solving boundary value problems integral and integro differential equations using gegenbauer integration matrices we introduce a hybrid gegenbauer ultraspherical integration method hgim for solving boundary value problems bvps integral and integro differential equations the proposed approach recasts the original problems into their integral formulations which are then discretized into linear systems of algebraic equations using gegenbauer integration matrices gims the resulting linear systems are well conditioned and can be easily solved using standard linear system solvers a study on the error bounds of the proposed method is presented and the spectral convergence is proven for two point bvps tpbvps comparisons with other competitive methods in the recent literature are included the proposed method results in an efficient algorithm and spectral accuracy is verified using eight test examples addressing the aforementioned classes of problems the proposed method can be applied on a broad range of mathematical problems while producing highly accurate results the developed numerical scheme provides a viable alternative to other solution methods when high order approximations are required using only a relatively small number of solution nodes

a priori and a posteriori analysis of finite volume discretizations of darcy s equations this paper is devoted to the numerical analysis of some finite volume discretizations of darcy s equations we propose two finite volume schemes on unstructured meshes and prove their equivalence with either conforming or nonconforming finite element discrete problems this leads to optimal a priori error estimates in view of mesh adaptivity we exhibit residual type error indicators and prove estimates which allow to compare them with the error in a very accurate way

applications of field theoretic renormalization group methods to reaction diffusion problems we review the application of field theoretic renormalization group rg methods to the study of fluctuations in reaction diffusion problems we first investigate the physical origin of universality in these systems before comparing rg methods to other available analytic techniques including exact solutions and smoluchowski type approximations starting from the microscopic reaction diffusion master equation we then pedagogically detail the mapping to a field theory for the single species reaction ka a k we employ this particularly simple but non trivial system to introduce the field theoretic rg tools including the diagrammatic perturbation expansion renormalization and callan symanzik rg flow equation we demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative dc d expansions with respect to the upper critical dimension dc with these basics established we then provide an overview of more sophisticated applications to multiple species reactions disorder effects l evy flights persistence problems and the influence of spatial boundaries we also analyse field theoretic approaches to non equilibrium phase transitions separating active from absorbing states we focus particularly on the generic directed percolation universality class as well as on the most prominent exception to this class even offspring branching and annihilating random walks finally we summarize the state of the field and present our perspective on outstanding problems for the future

superdiffusive transport due to random velocity fields the motion of a random walk in a medium containing random but spatially correlated velocity fields is discussed this type of disorder generally leads to superdiffusive behavior in which the mean square displacement x2 t grows faster than linearly with time for a two dimensional medium with layers of random velocities in the x direction x2 t is found to increase as t2v with 2v 32 the probability distribution of displacements appears to fit the form p x t t 34 exp x t34 with 1 7 however a lifshitz argument for the tail of the distribution suggests that 43 this discrepancy is yet to be fully resolved we also discuss some intriguing properties in a model with isotropic velocity fields in two dimensions we find v 43 while 3 these values obey the general scaling relation 1 v 1

adaptive finite element methods for elliptic equations with non smooth coefficients nous considerons une equation elliptique du second ordre a coefficients discontinus ou anisotropes dans un domaine borne en dimension 2 ou 3 et sa discretisation par elements finis le but de cet article est de demontrer des estimations d erreur a priori et a posteriori dans une norme appropriee qui soient independantes de la variation des coefficients

integration preconditioning matrix for ultraspherical pseudospectral operators in this paper we give an integration rectangular preconditioner for ultraspherical pseudospectral operators we present an explicit solution of matrix equations with the structure provided by the pseudospectral discretization of the nth order differential operator subject to various types of boundary conditions this occurs in two steps in the first step we introduce an integration rectangular preconditioner in the second step we use a new strategy to impose the boundary conditions on the condition that the coefficients matrix has a known inverse computational examples are given

