the eigenvalues of second order spectral differentiation matrices the eigenvalues of the pseudospectral second derivative matrix with homogeneous dirichlet boundary conditions are important in many applications of spectral methods this paper investigates some of their properties numerical results show that a certain fraction of the eigenvalues approximate the eigenvalues of the continuous operator very accurately but the errors in the remaining ones are large it is demonstrated that the inaccurate eigenvalues correspond to those eigenfunctions of the continuous operator that cannot be resolved by polynomial interpolation in the spectral grid in particular it is proved that 7r points on average per wavelength are sufficient for successful interpolation of the eigenfunctions of the continuous operator in a chebyshev distribution of nodes and six points per wavelength for a uniform distribution these results are in agreement with the observed fractions of accurate eigenvalues by using the characteristic polynomial a bound on the spectral radius of the differentiation matrix is derived that is accurate to 2 or better the effect of filtering on the eigenvalues is studied numerically

extension operators on tensor product structures in two and three dimensions in this paper a uniformly elliptic second order boundary value problem in two dimensions is discretized by the p version of the finite element method an inexact dirichlet dirichlet domain decomposition preconditioner for the system of linear algebraic equations is investigated the ingredients of such a preconditioner are a preconditioner for the schur complement a preconditioner for the subdomains and an extension operator operating from the edges of the elements into their interior using methods of multiresolution analysis we propose a new method to compute the extension efficiently we prove that this type of extension is optimal i e the h 1 omega norm of the extended function is bounded by the h 0 5 partial omega norm of the given function numerical experiments show the optimal performance of the described extension

finite element preconditioning for spectral multigrid methods abstract the good performance of finite element preconditioning for the algebraic spectral systems is demonstrated different types of multigrid strategies are compared the most efficient strategy consists of a pure richardson relaxation preconditioned by one v cycle of a finite element multigrid method the good smoothing properties of line relaxation become obvious the computational work for the various methods is estimated

preconditioned minimal residual methods for chebyshev spectral calculations abstract the problem of preconditioning the pseudospectral chebyshev approximation of an elliptic operator is considered the numerical sensitivity to variations of the coefficients of the operator are investigated for two classes of preconditioning matrices one arising from finite differences the other from finite elements the preconditioned system is solved by a conjugate gradient type method and by a dufort frankel method with dynamical parameters the methods are compared on some test problems with the richardson method 13 and with the minimal residual richardson method 21

chebyshev pseudospectral solution of second order elliptic equations with finite element preconditioning abstract a new chebyshev pseudospectral algorithm for second order elliptic equations using finite element preconditioning is proposed and tested on various problems bilinear and biquadratic lagrange elements are considered as well as bicubic hermite elements the numerical results show that bilinear elements produce spectral accuracy with the minimum computational work l shaped regions are treated by a subdomain approach

bubble stabilization of spectral legendre methods for the advection diffusion equation abstract an advection diffusion model problem is discretized by a spectral legendre method augmented by trial test functions with local support bubbles the numerical analysis shows that the correction is equivalent to adding a spectrally accurate streamline upwind stabilization term the performances of two strategies for tuning the bubble stabilization are investigated

a two dimensional adaptive pseudo spectral method we develop a two dimensional adaptive pseudo spectral procedure which is capable of improving the approximation of functions which are rapidly varying in two dimensions the method is based on introducing two dimensional coordinate transformations chosen to minimize certain functionals of the solution to be approximated the method is illustrated by numerical computation of the solutions to a system of reaction diffusion equations modeling the gasless combustion of a solid fuel spatio temporal patterns are computed as a parameter related to the activation energy is increased above a critical value c the spatial patterns are characterized by a very rapid variation in the direction of the axis of the cylinder together with a standing wave pattern in the direction of the azimuthal angle for small values of c the solutions exhibit a nearly sinusoidal dependence in both time and as is increased further relaxation oscillations in both time and occur beyond a critical value of stable time periodic solutions are no longer found and the solution exhibits a quasi periodic time dependence

adaptive spectral methods with application to mixing layer computations this paper reports some experiments on the use of adaptive chebyshev pseudospectral methods for compressible mixing layer computations different functionals measuring the optimality of the polynomial approximation are discussed and compared in particular we address the problem of the practical computation of the various functionals the utility of the self adaptive method is then demonstrated by some examples from compressible mixing layer calculations

positivity of character subbundles and minimum principle for noncompact group actions based on the results of berndtsson on the positivity of holomorphic vector bundles associated to holomorphic fibration we prove that under the setting of the compact lie group actions the character subbundles of such bundles are holomorphic vector bundles and positive in the sense of nakano the second main purpose of the present paper is to establish berndtsson s integral form of kiselman s minimum principle in the setting of the noncompact lie group actions

