dynamical noether invariants for time dependent nonlinear systems dynamical invariants are derived for time dependent systems with nonlinear equations of motion including nonharmonic damped systems the concept of a dynamical algebra is discussed and its utility for the construction of dynamical invariants for nonharmonic systems is demonstrated finally we show the existence of dynamical invariants for some nonlinear quantum systems

harnack estimates and extinction profile for weak solutions of certain singular parabolic equations we establish an intrinsic harnack estimate for nonnegative weak solutions of the singular equation 1 1 below for m in the optimal range n 2 n 1 intrinsic means that due to the singularity the space time dimensions in the parabolic geometry must be rescaled by a factor determined by the solution itself consequences are sharp supestimates on the solutions and decay rates as t approaches the extinction time analogous results are shown for p laplacian type equations

optimized double sweep schwarz method by complete radiation boundary conditions we present an optimized double sweep nonoverlapping schwarz method for solving the helmholtz equation in semi infinite waveguides the domain is decomposed into nonoverlapped layered subdomains along the axis of the waveguide and local wave propagation problems equipped with complete radiation conditions for high order absorbing boundary conditions are solved forward and backward sequentially for communication between subdomains neumann data of local solutions in one domain are transferred to the neighboring subdomain in the forward direction and dirichlet data are exploited in the backward direction the complete radiation boundary conditions enable us to not only minimize reflection coefficients for most important modes in an optimal way but also find neumann data without introducing errors that would be produced if finite difference formulas were used for computing neumann data the convergence of the double sweep schwarz method is proved and numerical experiments using it as a preconditioner are presented to confirm the convergence theory

double sweep preconditioner for optimized schwarz methods applied to the helmholtz problem this paper presents a preconditioner for non overlapping schwarz methods applied to the helmholtz problem starting from a simple analytic example we show how such a preconditioner can be designed by approximating the inverse of the iteration operator for a layered partitioning of the domain the preconditioner works by propagating information globally by concurrently sweeping in both directions over the subdomains and can be interpreted as a coarse grid for the domain decomposition method the resulting algorithm is shown to converge very fast independently of the number of subdomains and frequency the preconditioner has the advantage that like the original schwarz algorithm it can be implemented as a matrix free routine with no additional preprocessing

corner treatments for high order local absorbing boundary conditions in high frequency acoustic scattering this paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal computational domains for the finite element solution of high frequency acoustic scattering problems while high order absorbing boundary conditions habcs are accurate for smooth fictitious boundaries the precision of the solution drops in the presence of corners if no specific treatment is applied we present and analyze two strategies to preserve the accuracy of pade type habcs at corners first by using compatibility relations derived for right angle corners and second by regularizing the boundary at the corner we show that the former strategy is well adapted to right corners and efficient for nearly right corners while the later is better for very obtuse corners numerical results are proposed to analyze and compare the approaches for two and three dimensional problems

sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities the goal of this paper is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and in self similar variables the optimal convergence rates to barenblatt self similar profiles and their generalizations it relies on the identification of the optimal constants in some related hardy poincare inequalities and concludes a long series of papers devoted to generalized entropies functional inequalities and rates for nonlinear diffusion equations

an energetic variational approach for the cahn hilliard equation with dynamic boundary condition model derivation and mathematical analysis the cahn hilliard equation is a fundamental model that describes phase separation processes of binary mixtures in recent years several types of dynamic boundary conditions have been proposed in order to account for possible short range interactions of the material with the solid wall our first aim in this paper is to propose a new class of dynamic boundary conditions for the cahn hilliard equation in a rather general setting the derivation is based on an energetic variational approach that combines the least action principle and onsager s principle of maximum energy dissipation one feature of our model is that it naturally fulfills three important physical constraints conservation of mass dissipation of energy and force balance relations next we provide a comprehensive analysis of the resulting system of partial differential equations under suitable assumptions we prove the existence and uniqueness of global weak strong solutions to the initial boundary value problem with or without surface diffusion furthermore we establish the uniqueness of the asymptotic limit as t to infty and characterize the stability of local energy minimizers for the system

