacobi weighted besov spaces we analyze the convergence of the h p version of finite element solutions on quasi uniform meshes and the lower and upper bounds of errors for elliptic problems on polygons both lower and upper bounds are proved to be optimal in h and p which leads to the optimal convergence of the h p version of the finite element method with quasi uniform meshes for elliptic problems on polygons the results proved for the h p version include the h version with quasi uniform meshes and the p version with quasi uniform degrees as two special cases

mixed spectral method for three dimensional exterior problems using spherical harmonic and generalized laguerre functions in this paper we develop the mixed spectral method for three dimensional exterior problems using spherical harmonic and generalized laguerre functions some basic approximation results are established the mixed spectral schemes are proposed for two model problems their convergences are proved numerical results demonstrate the efficiency of this new approach

approximation of some diffusion evolution equations in unbounded domains by hermite functions spectral and pseudospectral approximations of the heat equation are analyzed the solution is represented in a suitable basis constructed with hermite polynomials stability and convergence estimates are given and numerical tests are discussed

a discontinuity capturing crosswind dissipation for the finite element solution of the convection diffusion equation abstract to avoid the local oscillations that still remain using the streamline upwind petrov galerkin formulation for the scalar convection diffusion equation the introduction of a nonlinear crosswind dissipation is proposed it is shown that the method is less overdiffusive than other discontinuity capturing techniques and has better numerical behavior the design of the crosswind diffusion is based on the study of the discrete maximum principle for some simple cases

a rational approximation and its applications to nonlinear partial differential equations on the whole line an orthogonal system of rational functions is discussed some inverse inequalities imbedding inequalities and approximation results are obtained two model problems are considered the stabilities and convergences of proposed rational spectral schemes and rational pseudospectral schemes are proved the techniques used in this paper are also applicable to other problems on the whole line numerical results show the efficiency of this approach

an accurate fic fem formulation for the 1d advection diffusion reaction equation abstract in this paper we present an accurate stabilized fic fem formulation for the 1d advection diffusion reaction equation in the exponential and propagation regimes using two stabilization parameters both the steady state and transient solutions are considered the stabilized formulation is based on the standard galerkin fem solution of the governing differential equations derived via the finite increment calculus fic method the steady state problem is considered first the optimal value of the two stabilization parameters ensuring an exact nodal fem solution using uniform meshes of linear 2 noded elements is obtained in the absence of the absorption term the formulation simplifies to the standard one parameter petrov galerkin method for the advection diffusion problem for the diffusion reaction case one stabilization parameter is just needed and the diffusion type stabilization term is identical to that obtained by felippa and onate 2007 using a variational fic approach a procedure for computing the stabilization parameters for the transient problem is proposed the accuracy of the new fic fem formulation is demonstrated in the solution of steady state and transient 1d advection diffusion radiation problems for a the range of physical parameters and boundary conditions

accurate fic fem formulation for the multidimensional steady state advection diffusion absorption equation abstract in this paper we present an accurate stabilized fic fem formulation for the multidimensional steady state advection diffusion absorption equation the stabilized formulation is based on the galerkin fem solution of the governing differential equations derived via the finite increment calculus fic method using two stabilization parameters the value of the two stabilization parameters ensuring an accurate nodal fem solution using uniform meshes of linear elements is obtained from the optimal values for the 1d problem the accuracy of the new fic fem formulation is demonstrated in the solution of 2d steady state advection diffusion absorption problems for a range of physical parameters and boundary conditions

finite element formulation for convective diffusive problems with sharp gradients using finite calculus abstract a finite element method fem for steady state convective diffusive problems presenting sharp gradients of the solution both in the interior of the domain and in boundary layers is presented the necessary stabilization of the numerical solution is provided by the finite calculus fic approach the fic method is based in the solution by the galerkin fem of a modified set of governing equations which include characteristic length parameters it is shown that the fic balance equation for the multidimensional convection diffusion problem written in the principal curvature axes of the solution introduces an orthotropic diffusion which stabilizes the numerical solution both in smooth regions as well in the vicinity of sharp gradients the dependence of the stabilization terms with the principal curvature directions of the solution makes the method non linear details of the iterative scheme to obtain stabilized results are presented together with examples of application which show the efficiency and accuracy of the approach

generalized laguerre interpolation and pseudospectral method for unbounded domains in this paper error estimates for generalized laguerre gauss type interpolations are derived in nonuniformly weighted sobolev spaces weighted with omega alpha beta x x alpha e beta x alpha 1 beta 0 generalized laguerre pseudospectral methods are analyzed and implemented two model problems are considered the proposed schemes keep spectral accuracy and with suitable choice of basis functions lead to sparse and symmetric linear systems

hermite pseudospectral approximations an error estimate abstract we find an error bound for the pseudospectral approximation of a function in terms of hermite functions h n x e x 2 h n x in certain weighted sobolev spaces we use properties of hermite polynomials as well as an asymptotic expression for their largest zeroes to achieve the mentioned estimate

