properties of some weighted sobolev spaces and application to spectral approximations the weighted sobolev spaces on a square whose weight is a given power of the product of the distances to the edges are introduced trace theorems related to these spaces are proved then a regularity result for the dirichlet problem for the laplace operator is proved next several projection operators with polynomial values are considered for which approximation results in weighted norms are stated finally a collocation spectral method for the dirichlet problem for the laplace operator with inhomogeneous boundary conditions is analyzed this work includes new results on chebyshev approximation

the use of modulated splines for the reconstruction of band limited signals a class of kernels of short duration is developed for interpolating band limited signals the class consists of modulated splines and utilizes the rapid attenuation of their fourier transform the design is based on a minimax mean square error criterion

sensitivity analysis and stochastic simulations of non equilibrium plasma flow we study the parametric uncertainties involved in plasma flows and apply stochastic sensitivity analysis to rank the importance of all inputs to guide large scale stochastic simulations specifically we employ different gradient based sensitivity methods namely morris multi element probabilistic collocation method on sparse grids quasi monte carlo and monte carlo methods these approaches go beyond the standard one at a time sensitivity analysis and provide a measure of the non linear interaction effects for the uncertain inputs the objective is to perform systematic stochastic simulations of plasma flows treating only as stochastic processes the inputs with the highest sensitivity index hence reducing substantially the computational cost two plasma flow examples are presented to demonstrate the capability and efficiency of the stochastic sensitivity analysis the first one is a two fluid model in a shock tube whereas the second one is a one fluid two temperature model in flow past a cylinder copyright 2009 john wiley sons ltd

polynomial interpolation results in sobolev spaces abstract in this paper we derive new results related to the lagrange polynomial interpolation on the nodes of gauss type quadrature formulas in particular we obtain asymptotic error estimates of the same order as the distance to the best fit in the polynomial space

analysis of spectral projectors in one dimensional domains in this paper we analyze a class of projection operators with values in a subspace of polynomials these projection operators are related to the hubert spaces involved in the numerical analysis of spectral methods they are in the first part of the paper the standard sobolev spaces and in the second part some weighted sobolev spaces the weight of which is related to the orthogonality relation satisfied by the chebyshev polynomials these results are used to study the approximation of a model fourth order problem

sampling theory for not necessarily band limited functions a historical overview shannon s sampling theorem is one of the most powerful results in signal analysis the aim of this overview is to show that one of its roots is a basic paper of de la vallee poussin of 1908 the historical development of sampling theory from 1908 to the present especially the matter dealing with not necessarily band limited functions which includes the duration limited case actually studied in 1908 is sketched emphasis is put on the study of error estimates as well as on the delicate point wise behavior of sampling sums at discontinuity points of the signal to be reconstructed

convergence analysis of spectral collocation methods for a singular differential equation solutions of partial differential equations with coordinate singularities often have special behavior near the singularities which forces them to be smooth special treatment for these coordinate singularities is necessary in spectral approximations in order to avoid degradation of accuracy and efficiency it has been observed numerically in the past that for a scheme to attain high accuracy it is unnecessary to impose all the pole conditions the constraints representing the special solution behavior near singularities in this paper we provide a theoretical justification for this observation specifically we consider an existing approach which uses a pole condition as the boundary condition at a singularity and solves the reformulated boundary value problem with a commonly used gauss lobatto collocation scheme spectral convergence of the legendre and chebyshev collocation methods is obtained for a singular differential equation arising from polar and cylindrical geometries

on error estimates for galerkin spectral discretizations for parabolic problems with nonsmooth initial data we analyze the legendre and chebyshev spectral galerkin semi discretizations of a one dimensional homogeneous parabolic problem with non constant coefficients we present error estimates for both smooth and nonsmooth data in the chebyshev case a limit in the order of approximation is established on the contrary in the legendre case we find an arbitrary high order of convegence