domain decomposition method for harmonic wave propagation a general presentation in this paper we give a general presentation of non overlapping domain decomposition methods for harmonic wave propagation models our abstract framework leads to concise convergence proofs and contains some recent developments about the use of non local transmission conditions it also includes applications to acoustic electromagnetic or elastic waves as well as the treatment of space discretization

optimized schwarz methods without overlap for the helmholtz equation the classical schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel convergence is achieved through overlap of the subdomains we study in this paper a variant of the schwarz method which converges without overlap for the helmholtz equation we show that the key ingredients for such an algorithm are the transmission conditions we derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps these conditions are however nonlocal in nature and we introduce local approximations which we optimize for performance of the schwarz method this leads to an algorithm in the class of optimized schwarz methods we present an asymptotic analysis of the optimized schwarz method for two types of transmission conditions robin conditions and transmission conditions with second order tangential derivatives numerical results illustrate the effectiveness of the optimized schwarz method on a model problem and on a problem from industry

a sweeping preconditioner for time harmonic maxwell s equations with finite elements this paper is concerned with preconditioning the stiffness matrix resulting from finite element discretizations of maxwell s equations in the high frequency regime the moving pml sweeping preconditioner first introduced for the helmholtz equation on a cartesian finite difference grid is generalized to an unstructured mesh with finite elements the method dramatically reduces the number of gmres iterations necessary for convergence resulting in an almost linear complexity solver numerical examples including electromagnetic cloaking simulations are presented to demonstrate the efficiency of the proposed method

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

electromagnetic modeling of high magnetic contrast media using calderon preconditioning in this contribution we present a calderon preconditioner for a novel single source equation to efficiently model electromagnetic scattering problems involving high magnetic contrasts through analysis of the spectral properties of the system matrix after discretization it is shown that this formulation does not break down when high permeabilities are present which was an unresolved problem of the calderon preconditioned poggio miller chan harrington wu tsai method the adopted discretization scheme which involves rao wilton glisson and buffa christiansen basis functions allows for an easy integration in existing commercial method of moments software the efficiency and accuracy of the presented method is corroborated by numerical examples c 2018 elsevier ltd all rights reserved

a quasi optimal domain decomposition algorithm for the time harmonic maxwell s equations this paper presents a new non overlapping domain decomposition method for the time harmonic maxwell s equations whose effective convergence is quasi optimal these improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the magnetic to electric operator a convergence theorem of the algorithm is established and numerical results validating the new approach are presented

existence of many positive solutions of semilinear elliptic equations on annulus abstract we study the existence of many nonradial positive solutions in an annulus of rn it is an improvement of results in c v coffman a nonlinear boundary value problem with many positive solutions j differential equations 54 1984 429 437 an annulus is invariant under many group actions and these group actions act naturaly on the functional and leave the functional invariant therefore when we look for critical points of the functional we can restrict the functional to the manifold consisting of all fixed points under the group action we are going to use different group actions to obtain critical points and furthermore we can distinguish them

an improved surface radiation condition for high frequency acoustic scattering problems a new artificial boundary condition for two dimensional acoustic scattering problems is constructed in this paper it is designed from the principal symbol of the dirichlet to neumann map so that both the propagating and evanescent fields are properly modeled in the high frequency regime the new artificial boundary condition can be used as an on surface radiation condition for efficient approximation of the wave field the accuracy and efficiency of our method are demonstrated in numerical examples

stability of non isolated asymptotic profiles for fast diffusion the stability of asymptotic profiles of solutions to the cauchy dirichlet problem for fast diffusion equation fde for short is discussed the main result of the present paper is the stability of any asymptotic profiles of least energy it is noteworthy that this result can cover non isolated profiles e g those for thin annular domain cases the method of proof is based on the ojasiewicz simon inequality which is usually used to prove the convergence of solutions to prescribed limits as well as a uniform extinction estimate for solutions to fde besides local minimizers of an energy functional associated with this issue are characterized furthermore the instability of positive radial asymptotic profiles in thin annular domains is also proved by applying the ojasiewicz simon inequality in a different way

