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

spectral hp least squares finite element formulation for the navier stokes equations we consider the application of least squares finite element models combined with spectral hp methods for the numerical solution of viscous flow problems the paper presents the formulation validation and application of a spectral hp algorithm to the numerical solution of the navier stokes equations governing two and three dimensional stationary incompressible and low speed compressible flows the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity or velocity gradients as additional independent variables and the least squares method is used to develop the finite element model high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method spectral convergence of the l2 least squares functional and l2 error norms is verified using smooth solutions to the two dimensional stationary poisson and incompressible navier stokes equations numerical results for flow over a backward facing step steady flow past a circular cylinder three dimensional lid driven cavity flow and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation

direct minimization of the least squares spectral element functional part i direct solver this paper describes an equivalent but improved least squares formulation for the numerical approximation of the solution of partial differential equations instead of using variational analysis to impose the conditions for minimizing the residual the residuals are minimized directly thus leading to a method we will denote by direct minimization dm dm circumvents setting up the normal equations which consists of matrix matrix multiplications matrix matrix multiplications are expensive may lead to loss of accuracy and destroy the sparsity pattern present in the original system the condition number of the dm formulation is the square root of the condition number which would be obtained if variational analysis was employed an element by element procedure will be presented which allows for parallelization of dm a computational comparison between dm and the conventional least squares formulation based on variational analysis will be presented

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

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

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

compatibility conditions for the stokes system discretized in 2 d cartesian domains abstract the maximum number of independent numerical constraints that can be imposed on the velocity field is determined by identifying the compatibility conditions which arise between the discretized incompressibility constraint equations and the velocity boundary conditions this number does not depend on the chosen numerical scheme and defines the maximum number of pressure modes that are numerically accessible for the navier stokes problem discretized in 2 d cartesian domains it is shown that at most n 8 pressure modes can be evaluated n being the number of modes chosen for each velocity component the eight out of access pressure modes are the so called spurious pressure modes

the stability of pseudospectral chebyshev methods abstract the stability of pseudospectral chebyshev methods is demonstrated for parabolic and hyperbolic problems with variable coefficients the choice of collocation points is discussed numerical examples are given for the case of variable coefficient hyperbolic equations author

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

mass and momentum conservation of the least squares spectral collocation method for the navier stokes equations from the literature it is known that the least squares spectral element method lssem for the stationary stokes equations performs poorly with respect to mass conservation but compensates this lack by a superior conservation of momentum furthermore it is known that the least squares spectral collocation method lsscm leads to superior conservation of mass and momentum for the stationary stokes equations in the present paper we consider mass and momentum conservation of the lsscm for time dependent stokes and navier stokes equations we observe that the lsscm leads to improved conservation of mass and momentum for these problems furthermore the lsscm leads to the well known time dependent profiles for the velocity and the pressure profiles to obtain these results we use only a few elements each with high polynomial degree avoid normal equations for solving the overdetermined linear systems of equations and introduce the clenshaw curtis quadrature rule for imposing the average pressure to be zero furthermore we combined the transformation of gordon and hall transfinite mapping with the least squares spectral collocation scheme to discretize the internal flow problems

on jordan holder series of some locally analytic representations let g be a split reductive p adic group this paper is about the jordan holder series of locally analytic g representations which are induced from locally algebraic representations of a parabolic subgroup p g we construct for every representation m of lie g in the bgg category o which is equipped with an algebraic p action and for every smooth p representation v a locally analytic representation fg p m v of g this gives rise to a bi functor to the category of locally analytic representations we prove that it is exact and give a criterion for the topological irreducibility of fg p m v in terms of m and v

splitting techniques for the pseudospectral approximation of the unsteady stokes equations a pseudospectral or collocation approximation of the unsteady stokes equations is considered using the uzawa algorithm the spectral system is decoupled into a helmholtz equation and an equation with a pseudo laplace operator it is proven that the eigenvalues of the pseudo laplacian are real and negative except one zero eigenvalue belonging to the constant mode the pseudospectral method avoids spurious modes since the pressure is approximated with lower order two degrees lower polynomials than the velocity only one grid no staggered grids with the standard chebyshev gauss lobatto nodes is used for discretization for the iterative solution of the spectral system an effective preconditioner is necessary here effective finite difference preconditioners for the spectral pseudo laplacian are presented

