singular perturbations of first order hyperbolic systems with stiff source terms abstract this work develops a singular perturbation theory for initial value problems of nonlinear first order hyperbolic systems with stiff source terms in several space variables it is observed that under reasonable assumptions many equations of classical physics of that type admit a structural stability condition this condition is equivalent to the well known subcharacteristic condition for one dimensional 2 2 systems and the well known time like condition for one dimensional scalar second order hyperbolic equations with a small positive parameter multiplying the highest derivatives under this stability condition we construct formal asymptotic approximations of the initial layer solution to the nonlinear problem furthermore assuming some regularity of the solutions to the limiting inner problem and the reduced problem we prove the existence of classical solutions in the uniform time interval where the reduced problem has a smooth solution and justify the validity of the formal approximations in any fixed compact subset of the uniform time interval the stability condition seems to be a key to problems of this kind and can be easily verified moreover this presentation unifies and improves earlier works for some specific equations

on singular singularly perturbed initial value problems singularly perturbed systems of ordinary differential equations are considered for which the reduced problem has an infinite family of solutions recent approaches to such problems are compared an

modelling and numerical solution of elastoacoustic vibrations with interface damping a finite element method is applied to compute the vibrations of an elastoacoustic system subject to an external harmonic excitation the effect of a thin layer of noise damping material between the fluid and the solid is taken into account by relaxing the interface kinematic condition of perfect contact we consider the non linear eigenvalue problem arising from the free vibration problem for the damped system for the case of a fluid in a rectangular rigid cavity with one absorbing wall we deduce the dispersion equation which allows computing analytically its eigenvalues and eigenmodes and compare the numerical solution with them finally we apply the finite element method to assess the effect of introducing a real viscoelastic material to damp the elastoacoustic vibrations of a coupled system copyright 1999 john wiley sons ltd

daes arising from traveling wave solutions of pdes the study of traveling waves for explicit and implicit pdes can sometimes result in differential algebraic equations daes instead of ordinary differential equations the advantages of using daes is discussed as are the implications of dae theory for the study of traveling waves a specific type of traveling wave that connects equilibria is examined in more detail specific examples are given

finite element analysis of a quadratic eigenvalue problem arising in dissipative acoustics a quadratic eigenvalue problem arising in the determination of the vibration modes of an acoustic fluid contained in a cavity with absorbing walls is considered the problem is shown to be equivalent to the spectral problem for a noncompact operator and a thorough spectral characterization is given a numerical discretization based on raviart thomas finite elements is analyzed the method is proved to be free of spurious modes and to converge with optimal order implementation issues and numerical experiments confirming the theoretical results are reported

finite element vibration analysis of fluid solid systems without spurious modes this paper deals with the finite element approximation of the vibration modes of a problem with fluid structure interaction displacement variables are used for both the fluid and the solid to avoid the typical spurious modes of this formulation we introduce a nonconforming discretization error estimates for the approximation of eigenvalues and eigenvectors are given

external finite element approximations of eigenfunctions in the case of multiple eigenvalues abstract the paper deals with the finite element analysis of second order elliptic eigenvalue problems when the approximate domains h are not subdomains of the original domain r 2 the considerations are restricted to piecewise linear approximations special attention is devoted to the convergence of approximate eigenfunctions in the case of multiple exact eigenvalues as yet the approximate solutions have been compared with linear combinations of exact eigenfunctions with coefficients depending on the mesh parameter h we avoid this disadvantage