an accurate solution of parabolic equations by expansion in ultraspherical polynomials abstract an ultraspherical expansion technique is applied to numerically obtain the solution of the third boundary value problem for a linear partial differential equation in a one space variable the method depends basically on the fact that an expansion in a series of ultraspherical polynomials c n for the function and its space derivatives occuring in the partial differential equation is assumed the coefficients of expansion are then determined by reducing the partial differential equation with its boundary and initial conditions to a system of ordinary differential equations for these coefficients this system may be solved analytically or numerically in a step by step manner the expansion for the function enables one to get the sought for approximation for any possible value of the parameter 1 2 numerical experiments show that more accurate results may be obtained by taking to be small and negative the method in its present form may be considered as a generalization of that of dew and scraton the extension of the method to the polar type equations u t p 2 u x 2 k u x x is also considered

on the a stability of runge kutta collocation methods based on orthogonal polynomials some results about the a stability of runge kutta collocation methods based on the zeros and extrema of the ultraspherical or gegenbauer polynomials are presented these methods are implicit and the a stability is an important property usually demanded the routh hurwitz algorithm is used to establish the main results about the a stability of the methods depending on the parameter lambda of the polynomials for low degree methods some analytical results are presented while for higher degree an open problem based on numerical tests is proposed

on the errors incurred calculating derivatives using chebyshev polynomials abstract the severe errors associated with the computation of derivatives of functions approximated by chebyshev polynomials are investigated when using standard chebyshev transform methods it is found that the maximum error in the computed first derivative grows as n2 where n 1 is the number of chebyshev polynomials used to approximate the function the source of the error is found to be magnification of roundoff error by the recursion equation which links coefficients of a function to those of its derivative tight coupling between coefficients enables propagation of errors from high frequency to low frequency modes matrix multiplication techniques exhibit errors of the same order of magnitude however standard methods for computing the matrix elements are shown to be ill conditioned and to magnify the differentiation errors by an additional factor of n2 for both the transform and matrix methods the errors are found to be most severe near the boundaries of the domain where they grow as 1 x 2 1 2 as x approaches 1 comparisons are made with the errors associated with derivatives of functions approximated by fourier series in which case it is reported that the errors only grow linearly with n and are evenly distributed throughout the domain a method for reducing the error is discussed

a strategy for choosing gegenbauer reconstruction parameters for numerical stability the gegenbauer reconstruction method was first proposed in 1992 but in early studies no attempts were made to optimize the relevant parameters of this method these parameters were allowed to grow proportionally with the number of nodes which in many cases resulted in exponential convergence for a selected range of the proportionality constants early studies also made clear that very large error bounds could be expected if these key parameters were not chosen carefully subsequent studies then pointed out that although unrelated to the method s analytically predictable domains of poor accuracy round off errors could also sabotage the method s accuracy the challenge of successfully implementing a gegenbauer reconstruction then rests on understanding the performance trade offs we can expect when choosing the key parameters in accordance with different objectives in this study we propose a new strategy for choosing optimal parameters in the chebyshev gegenbauer reconstruction method specifically to achieve numerical stability this strategy is based on asymptotic analysis as well as minimization problems in one and two dimensions the effectiveness of our approach which could also be applied to a wider selection of polynomials is then illustrated with results from numerical experiments

anomalous diffusion in a unidirectional random velocity field with long range correlations the anomalous diffusion in a unidirectional random velocity field with long range correlations is analysed in directions transverse to the direction of the field or along the field direction critical values of the exponents which characterize the power like falloff of the correlations in the transverse and longitudinal directions and the critical dimension of space are determined the anomalous dimension of the longitudinal diffusion coefficient is also calculated in the first order of the in expansion for several cases of a long range correlated random velocity field

spectral integration and two point boundary value problems a numerical method for two point boundary value problems with constant coefficients is developed which is based on integral equations and the spectral integration matrix for chebyshev nodes the method is stable achieves superalgebraic convergence and requires o n log n operations where n is the number of nodes in the discretization although stable spectral methods have been constructed in the past they have generally been based on reformulating the recurrence relations obtained through spectral differentiation in an attempt to avoid the ill conditioning introduced by that process