least squares finite element formulations for viscous incompressible and compressible fluid flows abstract we present least squares based finite element formulations for the numerical solution of viscous fluid flows governed by the navier stokes equations as an alternate approach to the well known weak form galerkin finite element formulations the use of least squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise and the resulting linear algebraic problem will have a symmetric positive definite coefficient matrix we address the issue of norm equivalence of the least squares functional and its implications on the resulting finite element model in particular we develop an understanding of the compromise that must exist between the optimality and practicality of the finite element model the latter measured in terms of c k regularity across inter element boundaries we show through numerical examples that for the traditional c 0 basis such a compromise is possible when high p levels are used to span the finite element spaces when a low p level solution is desired guidelines are presented to obtain a reliable least squares collocation solution numerical examples are presented to demonstrate the high and low p level approaches these include incompressible flow past two circular cylinders in a side by side arrangement for gap sizes s d 2 0 and 0 85 incompressible flow past a square cylinder and subsonic transonic and supersonic compressible flow past a circular cylinder in addition we present a discontinuous least squares formulation where c k regularity across inter element boundaries is enforced in a weak sense through the least squares functional allowing for h and p type non conformities in the computational domain

high order direct stokes solvers with or without temporal splitting numerical investigations of their comparative properties a recently proposed direct stokes solver which decouples the velocity and pressure operators without calling for a temporal scheme is numerically analyzed in comparison first with the splitting scheme proposed by g karniadakis m israeli and s orszag in j comput phys 97 1991 pp 414 443 and with the unique grid bbb p n bbb p n 2 uzawa approach for the space accuracy and computational costs the chebyshev collocation approximation is used to analyze the spectra of the continuous temporal evolution operators and their discrete time versions from the first to the fourth order an explicit boundary condition is also involved in the proposed stokes solver and it is numerically shown that the trace of bf delta bf u on the boundary must be evaluated through its bf nabla times bf nabla times contribution only otherwise the ellipticity is lost before proceeding to the time discretization the explicit evaluation of the rotational boundary term does not prevent the first and second order in time schemes from being unconditionally stable while the schemes built at the next two higher orders are limited by the usual explicit cal o n 4 stability criterion on the time step the proposed bf u p decoupling gives this limitation a less restrictive coefficient and supplies the expected temporal orders on the whole explored range of time step sizes the effective accuracy obtained for the navier stokes 2d solutions has been measured for the re 1000 lid driven cavity problem and has been found equivalent to what is supplied by the much more expensive uzawa decoupling method

splitting techniques with staggered grids for the navier stokes equations in the 2d case abstract a pseudo spectral approximation for the navier stokes equations in the 2d case is presented using a new splitting technique based on the uzawa algorithm the system is decoupled into helmholtz equations for the velocity and an equation with the pseudo laplacian for the pressure staggered grids with gauss and gauss lobatto nodes are employed preconditioning with finite differences is considered by extrapolation a stable second order method in time for the velocity and at least a first order method for the pressure can be achieved

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

on the breuil mezard conjecture we give a new local proof of the breuil mezard conjecture for 2 dimensional representations of the absolute galois group of qp when p 5 and the representation has scalar endomorphisms

least squares spectral elements applied to the stokes problem least squares spectral element methods are based on two important and successful numerical methods spectral hp element methods and least squares finite element methods in this respect least squares spectral element methods seem very powerful since they combine the generality of finite element methods with the accuracy of the spectral methods and also have the theoretical and computational advantages of the least squares methods these features make the proposed method a competitive candidate for the solution of large scale problems arising in scientific computing in order to demonstrate its competitiveness the method has been applied to an analytical problem and the theoretical optimal and suboptimal a priori estimates have been confirmed for various boundary conditions moreover the exponential convergence rates typical for a spectral element discretization have also been confirmed the comparison with the classical galerkin spectral element method revealed that the least squares spectral element method is as accurate as the galerkin method for the smooth model problem