linear invariants of complex manifolds and their plurisubharmonic variations abstract for a bounded domain d and a real number p 0 denote by a p d the linear space of l p integrable holomorphic functions on d equipped with the l p pseudonorm we prove that two bounded hyperconvex domains d 1 c n and d 2 c m are biholomorphic in particular n m if there is a linear isometry between a p d 1 and a p d 2 for some 0 p 2 the same result holds for p 2 p 2 4 provided that the p bergman kernels on d 1 and d 2 are exhaustive with this as a motivation we show that for all p 0 the p bergman kernel on a strongly pseudoconvex domain with c 2 boundary or a simply connected homogeneous regular domain is exhaustive these results show that spaces of pluricanonical sections of complex manifolds equipped with canonical pseudonorms are important linear invariants of complex manifolds the second part of the present work devotes to studying variations of these invariants we show that the direct image sheaf of the twisted relative pluricanonical bundle associated to a holomorphic family of stein manifolds or compact kahler manifolds is positively curved with respect to the canonical singular finsler metric

an efficient spectral method for ordinary differential equations with rational function coefficients the authors present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients they employ expansions in terms of a large class of orthogonal polynomial families including all the classical orthogonal polynomials these families obey a simple 3 term recurrence relation for differentiation which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse the inverse is an integration operator for the family and it is simply the tridiagonal coefficient matrix for the recurrence since in these families convolution operators i e matrix representations of multiplication by a function are banded for polynomials the authors are able to obtain a banded representation for linear differential operators with rational coefficients this leads to a method of solution of initial or boundary value problems that besides having an operation count that scales linearly with the order of truncation n is computationally well conditioned among the applications considered is the use of rational maps for the resolution of sharp interior layers 19 refs 2 figs 3 tabs

the spectrum of the chebyshev collocation operator for the heat equation it is shown that the eigenvalues of the pseudospectral chebyshev second derivative operator with separated boundary conditions are real negative and distinct an application to the full potential equation is also presented

a chebyshev collocation algorithm for the solution of advection diffusion equations an original method to precondition chebyshev collocated advection diffusion operators is proposed the preconditioner based on an upwind finite difference approximation is tested on one and two dimensional problems the preconditioner behaves efficiently for advection dominated cases and produces rates of convergence independent of the number of nodes finally an ad hoc designed multi domain strategy is introduced

three dimensional fully spectral numerical method for mantle convection with depth dependent properties a semi implicit fully spectral collocation method for the simulation of three dimensional mantle convection with depth dependent thermodynamic and transport properties is presented the variable property navier stokes equation expressed in terms of the primitive variable velocity and pressure is solved with the mass continuity and temperature equations the periodic horizontal boundary conditions allow a fourier expansion for the two horizontal directions the stress free impermeable isothermal boundary conditions along with the depth dependent coefficients are handled with a chebyshev expansion in the vertical direction in the limit of an infinite prandtl number appropriate to mantle convection the inertial terms in the momentum equation are unimportant in this case an explicit solution of a poisson equation for pressure can be avoided instead a fourth order equation for vertical velocity can be solved simultaneous imposition of both impermeable and continuity boundary conditions during the vertical velocity evaluation is discussed the pressure distributions on the top and bottom bounding planes were determined by means of an influence matrix technique the numerical method employed here avoids time splitting errors and enforces velocity boundary conditions and continuity over the entire domain including the boundaries to machine accuracy strongy time dependent three dimensional solutions up to a surface rayleigh number of 1 107 have been obtained strong upwellings pulsating chaotically are formed by the collective merging of cylindrical plumes

an algorithm for the evaluation of multidimensional direct and inverse discrete chebyshev transforms abstract a method is presented for the efficient computation of n d n dimensional imbricated sums of direct and inverse discrete chebyshev transforms the algorithm consists essentially in the generalization of well known onedimensional procedures which are fitted together in order to perform the pre and post processing of the chebyshev coefficients array

preconditioning legendre spectral collocation methods for elliptic problems i finite difference operators in 1979 orszag and morchoisne independently proposed a finite difference preconditioning of the chebyshev collocation discretization of the poisson equation over the years there has been intensive research both experimental and theoretical on the finite difference and finite element preconditioning of both chebyshev and legendre spectral collocation methods for this problem in this work we present results on a frequently employed variant of finite element preconditioning of legendre spectral collocation methods for the helmholtz equation with dirichlet boundary conditions we show that there exist two positive constants 0 0 and w n m a n m u u ell 2 w n m m 1 n m s n m u u ell 2 leq lambda 1 where w n m rm diagonal w k tilde w j the diagonal matrix of legendre gauss lobatto lgl weights and a n m is the collocation matrix m n m is the mass matrix and s n m is the stiffness matrix of the finite element method these estimates imply that the eigenvalues gamma k of s 1 n m m n m a n m satisfy rm re gamma k geq lambda 0 0 gamma k leq lambda 1 these results depend on a thorough study of the ratio 2w k x k 1 x k 1 where x k are lgl collocation points