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

error estimates for spectral and pseudospectral approximations of hyperbolic equations the linear one dimensional advection equation with variable coefficients and nonperiodic boundary conditions is considered we investigate some spectral and pseudospectral collocation approximations of jacobi and legendre type in the latter case the boundary condition is implicitly treated stability and convergence results in weighted norms are given some precise estimates of the rate of convergence in terms of the regularity of the continuous solution are derived

spectral and pseudo spectral approximations of the navier stokes equations galerkin and collocation approximations by trigonometric polynomials to the stationary navier stokes equations with periodic boundary conditions are analyzed stability results and optimal rates of convergence for both velocity and pressure in sobolev spaces are proven essential tools to carry out this analysis are some abstract results concerning general polynomial approximations to nonsingular solutions of nonlinear problems brezzi rappag raviart numer math 36 1980 pp 1 25 maday quarteroni rairo numer anal 1980

generalized hermite spectral method and its applications to problems in unbounded domains in this paper we develop a spectral method based on generalized hermite functions with weight chi x equiv1 we also establish some basic results on generalized hermite orthogonal approximations which play an important role in spectral methods as examples the generalized ginzburg landau equation in a population problem and an elliptic equation with a harmonic potential are considered related spectral schemes are proposed and their convergence is proved numerical results demonstrate the spectral accuracy of this approach

chebyshev rational spectral and pseudospectral methods on a semi infinite interval a weighted orthogonal system on the half line based on the chebyshev rational functions is introduced basic results on chebyshev rational approximations of several orthogonal projections and interpolations are established to illustrate the potential of the chebyshev rational spectral method a model problem is considered both theoretically and numerically error estimates for the chebyshev rational spectral and pseudospectral methods are established preliminary numerical results agree well with the theoretical estimates and demonstrate the effectiveness of this approach

generalized hermite spectral method matching asymptotic behaviors in this paper we propose the generalized hermite spectral method by using a family of new generalized hermite functions which are mutually orthogonal with the weight function 1 x 2 c c being an arbitrary real number we establish the basic results on the corresponding orthogonal approximation and interpolation which simulate the asymptotic behaviors of approximated functions at infinity reasonably as examples of applications the spectral schemes are provided for two model problems numerical results demonstrate their spectral accuracy in space

the discontinuous enrichment method we propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it the enrichment contains free space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field continuity of the enrichment across element interfaces is enforced weakly by lagrange multipliers in this manner we expect to attain high coarse mesh accuracy without significant degradation of conditioning assuring good performance of the computation at any mesh resolution examples of application to acoustics and advection diffusion are presented

combined hermite spectral finite difference method for the fokker planck equation the convergence of a class of combined spectral finite difference methods using hermite basis applied to the fokker planck equation is studied it is shown that the hermite based spectral methods are convergent with spectral accuracy in weighted sobolev space numerical results indicating the spectral convergence rate are presented a velocity scaling factor is used in the hermite basis and is shown to improve the accuracy and effectiveness of the hermite spectral approximation with no increase in workload some basic analysis for the selection of the scaling factors is also presented

possibilities of finite calculus in computational mechanics abstract the expression finite calculus refers to the derivation of the governing differential equations in mechanics by invoking balance of fluxes forces etc in a domain of finite size the governing equations resulting from this approach are different from those of infinitessimal calculus theory and they incorporate new terms depending on the dimensions of the balance domain the new modified equations allow to derive naturally stabilized numerical schemes using finite element finite difference finite volume or meshless methods the paper briefly discusses the possibilities of the modified governing equations derived via the finite calculus technique for the numerical solution of convection diffusion problems incompressible flow and incompressible solid mechanic problems and strain localization problems

finite element methods for first order hyperbolic systems with particular emphasis on the compressible euler equations a petrov galerkin finite element formulation is presented for first order hyperbolic systems of conservation laws with particular emphasis on the compressible euler equations applications of the methodology are made to one and two dimensional steady and unsteady flows with shocks results obtained suggest the potential of the type of methods developed

