fourier chebyshev spectral method for the two dimensional navier stokes equations a fourier chebyshev spectral scheme is proposed for the two dimensional navier stokes equations to avoid the difficulty of choosing the trial function space satisfying the incompressible condition the pressure is evaluated by solving a poisson equation the generalized stability and convergence are analyzed strictly the numerical results show the high accuracy of this method

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

spline based deforming ellipsoids for interactive 3d bioimage segmentation we present a new fast active contour model a k a snake for image segmentation in 3d microscopy we introduce a parametric design that relies on exponential b spline bases and allows us to build snakes that are able to reproduce ellipsoids we design our bases to have the shortest possible support subject to some constraints thus computational efficiency is maximized the proposed 3d snake can approximate blob like objects with good accuracy and can perfectly reproduce spheres and ellipsoids irrespective of their position and orientation the optimization process is remarkably fast due to the use of gauss theorem within our energy computation scheme our technique yields successful segmentation results even for challenging data where object contours are not well defined this is due to our parametric approach that allows one to favor prior shapes in addition this paper provides a software that gives full control over the snakes via an intuitive manipulation of few control points

wavelets fractals and radial basis functions wavelets and radial basis functions rbfs lead to two distinct ways of representing signals in terms of shifted basis functions rbfs unlike wavelets are nonlocal and do not involve any scaling which makes them applicable to nonuniform grids despite these fundamental differences we show that the two types of representation are closely linked together through fractals first we identify and characterize the whole class of self similar radial basis functions that can be localized to yield conventional multiresolution wavelet bases conversely we prove that for any compactly supported scaling function spl phi x there exists a one sided central basis function spl rho sub x that spans the same multiresolution subspaces the central property is that the multiresolution bases are generated by simple translation of spl rho sub without any dilation we also present an explicit time domain representation of a scaling function as a sum of harmonic splines the leading term in the decomposition corresponds to the fractional splines a recent continuous order generalization of the polynomial splines

locally supported rational spline wavelets on a sphere in this paper we construct certain continuous piecewise rational wavelets on arbitrary spherical triangulations giving explicit expressions of these wavelets our wavelets have small support a fact which is very important in working with large amounts of data since the algorithms for decomposition compression and reconstruction deal with sparse matrices we also give a quasi interpolant associated to a given triangulation and study the approximation error some numerical examples are given to illustrate the efficiency of our wavelets

how to get around a simple quadratic fold we analyse from a theoretical point of view a computational technique previously introduced 2 5 for tracing branches of solutions of nonlinear equations near simple quadratic folds

convergence cones near bifurcation the performance of predictor solver continuation methods near bifurcation points is considered a general theory is developed for the convergence of two commonly used solvers namely newton s method and the chord method near singular points of a nonlinear equation this theory provides a uniform treatment of continuation for simple bifurcation multiple bifurcation and multiple limit point bifurcation problems for these types of bifurcation it is shown that there are conical domains of attraction for the above iterative methods centred on solution branches which correspond to isolated roots of the algebraic bifurcation equations

error estimates for scattered data interpolation on spheres we study sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the n sphere the interpolation knots are scattered our approach partly follows the general theory of golomb and weinberger and related estimates these error estimates are then based on series expansions of smooth functions in terms of spherical harmonics the markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics

an analysis of upwind schemes for the navier stokes equations we state a general result on approximation of branches of nonsingular solutions of nonlinear problems the abstract theory is then applied to the study of two mixed finite element schemes for the navier stokes equations using an upwind discretization of the convective terms

unified and extended form of three types of splines the three types refer to polynomial trigonometric and hyperbolic splines in this paper we unify and extend them by a new kind of spline ue spline for short defined over the space cos wt sin wt 1 t t l where l is an arbitrary nonnegative integer w is a frequency sequence w i a i a i r existing splines such as usual polynomial b splines cb splines hb splines nuat splines ah splines fb splines and the third form fb splines etc are all special cases of ue splines ue splines inherit most properties of usual polynomial b splines and enjoy some other advantageous properties for modelling they can exactly represent classical conics the catenary the helix and even the eight curve a kind of snake like curves etc

wavelets bases adapted to a self similar quasicrystal given any self similar quasicrystal in rn with inflation 1 we construct bases of l2 rn having the following structure nj 2 jx j z where the mother wavelets are smooth and with exponential decay or compact support we also show that wavelets constitute a relatively compact set in some sobolev space and that they depend continuously on when is equipped with an appropriate topology