numerical bifurcation and stability analysis of solitary pulses in an excitable reaction diffusion medium abstract we present a systematic computer assisted study of the bifurcations and instabilities of solitary pulses in an excitable medium capable of displaying both stable pulse propagation and spatiotemporally chaotic dynamics over intervals of parameter space the reaction diffusion model used is of the activator inhibitor type only the activator diffuses in this medium the control parameters are the ratio of time scales of the activator and inhibitor dynamics and the excitation threshold this study focuses on travelling pulses their domain of existence and the bifurcations that render them unstable these pulses are approximated as a homoclinic orbits in a travelling wave ode frame and b as solutions of the full partial differential equation pde with periodic boundary conditions in large domains a variety of bifurcations in the travelling wave ode frame are observed including heteroclinic loops so called t points a r champneys and y a kuznetsov numerical detection and continuation of codimension 2 homoclinic bifurcations int j bif chaos 4 1994 785 h kokobu homoclinic and heteroclinic bifurcations of vectorfields japan j appl math 5 1988 455 instabilities in the full pde frame include both hopf bifurcations to modulated travelling waves involving the discrete pulse spectrum as well as transitions involving the continuous spectrum such as the so called backfiring transition m bar m hildebrand m eiswirth m falcke h engel and m neufeld chemical turbulence and standing waves in a surface reaction model the influence of global coupling and wave instabilities chaos 4 1994 499 the stability of modulated pulses is computed through numerical floquet analysis and a cascade of period doubling bifurcations is observed as well as certain global bifurcations these results corroborated by observations from direct numerical integration provide a skeleton around which many features of the overall complex spatiotemporal dynamics of the pde are organized

finite element approximation of spectral problems with neumann boundary conditions on curved domains this paper deals with the finite element approximation of the spectral problem for the laplace equation with neumann boundary conditions on a curved nonconvex domain convergence and optimal order error estimates are proved for standard piecewise linear continuous elements on a discrete polygonal domain h in tile framework of the abstract spectral approximation theory

a posteriori error estimates in finite element acoustic analysis we present an a posteriori error estimator for the approximations of the acoustic vibration modes obtained by a finite element method which does not present spurious or circulation modes for non zero frequencies we prove that the proposed estimator is equivalent to the error in the approximation of the eigenvectors up to higher order terms with constants independent of the eigenvalues numerical results for some test examples are presented which show the good behavior of the estimator when it is used as local error indicator for adaptive refinement

finite element analysis of compressible and incompressible fluid solid systems this paper deals with a finite element method to solve interior fluid structure vibration problems valid for compressible and incompressible fluids it is based on a displacement formulation for both the fluid and the solid the pressure of the fluid is also used as a variable for the theoretical analysis yielding a well posed mixed linear eigenvalue problem lowest order triangular raviart thomas elements are used for the fluid and classical piece wise linear elements for the solid transmission conditions at the fluid solid interface are taken into account in a weak sense yielding a nonconforming discretization the method does not present spurious or circulation modes for nonzero frequencies convergence is proved and error estimates independent of the acoustic speed are given for incompressible fluids a convenient equivalent stream function formulation and a post process to compute the pressure are introduced

finite element solution of incompressible fluid structure vibration problems in this paper we solve an eigenvalue problem arising from the computation of the vibrations of a coupled system incompressible fluid elastic structure in absence of external forces we use displacement variables for both the solid and the fluid but the fluid displacements are written as curls of a stream function classical linear triangular finite elements are used for the solid displacements and for the stream function in the fluid the kinematic transmission conditions at the fluid solid interface are taken into account in a weak sense by means of a lagrange multiplier the method does not present spurious or circulation modes for non zero frequencies numerical results are given for some test cases 1997 by john wiley sons ltd

finite element analysis of the vibration problem of a plate coupled with a fluid we consider the approximation of the vibration modes of an elastic plate in contact with a compressible fluid the plate is modelled by reissner mindlin equations while the fluid is described in terms of displacement variables this formulation leads to a symmetric eigenvalue problem reissner mindlin equations are discretized by a mixed method the equations for the fluid with raviart thomas elements and a non conforming coupling is used on the interface in order to prove that the method is locking free we consider a family of problems one for each thickness t 0 and introduce appropriate scalings for the physical parameters so that these problems attain a limit when t to 0 we prove that spurious eigenvalues do not arise with this discretization and we obtain optimal order error estimates for the eigenvalues and eigenvectors valid uniformly on the thickness parameter t finally we present numerical results confirming the good performance of the method

