a hybridizable discontinuous galerkin method for modeling fluid structure interaction abstract this work presents a novel application of the hybridizable discontinuous galerkin hdg finite element method to the multi physics simulation of coupled fluid structure interaction fsi problems recent applications of the hdg method have primarily been for single physics problems including both solids and fluids which are necessary building blocks for fsi modeling utilizing these established models hdg formulations for linear elastostatics a nonlinear elastodynamic model and arbitrary lagrangian eulerian navier stokes are derived the elasticity formulations are written in a lagrangian reference frame with the nonlinear formulation restricted to hyperelastic materials with these individual solid and fluid formulations the remaining challenge in fsi modeling is coupling together their disparate mathematics on the fluid solid interface this coupling is presented along with the resultant hdg fsi formulation verification of the component models through the method of manufactured solutions is performed and each model is shown to converge at the expected rate the individual components along with the complete fsi model are then compared to the benchmark problems proposed by turek and hron 1 the solutions from the hdg formulation presented in this work trend towards the benchmark as the spatial polynomial order and the temporal order of integration are increased

a singular quasilinear parabolic problem in one space dimension abstract of concern are mixed initial boundary value problems of the form u t t x u u t6x 2 u t6x 2 0 x 1 0 t t where t x p q 0 x 0 for 0

the motion of a vortex on a closed surface of constant negative curvature the purpose of this work is to present an algorithm to determine the motion of a single hydrodynamic vortex on a closed surface of constant curvature and of genus greater than one the algorithm is based on a relation between the laplace beltrami green function and the heat kernel the algorithm is used to compute the motion of a vortex on the bolza surface this is the first determination of the orbits of a vortex on a closed surface of genus greater than one the numerical results show that all the 46 vortex equilibria can be explicitly computed using the symmetries of the bolza surface some of these equilibria allow for the construction of the first two examples of infinite vortex crystals on the hyperbolic disc the following theorem is proved a weierstrass point of a hyperellitic surface of constant curvature is always a vortex equilibrium

an algorithm for the computation of eigenvalues spectral zeta functions and zeta determinants on hyperbolic surfaces we present a rigorous scheme that makes it possible to compute eigenvalues of the laplace operator on hyperbolic surfaces within a given precision the method is based on an adaptation of the method of particular solutions to the case of locally symmetric spaces and on explicit estimates for the approximation of eigenfunctions on hyperbolic surfaces by certain basis functions it can be applied to check whether or not there is an eigenvalue in an e neighborhood of a given number 0 this makes it possible to find all the eigenvalues in a specified interval up to a given precision with rigorous error estimates the method converges exponentially fast with the number of basis functions used combining the knowledge of the eigenvalues with the selberg trace formula we are able to compute values and derivatives of the spectral zeta function again with error bounds as an example we calculate the spectral determinant and the casimir energy of the bolza surface and other surfaces

high order methods for the approximation of the incompressible navier stokes equations in a moving domain in this paper we address the numerical approximation of the incompressible navier stokes equations in a moving domain by the spectral element method and high order time integrators we present the arbitrary lagrangian eulerian ale formulation of the incompressible navier stokes equations and propose a numerical method based on the following kernels a lagrange basis associated with fekete points in the spectral element method context bdf time integrators an ale map of high degree and an algebraic linear solver in particular the high degree ale map is appropriate to deal with a computational domain whose boundary is described with curved elements finally we apply the proposed strategy to a test case c 2011 elsevier b v all rights reserved

perturbative analysis of the method of particular solutions for improved inclusion of high lying dirichlet eigenvalues the dirichlet eigenvalue or drum problem in a domain omega subset mathbb r 2 becomes numerically challenging at high eigenvalue frequency e in this regime the method of particular solutions mps gives spectral accuracy for many domain shapes it requires a number of degrees of freedom scaling as sqrt e the number of wavelengths on the boundary in contrast to direct discretization for which this scaling is e our main result is an inclusion bound on eigenvalues that is a factor o sqrt e tighter than the classical bound of moler payne and that is optimal in that it reflects the true slopes of curves appearing in the mps we also present an mps variant that cures a normalization problem in the original method while evaluating basis functions only on the boundary this method is efficient at high frequencies where we show that in practice our inclusion bound can give three extra digits of eigenvalue accuracy with no extra effort

on largeness and multiplicity of the first eigenvalue of finite area hyperbolic surfaces we apply topological methods to study the smallest non zero number lambda 1 in the spectrum of the laplacian on finite area hyperbolic surfaces for closed hyperbolic surfaces of genus two we show that the set s in mathcal m 2 lambda 1 s frac 1 4 is unbounded and disconnects the moduli space mathcal m 2 using this for genus g ge 3 we show the existence of eigenbranches that start as lambda 1 and eventually becomes frac 1 4