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

a borsuk ulam theorem for orthogonal tk and zpr actions and applications abstract a version if the borsuk ulam theorem is proved for the torus tk and p torus zpr we deal with the case of a nonempty fixed point set of groups on the sphere of representation as an application a geometrical index theory is introduced for these groups the index is used to obtain the existence of multiple critical points for functions invariant under tk or zpr action

on algebraic problems behind the brouwer degree of equivariant maps abstract given a finite group g and two unitary g representations v and w possible restrictions on topological degrees of equivariant maps between representation spheres s v and s w are usually expressed in a form of congruences modulo the greatest common divisor of lengths of orbits in s v denoted by v effective applications of these congruences is limited by answers to the following questions i under which conditions is v 1 and ii does there exist an equivariant map with the degree easy to calculate in the present paper we address both questions we show that v 1 for each irreducible non trivial c g module if and only if g is solvable this provides a new solvability criterion for finite groups for non solvable groups we use 2 transitive actions to construct complex representations with non trivial characteristic regarding the second question we suggest a class of norton algebras without 2 nilpotents giving rise to equivariant quadratic maps which admit an explicit formula for the degree

an optimization based domain decomposition method for pdes with random inputs an optimization based domain decomposition method for stochastic elliptic partial differential equations is presented the main idea of the method is a constrained optimization problem for which the minimization of an appropriate functional forces the solutions on the two subdomains to agree on the interface the constraints are the stochastic partial differential equations the existence of optimal solutions for the stochastic optimal control problem is shown as is the convergence to the exact solution of the given problem we prove the existence of a lagrange multiplier and derive an optimality system from which solutions of the domain decomposition problem may be determined finite element approximations to the solutions of the optimality system are defined and analyzed with respect to both spatial and random parameter spaces then the results of some numerical experiments are given to confirm theoretical error estimate results

on compactly supported spline wavelets and a duality principle let v 1 v 0 v 1 be a multiresolution analysis of l 2 generated by the mth order b spline n m x we exhibit a compactly supported basic wavelet m x that generates the corresponding orthogonal complementary wavelet subspaces w 1 w 0 w 1 consequently the two finite sequences that describe the two scale relations of n m x and m x in terms of n m 2x j j z yield an efficient reconstruction algorithm to give an efficient wavelet decomposition algorithm based on these two finite sequences we derive a duality principle which also happens to yield the dual bases n m x j and 59 m x j relative to n m x j and m x j respectively

on the reconstruction of height functions and terrain maps from dense range data this paper describes a method for combining multiple dense range images to create surface reconstructions of height functions height functions are a special class of three dimensional 3 d surfaces where one 3 d coordinate is a function of the other two they are relevant for application domains such as terrain modeling or two and half dimensional surface reconstruction dense range maps are produced by either a range measuring device combined with a scanning mechanism or a triangulation scheme such as active or passive stereo the proposed method follows from a statistical formulation that characterizes the optimal surface estimate as the one that maximizes the posterior probability conditional on the input data and prior information about the application domain because the domain of the reconstruction is a two dimensional 2 d scalar function the optimal surface can be expressed as an image and the variational form of that optimization produces a 2 d partial differential equation pde the pde consists of two parts a first order data term and a second order smoothing term thus optimal surface reconstruction is formulated as the solution to a second order nonlinear pde on an image which is related to the family of pde based image processing algorithms in the literature this paper presents the theory for reconstruction and some particular aspects of the numerical implementation it also analyzes results on both synthetic and real data sets which show a 75 95 reduction of the rms sensor error

computational methods for bifurcation problems with symmetries with special attention to steady state and hopf bifurcation points we show how group theoretical methods can be employed to utilize the symmetry of a bifurcation problem in numerical computations we extend the approach by werner 1988 by presenting methods for the detection of bifurcation points and the computation of multiple hopf points the essential numerical point is the utilization of certain reduced instead of full systems involving appropriate subgroups of the underlying symmetry group the group theoretical tool is an a priori knowledge of the interaction of certain subgroups 0 and of at possibly multiple steady state or hopf bifurcation points minimal 0 breaking bifurcation we introduce a bifurcation graph which shows graphically this a priori information its edges represent possible symmetry breaking bifurcations our analysis follows the lines of golubitsky stewart and schaeffer 1988 but it is aimed to numerical applications we have chosen a 4 box brusselator model in order to explain our notions and ideas and to discuss the numerical procedure