a domain decomposition method for discontinuous galerkin discretizations of helmholtz problems with plane waves and lagrange multipliers a nonoverlapping domain decomposition dd method is proposed for the iterative solution of systems of equations arising from the discretization of helmholtz problems by the discontinuous enrichment method this discretization method is a discontinuous galerkin finite element method with plane wave basis functions for approximating locally the solution and dual lagrange multipliers for weakly enforcing its continuity over the element interfaces the primal subdomain degrees of freedom are eliminated by local static condensations to obtain an algebraic system of equations formulated in terms of the interface lagrange multipliers only as in the feti h and feti dph dd methods for continuous galerkin discretizations this system of lagrange multipliers is iteratively solved by a krylov method equipped with both a local preconditioner based on subdomain data and a global one using a coarse space numerical experiments performed for two and three dimensional acoustic scattering problems suggest that the proposed dd based iterative solver is scalable with respect to both the size of the global problem and the number of subdomains copyright 2009 john wiley sons ltd

mathcal h matrices compressed multiplicative schwarz preconditioner for nonconformal fem bem ddm an mathcal h matrices compressed multiplicative schwarz ms preconditioner for the nonconformal finite element method fem boundary element method bem domain decomposition method ddm is proposed to analyze the electromagnetic radiation and scattering from periodic structures such as frequency selective surfaces fsss or antenna arrays to improve the overall convergence an ms preconditioner is proposed for the ddm matrix the hierarchical matrix mathcal h matrices technique is introduced into the preconditioned system to compress the dense matrices to further improve the efficiency moreover a finite element tearing and interconnecting like algorithm is adopted to eliminate the interior volumetric fem unknowns and the multilevel fast multipole algorithm is integrated to reduce the computational complexity of bem numerical examples are presented to validate the proposed method it is observed that this method exhibits superior performance and consumes far less memory and cpu time than the traditional implementation

the regularity chaos regularity transition in a periodically driven anharmonic oscillator abstract a complete analytical and numerical investigation of the classical phase space structure for monochromatically driven x n anharmonic oscillator is presented the existence of additional chaos to regularity transition for a family of considered hamiltonian systems have been found analytically numerical calculations of the critical amplitude of the external field confirm the theory

a quasi optimal non overlapping domain decomposition method for two dimensional time harmonic elastic wave problems abstract this article presents the construction of a new non overlapping domain decomposition method ddm for two dimensional elastic scattering problems the method relies on a high order transmission boundary condition tbc between sub domains which accurately approximates the exact dirichlet to neumann map first we explain the derivation of this new tbc in the context of a non overlapping ddm next a mode by mode convergence study for a model problem is presented which shows the new method to be quasi optimal i e with an optimal convergence rate for evanescent modes and an improved convergence rate for the other modes compared to the standard low order lysmer kuhlemeyer tbc finally the effectiveness of the new ddm is demonstrated in a finite element context by analyzing the behavior of the method on high frequency elastodynamic simulations

a robin type domain decomposition method with red black partition in this paper we present a new robin type nonoverlapping domain decomposition dd preconditioner the unknown variables to be solved in this preconditioned algebraic system are the robin transmission data on the interface which are different from the well known dd methods like substructuring nonoverlapping dd method and feti method through choosing suitable parameter on each subdomain boundary and using the tool of energy estimate for the second order elliptic problem we prove that the condition number of the preconditioned system is c 1 log frac h h 2 where h is the coarse mesh size and h is the fine mesh size moreover in this paper we always assume that there is a red black partition for the whole domain omega numerical results are given to illustrate the efficiency of our dd preconditioner

