numerical integration of non linear two point boundary value problems using iterated deferred corrections i abstract in several recent papers the idea has been proposed of using implicit runge kutta formulae for the approximate integration of non linear two point boundary value problems these formulae often take a very special form so that computational efficiencies are obtained in certain well defined circumstances in this paper we examine the relative merits of these different classes of formulae and discuss their efficient implementation in a deferred correction framework a theoretical comparison of these formulae is important firstly because they call for different computational tasks to be performed and secondly because they are not always applicable to exactly the same classes of problems the most efficient of these formulae are then compared with methods based on finite differences and with methods based on spline collocation for some simple smooth test functions on the basis of operation counts and numerical experimentation it is concluded that runge kutta methods can be generally competitive with the other methods considered

lower bounds of the topological entropy of maps of y abstract we give the best lower bound of the topological entropy of a continuous map of the space y z c z 3 0 1 into itself with 0 0 as a function of its set of periods

algebraic multigrid amg experiences and comparisons algebraic multigrid methods are designed for the solution of sparse linear systems of equations using multigrid principles in contrast to standard multigrid methods amg does not take advantage of the origin of a particular system of equations at hand nor does it exploit any underlying geometrical situation fully automatically and based solely on algebraic information contained in the given matrix amg constructs a sequence of grids and corresponding operators a special amg algorithm will be presented for a wide range of problems including certain problems which do not have a continuous background this algorithm yields an iterative method which exhibits a convergence behavior typical for multigrid methods

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

least squares spectral collocation for discontinuous and singular perturbation problems a least squares spectral collocation scheme for discontinuous problems is proposed for the first derivative operator the domain is decomposed in subintervals where the jumps are imposed at the discontinuities equal order polynomials are used on all subdomains for the discretization spectral collocation with chebyshev polynomials is employed fast fourier transforms are now available the collocation conditions and the interface conditions lead to an overdetermined system which can be efficiently solved by least squares the solution technique will only involve symmetric positive definite linear systems this approach is further extended to singular perturbation problems where least squares are used for stabilization by a suitable decomposition of the domain the boundary layer is well resolved numerical simulations confirm the high accuracy of our spectral least squares scheme

multigrid methods for combined finite difference and fourier problems abstract multigrid methods for combined finite difference and fourier pseudospectral problems are considered applications to two and three dimensional elliptic systems are given suitable relaxation schemes for isotropic and anisotropic problems are presented numerical results demonstrate the efficiency of these methods

scrambled sets of continuous maps of 1 dimensional polyhedra let k be a 1 dimensional simplicial complex in r3 without isolated vertexes x iki be the polyhedron of k with the metric dk induced by k and f x x be a continuous map in this paper we prove that if k is finite then the interior of every scrambled set of f in x is empty we also show that if k is an infinite complex then there exist continuous maps from x to itself having scrambled sets with nonempty interiors and if x r or r then there exist c maps of x with the whole space x being a scrambled set

on the use of nurbs based discretizations in the scaled boundary finite element method for wave propagation problems abstract we discuss the application of non uniform rational b splines nurbs in the scaled boundary finite element method sbfem for the solution of wave propagation problems at rather high frequencies we focus on the propagation of guided waves along prismatic structures of constant cross section comparisons are made between nurbs based discretizations and high order spectral elements in terms of the achievable convergence rates we find that for the same order of shape functions nurbs can lead to significantly smaller errors compared with lagrange polynomials the difference becomes particularly important at very high frequencies where spectral elements are prone to instabilities furthermore we analyze the behavior of nurbs for the discretization of curved boundaries where the benefit of exact geometry representation becomes crucial even in the low frequency range

an adaptive uniformly convergent numerical method for a semilinear singular perturbation problem a singularly perturbed semilinear two point boundary value problem is considered without any restriction on its turning points a difference scheme is presented for solving this problem on an arbitrary locally quasi uniform mesh it is shown that the solution of the scheme is first order accurate uniformly in the perturbation parameter in a discrete l 1 norm numerical results are presented for an adaptive method based on the scheme

sharp regularity for certain nilpotent group actions on the interval according to the classical plante thurston theorem all nilpotent groups of c 2 diffeomorphisms of the closed interval are abelian using techniques coming from the works of denjoy and pixton farb and franks constructed a faithful action by c 1 diffeomorphisms of 0 1 for every finitely generated torsion free non abelian nilpotent group in this work we give a version of this construction that is sharp in what concerns the holder regularity of the derivatives half of the proof relies on results on random paths on heisenberg like groups that are interesting by themselves

analysis of the kleiser schumann method the kleiser schumann algorithm for the approximation of the stokes problem by fourier legendre polynomials is analized stability when the degree of the polynomials increases is established whereas error estimates in sobolev spaces are proven