efficient estimation of neural weights by polynomial approximation it has been known for some years that the uniform density problem for forward neural networks has a positive answer any real valued continuous function on a compact subset of r sup d can be uniformly approximated by a sigmoidal neural network with one hidden layer we design here algorithms for efficient uniform approximation by a certain class of neural networks with one hidden layer which we call nearly exponential this class contains e g all networks with the activation functions 1 1 e sup t tanh t or e sup t spl and 1 in their hidden layers the algorithms flow from a theorem stating that such networks attain the order of approximation o n sup 1 d d being dimension and n the number of hidden neurons this theorem in turn is a consequence of a close relationship between neural networks of nearly exponential type and multivariate algebraic and exponential polynomials the algorithms need neither a starting point nor learning parameters they do not get stuck in local minima and the gain in execution time relative to the backpropagation algorithm is enormous the size of the hidden layer can be bounded analytically as a function of the precision required

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

applications of optimally local interpolation to interpolatory approximants and compactly supported wavelets the objective of this paper is to introduce a general scheme for the construction of interpolatory approximation formulas and compactly supported wavelets by using spline functions with arbitrary nonuniform knots both construction procedures are based on certain optimally local interpolatory fundamental spline functions which are not required to possess any approximation property

analysis of iterative methods for the steady and unsteady stokes problem application to spectral element discretizations a new and detailed analysis of the basic uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady stokes operator is presented the paper focuses on the following new aspects explicit construction of the uzawa pressure operator spectrum for a semiperiodic model problem general relationship of the convergence rate of the uzawa procedure to classical inf sup discretization analysis and application of the method to high order variational discretization

une methode numerique pour le calcul des points de retournement application a un probleme aux limites non lineaire an iterative method is presented for determining a turning point in a branch of solutions of an algebraic system of equations depending on a parameter one shows that the convergence is superlinear and that this method may constitute a complement for the newton method with a finite element method numerical results are given testing the process on the discrete version of a mildly nonlinear boundary value problem to determine the turning point in the positive solution branch

fitting scattered data on spherelike surfaces using tensor products of trigonometric and polynomial splines a method is presented for fitting a function defined on a general smooth spherelike surfaces given measurements on the function at a set of scattered points lying ons the approximating surface is constructed by mapping the surface onto a rectangle and using a tensor product of polynomial splines with periodic trigonometric splines the use of trigonometric splines allows a convenient solution of the problem of assuring that the resulting surface is continuous and has continuous tangent planes at all points ons two alternative algorithms for computing the coefficients of the tensor fit are presented one based on global least squares and the other on the use of local quasi interpolators the approximation order of the method is established and the numerical performance of the two algorithms is compared

robust regression of scattered data with adaptive spline wavelets a coarse to fine data fitting algorithm for irregularly spaced data based on boundary adapted adaptive tensor product semi orthogonal spline wavelets has been proposed in castano and kunoth 2003 this method has been extended in castano and kunoth 2005 to include regularization in terms of sobolev and besov norms in this paper we develop within this least squares approach some statistical robust estimators to handle outliers in the data our wavelet scheme yields a numerically fast and reliable way to detect outliers

quadratic spline wavelets with arbitrary simple knots on the sphere in this paper we extend the method for fitting functions on the sphere described in lyche and schumaker siam j sci comput 22 2 2000 724 to the case of nonuniform knots we present a multiresolution method leading to functions on the sphere which is based on tensor products of quadratic polynomial splines and trigonometric splines of order three with arbitrary simple knot sequences we determine the decomposition and reconstruction matrices corresponding to the polynomial and trigonometric spline spaces we describe the tensor product decomposition and reconstruction algorithms in matrix forms which are convenient for the compression of surfaces we give the different steps of computer implementation and finally we present a test example by using two knot sequences a uniform one and a sequence of chebyshev points