deformations of global symmetries in the extended antifield formalism it is outlined how deformations of field theoretical rigid symmetries can be constructed and classified by cohomological means in the extended antifield formalism special attention is devoted to deformations referring only to a subset of the rigid symmetries of a given model and leading to a nontrivial extension of the graded lie algebra associated with that subset the method is illustrated for a d 4 n 2 supersymmetric model where the central extension of the supersymmetry algebra emerges via a deformation deformations of gauge fixed actions with a brst symmetry are discussed too and illustrated by the curci ferrari model

a fast algorithm for chebyshev fourier and sinc interpolation onto an irregular grid a chebyshev or fourier series may be evaluated on the standard collocation grid by the fast fourier transform fft unfortunately the fft does not apply when one needs to sum a spectral series at n points which are spaced irregularly the cost becomes o n2 operations instead of the ffts o n log n this sort of off grid interpolation is needed by codes which dynamically readjust the grid every few time steps to resolve a shock wave or other narrow features it is even more crucial to semi lagrangian spectral algorithms for solving convection diffusion and navier stokes problems because off grid interpolation must be performed several times per time step in this work we describe an alternative algorithm the first step is to pad the set of spectral coefficients an with zeros and then take an fft of length 3n to interpolate the chebyshev series to a very fine grid the second step is to apply either the mth order euler sum acceleration or 2m 1 point lagrangian interpolation to approximate the sum of the series on the irregular grid we show that both methods yield full precision with m n allowing an order of magnitude reduction in cost with no loss of accuracy

parallel algorithms with local fourier basis abstract we present a nonoverlapping domain decomposition method with local fourier basis applied to a model problem in liquid flames the introduction of domain decomposition techniques in this paper is for numerical and parallel efficiency purposes when one requires a large number of grid points to catch complex structures we obtain then a high order accurate domain decomposition method that allows us to generalize our previous work on the use of local fourier basis to solve combustion problems with nonperiodic boundary conditions m garbey and d tromeur dervout j comput phys 145 316 1998 local fourier basis methodology fully uses the superposition principle to split the searched solution in a numerically computed part and an analytically computed part our present methodology generalizes the israeli et al 1993 j sci comput 8 135 method which applies domain decomposition with local fourier basis to the helmholtz s problem in the present work several new difficulties occur first the problem is unsteady and nonlinear which makes the periodic extension delicate to construct in terms of stability and accuracy second we use a streamfunction biharmonic formulation of the incompressible navier stokes equation in two space dimensions the application of domain decomposition with local fourier basis to a fourth order operator is more difficult to achieve than for a second order operator a systematic investigation of the influence of the method s parameters on the accuracy is done a detail parallel mimd implementation is given we give an a priori estimate that allows the relaxation of the communication between processors for the interface problem treatment results on nonquasi planar complex frontal polymerization illustrate the capability of the method

homological perturbation theory and the algebraic structure of the antifield antibracket formalism for gauge theories the algebraic structure of the antifield antibracket formalism for both reducible and irreducible gauge theories is clarified this is done by using the methods of homological perturbation theory hpt a crucial ingredient of the construction is the koszul tate complex associated with the stationary surface of the classical extremals the koszul tate differential acts on the antifields and is graded by the antighost number it provides a resolution of the algebraa of functions defined on the stationary surface namely it is acyclic except at degree zero where its homology group reduces toa acyclicity only holds because of the introduction of the ghosts of ghosts and provides an alternative criterion for what is meant by a proper solution of the master equation the existence of the brst symmetry follows from the techniques of hpt the classical lagrangian brst cohomology is completely worked out and shown to be isomorphic with the cohomology of the exterior derivative along the gauge orbits on the stationary surface the algebraic structure of the formalism is identical with the structure of the hamiltonian brst construction the role played there by the constraint surface is played here by the stationary surface only elementary quantum questions general properties of the measure are addressed

a gpu parallelized spectral method for elliptic equations in rectangular domains we design and implement a polynomial based spectral method on graphic processing units gpus the key to success lies in the seamless integration of the matrix diagonalization technique and the new generation cuda tools the method is applicable to elliptic equations in rectangular domains with general boundary condition we show remarkable speedups of up to 15 times in the 2 d case and more than 35 times in the 3 d case

pseudospectral least squares method for the second order elliptic boundary value problem the least squares legendre and chebyshev pseudospectral methods are presented for a first order system equivalent to a second order elliptic partial differential equation continuous and discrete homogeneous least squares functionals using legendre and chebyshev weights are shown to be equivalent to the h1 omega norm and chebyshev weighted div curl norm over appropriate polynomial spaces respectively the spectral error estimates are derived the block diagonal finite element preconditioner is developed for the both cases several numerical tests are demonstrated on the spectral discretization errors and on performances of the finite element preconditioner