a mapped polynomial method for high accuracy approximations on arbitrary grids the focus of this paper is the approximation of analytic functions on compact intervals from their pointwise values on arbitrary grids we introduce a new method for this problem based on mapped polynomial approximation by careful selection of the mapping parameter we ensure both high accuracy of the approximation and an asymptotically optimal scaling of the polynomial degree with the grid spacing as we explain efficient implementation of this method can be achieved using nonuniform fast fourier transforms numerical results demonstrate the efficiency and accuracy of this approach

accuracy enhancement for higher derivatives using chebyshev collocation and a mapping technique a new method is investigated to reduce the roundoff error in computing derivatives using chebyshev collocation methods by using a grid mapping derived by kosloff and tal ezer and the proper choice of the parameter alpha the roundoff error of the k th derivative can be reduced from o n 2k to o n lge k where epsilon is the machine precision and n is the number of collocation points this drastic reduction of roundoff error makes mapped chebyshev methods competitive with any other algorithm in computing second or higher derivatives with large n several other aspects of the mapped chebyshev differentiation matrix are also studied revealing that the mapped chebyshev methods require much less than pi points to resolve a wave the eigenvalues are less sensitive to perturbation by roundoff error and larger time steps can be used for solving pdes all these advantages of the mapped chebyshev methods can be achieved while maintaining spectral accuracy

differentiation by integration with jacobi polynomials in this paper the numerical differentiation by integration method based on jacobi polynomials originally introduced by mboup et al 19 20 is revisited in the central case where the used integration window is centered such a method based on jacobi polynomials was introduced through an algebraic approach 19 20 and extends the numerical differentiation by integration method introduced by lanczos 1956 21 the method proposed here rooted in 19 20 is used to estimate the nth email protected n order derivative from noisy data of a smooth function belonging to at least c n 1 q email protected n in 19 20 where the causal and anti causal cases were investigated the mismodelling due to the truncation of the taylor expansion was investigated and improved allowing a small time delay in the derivative estimation here for the central case we show that the bias error is o h q 2 where h is the integration window length for email protected c n q 2 in the noise free case and the corresponding convergence rate is o d q 1 n 1 q where d is the noise level for a well chosen integration window length numerical examples show that this proposed method is stable and effective

a chebyshev technique for solving nonlinear optimal control problems a numerical technique for solving nonlinear optimal control problems is introduced the state and control variables are expanded in the chebyshev series and an algorithm is provided for approximating the system dynamics boundary conditions and performance index application of this method results in the transformation of differential and integral expressions into systems of algebraic or transcendental expressions in the chebyshev coefficients the optimum condition is obtained by applying the method of constrained extremum for linear quadratic optimal control problems the state and control variables are determined by solving a set of linear equations in the chebyshev coefficients applicability is illustrated with the minimum time and maximum radius orbit transfer problems

error indicators for the mortar finite element discretization of the laplace equation the mortar technique turns out to be well adapted to handle mesh adaptivity in finite elements since it allows for working with nonnecessarily compatible discretizations on the elements of a nonconforming partition of the initial domain the aim of this paper is to extend the numerical analysis of residual error indicators to this type of methods for a model problem and to check their efficiency thanks to some numerical experiments