a rapidly converging domain decomposition method for the helmholtz equation a new domain decomposition method is introduced for the heterogeneous 2 d and 3 d helmholtz equations transmission conditions based on the perfectly matched layer pml are derived that avoid artificial reflections and match incoming and outgoing waves at the subdomain interfaces we focus on a subdivision of the rectangular domain into many thin subdomains along one of the axes in combination with a certain ordering for solving the subdomain problems and a gmres outer iteration when combined with multifrontal methods the solver has near linear cost in examples due to very small iteration numbers that are essentially independent of problem size and number of subdomains it is to our knowledge only the second method with this property next to the moving pml sweeping method

rational square root approximations for parabolic equation algorithms in this article stable pade approximations to the function 1 z are derived by choosing a branch cut in the negative half plane the pade coefficients are complex and may be derived analytically to arbitrary order from the corresponding real coefficients associated with the principal branch defined by z 1 i z 0 i z denotes the imaginary part of z the characteristics of the corresponding square root approximation are illustrated for various segments of the complex plane in particular for waveguide problems it is shown that an increasingly accurate representation may be obtained of both the evanescent part of the mode spectrum for the acoustic case and the complex mode spectrum for the elastic case an elastic parabolic equation algorithm is used to illustrate the application of the new pade approximations to a realistic ocean environment including elasticity in the ocean bottom

non overlapping domain decomposition method for a nodal finite element method a new approach is proposed for constructing nonoverlapping domain decomposition procedures for solving a linear system related to a nodal finite element method it applies to problems involving either positive semi definite or complex indefinite local matrices the main feature of the method is to preserve the continuity requirements on the unknowns and the finite element equations at the nodes shared by more than two subdomains and to suitably augment the local matrices we prove that the corresponding algorithm can be seen as a converging iterative method for solving the finite element system and that it cannot break down each iteration is obtained by solving uncoupled local finite element systems posed in each subdomain and in contrast to a strict domain decomposition method is completed by solving a linear system whose unknowns are the degrees of freedom attached to the above special nodes

the new mass conserving s ddm scheme for two dimensional parabolic equations with variable coefficients abstract in the article a new and efficient mass conserving operator splitting domain decomposition method s ddm is proposed and analyzed for solving two dimensional variable coefficient parabolic equations with reaction term the domain is divided into multiple non overlapping block divided subdomains on each block divided subdomain the interface fluxes are first computed explicitly by local multi point weighted schemes and the solutions in the interior of subdomain are computed by the one directional operator splitting implicit schemes at each time step the scheme is proved to satisfy mass conservation over the whole domain of domain decomposition by combining with some auxiliary lemmas and applying the energy method we analyze theoretically the stability of our scheme and prove it to have second order accuracy in space step in the l2 norm numerical experiments are performed to illustrate its accuracy conservation stability efficiency and parallelism our scheme not only keeps the excellent advantages of the non overlapping domain decomposition and the operator splitting technique but also preserves the global mass

a quasi optimal non overlapping domain decomposition algorithm for the helmholtz equation this paper presents a new non overlapping domain decomposition method for the helmholtz equation whose effective convergence is quasi optimal these improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the dirichlet to neumann operator a convergence theorem of the algorithm is established and numerical results validating the new approach are presented in both two and three dimensions

dispersion analysis and error estimation of galerkin finite element methods for the helmholtz equation when applying numerical methods for the computation of stationary waves from the helmholtz equation one obtains numerical waves that are dispersive also in non dispersive media the numerical wave displays a phase velocity that depends on the parameter k of the helmholtz equation in dispersion analysis the phase difference between the exact and the numerical solutions is investigated in this paper the authors recent result on the phase difference for one dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic it is then shown that previous error estimates in h1 norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only on the other hand recently proven error estimates for constant resolution contain a pollution term with certain assumptions on the exact solution this term is of the order of the phase difference thus a link is established between the results of dispersion analysis and the results of numerical analysis throughout the paper the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems special attention is given to the performance of the galerkin method with a higher order of polynomial approximation p h p version