curvature of vector bundles associated to holomorphic fibrations let l be a semi positive line bundle over a kahler manifold x fibered over a complex manifold y assuming the fibers are compact and nonsingular we prove that the hermitian vector bundle e over y whose fibers over points y are the spaces of global sections over xy to l kx y endowed with the l2 metric is semi positive in the sense of nakano we also discuss various applications among them a partial result on a conjecture of griffiths on the positivity of ample bundles

stabilization of spectral methods by finite element bubble functions abstract we show that a standard spectral collocation scheme can be stabilized by adding extra trial test functions with local support bubbles applications are given to a scalar advection diffusion problem and to the stokes problem

asymptotic numerical study of supersensitivity for generalized burgers equations this article addresses some asymptotic and numerical issues related to the solution of burgers equation varepsilon u xx u t u u x 0 on 1 1 subject to the boundary conditions u 1 1 delta u 1 1 and its generalization to two dimensions varepsilon delta u u t u u x u u y 0 on 1 1 times pi pi subject to the boundary conditions u x 1 1 delta u x 1 1 with 2 pi periodicity in y the perturbation parameters delta and varepsilon are arbitrarily small positive and independent when they approach 0 they satisfy the asymptotic order relation delta o s rm e a varepsilon for some constant a in 0 1 the solutions of these convection dominated viscous conservation laws exhibit a transition layer in the interior of the domain whose position as t to infty is supersensitive to the boundary perturbation algorithms are presented for the computation of the position of the transition layer at steady state the algorithms generalize to viscous conservation laws with a convex nonlinearity and are scalable in a parallel computing environment

global properties of pseudospectral methods abstract we apply polynomial interpolation methods both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations the derivative matrix for a general sequence of collocation points is explicitly constructed we explore the effect of several factors on the performance of these methods we show that global methods cannot be interpreted in terms of local methods for example the accuracy of the approximation differs when large gradients of the function occur near the center of the region or in the vicinity of the boundary this difference does not depend on the density of the collocation points near the boundaries hence intuition based on finite difference methods can lead to false results

galerkin alternating direction methods for nonrectangular regions using patch approximations galerkin alternating direction procedures are considered for the nonlinear parabolic equation u t sum i j 1 n frac partial partial xi i left tilde a ij xi u frac partial u partial xi j right sum i 1 n tilde b i xi u frac partial u partial xi i f xi t u which is given on a nonrectangular curved region we use isoparametric elements and an approximation to the jacobian of the isoparametric map which is based on patches of finite elements with these procedures a multidimensional problem can be solved as a series of one dimensional problems optimal order l 2 and h 1 error estimates as well as optimal order work estimates are derived

the exponential accuracy of fourier and chebyshev differencing methods it is shown that when differencing analytic functions using the pseudospectral fourier or chebyshev methods the error committed decays to zero at an exponential rate

pseudospectral solution of near singular problems using numerical coordinate transformations based on adaptivity the work presented here describes a method of coordinate transformation that enables spectral methods to be applied efficiently to differential problems with steep solutions the approach makes use of the adaptive finite difference method presented by huang and sloan siam j sci comput 15 1994 pp 776 797 this method is applied on a coarse grid to obtain a rough approximation of the solution and a suitable adapted mesh the adaptive finite difference solution permits the construction of a smooth coordinate transformation that relates the computational space to the physical space the map between the spaces is based on chebyshev polynomial interpolation finally the standard pseudospectral ps method is applied to the transformed differential problem to obtain highly accurate nonoscillatory numerical solutions numerical results are presented for steady problems in one and two space dimensions

error analysis for mapped legendre spectral and pseudospectral methods a general framework is introduced to analyze the approximation properties of mapped legendre polynomials and of interpolations based on mapped legendre gauss lobatto points optimal error estimates featuring explicit expressions on the mapping parameters for several popular mappings are derived these results not only play an important role in numerical analysis of mapped legendre spectral and pseudospectral methods for differential equations but also provide quantitative criteria for the choice of parameters in these mappings

relaxation schemes for spectral multigrid methods the effectiveness of relaxation schemes for solving the systems of algebraic equations which arise from spectral discretizations of elliptic equations is examined iterative methods are an attractive alternative to direct methods because fourier transform techniques enable the discrete matrix vector products to be computed almost as efficiently as for corresponding but sparse finite difference discretizations preconditioning is found to be essential for acceptable rates of convergence preconditioners based on second order finite difference methods are used a comparison is made of the performance of different relaxation methods on model problems with a variety of conditions specified around the boundary the investigations show that iterations based on incomplete lu decompositions provide the most efficient methods for solving these algebraic systems