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

diffeomorphism groups of critical regularity let m be a circle or a compact interval and let alpha k tau ge 1 be a real number such that k lfloor alpha rfloor we write mathrm diff alpha m for the group of orientation preserving c k diffeomorphisms of m whose kth derivatives are holder continuous with exponent tau we prove that there exists a continuum of isomorphism types of finitely generated subgroups g le mathrm diff alpha m with the property that g admits no injective homomorphisms into bigcup beta alpha mathrm diff beta m we also show the dual result there exists a continuum of isomorphism types of finitely generated subgroups g of bigcap beta alpha mathrm diff beta m with the property that g admits no injective homomorphisms into mathrm diff alpha m the groups g are constructed so that their commutator groups are simple we give some applications to smoothability of codimension one foliations and to homomorphisms between certain continuous groups of diffeomorphisms for example we show that if alpha ge 1 is a real number not equal to 2 then there is no nontrivial homomorphism mathrm diff alpha s 1 rightarrow bigcup beta alpha mathrm diff beta s 1 finally we obtain an independent result that the class of finitely generated subgroups of mathrm diff 1 m is not closed under taking finite free products

black box multigrid one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in this paper we investigate such a situation for the multigrid method the end result is code boxmg in which one need only specify the logically rectangular positive definite matrix problem boxmg does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution

a direct solver for the least squares spectral collocation system on rectangular elements for the incompressible navier stokes equations a least squares spectral collocation scheme for the stokes and incompressible navier stokes equations is proposed the original domain is decomposed into quadrilateral subelements and on the element interfaces continuity of the functions is enforced in the least squares sense the collocation conditions and the interface conditions lead to overdetermined systems these systems are directly solved by qr decomposition of the underlying matrices by numerical simulations it is shown that the direct method leads to better results than the approach with normal equations furthermore it is shown that the condition numbers can be reduced by introducing the clenshaw curtis quadrature rule for imposing the average pressure to be zero finally our scheme is successfully applied to the regularized and lid driven cavity flow problems

defect correction for the advection diffusion equation abstract a defect correction method for the advection diffusion equation is presented the discretization is performed by 2nd order finite difference schemes schemes where the 2nd order upstream scheme is combined with the standard central scheme for preconditioning the usual first order upstream scheme is employed the defect correction iteration is used for relaxation inside a multigrid procedure it is shown that the smoothing analysis yields rather pessimistic results with rates of about 0 7 0 8 in the practical computation the discretization error is reached in 2 v cycles numerical results are presented which demonstrate the high efficiency of our treatment

improved lebesgue constants on the triangle new sets of points with improved lebesgue constants in the triangle are calculated starting with the fekete points a direct minimization process for the lebesgue constant leads to better results the points and corresponding quadrature weigths are explicitly given it is quite surprising that the optimal points are not symmetric the points along the boundary of the triangle are the 1d gauss lobatto points for all degrees our points yield the smallest lebesgue constants currently known numerical examples are presented which show the improved interpolation properties of our nodes

subgrid stabilized defect correction methods for the navier stokes equations we consider the synthesis of a recent subgrid stabilization method with defect correction methods the combination is particularly efficient and combines the best algorithmic features of each we prove convergence of the method for a fixed number of corrections as the mesh size goes to zero and derive parameter scalings from the analysis we also present some numerical tests which both verify the theoretical predictions and illustrate the method s promise

the p and hp versions of the finite element method for problems with boundary layers we study the uniform approximation of boundary layer functions exp x d for x 0 1 d 0 1 by the p and hp versions of the finite element method for the p version with fixed mesh we prove super exponential convergence in the range p 1 2 e 2d we also establish for this version an overall convergence rate of o p 1 ln p in the energy norm error which is uniform in d and show that this rate is sharp up to the ln p term when robust estimates uniform in d 0 1 are considered for the p version with variable mesh i e the hp version we show that exponential convergence uniform in d 0 1 is achieved by taking the first element at the boundary layer to be of size o pd numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates even when few degrees of freedom are used and when d is as small as e g 10 8 they also illustrate the superiority of the hp approach over other methods including a low order h version with optimal exponential mesh refinement the estimates established in this paper are also applicable in the context of corresponding spectral element methods