boundary integral multi trace formulations and optimised schwarz methods in the present contribution we consider helmholtz equation with material coefficients being constants in each subdomain of a geometric partition of the propagation medium discarding the presence of junctions and we are interested in the numerical solution of such a problem by means of local multi trace boundary integral formulations local mtf for a one dimensional problem and configurations with two subdomains it has been recently established that applying a jacobi iterative solver to local mtf is exactly equivalent to an optimised schwarz method osm with a non local impendance in the present contribution we show that this correspondance still holds in the case where the subdomain partition involves an arbitrary number of subdomains from this we deduce that the depth of the adjacency graph of the subdomain partition plays a critical role in the convergence of linear solvers applied to local mtf we prove it for the case of homogeneous propagation medium and show through numerical evidences that this conclusion still holds for heterogeneous media our study also shows that considering variants of local mtf involving a relaxation parameter there is a fixed value of this relaxation parameter that systematically leads to optimal speed of convergence for linear solvers

continued fraction absorbing boundary conditions for convex polygonal domains continued fraction absorbing boundary conditions cfabcs are highly effective boundary conditions for modelling wave absorption into unbounded domains they are based on rational approximation of the exact dispersion relationship and were originally developed for straight computational boundaries in this paper cfabcs are extended to the more general case of polygonal computational domains the key to the current development is the surprising link found between the cfabcs and the complex co ordinate stretching of perfectly matched layers pmls this link facilitates the extension of cfabcs to oblique corners and thus to polygonal domains it is shown that the proposed cfabcs are easy to implement expected to perform better than pmls and are effective for general polygonal computational domains in addition to the derivation of cfabcs a novel explicit time stepping scheme is developed for efficient numerical implementation numerical examples presented in the paper illustrate that effective absorption is attained with a negligible increase in the computational cost for the interior domain although this paper focuses on wave propagation its theoretical development can be easily extended to the more general class of problems where the governing differential equation is second order in space with constant coefficients copyright 2005 john wiley sons ltd

positivity local smoothing and harnack inequalities for very fast diffusion equations abstract we investigate qualitative properties of local solutions u t x 0 to the fast diffusion equation t u u m m with m 1 corresponding to general nonnegative initial data our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of the form 0 t with r d they combine into forms of new harnack inequalities that are typical of fast diffusion equations such results are new for low m in the so called very fast diffusion range precisely for all m m c d 2 d the boundedness statements are true even for m 0 while the positivity ones cannot be true in that range

a coarse space for heterogeneous helmholtz problems based on the dirichlet to neumann operator the helmholtz equation governing wave propagation and scattering phenomena is difficult to solve numerically its discretization with piecewise linear finite elements results in typically large linear systems of equations the inherently parallel domain decomposition methods constitute hence a promising class of preconditioners an essential element of these methods is a good coarse space here the helmholtz equation presents a particular challenge as even slight deviations from the optimal choice can be devastating in this paper we present a coarse space that is based on local eigenproblems involving the dirichlet to neumann operator our construction is completely automatic ensuring good convergence rates without the need for parameter tuning moreover it naturally respects local variations in the wave number and is hence suited also for heterogeneous helmholtz problems the resulting method is parallel by design and its efficiency is demonstrated on 2d homogeneous and heterogeneous numerical examples

narrow tube approximation of semiclassical quasi energies application to the weakly nonlinear duffing oscillator an exact transformation of semiclassical quantization conditions determining quantal quasi energies of time periodic hamiltonian systems is suggested for motion on vortex tubes centred at stable periodic orbits the classical quasi energy of the underlying periodic orbit separates out as a single term which is a lower bound for the quantized quasi energies a particular approximation involving the classical lewis invariant for calculating semiclassical quasi energies near the centre of a perturbed linear response is discussed in some detail

phase reduction models for improving the accuracy of the finite element solution of time harmonic scattering problems i general approach and low order models this paper introduces a new formulation of high frequency time harmonic scattering problems in view of a numerical finite element solution it is well known that pollution error causes inaccuracies in the finite element solution of short wave problems to partially avoid this precision problem the strategy proposed here consists in firstly numerically computing at a low cost an approximate phase of the exact solution through asymptotic propagative models secondly using this approximate phase a slowly varying unknown envelope is introduced and is computed using coarser mesh grids the global procedure is called phase reduction in this first paper the general theoretical procedure is developed and low order propagative models are numerically investigated in detail improved solutions based on higher order models are discussed showing the potential of the method for further developments