hp adaptive least squares spectral element method for hyperbolic partial differential equations this paper describes a hp adaptive spectral element formulation which is used to discretize the weak formulation obtained by minimizing the residuals in the l 2 norm the least squares error indicator will be briefly discussed refinement of the numerical approximation is based on an estimate of the regularity of the underlying exact solution if the underlying exact solution is sufficiently smooth polynomial enrichment is employed in areas with limited regularity h refinement is used for this purpose the sobolev regularity is estimated functionally and geometrically non conforming neighbouring elements are patched together using so called mortar elements results of this approach are compared to uniform h and p refinement for a linear advection equation

fourth order time stepping for stiff pdes on the sphere we present in this paper algorithms for solving stiff pdes on the unit sphere with spectral accuracy in space and fourth order accuracy in time these are based on a variant of the double fourier sphere method in coefficient space with multiplication matrices that differ from the usual ones and implicit explicit time stepping schemes operating in coefficient space with these new matrices allows one to use a sparse direct solver avoids the coordinate singularity and maintains smoothness at the poles while implicit explicit schemes circumvent severe restrictions on the time steps due to stiffness a comparison is made against exponential integrators and it is found that implicit explicit schemes perform best implementations in matlab and chebfun make it possible to compute the solution of many pdes to high accuracy in a very convenient fashion

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

invariant distributions on p adic analytic groups let p be a prime number let l be a finite extension of the field qp of p adic numbers let k be a spherically complete extension field of l and let g be the group of l rational points of a split reductive group over l we derive several explicit descriptions of the center of the algebra d g k of locally analytic distributions on g with values in k the main result is a generalization of an isomorphism of harish chandra which connects the center of d g k with the algebra of weyl invariant centrally supported distributions on a maximal torus of g this isomorphism is supposed to play a role in the theory of locally analytic representations of g as studied by p schneider and j teitelbaum

n cohomology of simple highest weight modules on walls and purity letg p b be respectively a complex connected linear algebraic semisimple group a parabolic subgroup and a borel subgroup the first main result is the following theorem let be a pure complex ong b smooth with respect to bruhat cells then its restriction to anyp orbit is pure as well of the same weight as a consequence we are able to compute then cohomology of simple highest weight modules on walls

space time coupled spectral hp least squares finite element formulation for the incompressible navier stokes equations we consider least squares finite element models for the numerical solution of the non stationary navier stokes equations governing viscous incompressible fluid flows the paper presents a formulation where the effects of space and time are coupled resulting in a true space time least squares minimization procedure as opposed to a space time decoupled formulation where a least squares minimization procedure is performed in space at each time step the formulation is first presented for the linear advection diffusion equation and then extended to the navier stokes equations the formulation has no time step stability restrictions and is spectrally accurate in both space and time to allow the use of practical c0 element expansions in the resulting finite element model the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity as an additional independent variable and the least squares method is used to develop the finite element model of the governing equations high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method in matrix free form spectral convergence of the l2 least squares functional and l2 error norms in space time is verified using a smooth solution to the two dimensional non stationary incompressible navier stokes equations numerical results are presented for impulsively started lid driven cavity flow oscillatory lid driven cavity flow transient flow over a backward facing step and flow around a circular cylinder the results demonstrate the predictive capability and robustness of the proposed formulation even though the space time coupled formulation is emphasized we also present the formulation and numerical results for least squares space time decoupled finite element models the numerical results show that the space time coupled formulation has superior predictive capabilities for flows demanding high space time resolution exemplified here by the transient flow over a backward facing step

conservation of mass and momentum of the least squares spectral collocation scheme for the stokes problem from the literature it is known that spectral least squares schemes perform poorly with respect to mass conservation and compensate this lack by a superior conservation of momentum this should be revised since the here presented new least squares spectral collocation scheme leads to an outstanding performance with respect to conservation of momentum and mass the reasons can be found in using only a few elements each with high polynomial degree avoiding normal equations for solving the overdetermined linear systems of equations and by introducing the clenshaw curtis quadrature rule for imposing the average pressure to be zero furthermore we combined the transformation of gordon and hall transfinite mapping with our least squares spectral collocation scheme to discretize the internal flow problems