spectral collocation domain decomposition method for viscoelastic flow simulations in model porous geometries abstract a domain decomposition spectral collocation ddsc method is developed for the solution of steady state non linear viscoelastic flow problems it is applied in simulations of viscoelastic flows through model porous media represented by arrays of cylinders the technique uses a splitting of the computational domain in sub elements in a fashion similar to finite elements at each element the solution is represented by using high order orthogonal polynomials static condensation is used for the solution of the linearized system of equations resulting from the application of newton s method this approach is suggested by the limited connectivity between neighboring elements mesh converged solutions have been obtained for the flow of an oldroyd b fluid through a square array of cylinders up to we 5 0 and for the flow past a single row of cylinders up to to we 4 0

an iterative procedure with interface relaxation for domain decomposition methods a domain decomposition method for second order elliptic problems is considered an iterative procedure that reduces the problem to a sequence of mixed boundary value problems on each subdomain is proposed at each iteration a relaxation is accomplished at the subdomain interfaces in several circumstances a value of the relaxation parameter that yields exact convergence in a finite number of iterations is explicitly found moreover when such a value is not available an appropriate strategy for the automatic selection of the relaxation parameter at each iteration is indicated and analyzed this iterative method is then applied to the spectral collocation approximation of the differential problem the same kind of convergence results are proven many numerical experiments show the effectiveness of the method proposed here

spectral collocation schemes on the unit disc two spectral collocation schemes on the unit disc are presented the first one is based on the mapping of gordon and hall here the unit square is directly mapped onto the unit disc by means of an interpolation technique unlike other poisson solvers on the unit disc no polar coordinates are involved hence the usual problems with the singularity of polar coordinates are avoided this is also shown for more complex geometries the second method is based on a diameter approach where the collocation nodes are no more clustering in the center numerical results are presented which demonstrate the high accuracy of our new spectral collocation schemes

a spectral multigrid method for the stokes problem in streamfunction formulation abstract suitable spectral multigrid components for the stokes problem in streamfunction formulation are presented we split the stokes problem into a system of two equations with the laplace operator we discretize by means of a collocation or pseudo spectral method preconditioning with finite differences is employed numerical results for the spectral multigrid method are presented

a triangular spectral element method applications to the incompressible navier stokes equations abstract encouraged by the success of spectral elements methods in computational fluid dynamics and p type finite element methods in structural mechanics we wish to extend these ideas to solving high order polynomial approximations on triangular domains as the next generation of spectral element solvers we introduce here a complete formulation using a modal basis which has been implemented in a new code ne t r the new basis has the following properties jacobi polynomials of mixed weights semi orthogonality hierarchical structure generalized tensor warped product variable order and a new apex co ordinate system allowing automated integration with gaussian quadrature we have discussed the formulation using a matrix notation which allows for an easy interpretation of the forward and backward transformations we use this notation to formulate the linear advection and helmholtz equations in an efficient manner and show that we recover the following properties well conditioned matrices asymptotic operation count of o n 3 scaling o n 2 of the spectral radius of the weak convective operator and exponential convergence using polynomials up to 40 having constructed and numerically analysed these equations we are then able to solve the incompressible navier stokes equations using a high order splitting scheme we demonstrate a variety of results showing exponential convergence using both deformed and straight triangular subdomains for both the stokes and navier stokes problems

the asymptotic chebyshev coefficients for functions with logarithmic endpoint singularities mappings and singular basis functions abstract when a function is singular at the ends of its expansion interval its chebyshev coefficients a n converge very poorly we analyze three numerical strategies for coping with such singularities of the form 1 x k log 1 x and in the process make some modest additions to the theory of chebyshev expansions the first two numerical methods are the convergence improving changes of coordinate x sin 2 y and x tanh ly 1 y 2 1 2 we derive the asymptotic chebyshev coefficients in the limit n for both mappings and for the original untransformed chebyshev series for the original function the asymptotic approximation for general k is augmented by the exact chebyshev coefficients for integer k numerical tests show that the sine mapping is excellent for k 1 increasing the rate of convergence to b n o 1 n 4 k 1 although the tanh transformation is guaranteed to be better for sufficiently large n we offer both theoretical and numerical evidence to explain why the sine mapping is usually better in practice sufficiently large n is usually huge instead of mapping one may use a third strategy supplementing the chebyshev polynomials with singular basis functions simple experiments show that this approach is also successful

a fast solver for elliptic boundary value problems in the square abstract we approximate the solution of advection diffusion equations by collocation at a special grid related to the differential operator and the classical legendre grid