the ermakov lewis invariants for coupled linear oscillators we consider n coupled linear oscillators with time dependent coefficients an exact complex amplitude real phase decomposition of the oscillatory motion is constructed this decomposition is further used to derive n exact constants of motion which generalize the so called ermakov lewis invariant of a single oscillator in the floquet problem of periodic oscillator coefficients we discuss the existence of periodic complex amplitude functions in terms of existing floquet solutions

a hybrid finite element and integral equation domain decomposition method for the solution of the 3 d scattering problem abstract a domain decomposition method ddm is presented for the solution of the time harmonic electromagnetic scattering problem by inhomogeneous 3 d objects the computational domain is partitioned into concentric subdomains on the interfaces of which robin type transmission conditions are prescribed on the outer boundary terminating the computational domain the radiation condition is accounted for by employing an integral equation ie formulation the ddm decouples the interior problems that correspond to the solution of maxwell s equations inside each subdomain and are formulated by using a finite element method from the exterior problem solved by employing the ie it has been shown that the solutions of this ddm algorithm converge to those of the original problem a particular ie is used that allows the implementation of a very simple and fully iterative solver the main advantage offered by this technique is a reduction in memory requirements various numerical examples are presented that illustrate its potential

an analysis of the bem fem non overlapping domain decomposition method for a scattering problem in this paper we analyze the bem fem non overlapping domain decomposition method introduced in boubendir techniques de decomposition de domaine et methode d equations integrales ph d thesis insa toulouse france 2002 and improved in boubendir et al a coupling bem fem method using a feti like domain decomposition method in proceedings of waves 2005 providence ri 2005 pp 188 190 and bendali et al a feti like domain decomposition method for coupling fem and bem in large size problems of acoustic scattering to appear the transmission conditions used in this method introduce a quantity that prevents the approach of despres methodes de decomposition de domaine pour les problemes de propagation d ondes en regime harmonique le theoreme de borg pour l equation de hill vectorielle ph d thesis paris vi university france 1991 to establish convergence to be adapted however we show that convergence can be established when the geometry allows for a decomposition of the solution into propagating and evanescent portions with a methodology based on modal analysis here we exemplify this in the case of circular cylindrical geometries where the derivations can be based on properties of bessel functions

time dependent normal form hamiltonian for dynamical equilibria a new time dependent normal form procedure for dynamical equilibria undergoing parametric excitation of one dimensional 1d hamiltonian systems is developed with the method of lie transforms the expansion is based on the lewis invariant for the linearized motion the time dependent hamiltonian normal form reduces smoothly to the usual representation in the autonomous limit illustrative examples of the formalism are focused on time periodic systems and the dynamics of hamiltonian switching

a perfectly matched layer for the helmholtz equation in a semi infinite strip the perfectly matched layer pml has become a widespread technique for preventing reflections from far field boundaries for wave propagation problems in both the time dependent and frequency domains we develop a discretization to solve the helmholtz equation in an infinite two dimensional strip we solve the interior equation using high order finite differences schemes the combined helmholtz pml problem is then analyzed for the parameters that give the best performance we show that the use of local high order methods in the physical domain coupled with a specific second order approximation in the pml yields global high order accuracy in the physical domain we discuss the impact of the parameters on the effectiveness of the pml numerical results are presented to support the analysis