stable flatness of nonarchimedean hyperenveloping algebras let l be a completely valued nonarchimedean field and g a finite dimensional lie algebra over l we show that its hyperenveloping algebra f g agrees with its arens michael envelope and furthermore is a stably flat completion of its universal enveloping algebra as an application we prove that the taylor relative cohomology for the locally convex algebra f g is naturally isomorphic to the lie algebra cohomology of g

least squares finite element method for the stokes problem with zero residual of mass conservation in this paper the simulation of incompressible flow in two dimensions by the least squares finite element method lsfem in the vorticity velocity pressure version is studied in the lsfem the equations for continuity of mass and momentum and a vorticity equation are minimized on a discretization of the domain of interest a problem is these equations are minimized in a global sense thus this method may not enforce that div underline u 0 at every point of the discretization in this paper a modified lsfem is developed which enforces near zero residual of mass conservation i e div underline u is nearly zero at every point of the discretization this is accomplished by adding an extra restriction in the divergence free equation through the lagrange multiplier strategy in this numerical method the inf sup or say lbb condition is not necessary and the matrix resulting from applying the method on a discretization is symmetric the uniqueness of the solution and the application of the conjugate gradient method are also valid numerical experience is given in simulating the flow of a cylinder with diameter 1 moving in a narrow channel of width 1 5 results obtained by the lsfem show that mass is created or destroyed at different points in the interior of discretization the results obtained by the modified lsfem show the mass is nearly conserved everywhere

a stable penalty method for the compressible navier stokes equations i open boundary conditions the purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible navier stokes equations in three spatial dimensions the treatment uses the conservation form of the navier stokes equations and utilizes linearization and localization at the boundaries based on these variables the proposed boundary conditions are applied through a penalty procedure thus ensuring correct behavior of the scheme as the reynolds number tends to infinity the versatility of this method is demonstrated for the problem of a compressible flow past a circular cylinder

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

chebyshev pseudospectral multi domain technique for viscous flow calculation abstract a novel chebyshev pseudospectral multi domain technique is introduced for the numerical solution of the navier stokes equations in the primitive variable formulation careful consideration is given to the proper interface condition in the multi domain method which decomposes the solution domain into subdomains by overlapping one grid point a finite difference approximation is used for the time discretization which transforms the system to a set of coupled equations which is then solved by an efficient computational boundary cb method in the cb method the pressure neumann boundary conditions are simply obtained by re arranging the discrete algebraic equations resulting from chebyshev pseudospectral multi domain approximation for the space variables to demonstrate the effectiveness of this technique we present some accurate computational results for calculating the driven cavity flows with re up to 10 000 and for the flow past a circular cylinder with re up to 100 the results for both applications are in good agreement with those of previous researchers

spectral analysis of hermite cubic spline collocation systems in this paper we present an eigenvalue analysis of the first order and second order hermite cubic spline collocation differentiation matrices with arbitrary collocation points some important features are explored and compared with some other discrete methods such as finite difference methods

boundary conditions in chebyshev and legendre methods we discuss two different ways of treating non dirichlet boundary conditions in chebyshev and legendre collocation methods for second order differential problems an error analysis is provided the effect of preconditioning the corresponding spectral operators by finite difference matrices is also investigated

on the least squares method abstract it is shown that the theoretical basis of the general least squares method is the bounded inverse theorem this explains why the least squares finite element method lsfem within one mathematical computational framework without any special treatment can provide numerical solutions for all types of partial differential equations error estimates of lsfem for general and elliptic first order systems of partial differential equations are established as an example the incompressible stokes equations are considered the principal part of the stokes equations in the first order velocity pressure vorticity formulation consists of two div curl systems this knowledge is employed to derive all permissible combinations of non standard boundary conditions for the stokes equations and to show that under these conditions the stokes operator is bounded below in the full h1 sense and hence the corresponding lsfem is optimal the numerical results are also given to support this conclusion

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