a semi implicit spectral method for compressible convection of rotating and density stratified flows in cartesian geometry in this paper we have described a stratified semi implicit spectral method to study compressible convection in cartesian geometry the full set of compressible hydrodynamic equations are solved in conservative forms the numerical scheme is accurate and efficient based on fast fourier sin cos spectral transforms in the horizontal directions chebyshev spectral transform or second order finite difference scheme in the vertical direction and second order semi implicit scheme in time marching of linear terms we have checked the validity of both the fully pseudo spectral scheme and the mixed finite difference pseudo spectral scheme by studying the onset of compressible convection the difference of the critical rayleigh number between our numerical result and the linear stability analysis is within two percent besides we have computed the mach numbers with different rayleigh numbers in compressible convection it shows good agreement with the numerical results of finite difference methods and finite volume method this model has wide application in studying laminar and turbulent flow illustrative examples of application on horizontal convection gravity waves and long lived vortex are given in this paper

spectral collocation on triangular elements we consider the poisson problem on a segment of the unit disc and on triangles on the segment we transform the poisson problem by means of polar coordinates in these new coordinates we have a problem in a rectangle which can easily be mapped onto the square here standard chebyshev collocation techniques can be applied then the segment is mapped onto a triangle where the same spectral scheme may be used by numerical tests we observed the expected high spectral accuracy due to the corner singularity a singular behaviour of the solution can be expected here we improved the accuracy by auxiliary mapping techniques further it is shown that finite difference preconditioning can be successfully applied in order to construct an efficient iterative solver finally a domain decomposition technique is applied to the patching of a rectangular and a triangular element

conjugate gradient like algorithms for solving nonsymmetric linear systems this paper presents a unified formulation of a class of the conjugate gradient like algorithms for solving nonsymmetric linear systems the common framework is the petrov galerkin method on krylov subspaces we discuss some practical points concerning the methods and point out some of the interrelations between them 1 introduction in the recent few years a large number of generalizations of the conjugate gradient and conjugate residual methods which are very successful in solving symmetric positive definite linear systems have been proposed for solving nonsymmetric linear systems 3 6 5 7 12 in this paper we present an abstract framework which includes most of these methods and many new ones our goal is to understand the relationships among the methods and to synthesize consider the general linear system

spectral methods and a maximum principle etude de diverses approximations spectrales de tchebychev d un probleme de couche limite pour un operateur d helmholtz et d advection diffusion comportement des solutions spectrales dans l espace des frequences et dans l espace physique estimations d erreur

uniform high order difference schemes for a singularly perturbed two point boundary value problem on construit une famille de schemas aux differences finies uniformement precis pour le probleme du type eu a x u b x u f x on donne une analyse complete de l erreur de discretisation basee sur les resultats de stabilite on presente des essais numeriques

a defect correction method for unsteady conduction convection problems ii time discretization in this paper a fully discrete defect correction mixed finite element method mfem for solving the non stationary conduction convection problems in two dimension which is leaded by combining the back euler time discretization with the two step defect correction in space is presented in this method we solve the nonlinear equations with an added artificial viscosity term on a finite element grid and correct these solutions on the same grid using a linearized defect correction technique the stability and the error analysis are derived the theory analysis shows that our method is stable and has a good convergence property some numerical results are also given which show that this method is highly efficient for the unsteady conduction convection problems

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

jacobi spectral galerkin method for the integrated forms of second order differential equations abstract efficiently direct solvers based on the jacobi galerkin method for the integrated forms of second order elliptic equations in one and two space variables are presented they are based on appropriate base functions for the galerkin formulation which lead to discrete systems with specially structured matrices that can be efficiently inverted the homogeneous dirichlet boundary conditions are satisfied exactly by expanding the unknown variable into a polynomial basis of functions which are built upon the jacobi polynomials the direct solution algorithm here developed for the homogeneous dirichlet problem in two dimensions relies upon a diagonalization process numerical results indicating the high accuracy and effectiveness of these algorithms are presented

spectral schemes on triangular elements abstract the poisson problem with homogeneous dirichlet boundary conditions is considered on a triangle the transformation from square to triangle is realized by mapping an edge of the square onto a corner of the triangle then standard chebyshev collocation techniques can be implemented numerical experiments demonstrate the expected high spectral accuracy for smooth solutions furthermore it is shown that finite difference preconditioning can be successfully employed to construct an efficient iterative solver then the convection diffusion equation is considered here finite difference preconditioning with central differences leads to instability however using the first order upstream scheme we obtain a stable method finally a domain decomposition technique is applied to the patching of rectangular and triangular elements