finite dimensional approximation of nonlinear problems part ii limit points note univ paris 6 f 75230 paris 05 france ecole polytech ctr math appl cnrs equipe rech 747 f 91128 palaiseau france univ pavia cnr ist anal numer i 27100 pavia italy brezzi f univ pavia cnr ist math appl i 27100 pavia italy isi document delivery no lr761times cited 54cited reference count 13 reference asn article 1981 002doi 10 1007 bf01396184 record created on 2006 08 24 modified on 2017 05 12

error estimates for the spectral approximation of the nonstationary stokes problem this paper analyzes a discretization of the nonstationary stokes problem with periodic nonperiodic boundary conditions fourier legendre spectral approximations are considered with respect to the space variables while the time discretization is based on the second order crank nicolson scheme optimal convergence estimates are proved for the velocity field

divergence free velocity fields in nonperiodic geometries the influence matrix method of enforcing incompressibility in pseudospectral simulations of fluid dynamics as described by kleiser and schumann for channel flow is generalized to other geometries a formalism of projection and matrix operators is introduced in which the influence matrix method is shown to be an application of the classic sherman morrison woodbury formula of numerical linear algebra special attention is paid to the tau correction applications to cartesian geometries illustrate the concepts and highlight the role of symmetry a coded implementation in a cylindrical geometry requiring special treatment of coordinate singularities is used to investigate properties of the influence matrix and to provide estimates of timings

arithmetic results on orbits of linear groups the first four authors acknowledge the support of an arc discovery grant the first author acknowledges the support of an australian research fellowship the third author acknowledges the support of a federation fellowship the fifth author acknowledges the support of the nsf grants dms 0901241 and dms 1201374

spline prewavelets for non uniform knots in this paper we develop a framework suitable for performing a multiresolution analysis using univariate spline spaces of arbitrary degree and with non uniform knot sequences to this end we show among other things the existence of compactly supported prewavelets and of prewavelets that are globally supported but decay exponentially in each case we obtain a decomposition of a fine spline space as a sum of a coarse spline space plus a spline space spanned by prewavelets

automatic generation of rbf networks using wavelets abstract learning can be viewed as mapping from an input space to an output space examples of these mappings are used to construct a continuous function that approximates the given data and generalizes for intermediate instances radial basis function rbf networks are used to formulate this approximating function a novel method is introduced that automatically constructs a generalized radial basis function grbf network for a given mapping and error bound this network is shown to be the smallest network within the error bound for the given mapping the integral wavelet transform is used to determine the parameters of the network simple one dimensional examples are used to demonstrate how the network constructed using the transform is superior to that constructed using standard ad hoc optimization techniques the paper concludes with the automatic generation of grbf networks for a multi dimensional problem namely real time 3d object recognition and pose estimation the results of this application are favorable

spectral method solution of the stokes equations on nonstaggered grids abstract the stokes equations are solved using spectral methods with staggered and nonstaggered grids numerous ways to avoid the problem of spurious pressure modes are presented including new techniques using the pseudospectral method and a method solving the weak form of the governing equations a variation on the spectral element method developed by patera the pseudospectral methods using nonstaggered grids are simpler to implement and have comparable or better accuracy than the staggered grid formulations three test cases are presented a formulation with an exact solution a formulation with homogeneous boundary conditions and the driven cavity problem the solution accuracy is shown to be greatly improved for the driven cavity problem when the analytical solution of the singular flow behavior in the upper corners is separated from the computational solution

analysis of the combined finite element and fourier interpolation we consider lagrange interpolation involving trigonometric polynomials of degree n in one space direction and piecewise polynomials over a finite element decomposition of mesh size h in the other space directions we provide error estimates in non isotropic sobolev norms depending additively on the parametersh andn an application to the convergence analysis of an elliptic problem with some numerical results is given

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

a spectral collocation method for the navier stokes equations abstract a fourier chebyshev spectral method for the incompressible navier stokes equations is described it is applicable to a variety of problems including some with fluid properties which vary strongly both in the normal direction and in time in this fully spectral algorithm a preconditioned iterative technique is used for solving the implicit equations arising from semi implicit treatment of pressure mean advection and vertical diffusion terms the algorithm is tested by applying it to hydrodynamic stability problems in channel flow and in external boundary layers with both constant and variable viscosity