fast accurate and small scale direct trajectory optimization using a gegenbauer transcription method this paper reports a novel direct gegenbauer ultraspherical transcription method gtm for solving continuous time optimal control oc problems ctocps with linear nonlinear dynamics and path constraints in elgindy et al 2012 1 we presented a gtm for solving nonlinear ctocps directly for the state and the control variables and the method was tailored to find the best path for an unmanned aerial vehicle mobilizing in a stationary risk environment this article extends the gtm to deal further with problems including higher order time derivatives of the states by solving the ctocp directly for the control u t and the highest order time derivative x n t n z the state vector and its derivatives up to the n 1 th order derivative can then be stably recovered by successive integration moreover we present our solution method for solving linear quadratic regulator lqr problems as we aim to cover a wider collection of ctocps with the concrete aim of comparing the efficiency of the current work with other classical discretization methods in the literature the proposed numerical scheme fully parameterizes the state and the control variables using gegenbauer expansion series for problems with various order time derivatives of the state variables arising in the cost function dynamical system or path terminal constraints the gtm seeks to fully parameterize the control variables and the highest order time derivatives of the state variables the time horizon is mapped onto the closed interval 0 1 the dynamical system characterized by differential equations is transformed into its integral formulation through direct integration the resulting problem on the finite interval is then transcribed into a nonlinear programming nlp problem through collocation at the gegenbauer gauss gg points the integral operations are approximated by optimal gegenbauer quadratures in a certain optimality sense the reduced nlp problem is solved in the gegenbauer spectral space and the state and the control variables are approximated on the entire finite horizon the proposed method achieves discrete solutions exhibiting exponential convergence using relatively small scale number of collocation points the advantages of the proposed direct gtm over other traditional discretization methods are shown through four well studied oc test examples the present work is a major breakthrough in the area of computational oc theory as it delivers significantly more accurate solutions using considerably smaller numbers of collocation points states and controls expansion terms moreover the gtm produces very small scale nlp problems which can be solved very quickly using the modern nlp software

an improved pseudospectral method for fluid dynamics boundary value problems abstract pseudospectral methods are commonly used to obtain accurate solutions to initial boundary value problems especially with second or higher order derivatives in space the corresponding differentiation matrices tend to have large spurious eigenvalues leading for example to severe time step restrictions in case of explicit time stepping methods we introduce here a new procedure for incorporating boundary conditions which reduces spurious eigenvalues by an order of magnitude or more also allowing more freedoms in the choice of grid point locations

renormalization group analysis of superdiffusion in random velocity fields field theoretic renormalization group theory is applied to the analysis of the asymptotic behaviour of diffusion in random velocity fields each nonvanishing component of which is independent of the coordinate along the direction of that component below the critical dimension dc 3 the corresponding field theory is shown to have a nontrivial infrared stable fixed point of the renormalization group which controls the long time large distance behaviour of the model the anomalous dimension nu has been found exactly in the epsilon 3 d expansion and the leading logarithmic corrections to the normal diffusion at d 3 have been determined for all d 1 or d or d where d is the number of components of the drift field

superdiffusion in random velocity fields abstract stochastic transport in a medium containing random but spatially correlated velocity fields is discussed this type of disorder generally leads to superdiffusive behavior in which the mean square displacement of a random walk x2 t grows faster than linearly with time for a two dimentional layered medium with y dependent random velocities in the x direction ux y x2 x t2 with v 3 4 and with strong sample to sample fluctuations the probability distribution of displacements averaged over environments takes a non gaussian scaling form at large time p x t 3 4 x t 3 4 where u exp u for u 1 with 4 3 for an isotropic two dimensional medium with ux y having the same statistical properties we find v 2 3 and 1 1 3 for the layered medium the moments of the time for a random walker to first reach a distance x in the longitudinal direction increases as t n 1 n x 4 3 possibly modified by logarithmic corrections however the probability that the walk has not reached a distance x in a time t decreases asymptotically as e t with x2 indicating that more than a single time scale is needed to account for first passage properties

analysis and application of fourier gegenbauer method to stiff differential equations the fourier gegenbauer fg method introduced by gottlieb shu solomonoff and vandeven icase report 92 4 hampton va 1992 is aimed at removing the gibbs phenomenon that is recovering the point values of a nonperiodic function from its fourier coefficients in this paper we discuss some numerical aspects of the fg method related to its em pseudospectral implementation in particular we analyze the behavior of the gegenbauer series with a moderate several hundred number of terms suitable for computations we also demonstrate the ability of the fg method to get a spectrally accurate approximation on small subintervals for rapidly oscillating functions or functions having steep profiles bearing on the previous analysis we suggest a high order spectral fourier method for the solution of nonperiodic differential equations it includes a polynomial subtraction technique to accelerate the convergence of the fourier series and the fg algorithm to evaluate derivatives on the boundaries of nonperiodic functions the present hybrid fourier gegenbauer hfg method possesses better resolution properties than the original fg method the precision of this method is demonstrated by solving stiff elliptic problems with steep solutions