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

semicirculant preconditioning of elliptic operators in this work we consider the semicirculant preconditioning of elliptic differential operators of the form lu epsilon delta u au x bu y cu in two cases 0 epsilon ll 1 and epsilon equiv 1 the paper numer math 81 1998 pp 211 249 provided extremely interesting and useful results in the first case on the other hand those appear to contradict basic results on preconditioning given in siam j numer anal 27 1990 pp 656 694 we reobtain the results of numer math 81 1998 pp 211 249 by a new approach which we believe to be more transparent we also clarify the situation regarding the apparent contradiction with siam j numer anal 27 1990 pp 656 694 finally we describe the distribution of the preconditioned eigenvalues in the uniformly elliptic case epsilon equiv 1

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

line relaxation for spectral multigrid methods abstract spectral multigrid methods for elliptic problems with dirichlet and periodic boundary conditions are examined the multigrid convergence is improved by certain line relaxation techniques which are suggested by a smoothing analysis furthermore several choices of relaxation parameters are compared numerical results are presented showing the gain in efficiency and accuracy over previous methods

spectral methods with sparse matrices spectral methods employ global polynomials for approximation hence they give very accurate approximations for smooth solutions unfortunately for dirichlet problems the matrices involved are dense and have condition numbers growing aso n 4 for polynomials of degree n in each variable we propose a new spectral method for the helmholtz equation with a symmetric and sparse matrix whose condition number grows only aso n 2 certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system numerical results are presented which show that we have probably found the most effective solver for spectral systems

a new method of stabilization for singular perturbation problems with spectral methods a stabilized treatment of singular perturbation problems with spectral methods is presented the stabilization is achieved by introducing one additional collocation point the resulting overdetermined system leads to the normal equations which yield a stabilized system numerical results in one and two dimensions are presented which show the improved condition number of the stabilized spectral system

a fully spectral collocation approximation for multi dimensional fractional schrodinger equations a shifted legendre collocation method in two consecutive steps is developed and analyzed to numerically solve one and two dimensional time fractional schrodinger equations tfses subject to initial boundary and non local conditions the first step depends mainly on shifted legendre gauss lobatto collocation sl gl c method for spatial discretization an expansion in a series of shifted legendre polynomials for the approximate solution and its spatial derivatives occurring in the tfse is investigated in addition the legendre gauss lobatto quadrature rule is established to treat the nonlocal conservation conditions thereby the expansion coefficients are then determined by reducing the tfse with its nonlocal conditions to a system of fractional differential equations sfdes for these coefficients the second step is to propose a shifted legendre gauss radau collocation sl gr c scheme for temporal discretization to reduce such system into a system of algebraic equations which is far easier to be solved the proposed collocation scheme both in temporal and spatial discretizations is successfully extended to solve the two dimensional tfse numerical results are carried out to confirm the spectral accuracy and efficiency of the proposed algorithms by selecting relatively limited legendre gauss lobatto and gauss radau collocation nodes we are able to get very accurate approximations demonstrating the utility and high accuracy of the new approach over other numerical methods

rigidity for c 1 actions on the interval arising from hyperbolicity i solvable groups we consider abelian by cyclic groups for which the cyclic factor acts by hyperbolic automorphisms on the abelian subgroup we show that if such a group acts faithfully by c 1 diffeomorphisms of the closed interval with no global fixed point at the interior then the action is topologically conjugate to that of an affine group moreover in case of non abelian image we show a rigidity result concerning the multipliers of the homotheties despite the fact that the conjugacy is not necessarily smooth some consequences for non solvable groups are proposed in particular we give new proofs examples yielding the existence of finitely generated locally indicable groups with no faithful action by c 1 diffeomorphisms of the interval

a triangular spectral element method for elliptic and stokes problems abstract in this paper we study a triangular spectral element method based on a one to one mapping between the rectangle and the triangle we construct a new approximation space where the integral singularity brought by the mapping can be removed in a naive and stable way we build aquasi interpolation triangular spectral element approximation and analyze its approximation error based on this quasi interpolation spectral element approximation we put forward a new triangular spectral element method for the elliptic problems we present the approximation scheme analyze the convergence and do some experiments to test the effectiveness at last we implement this triangular spectral element method to solve the steady stokes problem