on nonlinear singularly perturbed initial value problems the study of singularly perturbed initial value problems for nonlinear systems of ordinary differential equations parallels the analysis underlying the development of numerical algorithms for obtaining solutions to systems of stiff differential equations this paper seeks to emphasize the advantages of combining these two substantial research efforts it develops insight and intuition based on a sequence of solvable model problems and it relates a variety of literature scattered throughout asymptotic and numerical analyses stability and control theory and specific topics in applied mathematical modeling

finite element solution of nonlinear elliptic problems the study of the finite element approximation to nonlinear second order elliptic boundary value problems with mixed dirichlet neumann boundary conditions is presented in the discretization variational crimes are commited approximation of the given domain by a polygonal one numerical integration with the assumption that the corresponding operator is strongly monotone and lipschitz continuous and that the exact solutionu h1 the convergence of the method is proved under the additional assumptionu h2 the rate of convergenceo h is derived without the use of green s theorem

on the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form in the approximation of linear elliptic operators in mixed form it is well known that the so called inf sup and ellipticity in the kernel properties are sufficient and in a sense to be made precise necessary in order to have good approximation properties and optimal error bounds one might think in the spirit of mercier osborn rappaz raviart and in consideration of the good behavior of commonly used mixed elements like raviart thomas or brezzi douglas marini elements that these conditions are also sufficient to ensure good convergence properties for eigenvalues in this paper we show that this is not the case in particular we present examples of mixed finite element approximations that satisfy the above properties but exhibit spurious eigenvalues such bad behavior is proved analytically and demonstrated in numerical experiments we also present additional assumptions fulfilled by the commonly used mixed methods already mentioned which guarantee optimal error bounds for eigenvalue approximations as well

upper bounds on the error probabilities and asymptotic error exponents in quantum multiple state discrimination we consider the multiple hypothesis testing problem for symmetric quantum state discrimination between r given states 1 r by splitting up the overall test into multiple binary tests in various ways we obtain a number of upper bounds on the optimal error probability in terms of the binary error probabilities these upper bounds allow us to deduce various bounds on the asymptotic error rate for which it has been hypothesized that it is given by the multi hypothesis quantum chernoff bound or chernoff divergence c 1 r as recently introduced by nussbaum and szkola in analogy with salikhov s classical multi hypothesis chernoff bound this quantity is defined as the minimum of the pairwise binary chernoff divergences minj kc j k it was known already that the optimal asymptotic rate must lie between c 3 and c and that for certain classes of sets of states the bound is actually achieved it was known to be achieved in particular when the state pair that is closest together in chernoff dive

mixed finite element methods in fluid structure systems the eigenvalue problem describing the frequencies of a fluid vibrating in a rigid cavity or within moving boundaries is considered based on the method of lagrange multipliers a three field mixed formulation is introduced in order to avoid the spurious circulating modes stability and optimal error bounds are proved for two choices of finite element spaces

the combined effect of numerical integration and approximation of the boundary in the finite element method for eigenvalue problems the paper deals with the finite element analysis of second order elliptic eigenvalue problems when the approximate domains omega h are not subdomains of the original domain omega subset r 2 and when at the same time numerical integration is used for computing the involved bilinear forms the considerations are restricted to piecewise linear approximations the optimum rate of convergence o h 2 for approximate eigenvalues is obtained provided that a quadrature formula of first degree of precision is used in the case of a simple exact eigenvalue the optimum rate of convergence o h for approximate eigenfunctions in the h 1 omega h norm is proved while in the l 2 omega h norm an almost optimum rate of convergence i e near to o h 2 is achieved in both cases a quadrature formula of first degree of precision is used quadrature formulas with degree of precision equal to zero are also analyzed and in the case when the exact eigenfunctions belong only to h 1 omega the convergence without the rate of convergence is proved in the case of a multiple exact eigenvalue the approximate eigenfunctions are compard in contrast to standard considerations with linear combinations of exact eigenfunctions with coefficients not depending on the mesh parameter h