galerkin alternating direction methods for a kind of nonlinear hyperbolic equations on nonrectangular regions patch approximation and an approximation to the jacobian of the isoparametric map are applied to discuss a three level and a four level galerkin alternating direction procedures for a kind of nonlinear hyperbolic equation which is given on a nonrectangular curved region with these procedures multi dimensional problems can be solved as a series of one dimensional problems and the l 2 norm error estimates are derived

mappings and accuracy for chebyshev pseudo spectral approximations abstract the effect of mappings on the approximation by chebyshev collocation of functions which exhibit localized regions of rapid variation is studied a general strategy is introduced whereby mappings are adaptively constructed which map specified classes of rapidly varying functions into low order polynomials a particular family of mappings constructed in this way is tested on a variety of rapidly varying functions similar to those occurring in applications it is shown that the mapped function can be approximated much more accurately by chebyshev polynomial approximations than the original function the effect on the approximation of introducing subdomains is also studied the accuracy of the pseudo spectral approximation is very sensitive to the location of the interface although this sensitivity is reduced when mappings are employed within the subdomains

a 2d chebyshev differential operator for the elastic wave equation abstract this work analyses the performance of a two dimensional chebyshev differential operator for solving the elastic wave equation the technique allows the implementation of non periodic boundary conditions at the four boundaries of the numerical mesh which requires a special treatment of these conditions based on one dimensional characteristics in addition spatial grid adaptation by appropriate one dimensional coordinate mappings allows a more accurate modeling of complex media and reduction of the computational cost by controlling the minium grid spacing the examples illustrate the ability of the method to simulate rayleigh waves around a corner and adapt the mesh to the model geometry in addition a domain decomposition example shows how the boundary treatment handles wave propagation from one mesh to another mesh

on the convergence of finite difference schemes for parabolic equations with variable coefficients we study the convergence of the finite difference schemes for the first initial boundary value problem for linear second order parabolic equations with variable coefficients using the bilinear version of the bramble hilbert lemma we obtain estimate of convergence in discretew 2 1 1 2 norm compatible with the smoothness of generalized solutionu w 2 2 q 1 3 and coefficients of equation

chebyshev 3 d spectral and 2 d pseudospectral solvers for the helmholtz equation two chebyshev solvers are presented for the linear helmholtz equation the first algorithm is a 3 d direct spectral solver based on a diagonalization technique whilst the second performs an iterative pseudospectral 2 d calculation with finite difference preconditioning both techniques handle general nonhomogeneous boundary conditions computing times and accuracies of the two methods are compared

structure and dynamics of modulated traveling waves in cellular flames we describe the formation and evolution of spatial and temporal patterns in cylindrical premixed flames we consider the cellular regime le 1 where the lewis number le is the ratio of thermal to mass diffusivity of a deficient component of the combustible mixture a transition from stationary axisymmetric flames to stationary cellular flames is predicted analytically if le is decreased below a critical value we present the results of numerical computations to show that as le is further decreased with all other parameters fixed traveling waves tws along the flame front arise via an infinite period bifurcation which breaks the reflection symmetry of the cellular array upon further decreasing le we find the development of different kinds of periodically modulated traveling waves mtws as well as a branch of quasiperiodically modulated traveling waves qpmtws these transitions are accompanied by the development of different spatial and temporal symmetries including period doublings and period halvings in appropriate coordinate systems we also observe the apparently chaotic temporal behavior of a disordered cellular pattern involving creation and annihilation of cells we analytically describe the stability of the tw solution near its onset using suitable phase amplitude equations within this framework one of the mtws can be identified as a localized wave traveling through an underlying stationary spatially periodic structure

spectral multigrid techniques for the navier stokes equations abstract we present a spectral multigrid method for the effective solution of the navier stokes equations for incompressible fluids here we consider the vorticity streamfunction formulation discretization in time is performed by a second order semi implicit scheme where the diffusive term is treated in an implicit manner while the convective term is evaluated explicitly for the discretization of the resulting stokes type problem we introduce a new spectral collocation method this scheme is highly accurate and allows efficient solution by spectral multigrid methods we propose an efficient finite difference preconditioner numerical results are presented which substantiate the usefulness of our concept

a multidomain spectral collocation method for the stokes problem we propose a multidomain spectral collocation scheme for the approximation of the two dimensional stokes problem we show that the discrete velocity vector field is exactly divergence free and we prove error estimates both for the velocity and the pressure