spectral multigrid methods for elliptic equations ii a detailed description of spectral multigrid methods is provided this includes the interpolation and coarse grid operators for both periodic and dirichlet problems the spectral methods for periodic problems use fourier series and those for dirichlet problems are based upon chebyshev polynomials an improved preconditioning for dirichlet problems is given numerical examples and practical advice are included

a triangular spectral element method using fully tensorial rational basis functions a rational approximation in a triangle is proposed and analyzed in this paper the rational basis functions in the triangle are obtained from the polynomials in the reference square through a collapsed coordinate transform optimal error estimates for the l 2 and h 1 0 orthogonal projections are derived with upper bounds expressed in the original coordinates in the triangle it is shown that the rational approximation is as accurate as the polynomial approximation in the triangle illustrative numerical results which are in agreement with the theoretical estimates are presented

on topological entropy of commuting tree maps abstract let t be a tree with s ends and f g be continuous maps from t to t with f g g f in this note we show that if there exists a positive integer m 2 such that gcd m l 1 for any 2 l s and f g share a periodic point which is a k m periodic point of f for some positive integer k then the topological entropy of f g is positive

defect correction for convection dominated flow a defect correction scheme with the first order upwind preconditioner is considered by fourier analysis the preconditioning properties for the second order upwind scheme the central scheme and spectral methods are examined since the eigenvalues of the preconditioned operator are complex the generalized minimal residual iteration is used for the iterative solution this procedure is applied to the boussinesq flow problem in vorticity streamfunction formulation numerical results are presented for increasing rayleigh numbers

sur la dynamique unidimensionnelle en regularite intermediaire using probabilistic methods we prove new rigidity results for groups and pseudo groups of diffeomorphisms of one dimensional manifolds with intermediate regularity class i e between c 1 and c 2 in particular we show some generalizations of denjoy theorem and the classical kopell lemma for abelian groups these techniques are also applied to the study of codimension 1 foliations for instance we obtain several generalized versions of sacksteder theorem in class c 1 we conclude with some remarks about the stationary measure

efficient spectral galerkin algorithms for direct solution of second order equations using ultraspherical polynomials it is well known that spectral methods tau galerkin collocation have a condition number of o n4 n is the number of retained modes of polynomial approximations this paper presents some efficient spectral algorithms which have a condition number of o n2 based on the ultraspherical galerkin methods for second order elliptic equations in one and two space variables the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with specially structured matrices that can be efficiently inverted the complexities of the algorithms are a small multiple of nd 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions

effective spectral approximations of convection diffusion equations abstract we propose two approaches for the spectral approximation of convection diffusion problems with thin layers the first is based upon stabilization methods of petrov galerkin type that generalize those proposed in recent years in the framework of finite elements the second is a heterogeneous domain decomposition which is extremely effective whenever layer location can be predicted a priori a third method stems from a combined approach in which a low order stabilization method is applied as a pre processor for the heterogeneous domain decomposition method

a 3d spectral multigrid method we consider a 3d spectral multigrid method for 3d elliptic problems with arbitrary anisotropies the smoothing method is a preconditioned richardson iteration where a pseudospectral discretization is used on the right hand side and a finite difference discretization on the left hand side in each step of this iteration a finite difference problem is solved approximately by plane relaxation sweeps the plane relaxation requires the solution of 2d finite difference problems these problems are solved approximately using 2d multigrid cycles

performance of gem2 on the ellpack problem population the pilot global element program gem2 mk1 0 has been used to solve the two dimensional problems in the ellpack population of partial differential equation test problems the performance achieved is summarized in this paper special attention is paid to those problems which are singular or near singular and for which comparisons have been published by houstis and rice 12

a fast adaptive numerical method for stiff two point boundary value problems we describe a robust adaptive algorithm for the solution of singularly perturbed two point boundary value problems many different phenomena can arise in such problems including boundary layers dense oscillations and complicated or ill conditioned internal transition regions working with an integral equation reformulation of the original differential equation we introduce a method for error analysis which can be used for mesh refinement even when the solution computed on the current mesh is underresolved based on this method we have constructed a black box code for stiff problems which automatically generates an adaptive mesh resolving all features of the solution the solver is direct and of arbitrarily high order accuracy and requires an amount of time proportional to the number of grid points

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

konvergenzaussagen fur kollokationsverfahren bei elliptischen randwertaufgaben we examine the convergence of collocation with polynomials for elliptic problems the general theory is imbedded in thel p theory from which known results of existence and regularity are adopted convergence results are deduced in the cases of periodic dirichlet and mixed boundary conditions it becomes obvious that the order of convergence essentially depends on the smoothness of the solution