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

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

high order gegenbauer integral spectral element method integrated with an adaptive chebyshev optimization strategy for solving linear singularly perturbed differential equations abstract we propose a high order adaptive spectral element method based on gegenbauer polynomial basis for solving singularly perturbed differential equations with dirichlet boundary conditions and a boundary layer at one end the method reduces the singularly perturbed boundary value problem into a single system of linear algebraic equations when the solution is smooth on the solution domain or two systems of linear algebraic equations when the solution has a thin boundary layer the core of the new adaptive strategy lies in constructing a mathematical model of the last few gegenbauer coefficients in the gegenbauer truncated series approximating the second order derivative of the solution which can be readily converted into a minimization problem in a least squares sense and solved using fast optimization techniques an error and convergence analysis of the method for sufficiently smooth solutions is presented a numerical test example with various parameter settings is presented to verify the accuracy effectiveness and applicability of the proposed method numerical comparisons with other rival methods in the literature are included to demonstrate further the power of the proposed method

multigrid for the mortar element method for p1 nonconforming element in this paper a multigrid algorithm is presented for the mortar element method for p1 nonconforming element based on the theory developed by bramble pasciak xu in 5 we prove that the w cycle multigrid is optimal i e the convergence rate is independent of the mesh size and mesh level meanwhile a variable v cycle multigrid preconditioner is constructed which results in a preconditioned system with uniformly bounded condition number

a new robust line search technique based on chebyshev polynomials abstract newton s method is an important and basic method for solving nonlinear univariate and unconstrained optimization problems in this study a new line search technique based on chebyshev polynomials is presented the proposed method is adaptive where it determines a descent direction at each iteration and avoids convergence to a maximum point approximations to the first and the second derivatives of a function using high order pseudospectral differentiation matrices are derived the efficiency of the new method is analyzed in terms of the most popular and widely used criterion in comparison with newton s method using seven test functions

automatic spectral collocation for integral integro differential and integrally reformulated differential equations automatic chebyshev spectral collocation methods for fredholm and volterra integral and integro differential equations have been implemented as part of the chebfun software system this system enables a symbolic syntax to be applied to numerical objects in order to pose and solve problems without explicit references to discretization the same objects can be used in matrix free iterative methods in linear algebra in order to avoid very large dense matrices or allow application to problems with nonsmooth coefficients as a further application of the ability to implement operator equations a method of greengard 1 for the recasting of differential equations as integral equations is generalized to mth order boundary value and generalized eigenvalue problems in the integral form large condition numbers associated with differentiation matrices in high order problems are avoided the ability to implement the recasting process generally follows from implementation of the operator expressions in chebfun the integral method also can be extended to first order systems although chebfun syntax does not currently allow easy implementation in this case

state control spectral chebyshev parameterization for linearly constrained quadratic optimal control problems in this paper we propose a computationally attractive numerical method for determining the optimal control of constrained linear dynamic systems with a quadratic performance the method is based upon constructing the mth degree interpolating polynomials using chebyshev nodes to approximate the control and the state vectors the system dynamics are collocated at chebyshev nodes the performance index is discretized by a cell averaging method the state and control inequality constraints are converted into algebraic inequalities through collocation at the nodes the linear quadratic optimal control problem is thereby transformed into a quadratic programming one simulation studies demonstrate computational advantages relative to a standard riccati method a classical chebyshev based method fourier based method and other methods in the literature