an exponentially convergent nonpolynomial finite element method for time harmonic scattering from polygons in recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems as with their close cousin the method of particular solutions high efficiency comes from using solutions to the helmholtz equation as basis functions we present and analyze such a method for the scattering of two dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions the solution is obtained by minimizing a least squares functional which we discretize in such a way that a matrix least squares problem is obtained we give computable exponential bounds on the rate of convergence of the least squares functional that are in very good agreement with the observed numerical convergence challenging numerical examples including a nonconvex polygon with several corner singularities and a cavity domain are solved to around 10 digits of accuracy with a few seconds of cpu time the examples are implemented concisely with mpspack a matlab toolbox for wave computations with nonpolynomial basis functions developed by the authors a code example is included

extremal properties of the determinant of the laplacian in the bergman metric on the moduli space of genus two riemann surfaces we study extremal properties of the determinant of the laplacian in the bergman metric on the moduli space of compact genus two riemann surfaces by a combination of analytical and numerical methods we identify four non degenerate critical points of this function and compute the signature of the hessian at these points the curve with the maximal number of automorphisms the burnside curve turns out to be the point of the absolute maximum our results agree with the mass formula for virtual euler characteristics of the moduli space a similar analysis is performed for bolza s strata of symmetric riemann surfaces of genus two

algebraic fractional step schemes with spectral methods for the incompressible navier stokes equations the numerical investigation of a recent family of algebraic fractional step methods for the solution of the incompressible time dependent navier stokes equations is presented these methods are improved versions of the yosida method proposed in a quarteroni f saleri a veneziani factorization methods for the numerical approximation of navier stokes equations comput methods appl mech engrg 188 1 3 2000 505 526 a quarteroni f saleri a veneziani j math pures appl 9 78 5 1999 473 503 and one of them the yosida4 method is proposed in this paper for the first time they rely on an approximate lu block factorization of the matrix obtained after the discretization in time and space of the navier stokes system yielding a splitting in the velocity and pressure computation in this paper we analyze the numerical performances of these schemes when the space discretization is carried out with a spectral element method with the aim of investigating the impact of the splitting on the global accuracy of the computation

ambient constructions for sasakian η einstein manifolds abstract the theory of ambient spaces is useful to define cr invariant objects such as cr invariant powers of the sub laplacian the p prime operator and q prime curvature however in general it is difficult to write down these objects in terms of the tanaka webster connection in this paper we give those explicit formulas for cr manifolds satisfying an einstein condition called sasakian einstein manifolds as an application we study properties of the first and the second variation of the total q prime curvature at sasakian einstein manifolds

a new approach to the analyticity of some classes of one parameter semigroups in weighted lpspaces we prove that the operatorau u with c 0 1 and x 0 in 0 1 generates an analyticco semigroup onlp 0 1 1 u lp 0 1 u p l1 0 1 1 p provided that some additional assumptions depending on the domain ofaare required for the main tools used here are j neuberger s sufficient condition for analyticity and the stein interpolation theorem

construction of a high order fluid structure interaction solver accuracy is critical if we are to trust simulation predictions in settings such as fluid structure interaction it is all the more important to obtain reliable results to understand for example the impact of pathologies on blood flows in the cardiovascular system in this paper we propose a computational strategy for simulating fluid structure interaction using high order methods in space and time first we present the mathematical and computational core framework life underlying our multi physics solvers life is a versatile library allowing for 1d 2d and 3d partial differential solves using h p type galerkin methods then we briefly describe the handling of high order geometry and the structure solver next we outline the high order space time approximation of the incompressible navier stokes equations and comment on the algebraic system and the preconditioning strategy finally we present the high order arbitrary lagrangian eulerian ale framework in which we solve the fluid structure interaction problem as well as some initial results

boundary quasi orthogonality and sharp inclusion bounds for large dirichlet eigenvalues we study eigenfunctions phi j and eigenvalues e j of the dirichlet laplacian on a bounded domain omega subset mathbb r n with piecewise smooth boundary we bound the distance between an arbitrary parameter e 0 and the spectrum e j in terms of the boundary l 2 norm of a normalized trial solution u of the helmholtz equation delta e u 0 we also bound the l 2 norm of the error of this trial solution from an eigenfunction both of these results are sharp up to constants hold for all e greater than a small constant and improve upon the best known bounds of moler payne by a factor of the wavenumber sqrt e one application is to the solution of eigenvalue problems at high frequency via for example the method of particular solutions in the case of planar strictly star shaped domains we give an inclusion bound where the constant is also sharp we give explicit constants in the theorems and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy the proof makes use of a new quasi orthogonality property of the boundary normal derivatives of the eigenmodes theorem 1 3 of interest in its own right namely the operator norm of the sum of rank 1 operators partial n phi j langle partial n phi j cdot rangle over all e j in a spectral window of width sqrt e a sum with about e n 1 2 terms is at most a constant factor independent of e larger than the operator norm of any one individual term

