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

a time indexed formulation of non preemptive single machine scheduling problems we consider the formulation of non preemptive single machine scheduling problems using time indexed variables this approach leads to very large models but gives better lower bounds than other mixed integer programming formulations we derive a variety of valid inequalities and show the role of constraint aggregation and the knapsack problem with generalised upper bound constraints as a way of generating such inequalities a cutting plane branch and bound algorithm based on these inequalities has been implemented computational experience on small problems with 20 30 jobs and various constraints and objective functions is presented

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

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

hodge cohomology of gravitational instantons we study the space of l2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type these metrics generalize the geometric structures at infinity of several different well known classes of metrics including asymptotically locally euclidean manifolds the known types of gravitational instantons and also poincare metrics on q rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in kahler manifolds the answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold the l2 signature formula implied by our result is closely related to the one proved by dai 25 and more generally by vaillant 67 and identifies dai s invariant directly in terms of intersection cohomology of differing perversities this work is also closely related to a recent paper of carron 12 and the forthcoming paper of cheeger and dai 17 we apply our results to a number of examples gravitational instantons among them arising in predictions about l2 harmonic forms in duality theories in string theory

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

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

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

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

some facets for an assignment problem with side constraints this paper presents a class of facet defining inequalities for an assignment problem with the additional constraints that specified variables are required to be equal to each other in a special case the complete polyhedral description is given

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

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

kahler parabolicity and the euler number of compact manifolds of non positive sectional curvature let m 2n be a compact riemannian manifold of non positive sectional curvature it is shown that if m 2n is homeomorphic to a kahler manifold then its euler number satisfies the inequality 1 n chi m 2n geq 0

the cohomology of the space of magnetic monopoles denote byx q the reduced space ofsu 2 monopoles of chargeq in 3 in this paper the cohomology ofx q the cohomology with compact supports ofx q and the image of the latter in the former are all calculated as representations of q which acts onx 2 this provides a non trivial lower bound for thel 2 cohomology ofx q which is compatible with some conjectures of sen it is also shown that granted some assumptions about the metric onx q itsl 2 cohomology does not exceed this bound in the situation referred to in the paper as the coprime case

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

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

sur la l 2 cohomologie des varietes a courbure negative we give a topological interpretation of the space of l2 harmonic forms on finite volume manifolds with sufficiently pinched negative curvature we give examples showing that this interpretation fails if the curvature is not sufficiently pinched and that our result is sharp with respect to the pinching constants the method consists first in comparing l2 cohomology with weighted l2 cohomology thanks to previuos works done by t ohsawa and then in identifying these weighted spaces

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

scheduling jobs of equal length complexity facets and computational results the following problem was originally motivated by a question arising in the automated assembly of printed circuit boards given are n jobs which have to be performed on a single machine within a fixed timespan 0 t subdivided into t unit length subperiods the processing time or length of each job equals p p n the processing cost of each job is an arbitrary function of its start time the problem is to schedule all jobs so as to minimize the sum of the processing costs this problem is proved to be np hard already for p 2 and 0 1 processing costs on the other hand when t np c with c constant the problem can be solved in polynomial time a partial polyhedral description of the set of feasible solutions is presented in particular two classes of facet defining inequalities are described for which the separation problem is polynomially solvable also we exhibit a class of objective functions for which the inequalities in the lp relaxation guarantee integral solutions finally we present a simple cutting plane algorithm and report on its performance on randomly generated problem instances

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

the hodge cohomology of a conformally compact metric on etudie le laplacien de hogde agissant sur des k formes differentielles pour une classe de varietes de riemann completes a courbure sectionnelle negative proche de l infini ces varietes ont des compactifications c sur lesquelles la metrique est conforme a une metrique lisse jusqu au bord avec un facteur conforme 2 une fonction definissante pour le bord

l 2 cohomology of hyperkahler quotients using an argument of jost and zuo we give a criterion which implies that the l 2 harmonic forms on a complete noncompact hyperkahler manifold lie in the middle dimension and are invariant under the isometry group this is applied to various examples and in particular gives a verification of some of the predictions of sen on monopole moduli spaces

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

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

the relation of time indexed formulations of single machine scheduling problems to the node packing problem the relation of time indexed formulations of nonpreemptive single machine scheduling problems to the node packing problem is established and then used to provide simple and intuitive alternate proofs of validity and maximality for previously known results on the facial structure of the scheduling problem previous work on the facial structure has focused on describing the convex hull of the set of feasible partial schedules schedules in which not all jobs have to be started the equivalence between the characteristic vectors of this set and those of the set of feasible node packings in a graph whose structure is determined by the parameters of the scheduling problem is established the main contribution of this paper is to show that the facet inducing inequalities for the convex hull of the set of feasible partial schedules that have integral coefficients and right hand side 1 or 2 are the maximal clique inequalities and the maximally and sequentially lifted 5 hole inequalities of the convex hull of the set of feasible node packings in this graph respectively

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

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

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

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 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

cohomologie l 2 des varietes qale we give a topological interpretation of the space of l 2 harmonis forms of some qale manifolds introduced by d joyce we introduce a analytical criterium which make possible the used of mayer vietoris sequence

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

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