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

numerical approximations using chebyshev polynomial expansions el gendi s method revisited we present numerical solutions for differential equations by expanding the unknown function in terms of chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema or zeros of the chebyshev polynomial of order n el gendi s method the solutions are exact at these points apart from round off computer errors and the convergence of other numerical methods used in solving the linear system of equations applications to initial value problems in time dependent quantum field theory and second order boundary value problems in fluid dynamics are presented

spectrally accurate solution of nonperiodic differential equations by the fourier gegenbauer method it is well known that the fourier partial sum of an analytic nonperiodic function supported on a finite interval converges slowly inside the interval and exhibits o 1 spurious oscillations near the boundaries the gibbs phenomenon an effective algorithm which allows one to completely overcome the gibbs phenomenon was developed in j comput appl math 43 1992 pp 81 92 the basic concept of this approach consists of the re expansion of the fourier partial sums into the rapidly convergent gegenbauer series in this paper we extend the fourier gegenbauer f g method of j comput appl math 43 1992 pp 81 92 to the evaluation of the spatial derivatives of a piecewise analytic function also we apply this method to the solution of nonperiodic boundary value problems although the derivatives of a discontinuous function are not in l2 the exponential convergence of the truncated gegenbauer series can be proved and the rate of convergence can be estimated the solution of differential equations is accomplished in two steps first a particular solution with arbitrary boundary conditions is constructed using the f g method this particular solution is then corrected to satisfy the prescribed boundary conditions of the problem by adding a proper linear combination of homogeneous solutions for boundary layer problems the intermediate particular solution has steep profiles near the boundaries these steep regions introduce a large error into the final solution which presumably has a smooth profile on the whole interval a method which compensates for this loss of accuracy by using the appropriately constructed homogeneous solutions is proposed

determining analyticity for parameter optimization of the gegenbauer reconstruction method the gegenbauer reconstruction method effectively eliminates the gibbs phenomenon and restores exponential accuracy to the approximations of piecewise smooth functions recent investigations show that its success depends upon choosing parameters in such a way that the regularization and the truncation error estimates are equally considered this paper shows that the underlying analyticity of the function in smooth regions plays a critical role in the regularization error estimate hence we develop a technique that first analyzes the behavior of the function in its regions of smoothness and then applies this knowledge to refine the regularization error estimate such refinement yields better parameter choices for the gegenbauer reconstruction method and is confirmed both by better accuracy and more robustness in the approximation of piecewise smooth functions

a preconditioned implementation of pseudospectral methods on arbitrary grids in this article a preconditioner for pseudospectral methods in terms of expansion coefficients on arbitrary grids is presented the preconditioner is first developed for application in ordinary differential equations and then is generalized for partial differential equations a fully discrete pseudospectral approximation is then used to solve bivariate problems this preconditioner in effect makes the pseudospectral methods insensitive to specific grid points the numerical results show the efficiency of the method

a posteriori error analysis of the fully discretized time dependent coupled darcy and stokes equations abstract we present an a posteriori error analysis of the fully discretized time dependent darcy and stokes equations that models laminar fluid flow over a porous medium in two or three dimensional connected open domains which are coupled via appropriate matching conditions on the interface the problem is discretized by the backward euler scheme in time and finite elements in space

integration matrix based on arbitrary grids with a preconditioner for pseudospectral method a matrix representation of integration for arbitrary grids is introduced suitable results are then obtained to be used along with differentiation matrix preconditioner to implement pseudospectral method on integro differential equations using arbitrary grids numerical examples are given to clarify the efficiency of the new method

a spectral collocation method based on integrated chebyshev polynomials for two dimensional biharmonic boundary value problems this paper reports a new spectral collocation method for numerically solving two dimensional biharmonic boundary value problems the construction of the chebyshev approximations is based on integration rather than conventional differentiation this use of integration allows i the imposition of the governing equation at the whole set of grid points including the boundary points and ii the straightforward implementation of multiple boundary conditions the performance of the proposed method is investigated by considering several biharmonic problems of first and second kinds more accurate results and higher convergence rates are achieved than with conventional differential methods

generation of higher order pseudospectral integration matrices a new explicit expression of the higher order pseudospectral integration matrices is presented using an explicit formula for computing iterated integrals of chebyshev polynomials applications to initial value problems boundary value problems linear integral and integro differential equations are presented the present numerical results are in satisfactory agreement with the exact solutions and show the advantages of this method to some other known methods