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

shape analysis of cubic trigonometric bezier curves with a shape parameter abstract for the cubic trigonometric polynomial curves with a shape parameter tb curves for short the effects of the shape parameter on the tb curve are made clear the shape features of the tb curve are analyzed the necessary and sufficient conditions are derived for these curves having single or double inflection points a loop or a cusp or be locally or globally convex the results are summarized in a shape diagram of tb curves which is useful when using tb curves for curve and surface modeling furthermore the influences of shape parameter on the shape diagram and the ability for adjusting the shape of the curve are shown by graph examples respectively

numerical solution to a mixed navier stokes darcy model by the two grid approach we study numerical methods for a coupled navier stokes darcy model which describes a fluid flow filtrating through porous media a decoupled and linearized two grid algorithm is proposed numerical analysis and experiments are presented to show the efficiency and effectiveness of the decoupled and linearized algorithm

spectral and pseudo spectral approximations of the navier stokes equations galerkin and collocation approximations by trigonometric polynomials to the stationary navier stokes equations with periodic boundary conditions are analyzed stability results and optimal rates of convergence for both velocity and pressure in sobolev spaces are proven essential tools to carry out this analysis are some abstract results concerning general polynomial approximations to nonsingular solutions of nonlinear problems brezzi rappag raviart numer math 36 1980 pp 1 25 maday quarteroni rairo numer anal 1980

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

a spectral numerical method for the navier stokes equations with applications to taylor couette flow a new spectral method for solving the incompressible navier stokes equations in a plane channel and between concentric cylinders is presented the method uses spectral expansions which inherently satisfy the boundary conditions and the continuity equation and yield banded matrices which are efficiently solved at each time step in addition the number of dependent variables is reduced resulting in a reduction in computer memory requirements several test problems have been computed for the channel flow and for flow between concentric cylinders including taylor couette flow with axisymmetric taylor vortices and wavy vortices in all cases agreement with available experimental and theoretical results is very good

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

multiquadric prewavelets on nonequally spaced knots in one dimension we identify univariate prewavelets on spaces spanned by translates of multiquadric functions and other radial basis functions with nonequally spaced centers or knots although the multiquadric function and its relations are our prime examples the theory is sufficiently broad to admit prewavelets from other radial basis function spaces as well

approximation of hopf bifurcation we make several assumptions on a nonlinear evolution problem ensuring the existence of a hopf bifurcation under a fairly general approximation condition we define a discrete problem which retains the bifurcation property and we prove an error estimate between the branches of exact and approximate periodic solutions

an optimization based domain decomposition method for the navier stokes equations an optimization based nonoverlapping domain decomposition method for the solution of the navier stokes equations is presented the crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains the constraints are the navier stokes equations in the subdomains with suitably chosen boundary conditions along the interfaces we show that solutions of the minimization problem exist and derive an optimality system from which these solutions may be determined finite element approximations of the solutions of the optimality system are examined the domain decomposition method is also reformulated as a nonlinear least squares problem and the results of some numerical experiments are given

quadratic dynamical systems and algebras abstract quadratic dynamical systems come from differential or discrete systems of the form q x or x k 1 q x k where q r n r n is homogeneous of degree 2 i e q x 2 q x for all r x r n defining a bilinear mapping r n r n r n by x y 1 2 q x y q x q y we view xy x y as a multiplication and thus consider a r n to be a commutative nonassociative algebra the quadratic systems are then studied with the general theme that the structure of the algebras helps determine the behavior of the solutions for example semisimple algebras give a decoupling of the original system into systems occurring in simple algebras and solvable algebras give solutions to differential systems via linear differential equations the general three dimensional example of the latter phenomenon is described there are many classical examples and the scope of quadratic systems is large every polynomial system can be embedded into a higher dimensional quadratic system such that solutions of the original system are obtained from the quadratic system for differential systems nilpotents of index 2 n 2 0 are equilibria and idempotents e 2 e give ray solutions the origin is never asymptotically stable and the existence of nonzero idempotents implies that the origin is actually unstable nonzero equilibria are not hyperbolic but can be studied by standard algebra techniques using nondegenerate bilinear forms as lyapunov functions periodic orbits lie on cones they cannot occur in dimension 2 or in power associative algebras no periodic orbit can be an attractor but limit cycles invariant cones can exist automorphisms of the algebra a leave equilibria periodic orbits and domains of attraction invariant also explicit solutions can be given by the action of automorphisms on an initial point the general three dimensional example of this is described thus if there are sufficient automorphisms hilbert s sixteenth problem in r 3 has the following answer if the periodic orbits of fixed period are isolated then there is only one cone of periodic solutions this cone is almost an attractor for discrete systems there are many similarities to the differential systems for example orbits can be given by automorphisms and again the general three dimensional example of this is described however distinctions become more obvious using algebras for example if the algebra a is nilpotent then for the differential system the solutions are unbounded but for the discrete system the trajectories iterate to zero in a also idempotents e 2 e are the fixed points for the discrete system and e is unstable if there exist suitable nilpotents n 2 0 the interplay between algebras and dynamical systems can solve old problems but more importantly it can create new opportunities in both areas