local brst cohomology in the antifield formalism i general theorems we establish general theorems on the cohomologyh s d of the brst differential modulo the spacetime exterior derivative acting in the algebra of localp forms depending on the fields and the antifields sources for the brst variations it is shown thath k s d is isomorphic toh k d in negative ghost degree k k 0 where is the koszul tate differential associated with the stationary surface the cohomology grouph1 d in form degreen is proved to be isomorphic to the space of constants of the motion thereby providing a cohomological reformulation of noether s theorem more generally the grouph k d in form degreen is isomorphic to the space ofn k forms that are closed when the equations of motion hold the groupsh k d k 2 are shown to vanish for standard irreducible gauge theories the grouph2 d is then calculated explicitly for electromagnetism yang mills models and einstein gravity the invariance of the groupsh k s d under the introduction of non minimal variables and of auxiliary fields is also demonstrated in a companion paper the general formalism is applied to the calculation ofh k s d in yang mills theory which is carried out in detail for an arbitrary compact gauge group

the arctan tan and kepler burgers mappings for periodic solutions with a shock front or internal boundary layer abstract many periodic solutions have internal regions of rapid change internal boundary layers shock waves and geophysical fronts are one class of examples a second class is composed of functions which decay rapidly away from a central peak or peaks spherical harmonics mathieu eigenfunctions prolate spheroidal wave functions and geophysical hough functions may all be locally approximated by hermite functions in the appropriate parameter range and decay exponentially fast outside a narrow subinterval similarly the large amplitude cnoidal waves of the korteweg devries equation are narrow isolated peaks which are well approximated by the sech 2 y form of the solitary wave in this article we show that a change of coordinate is a powerful tool for resolving such internal boundary layers in the first part we develop a general theory of mappings for the spherical harmonic cnoidal wave class of examples which decay rapidly away towards the edges of the spatial period the particular map y arc tan l tan x is a particularly effective choice four numerical examples show that this map and the fourier pseudospectral method are a good team in the second part we generalize the earlier theory to describe mappings which asymptote to a constant but non zero resolution at the ends of the periodicity interval we explain why the kepler burgers mapping is particularly suitable for shock and fronts

complex mapped matrix methods in hydrodynamic stability problems the ordinary differential equations governing the linear stability of inviscid flows contain singularities at real or complex points called critical latitudes which degrade the accuracy of standard numerical schemes however the use of a complex mapping prior to the numerical attack offers some respite this mapping shifts the computational domain to a contour in the complex plane to avoid the critical latitudes both quadratic and cubic complex maps are considered in some detail an analytic result for choosing the optimum quadratic complex map in the case of a single critical latitude is presented numerical results are given for two test problems and a barotropic vortex model a comparison is made between methods with and without these mappings the results show that the use of complex maps can lead to remarkably accurate solutions

poloidal toroidal decomposition in a finite cylinder the navier stokes equations in a finite cylinder are written in terms of poloidal and toroidal potentials in order to impose incompressibility regularity of the solutions is ensured in several ways first the potentials are represented using a spectral basis which is analytic at the cylindrical axis second the non physical discontinuous boundary conditions at the cylindrical corners are smoothed using a polynomial approximation to a steep exponential profile third the nonlinear term is evaluated in such a way as to eliminate singularities the resulting pseudo spectral code is tested using exact polynomial solutions and the spectral convergence of the coefficients is demonstrated our solutions are shown to agree with exact polynomial solutions and with previous calculations of axisymmetric vortex breakdown and of onset of non axisymmetric helical spirals parallelization by azimuthal wavenumber is shown to be highly effective

a dynamical pseudo spectral domain decomposition technique a dynamical spectral domain decomposition method is presented in each subdomain a transformation of coordinate is used both the locations of the interfaces and the parameters of the mappings are dynamically adapted by minimizing theh 2 norm of the calculated solution we show on some functions that the total norm of the chebyshev series depends on the location of the interfaces moreover there exists a minimum that defines the best location of the interface this defines a dynamical generation of chebyshev collocation points this numerical method is applied on partial differential equations and it is shown that both the overall accuracy and the matching at the interfaces are improved with respect to a fixed interface calculation this algorithm is then used for the numerical solution of the time dependent full navier stokes equations the solution technique consists in a fourier chebyshev collocation method combined with a matching method the computational domain is decomposed into subdomains in the vertical direction in each subdomain a coordinate transform is used and the locations of the interfaces are dynamically determined the elliptic problems coming from the viscous terms are solved by means of the chebyshev acceleration method density is matched with an upwind procedure whereas the velocities the temperature and the concentration are handled with the influence matrix method numerical examples are carried out on the compressible kelvin helmholtz and rayleigh taylor flows