the pseudospectral legendre method for discretizing optimal control problems this paper presents a computational technique for optimal control problems including state and control inequality constraints the technique is based on spectral collocation methods used in the solution of differential equations the system dynamics are collocated at legendre gauss lobatto points the derivative x spl dot t of the state x t is approximated by the analytic derivative of the corresponding interpolating polynomial state and control inequality constraints are collocated at legendre gauss lobatto nodes the integral involved in the definition of the performance index is discretized based on the gauss lobatto quadrature rule the optimal control problem is thereby converted into a mathematical programming program thus existing well developed optimization algorithms may be used to solve the transformed problem the method is easy to implement capable of handling various types of constraints and yields very accurate results illustrative examples are included to demonstrate the capability of the proposed method and a comparison is made with existing methods in the literature

mortar finite element discretization for the flow in a nonhomogeneous porous medium abstract darcy s equations model the flow of a viscous incompressible fluid in a rigid porous medium one of the parameters of the system depends on the permeability of the medium and when this medium is not homogeneous the variations of the parameter could be very high to handle this phenomenon we propose a discretization of the model that relies on the mortar finite element method indeed the idea is to construct a decomposition of the domain such that the permeability is constant on each element of the partition and to use independent meshes on the different subdomains we perform the a priori and a posteriori analysis of this discretization and present some numerical experiments which are in good coherency with the results of the analysis

optimal gegenbauer quadrature over arbitrary integration nodes this paper treats definite integrations numerically using gegenbauer quadratures the novel numerical scheme introduces the idea of exploiting the strengths of the chebyshev legendre and gegenbauer polynomials through a unified approach and using a unique numerical quadrature in particular the developed numerical scheme employs the gegenbauer polynomials to achieve rapid rates of convergence of the quadrature for the small range of the spectral expansion terms for a large scale number of expansion terms the numerical quadrature has the advantage of converging to the optimal chebyshev and legendre quadratures in the l norm and l 2 norm respectively the key idea is to construct the gegenbauer quadrature through discretizations at some optimal sets of points of the gegenbauer gauss gg type in a certain optimality sense we show that the gegenbauer polynomial expansions can produce higher order approximations to the definite integrals 1 x if x dx of a smooth function f x c 1 1 for the small range by minimizing the quadrature error at each integration point x i through a pointwise approach the developed gegenbauer quadrature can be applied for approximating integrals with any arbitrary sets of integration nodes exact integrations are obtained for polynomials of any arbitrary degree n if the number of columns in the developed gegenbauer integration matrix gim is greater than or equal to n the error formula for the gegenbauer quadrature is derived moreover a study on the error bounds and the convergence rate shows that the optimal gegenbauer quadrature exhibits very rapid convergence rates faster than any finite power of the number of gegenbauer expansion terms two efficient computational algorithms are presented for optimally constructing the gegenbauer quadrature we illustrate the high order approximations of the optimal gegenbauer quadrature through extensive numerical experiments including comparisons with conventional chebyshev legendre and gegenbauer polynomial expansion methods the present method is broadly applicable and represents a strong addition to the arsenal of numerical quadrature methods

on the coefficients of integrated expansions of ultraspherical polynomials the problem of expressing the coefficients of an expansion of orthogonal polynomials that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion is considered a formula is proved for the ultraspherical polynomials of which the chebyshev and legendre polynomials are important special cases

short note pade gegenbauer suppression of runge phenomenon in the diagonal limit of gegenbauer approximations boyd shows in 3 that for a function with singularities in the imaginary plane the gegenbauer expansion of the function may have runge type oscillations so large that all hope of convergence for the method is lost we consider the use of pade gegenbauer interpolants to resolve these oscillations we describe a fast interpolation based reconstruction for computing the pade gegenbauer approximants and demonstrate its excellent performance on test examples this method resolves the oscillations and restores accuracy to the gegenbauer approximation of a function with singularities in the imaginary plane without measuring the proximity of these off axis singularities