singular unlocking transition in the winfree model of coupled oscillators the winfree model consists of a population of globally coupled phase oscillators with randomly distributed natural frequencies as the coupling strength and the spread of natural frequencies are varied the various stable states of the model can undergo bifurcations nearly all of which have been characterized previously the one exception is the unlocking transition in which the frequency locked state disappears abruptly as the spread of natural frequencies exceeds a critical width viewed as a function of the coupling strength this critical width defines a bifurcation curve in parameter space for the special case where the frequency distribution is uniform earlier work had uncovered a puzzling singularity in this bifurcation curve here we seek to understand what causes the singularity using the poincar e lindstedt method of perturbation theory we analyze the locked state and its associated unlocking transition first for an arbitrary distribution of natural frequencies and then for discrete systems of n oscillators we confirm that the bifurcation curve becomes singular for a continuum uniform distribution yet find that it remains well behaved for any finite n suggesting that the continuum limit is responsible for the singularity

space time spectral method for a weakly singular parabolic partial integro differential equation on irregular domains abstract the spectral method is proposed for the partial integro differential equations with a weakly singular kernel on irregular domains the space discretization is based on the nodal spectral element method using the lagrange polynomials basis associated with the gauss lobatto legendre quadrature nodes also the model is discretized in time with the legendre spectral galerkin method the discretization leads to conversion of the problem to a sylvester matrix equation which can be solved efficiently by the qz algorithm gardiner et al 1992 the convergence of the method is proven by providing a priori l 2 error estimate numerical results illustrate the efficiency and spectral accuracy of the proposed method

localization of vibrations by structural irregularity an investigation of the localization phenomenon is presented for disrodered structural systems consisting of weakly coupled component systems emphasis is placed on the development of a perturbation method that allows one to obtain the localized modes of vibration of the disordered system from the modes of the individual subsystems the principal feature of this perturbation method is that it not only considers the variations of the parameters as perturbations but it also treats the small coupling between subsystems as a perturbation such a perturbation analysis is cost effective as compared to a global eigenvalue analysis of the entire system and proves to be very accurate moreover it provides physical insight into the localization phenomenon and allows one to formulate a criterion that predicts the occurrence of strongly localized modes results are obtained for a chain of single degree of freedom coupled oscillators but these can be readilt generalized to deal with chains of multi degree of freedom component systems

the schwarz algorithm for spectral methods recently the schwarz alternating method has been successfully coupled to spatial discretizations of spectral type in order to solve boundary value problems in complex geometries with infinite order accuracy in this paper a simple version of the method is considered a proof of its convergence is given in the energy norm exploiting the properties of discrete harmonic polynomials and a discrete maximum principle for spectral methods more general situations can be handled theoretically in one space dimension

spectral element discretization of the circular driven cavity part ii the bilaplacian equation this paper is devoted to the spectral element discretization of the laplace equation in a disk when provided with discontinuous boundary data relying on an appropriate variational formulation we propose a discrete problem and prove its convergence the use of weighted sobolev spaces to treat the discontinuity of the boundary conditions also allows for improving the order of convergence the results of the numerical experiments we present are in agreement with the theoretical ones

initialization and inner product computations of wavelet transform by interpolatory subdivision scheme the initialization of wavelet transforms and the inner product computations of wavelets with their derivatives are very important in many applications in this correspondence the interpolatory subdivision scheme iss is proposed to solve these problems efficiently we introduce a general procedure to compute the exact values of derivatives of the interpolatory fundamental function and then derive a fast recursive algorithm for the realization of the initialization and inner product evaluations error analysis of the algorithm and its comparison with other approaches are discussed numerical experiments demonstrate the high performance of the algorithm

uncertainty quantification via random domain decomposition and probabilistic collocation on sparse grids quantitative predictions of the behavior of many deterministic systems are uncertain due to ubiquitous heterogeneity and insufficient characterization by data we present a computational approach to quantify predictive uncertainty in complex phenomena which is modeled by partial differential equations with uncertain parameters exhibiting multi scale variability the approach is motivated by flow in random composites whose internal architecture spatial arrangement of constitutive materials and spatial variability of properties of each material are both uncertain the proposed two scale framework combines a random domain decomposition rdd and a probabilistic collocation method pcm on sparse grids to quantify these two sources of uncertainty respectively the use of sparse grid points significantly reduces the overall computational cost especially for random processes with small correlation lengths a series of one two and three dimensional computational examples demonstrate that the combined rdd pcm approach yields efficient robust and non intrusive approximations for the statistics of diffusion in random composites