two level domain decomposition methods with lagrange multipliers for the fast iterative solution of acoustic scattering problems we present two different but related lagrange multiplier based domain decomposition dd methods for solving iteratively large scale systems of equations arising from the finite element discretization of high frequency exterior helmholtz problems the proposed methods are essentially two distinct extensions of the regularized finite element tearing and interconnecting feti method to indefinite or complex problems the first method employs a single lagrange multiplier field to glue the local solutions at the subdomain interface boundaries the second method employs two lagrange multiplier fields for that purpose the key ingredients of both of these feti methods are the regularization of each subdomain matrix by a complex lumped mass matrix defined on the subdomain interface boundary and the preconditioning of the global interface problem by a coarse second level problem constructed with planar waves we show numerically that both methods are scalable with respect to the mesh size the subdomain size and the wavenumber but that the feti method with a single lagrange multiplier field labeled feti h h for helmholtz in this paper delivers superior computational performances we apply the feti h method to the parallel solution on a 24 processor origin 2000 of an acoustic scattering problem with a submarine shaped obstacle and report performance results that highlight the unique efficiency of this dd method for the solution of high frequency acoustic scattering problems

the łojasiewicz gradient inequality in the infinite dimensional hilbert space framework abstract we provide a reasonably optimal answer to the natural question of the conditions under which an analytic function on an infinite dimensional hilbert space satisfies the ojasiewicz gradient inequality

numerical accuracy of a pade type non reflecting boundary condition for the finite element solution of acoustic scattering problems at high frequency the present text deals with the numerical solution of two dimensional high frequency acoustic scattering problems using a new high order and asymptotic pade type artificial boundary condition the pade type condition is easy to implement in a galerkin least squares iterative finite element solver for arbitrarily convex shaped boundaries the method accuracy is investigated for different model problems and for the scattering problem by a submarine shaped scatterer as a result relatively small computational domains optimized according to the shape of the scatterer can be considered while yielding accurate computations for high frequencies copyright 2005 john wiley sons ltd

bayliss turkel like radiation conditions on surfaces of arbitrary shape this paper addresses the extension of the bayliss turkel second order radiation condition to an arbitrarily shaped surface the derivation is based mainly on the pseudo differential calculus as well as on the introduction of a criterion providing a precise handling of the approximation process involved in the derivation of the radiation condition the radiation condition then ranges among the most accurate of those of order two as a by product of the derivation almost all known radiation conditions of order less than or equal to two are recovered and their respective accuracies are compared

a two level domain decomposition method for the iterative solution of high frequency exterior helmholtz problems summary we present a lagrange multiplier based two level domain decomposition method for solving iteratively large scale systems of equations arising from the finite element discretization of high frequency exterior helmholtz problems the proposed method is essentially an extension of the regularized feti finite element tearing and interconnecting method to indefinite problems its two key ingredients are the regularization of each subdomain matrix by a complex interface lumped mass matrix and the preconditioning of the interface problem by an auxiliary coarse problem constructed to enforce at each iteration the orthogonality of the residual to a set of carefully chosen planar waves we show numerically that the proposed method is scalable with respect to the mesh size the subdomain size and the wavenumber we report performance results for a submarine application that highlight the efficiency of the proposed method for the solution of high frequency acoustic scattering problems discretized by finite elements

semiclassical narrow tube quantizations of time periodic hamiltonian systems an approximate semiclassical quantization procedure for time periodic hamiltonians is formulated which applies to motion near stable periodic orbits of the system corresponding to elliptic islands of the period map the theory involves canonical transformations and the use of a particular dynamical lewis invariant the basic assumption in the narrow tube approximation is the localization of the quantal wavefunction inside the elliptic islands

asymptotics of the fast diffusion equation via entropy estimates we consider non negative solutions of the fast diffusion equation ut um with m 0 1 in the euclidean space mathbb r d d 3 and study the asymptotic behavior of a natural class of solutions in the limit corresponding to t for m mc d 2 d or as t approaches the extinction time when m mc for a class of initial data we prove that the solution converges with a polynomial rate to a self similar solution for t large enough if m mc or close enough to the extinction time if m mc such results are new in the range m mc where previous approaches fail in the range mc m 1 we improve on known results