chebyshev collocation method and multi domain decomposition for navier stokes equations in complex curved geometries a general multidomain decomposition is proposed for the numerical solution of the 2d incompressible stationary navier stokes equations the solution technique consists in a chebyshev orthogonal collocation method preconditioned by a standard galerkin finite element technique the preconditioned system is then solved through a richardson procedure the domain of interest is decomposed into quadrilaterals curved when needed a gordon transfinite interpolation performs the curvilinear grid generation of the obtained simply connected planar subdomains the interface conditions naturally incorporated into the finite element approach relate neighbour subdomains through the normal jump of appropriate fluxes across internal boundaries where an integral form ofc 1 continuity is consequently achieved at convergence of the iterative processes the study of model stokes problems demonstrates that the current method still behaves spectrally in distorted geometries for curvilinear distortion a loss of several orders of magnitude is observed in the solution accuracy even when the distortion is very limited finally some results of flow simulation in a constricted channel are proposed to illustrate the abilities of the present method to treat navier stokes problems

a family of improved falkner type methods for oscillatory systems for solving general second order initial value problems u t f t u t u t the classical falkner methods can date back to falkner s work in 1936 in this paper we propose and study a family of improved falkner type methods for the oscillatory system u t m u t g t u t where g r rd rd in which the first derivative does not appear explicitly and m rd d is a symmetric positive semi definite matrix the new methods take into account the oscillatory structures of the problem and exactly integrate the unperturbed problem u t m u t 0 the global error bounds of the new methods are presented numerical experiments are performed to show that the new methods are more efficient than other effective methods appeared in the scientific literature

a pseudospectral method for solution of the three dimensional incompressible navier stokes equations abstract a new chebyshev pseudospectral technique based on the projection method that was previously applied by the authors to the solution of two dimensional incompressible navier stokes equations in primitive variables for nonperiodic boundary conditions is extended to solve the three dimensional navier stokes equations the crucial point of the method is the requirement that the continuity equation be satisfied everywhere in the domain on the boundaries as well as in the interior the key feature of the work presented in this paper is that the computer storage requirements of the full matrix inversion resulting from direct solution of the pressure poisson equation in three dimensions is greatly reduced by considering an eigenfunction decomposition the method was tested on a two dimensional driven cavity flow and the results were compared with those of the most accurate finite difference calculation the three dimensional driven cavity flow was then calculated at the same reynolds numbers as the two dimensional cases i e re 100 400 and 1000 in the calculated results three dimensional boundary effects were observed in all cases and became more apparent with increasing reynolds number

finite element preconditioning for legendre spectral collocation approximations to elliptic equations and systems spectral collocation approximations based on legendre gauss lobatto lgl points for helmholtz equations as well as for the linear elasticity system in rectangular domains are studied the collocati

spectral implementation of an adaptive moving mesh method for phase field equations phase field simulations have been extensively applied to modeling microstructure evolution during various materials processes however large scale simulations of three dimensional 3d microstructures are still computationally expensive among recent efforts to develop advanced numerical algorithms the semi implicit fourier spectral method is found to be particularly efficient for systems involving long range interactions as it is able to utilize the fast fourier transforms fft on uniform grids in this paper we report our recent progress in making grid points spatially adaptive in the physical domain via a moving mesh strategy while maintaining a uniform grid in the computational domain for the spectral implementation this approach not only provides more accurate treatment at the interfaces requiring higher resolution but also retains the numerical efficiency of the semi implicit fourier spectral method numerical examples using the new adaptive moving mesh semi implicit fourier spectral method are presented for both two and three space dimensional microstructure simulations and they are compared with those obtained by other methods by maintaining a similar accuracy the proposed method is shown to be far more efficient than the existing methods for microstructures with small ratios of interfacial widths to the domain size

siu s lemma optimal l 2 extension and applications to twisted pluricanonical sheaves we prove a general version of siu s lemma for plurisubharmonic functions with nontrivial multiplier ideal sheaves and use it to prove an optimal l 2 extension theorem and an optimal l frac 2 m extension theorem for kahler fibrations these results are used to prove the positivity of twisted relative pluricanonical bundles and their direct images for kahler fibrations and to answer a comparison question about singular metrics of twisted pluricanonical bundles

alternating direction multistep methods for parabolic problems iterative stabilization efficient multistep procedures for time stepping galerkin methods for parabolic partial differential equations are presented and analyzed the procedures involve using an alternating direction preconditioned iterative method for approximately solving the linear equations arising at each timestep in a discrete galerkin method optimal order convergence rates are obtained for the iterative methods work estimates of almost optimal order are obtained

boundary and interface conditions within a finite element preconditioner for spectral methods abstract the performances of a finite element preconditioner in the iterative solution of spectral collocation schemes for elliptic boundary value problems is investigated it is shown how to make the preconditioner cheap by adi iterations and how to take advantage of the finite element properties in enforcing neumann and interface conditions in the spectral schemes