polynomial approximation of divergence free functions etude de la meilleure approximation d une fonction sans divergence par un polynome algebrique ou trigonometrique sans divergence etablissement d une estimation optimale

a postprocessed galerkin method with chebyshev or legendre polynomials we present an approximate inertial manifold based postprocess to enhance chebyshev or legendre spectral galerkin methods we prove that the postprocess improves the order of convergence of the galerkin solution yielding the same accuracy as the nonlinear galerkin method numerical experiments show that the new method is computationally more efficient than galerkin and nonlinear galerkin methods new approximation results for chebyshev polynomials are presented

the poisson equation in axisymmetric domains with conical points this paper analyzes the effects of conical points on the rotation axis of axisymmetric domains r3 on the regularity of the fourier coefficients un n z of the solution u of the dirichlet problem for the poisson equation u f in the asymptotic behavior of the coefficients un near the conical points is carefully described and for f l2 it is proved that if the interior opening angle c at the conical point is greater than a certain critical angle then the regularity of the coefficient u0 will be lower than expected moreover it is shown that conical points on the rotation axis of the axisymmetric domain do not affect the regularity of the coefficients un n 0 an approximation of the critical angle is established numerically and a priori error estimate for the fourier finite element solutions in the norm of w21 is given

the dynamics of large scale arrays of coupled resonators abstract this work describes an analytical framework suitable for the analysis of large scale arrays of coupled resonators including those which feature amplitude and phase dynamics inherent element level parameter variation nonlinearity and or noise in particular this analysis allows for the consideration of coupled systems in which the number of individual resonators is large extending as far as the continuum limit corresponding to an infinite number of resonators moreover this framework permits analytical predictions for the amplitude and phase dynamics of such systems the utility of this analytical methodology is explored through the analysis of a system of n non identical resonators with global coupling including both reactive and dissipative components physically motivated by an electromagnetically transduced microresonator array in addition to the amplitude and phase dynamics the behavior of the system as the number of resonators varies is investigated and the convergence of the discrete system to the infinite n limit is characterized

the shannon sampling series and the reconstruction of signals in terms of linear quadratic and cubic splines the sine kernel function of the sampling series is replaced by spline functions having compact support all built up from the b splines m n the resulting generalized sampling series reduces to finite sums so that no truncation error occurs moreover the approximation error generally decreases more rapidly than for the classical series when w tends to infinity 1 w being the distance between the sampling points for the kernel function varphi t 5m 4 t 4m 5 t with support tfrac 5 2 tfrac 5 2 e g it decreases with order o w 4 provided the signal has a fourth order derivative for the classical series the order is o w 4 log w seven characteristic examples are treated in detail

prediction by samples from the past with error estimates covering discontinuous signals there are several reasons why the classical sampling theorem is rather impractical for real life signal processing first the sinc kernel is not very suitable for fast and efficient computation it decays much too slowly second in practice only a finite number n of sampled values are available so that the representation of a signal f by the finite sum would entail a truncation error which decreases rather slowly for n due to the first drawback third band limitation is a definite restriction due to the nonconformity of band and time limited signals further the samples needed extend from the entire past to the full future relative to some time t t0 this paper presents an approach to overcome these difficulties the sinc function is replaced by certain simple linear combinations of shifted b splines only a finite number of samples from the past need be available this deterministic approach can be used to process arbitrary not necessarily bandlimited nor differentiable signals and even not necessarily continuous signals best possible error estimates in terms of an lp average modulus of smoothness are presented several typical examples exhibiting the various problems involved are worked out in detail

on the zeros of the generalized euler frobenius laurent polynomial and reconstruction of cardinal splines of polynomial growth from local average samples abstract let h t be a nonnegative measurable function supported in 1 2 1 2 and m d t 1 2 1 2 1 2 1 2 1 2 1 2 t d 1 times be the central b spline of degree d we show that the roots of the generalized euler frobenius laurent polynomial defined by e h d z n z h m d n z n are simple negative and all are different from 1 as a consequence of this result we show that for every sequence y n n z r z of samples having polynomial growth and nonnegative measurable function h supported in 1 2 1 2 there is a unique spline f of degree d with polynomial growth satisfying f h n y n n z the presented work answers affirmatively the open problem posed in perez villalon and portal 2012 9