efficient preconditioning of the linearized navier stokes equations for incompressible flow we outline a new class of robust and efficient methods for solving subproblems that arise in the linearization and operator splitting of navier stokes equations we describe a very general strategy for preconditioning that has two basic building blocks a multigrid v cycle for the scalar convection diffusion operator and a multigrid v cycle for a pressure poisson operator we present numerical experiments illustrating that a simple implementation of our approach leads to an effective and robust solver strategy in that the convergence rate is independent of the grid robust with respect to the time step and only deteriorates very slowly as the reynolds number is increased

periodic orbit sum rules for the hadamard gutzwiller model abstract it is shown how a variety of periodic orbit sum rules can be used to extract information about a quantum mechanical system whose classical counterpart is completely chaotic from knowledge only of the classical system and vice versa the basis is the selberg trace formula an exact analogue for the hadamard gutzwiller model of the semiclassical periodic orbit theory of gutzwiller which relates the quantal energies to the lengths of the periodic orbits of the classical system statistical properties of the quantal energies in the low energy region are studied where we restrict ourselves to the level spacing and spectral rigidity

hyperbolic octagons and teichmuller space in genus 2 in this paper we study different models of the teichmuller space of compact hyperbolic surfaces with special emphasis on their construction by geodesic octagons such surfaces have become increasingly interesting due to their applications to the physical behavior of chaotic quantum systems they are on the one hand complex enough to show the relevant features and on the other hand they possess a simple mathematical structure allowing practical implementations we give a new description of teichmuller space and the mapping class group in terms of geometric data of the octagons this provides modellings based on the vertices and also on the generators of the associated isometry group in addition we explicitly connect our approach with other models currently used in the literature helling matrices fenchel nielsen parameters the result of the paper may be considered as a computational tool kit to work with a specific model and to relate it with the others

fast nonsymmetric iterations and preconditioning for navier stokes equations discretization and linearization of the steady state navier stokes equations gives rise to a nonsymmetric indefinite linear system of equations in this paper we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded independently of the mesh size used in the discretization we confirm and supplement these analytic results with a series of numerical experiments indicating that krylov subspace iterative methods for nonsymmetric systems display rates of convergence that are independent of the mesh parameter in addition we show that preconditioning costs can be kept small by using iterative methods for some intermediate steps performed by the preconditioner

a block preconditioner for high order mixed finite element approximations to the navier stokes equations an iterative solver for the linear system arising from a discretization by high order mixed finite elements for the linearized steady state navier stokes problem is presented two mixed finite elements are considered mathbf p k 1 p k and mathbf p k 2 p k the iterative solver is based on the gmres method with right preconditioning where the preconditioner has a block upper triangular form computational results show that the convergence rate seems to be independent of the grid size parameter h and the polynomial degree k but is slightly affected by the increasing of the reynolds number

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

from classical periodic orbits to the quantization of chaos we present a quantitative analysis of selberg s trace formula viewed as an exact version of gutzwiller s semiclassical periodic orbit theory for the quantization of classically chaotic systems two main applications of the trace formula are discussed in detail i the periodic orbit sum rules giving a smoothing of the quantal energy level density ii the selberg zeta function as a prototype of a dynamical zeta function defined as an euler product over the classical periodic orbits and its analytic continuation across the entropy barrier by means of a dirichlet series it is shown how the long periodic orbits can be effectively taken into account by a universal remainder term which is explicitly given as an integral over an orbit selection function numerical results are presented for the free motion of a point particle on compact riemann surfaces hadamard gutzwiller model which is the primary testing ground for our ideas relating quantum mechanics and classical mechanics in the case of strong chaos our results demonstrate clearly the crucial role played by the long periodic orbits an exact rule for quantizing chaos is derived for such systems where the dirichlet series representing the selberg zeta function converges on the critical line explicit formulae are given for the computation of the abscissae of absolute and conditional convergence respectively of these dynamical dirichlet series for the two riemann surfaces considered it turns out that one can cross the entropy barrier but that the critical line cannot be reached by a convergent dirichlet series it would seem that this is the main reason why the riemann siegel lookalike formula recently conjectured by m v berry and j p keating fails in generating the lower lying quantal energies for these strongly chaotic systems