positivity of direct images of positively curved volume forms we discuss kiselman berndtsson s minimum principle for plurisubharmonic functions in terms of the positivity of direct images of certain positively curved volume forms and generalize it to holomorphically convex manifolds with compact group actions with this generalization and other techniques we establish a minimum principle for positively curved volume forms from the point of view of geometric invariant theory on stein manifolds minimum principle with some noncompact group actions is also considered

chebyshev collocation solutions of the navier stokes equations using multi domain decomposition and finite element preconditioning abstract the steady navier stokes equations are solved using series of basis functions involving chebyshev polynomials the projection method is a collocation scheme a newton s linearization is performed in order to obtain a set of algebraic equations as the matrix system is ill conditioned the collocation technique is preconditioned by a standard galerkin finite element method using a 9 nodes lagrangian element which presents decisive advantages sparsity reduced condition number easy treatment of complicated geometries to handle nontrivial geometries in the collocation process a domain decomposition is set up the treatment of interface conditions is fully described several test problems like the regularized driven square cavity and the backward facing step are discussed to show the abilities of the present algorithm

an adaptive pseudo spectral method for reaction diffusion problems the spectral interpolation error was considered for both the chebyshev pseudo spectral and galerkin approximations a family of functionals i sub r u with the property that the maximum norm of the error is bounded by i sub r u j sub r where r is an integer and j is the degree of the polynomial approximation was developed these functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize i sub r u the number of collocation points is then chosen to maintain a prescribed error bound the method is illustrated by various examples from combustion problems in one and two dimensions

a symmetric characteristic fve method with second order accuracy for nonlinear convection diffusion problems the finite volume element fve methods used currently are essentially low order and unsymmetric in this paper by biquadratic elements and multistep methods we construct a second order fve scheme for nonlinear convection diffusion problem on nonuniform rectangular meshes to overcome the numerical oscillation we discretize the problem along its characteristic direction the choice of alternating direction strategy is critical in this paper which guarantees the high efficiency and symmetry of the discrete scheme optimal order error estimates in h 1 norm are derived and a numerical example is given at the end to confirm the usefulness of the method

an analytic approximation to the cardinal functions of gaussian radial basis functions on an infinite lattice gaussian radial basis functions rbfs have been very useful in computer graphics and for numerical solutions of partial differential equations where these rbfs are defined on a grid with uniform spacing h as translates of the master function f x a h exp a 2 h 2 x 2 where a is a user choosable constant unfortunately computing the coefficients of f x jh a h requires solving a linear system with a dense matrix it would be much more efficient to rearrange the basis functions into the equivalent lagrangian or cardinal basis because the interpolation matrix in the new basis is the identity matrix the cardinal basis c j x a h is defined by the set of linear combinations of the gaussians such that c j kh 1 when k j and c j kh 0 for all integers k j we show that the cardinal functions for the uniform grid are c j x h a c x h j a where c x a a 2 p sin px sinh a 2x the relative error is only about 4exp email protected 2 a 2 as demonstrated by the explicit second order approximation it has long been known that the error in a series of gaussian rbfs does not converge to zero for fixed a as h 0 but only to an error saturation proportional to exp email protected 2 a 2 because the error in our approximation to the master cardinal function c x a is the square of the error saturation there is no penalty for using our new approximations to obtain matrix free interpolating rbf approximations to an arbitrary function f x the master cardinal function on a uniform grid in d dimensions is just the direct product of the one dimensional cardinal functions thus in two dimensions c x y a a 4 p 2 sin px sin py sinh a 2x sinh a 2y we show that the matrix free interpolation can be extended to non uniform grids by a smooth change of coordinates

a consistent equilibrium in a cross section of an elastic plastic beam abstract a phenomenon of inequality of equilibrium and constitutive internal forces in a cross section of elastic plastic beams is common to many finite element formulations it is here discussed in a rate independent elastic plastic beam context and a possible treatment is presented the starting point of our discussion is reissners finite strain beam theory and its finite element implementation the questions of the consistency of interpolations for displacements and rotations and the related locking phenomena are fully avoided by considering the rotation function of the centroid axis of a beam as the only unknown function of the problem approximate equilibrium equations are derived by the use of the distribution theory in conjunction with the collocation method the novelty of our formulation is an inclusion of a balance function that measures the error between the equilibrium and constitutive bending moments in a cross section an advantage of the present approach is that the locations where the balance of equilibrium and constitutive moments should be satisfied can be prescribed in advance in order to minimize the error explicit analytical expressions are used for the constitutive forces for a rectangular cross section and bilinear constitutive law they are given in appendix a the comparison between the results of the two finite element formulations the one using consistent and the other inconsistent equilibrium in a cross section is shown for a cantilever beam subjected to a point load the problem of high curvature gradients in a plastified region is also discussed and solved by using an adapted collocation method in which the coordinate system is transformed such to follow high gradients of curvature