the asymptotic solution semiconductor device problem involving reverse bias a standard drift diffusion model for a one dimensional stationary semiconductor device is considered in the case of reverse bias the resulting boundary value problem is scaled and analyzed when two small physical parameters are of comparable size the limiting solution is obtained in both an interior depletion region and in the complementary intervals of zero space charge the challenging question of the structure of the corresponding outer expansions and of their corrections in interior transition and endpoint boundary layers of rapid change is also studied

displacement pressure based mixed finite element formulations for acoustic fluid structure interaction problems summary we present reliable finite element discretizations based on displacement pressure interpolations for the analysis of acoustic fluid structure interaction problems the finite element interpolations are selected using the inf sup condition and emphasis is given to the fact that the boundary conditions must satisfy the mass and momentum conservation we show that with our analysis procedure no spurious non zero frequencies are encountered as heretofore calculated with other displacement based discretizations 1997 by john wiley sons ltd

vibration analysis of fluid solid systems using a finite element displacement formulation this report presents a finite element solution for the vibration interaction between an inviscid fluid and a solid the equation of motion governing the inviscid fluid is expressed in terms of the displacements reduced integration is used when computing the element stiffness matrix contributed by the fluid in addition a projection is used on the element mass matrix in order to remove the spurious modes which result from the use of reduced integration

the order of convergence of eigenfrequencies in finite element approximations of fluid structure interaction problems in this paper we prove a double order for the convergence of eigenfrequencies in fluid structure vibration problems we improve estimates given recently for compressible and incompressible fluids to do this we extend classical results on finite element spectral approximation to nonconforming methods for noncompact operators

finite element approximation of spectral acoustic problems on curved domains this paper deals with the finite element approximation of the displacement formulation of the spectral acoustic problem on a curved non convex two dimensional domain convergence and error estimates are proved for raviart thomas elements on a discrete polygonal domain h inline equation in the framework of the abstract spectral approximation theory similar results have been previously proved only for polygonal domains numerical tests confirming the theoretical results are reported

an integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection diffusion by a steady incompressible velocity field are developed here under these conditions a rigorous stieltjes integral representation for effective diffusivity in turbulent transport is derived this representation is valid for all peclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the peclet number one consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all peclet numbers a nonlocal variational principle for effective diffusivity is developed along with applications to advection diffusion by random arrays of vortices a new class of rigorous examples is introduced these examples have an explicit stieltjes measure for the effective diffusivity furthermore the effective diffusivity behaves likek0 pe 1 2 in the limit of large peclet numbers wherek0 is the molecular diffusivity formal analogies with the theory of composite materials are exploited systematically

besov spaces on domains in rd we study besov spaces 5 lp i2 0 p q a oo on do mains i2 in rd we show that there is an extension operator w which is a bounded mapping from b lp u onto b lp rd this is then used to derive various properties of the besov spaces such as interpolation theorems for a pair of b lp il atomic decompositions for the elements of 5 lp f2 and a description of the besov spaces by means of spline approximation

error estimate in an isoparametric finite element eigenvalue problem the aim of this paper is to obtain an eigenvalue approximation for elliptic operators defined on a domain i2 with the help of isoparametric finite elements of degree k we prove that i xh 0 h2k provided the boundary of ii is well approximated which is the same estimate as the one obtained in the case of conforming finite elements

finite element computation of the vibration modes of a fluid solid system abstract in this paper we solve the interior elastoacoustic problem by a finite element method which does not present spurious or circulation modes for nonzero frequencies it consists of classical triangular lagrangian elements for the solid and lowest order triangular raviart thomas elements for the fluid transmission conditions at the fluid solid interface are taken into account in a weak sense numerical results for some test examples are presented which show the good performance of the proposed methodology