singular nonlinear parabolic boundary value problems in one space dimension global existence and uniqueness are established for the mixed initial boundary problem for the nonlinear parabolic equation u t x u x 2u x2 0 x 1 t 0 where x 0 x 0 for 0 x 1 and 01 0 x 1 dx the boundary conditions can be either linear e g dirichlet neumann or periodic or nonlinear in which case they take the form 1 j u j t j ux j t for j 0 1 where j is a maximal monotone graph in r r containing the origin

continuous interior penalty finite element method for the time dependent navier stokes equations space discretization and convergence this paper focuses on the numerical analysis of a finite element method with stabilization for the unsteady incompressible navier stokes equations incompressibility and convective effects are both stabilized adding an interior penalty term giving l 2 control of the jump of the gradient of the approximate solution over the internal faces using continuous equal order finite elements for both velocities and pressures in a space semi discretized formulation we prove convergence of the approximate solution the error estimates hold irrespective of the reynolds number and hence also for the incompressible euler equations provided the exact solution is smooth

reviving the method of particular solutions fox henrici and moler made famous a method of particular solutions for computing eigenvalues and eigenmodes of the laplacian in planar regions such as polygons we explain why their formulation of this method breaks down when applied to regions that are insufficiently simple and propose a modification that avoids these difficulties the crucial changes are to introduce points in the interior of the region as well as on the boundary and to minimize a subspace angle rather than just a singular value or a determinant similar methods may be used to improve other mesh free algorithms for a variety of computational problems

fourier tauberian theorems and applications abstract the main aim of the paper is to present a general version of the fourier tauberian theorem for monotone functions this result together with berezin s inequality allows us to obtain a refined version of the li yau estimate for the counting function of the dirichlet laplacian which implies the li yau estimate itself and at the same time the asymptotic results obtained by the variational method

analytic torsion and closed geodesics on hyperbolic manifolds pour une variete hyperbolique orientee fermee x et une representation orthogonale de 1 x on definit une fonction zeta de type ruelle r s pour re s grand comme le produit sur les geodesiques closes premieres des facteurs det i exp sl ou l est la longueur de on continue analytiquement r s et on calcule son terme principal en s 0

high order fluid structure interaction in 2d and 3d application to blood flow in arteries this paper addresses the numerical approximation of fluid structure interaction fsi problems through the arbitrary lagrangian eulerian ale framework high order methods and a dirichlet newmann approach for the coupling the paper is divided into two main parts the first part concerns the discretization method for the fsi problem we introduce an improved ale map capable of handling curved geometries in 2d and 3d in a unified manner that is based on a local differential operator we also propose a minimal continuous interior penalty cip stabilization term for the fluid discretization that accounts for a smaller computational effort while stabilizing the flow regime the second part is dedicated to validating our numerical strategy through a benchmark and some applications to blood flow in arteries

higher order transonic flutter predictions an order of accuracy convergence study is presented for a methodology to predict dynamic instabilities arising from fluid structure coupling the methodology models the fluid with compressible euler equations and the structure with timoshenko beam elements the tightly coupled equations of the two disciplines are linearized to formulate a stability eigenvalue problem which is rewritten in a form that is well known as the v g flutter solution method the rate of convergence is measured in the norm defined by the relative error in instability speed for polynomial orders from 1 to 4 this is done for the divergence mode at mach 0 7 and the flutter mode at mach 1 1 the influence of relative order of basis functions for structural and fluid disciplines is investigated on the rate of convergence the results show that for most cases the error converges at a rate of approximately h2p for a pth order polynomial a comparison of the computational cost shows that in comparison with the conventional linear inte

on the periodic orbits of a strongly chaotic system abstract a point particle sliding freely on a two dimensional surface of constant negative curvature hadamard gutzwiller model exemplifies the simplest chaotic hamiltonian system exploiting the close connection between hyperbolic geometry and the group su 1 1 1 we construct an algorithm symboliv dynamics which generates the periodic orbits of the system for the simplest compact riemann surface having as its fundamental group the octagon group we present an enumeration of more than 206 million periodic orbits for the length of the n th primitive periodic orbit we find a simple expression in terms of algebraic numbers of the form m 2 n m n n are governed by a particular beatty sequence which reveals a strange arithmetical structure of chaos knowledge of the length spectrum is crucial for quantization via the selberg trace formula periodic orbit theory which in turn is expected to unravel the mystery of quantum chaos