hp spectral element approximation for integral state constrained optimal control problems governed by harmonic equations abstract in this paper an integral state constrained optimal control problem governed by first bi harmonic equation and its h p spectral element approximation are considered we first present the optimality conditions of the continuous and discrete optimal control problems respectively then a priori error estimates of h p spectral element approximation for the optimal control problem are derived rigorously moreover a posteriori error estimates are established for the coupled state and control approximations in detail such estimators can be used to construct reliable adaptive methods for optimal control problems

fic fem formulation for the multidimensional transient advection diffusion absorption equation abstract in this paper we present a stabilized fic fem formulation for the multidimensional transient advection diffusion absorption equation the starting point is the non local form of the governing equations for the multidimensional transient advection diffusion absorption problems obtained via the finite increment calculus fic procedure the fic governing equations have a residual form that introduces a characteristic length vector that depends on streamline absorption and shock capturing stabilization parameters as well as on a characteristic element size that ensures a stabilized numerical solution using a standard galerkin fem the value of the stabilization parameters is obtained as an extension of the steady state form the accuracy of the fic fem formulation is verified in the solution of several transient advection diffusion absorption problems using regular meshes of 3 noded triangles and 4 noded quadrilaterals

the spectral methods for parabolic volterra integro differential equations in this paper we study the numerical solutions to parabolic volterra integro differential equations in one dimensional bounded and unbounded spatial domains in a bounded domain the given parabolic volterra integro differential equation is converted to two equivalent equations then a legendre collocation method is used to solve them and finally a linear algebraic system is obtained for an unbounded case we use the algebraic mapping to transfer the problem on a bounded domain and then apply the same presented approach for the bounded domain in both cases some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method

stabilization of incompressibility and convection through orthogonal sub scales in finite element methods two apparently different forms of dealing with the numerical instability due to the incompressibility constraint of the stokes problem are analyzed in this paper the first of them is the stabilization through the pressure gradient projection which consists of adding a certain least squares form of the difference between the pressure gradient and its l2 projection onto the discrete velocity space in the variational equations of the problem the second is a sub grid scale method whose stabilization effect is very similar to that of the galerkin least squares gls method for the stokes problem it is shown here that the first method can also be recast in the framework of sub grid scale methods with a particular choice for the space of sub scales this leads to a new stabilization procedure whose applicability to stabilize convection is also studied in this paper

the sinc collocation method for solving the thomas fermi equation a numerical technique for solving nonlinear ordinary differential equations on a semi infinite interval is presented we solve the thomas fermi equation by the sinc collocation method that converges to the solution at an exponential rate this method is utilized to reduce the nonlinear ordinary differential equation to some algebraic equations this method is easy to implement and yields very accurate results

exponentially accurate nonconforming least squares spectral element method for elliptic problems on unbounded domain abstract non conforming approximation methods are becoming increasingly popular because of the potential to apply to multi material and multi model analysis for both bounded and unbounded domains in this paper we present a least square approximation based method to solve the one or two dimensional elliptic problems on an unbounded domain the method gives exponential accuracy and shows superior performance when compared to other numerical methods differentiability estimates and the main stability estimate theorem using a non conforming spectral element method are also discussed the exponential convergence rate of the proposed method is also shown through rigorous error estimate and specific numerical examples

gegenbauer approximation and its applications to differential equations on the whole line abstract a gegenbauer approximation is discussed several imbedding inequalities and inverse inequalities are obtained some approximation results are given by variable transformation differential equations on the whole line are changed to certain equations on a finite interval gegenbauer polynomials are used for their numerical solutions the stabilities and convergences of proposed schemes are proved the main idea and techniques used in this paper are also applicable to other multiple dimensional problems in unbounded domains

legendre and chebyshev spectral approximations of burgers equation we give error estimates for both legendre and chebyshev spectral approximations of the steady state burgers problem u xx lambda uu x f 0 in i 1 1 u 1 u 1 0 to do that we prove some abstract approximation properties of orthogonal projection operators in some weighted sobolev spacesh s i for both legendre and chebyshev weights

a stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation abstract a stabilized finite element formulation for incompressible viscous flows is derived the starting point are the modified navier stokes equations incorporating naturally the necessary stabilization terms via a finite increment calculus fic procedure application of the standard finite element galerkin method to the modified differential equations leads to a stabilized discrete system of equations overcoming the numerical instabilities emanating from the advective terms and those due to the lack of compatibility between approximate velocity and pressure fields the fic method also provides a natural explanation for the stabilization terms appearing in all equations for both the navier stokes and the simpler stokes equations transient solution schemes with enhanced stabilization properties are also proposed finally a procedure for computing the stabilization parameters is presented

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100