asymptotic behavior of the nonlinear diffusion equation nt n 1nx x the asymptotic behavior of the equation nt ln n xx is studied on the finite interval 0 x 1 with the boundary conditions n 0 t n 1 t n0 and initial data n x 0 n0 we prove that asymptotically ln n x t n0 a exp 2t n0 21 2 sin x and also provide rigorous upper and lower bounds on the asymptotic amplitude a in terms of integrals of nonlinear functions of the initial data the rigorous bounds are compared to values of a obtained from computer experiments the lower bound l 23 2 exp li 1 q where li is the logarithmic integral is euler s constant and q 2 f n x 0 n0 1 sin x dx is found to be the best known estimate of a

nonlinear phase integral approximations of stationary waves in nonhomogeneous systems a recent normal form approximation for dynamical equilibria of one dimensional hamiltonian systems is shown to provide a phase integral wkb approximation to solutions of nonlinear differential equations in the present paper a restricted class of ordinary differential equations is considered the integrability of the truncated normal form allows for expressing the solutions as trigonometric expansions in terms of an amplitude and phase the method is applied to a dirichlet boundary value problem on the interval for n 3 and n 4 where the coefficient functions depend on an additional parameter as in the constant coefficient case we obtain approximate expressions for eigenvalues and eigensolutions near the linear limit the results show that the interpretative and the predictive power of the linear wkb solutions carry through to the nonlinear regime of small amplitude wavelike solutions we further analyse the mechanism by which the odd n nonlinearity in general causes a splitting of the linear eigenvalues in particular we discuss the singular threshold behaviour of the doubling mechanism for nonlinearities with n 3 if the coefficient functions become constants the doubling of eigenvalues corresponding to standing waves of odd numbers of nodes gradually disappears the method of approximation can be worked out similarly for any perturbing polynomial in

a simple unified approach to some convergence theorems of l simon abstract we give a new and easier proof for the convergences result of l simon 15 concerning global and bounded solutions of a heat equation with analytic nonlinearity by constructing a new liapunov functional for the elliptic equation we are able to prove the second convergence theorem of 15 in the same way as the first theorem the results are more general and can be applied to other problems such as fourth order equation and vector valued systems of gradient type

ebk quantization of quasi energies the semiclassical einstein brillouin keller vortex tube quantization for the quasi energies in time periodic hamiltonians is studied for typical systems when the vortex tubes of the quantizing tubes may have complicated structures in extended phase space numerical application to a periodically driven duffing quartic oscillator yields semiclassical quasi energies in excellent agreement with the exact quantum mechanical ones for the regular states

non overlapping domain decomposition algorithm based on modified transmission conditions for the helmholtz equation abstract a square root based transmission conditions domain decomposition method was recently introduced for the helmholtz equation it produces an effective algorithm where the convergence is independent of the wavenumber and the mesh discretization we modify here these conditions in order to guarantee well posedness of local problems and further improve the efficiency of the whole method numerical results in particular in the three dimensional case show significant reduction of the computational time needed in the iterative procedure while preserving the iteration number when compared with the original algorithm

3 d electromagnetic scattering computation in free space with the feti fdp2 method the electromagnetic dual primal finite element tearing and interconnecting feti dpem method is a nonoverlapping domain decomposition method developed for the finite element analysis of large scale electromagnetic problems where the corner edges are globally numbered this paper presents an extension of the feti dpem2 method named feti full dual primal feti fdp2 where more flexible robin type boundary conditions are imposed on the inner interfaces between subdomains as well as on the corner edges leading to a new interface problem its capacities are tested in the framework of a three dimensional 3 d free space scattering problem with a scattered field formulation and a computational domain truncated by perfectly mathed layers pml first we compare its accuracy with respect to other feti dpem2 methods and to a complete resolution of the fem problem thanks to a direct sparse solver we show that the convergence of iterative solvers is affected by the presence of the pml and can be accelerated by means of a more accurate approximation between adjacent subdomains of the dirichlet to neumann dtn operator the effectiveness of the iterative solvers are also considered for different test cases the advantages of the proposed feti fdp2 method combined with the associated dtn approximation is numerically demonstrated regardless the chosen working frequency or the iterative solvers