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

spectral element methods for large scale parallel navier stokes calculations abstract we analyze the computational complexity of a recently developed two level iteration scheme for spectral element solution of the time dependent incompressible navier stokes equations in complex domains we present several algorithmic advances which significantly enhance the scalability of this approach including implementation of an advanced combine operation for degrees of freedom on subdomain edges parallel solution of the fine grained coarse grid problem and implementation of local low order finite element preconditioners for the find grid problem timings on the 512 node intel delta machine show that the combined improvements lead to a fourfold reduction in navier stokes solution time for the particular case of a three dimensional boundary layer calculation consisting of one million degrees of freedom

spectral approximation of the periodic nonperiodic navier stokes equations in order to approximate the navier stokes equations with periodic boundary conditions in two directions and a no slip boundary condition in the third direction by spectral methods we justify by theoretical arguments an appropriate choice of discrete spaces for the velocity and the pressure the compatibility between these two spaces is checked via an infsup condition we analyze a spectral and a collocation pseudo spectral method for the stokes problem and a collocation pseudo spectral method for the navier stokes equations we derive error bounds of spectral type i e which behave likem wherem depends on the number of degrees of freedom of the method and represents the regularity of the data

a unified spectral method for fpdes with two sided derivatives part ii stability and error analysis abstract we present the stability and error analysis of the unified petrov galerkin spectral method developed in 1 for linear fractional partial differential equations with two sided derivatives and constant coefficients in any 1 d dimensional space time hypercube d 1 2 3 subject to homogeneous dirichlet initial boundary conditions specifically we prove the existence and uniqueness of the weak form and perform the corresponding stability and error analysis of the proposed method finally we perform several numerical simulations to compare the theoretical and computational rates of convergence

generalized inf sup conditions for chebyshev spectral approximation of the stokes problem a mixed problem and its approximation in an abstract framework are considered they are proved to be well posed if and only if several inf sup conditions are satisfied these results are applied to the stokes equations in a square formulated in chebyshev weighted sobolev spaces and their approximations two kinds of spectral discretizations are analyzed a galerkin method and a collocation method at the chebyshev nodes

wavelet approximations for first kind boundary integral equations on polygons an elliptic boundary value problem in the interior or exterior of a polygon is transformed into an equivalent first kind boundary integral equation its galerkin discretization with n degrees of freedom on the boundary with spline wavelets as basis functions is analyzed a truncation strategy is presented which allows to reduce the number of nonzero elements in the stiffness matrix from o n 2 to o n log n entries the condition numbers are bounded independently of the meshwidth it is proved that the compressed scheme thus obtained yields in o n log n 2 operations approximate solutions with the same asymptotic convergence rates as the full galerkin scheme in the boundary energy norm as well as in interior points numerical examples show the asymptotic error analysis to be valid already for moderate values of n

a multiresolution tensor spline method for fitting functions on the sphere we present the details of a multiresolution method which we proposed at the taormina wavelet conference in 1993 see l spline wavelets in wavelets theory algorithms and applications c chui l montefusco and l puccio eds academic press new york pp 197 212 which is suitable for fitting functions or data on the sphere the method is based on tensor products of polynomial splines and trigonometric splines and can be adapted to produce surfaces that are either tangent plane continuous or almost tangent plane continuous the result is a convenient compression algorithm for dealing with large amounts of data on the sphere we give full details of a computer implementation that is highly efficient with respect to both storage and computational cost we also demonstrate the performance of the method on several test examples