comparison results for multidimensional difference equations on compare les solutions u x de l equation aux differences non lineaire x u u x f x u x avec les solutions x et x des inegalites aux differences non lineaires correspondantes x n x f x x et x n x f x x

restarted gmres preconditioned by deflation this paper presents a new preconditioning technique for the restarted gmres algorithm it is based on an invariant subspace approximation which is updated at each cycle numerical examples show that this deflation technique gives a more robust scheme than the restarted algorithm at a low cost of operations and memory

a mass redistributed finite element method mr fem for acoustic problems using triangular mesh the accuracy of numerical results using standard finite element method fem in acoustic problems will deteriorate with increasing frequency due to the dispersion error such dispersion error depends on the balance between the stiffness and mass of discretization equation systems this paper reports an improved finite element method fem for solving acoustic problems by re distributing the mass in the mass matrix to tune the balance aiming to minimize the dispersion errors this is done by shifting the integration point locations when computing the entries of the mass matrix while ensuring the mass conservation the new method is verified through the detailed numerical error analysis and a strategy is also proposed for the best mass redistribution in terms of minimizing dispersion error the relative dispersion error of present mass redistributed finite element method mr fem is found to be much smaller than the fem solution in both theoretical prediction and numerical examination the present mr fem works well by using the linear triangular elements that can be generated automatically which enables automation in computation and saving computational cost in mesh generation numerical examples demonstrate the advantages of mr fem in comparison with the standard fem using the same triangular meshes and quadrilateral meshes

oscillation theorems for certain delay differential inequalities is oscillatory see i this change in the oscillatory behavior of a differential equation is obviously generated by the delay and its study is very interesting in applications for example oscillations caused by delays should be seriously taken into account in studying the motion of a controlled craft moving with increasing velocities where it is possible to have a sudden release of oscillations leading to instability cf minorsky 8 p 4181 414 0022 247x 85 3 00

a tensor product finite element method for the diffraction grating problem with transparent boundary conditions abstract we consider the diffraction grating problem in optics which has been modeled by a boundary value problem governed by a helmholtz equation with transparent boundary conditions a tensor product finite element method is proposed to numerically solve the problem an fft based matrix decomposition algorithm is developed to solve the linear system arising in the vertically layered medium case which can be used as a preconditioning technique for the general case numerical examples are presented to illustrate the accuracy and efficiency of the method

a novel multigrid based preconditioner for heterogeneous helmholtz problems an iterative solution method in the form of a preconditioner for a krylov subspace method is presented for the helmholtz equation the preconditioner is based on a helmholtz type differential operator with a complex term a multigrid iteration is used for approximately inverting the preconditioner the choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with fourier analysis multigrid analysis results are verified by numerical experiments high wavenumber helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner

volume and surface integral equations for electromagnetic scattering by a dielectric body we derive and analyze two equivalent integral formulations for the time harmonic electromagnetic scattering by a dielectric object one is a volume integral equation vie with a strongly singular kernel and the other one is a coupled surface volume system of integral equations with weakly singular kernels the analysis of the coupled system is based on standard fredholm integral equations and it is used to derive properties of the volume integral equation

oscillatory and asymptotic behavior of certain functional differential equations some new criteria for the oscillatory and asymptotic behavior of odd order functional equations of the form x n t p t x n 1 t q t f x g t are established

finite element solution of the helmholtz equation with high wave number part i the h version of the fem abstract the paper addresses the properties of finite element solutions for the helmholtz equation the h version of the finite element method with piecewise linear approximation is applied to a one dimensional model problem new results are shown on stability and error estimation of the discrete model in all propositions assumptions are made on the magnitude of hk only where k is the wavelength and h is the stepwidth of the fe mesh previous analytical results had been shown with the assumption that k 2 h is small for medium and high wavenumber these results do not cover the meshsizes that are applied in practical applications the main estimate reveals that the error in h 1 norm of discrete solutions for the helmholtz equation is polluted when k 2 h is not small the error is then not quasioptimal i e the relation of the fe error to the error of best approximation generally depends on the wavenumber k it is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution in the result of this analysis thorough and rigorous understanding of error behavior throughout the range of convergence is gained numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior the h p version of the fem is studied in part ii

a dispersion minimizing compact finite difference scheme for the 2d helmholtz equation in this paper we present a dispersion minimizing compact finite difference scheme for solving the 2d helmholtz equation which is a fourth order scheme the error between the numerical wavenumber and the exact wavenumber is analyzed and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters numerical results are presented to demonstrate the efficiency and accuracy of the compact finite difference scheme with refined parameters

nonlinear discrete inequalities with two variables and their applications abstract an error in the proof of theorem 1 by zheng et al k zheng y wu s zhong discrete nonlinear integral inequalities in two variables and their applications appl math comput 207 2009 140 147 is reported this paper gives the right proof under some additional condition application examples to show boundedness and uniqueness of solutions of a volterra type difference equation are also given

estimation of the dispersion error in the numerical wave number of standard and stabilized finite element approximations of the helmholtz equation an estimator for the error in the wave number is presented in the context of finite element approximations of the helmholtz equation the proposed estimate is an extension of the ideas introduced in steffens and d iez comput methods appl mech engng 2009 198 1389 1400 in the previous work the error assessment technique was developed for standard galerkin approximations here the methodology is extended to deal also with stabilized approximations of the helmholtz equation thus the accuracy of the stabilized solutions is analyzed including also their sensitivity to the stabilization parameters depending on the mesh topology the procedure builds up an inexpensive approximation of the exact solution using post processing techniques standard in error estimation analysis from which the estimate of the error in the wave number is computed using a simple closed expression the recovery technique used in steffens and diez comput methods appl mech engng 2009 198 1389 1400 is based in a polynomial least squares fitting here a new recovery strategy is introduced using exponential in a complex setup trigonometric local approximations respecting the nature of the solution of the wave problem copyright 2011 john wiley sons ltd

an example of explicit implementation strategy and preconditioning for the high order edge finite elements applied to the time harmonic maxwell s equations abstract in this paper we focus on high order finite element approximations of the electric field combined with suitable preconditioners to solve the time harmonic maxwell s equations in waveguide configurations the implementation of high order curl conforming finite elements is quite delicate especially in the three dimensional case here we explicitly describe an implementation strategy which has been embedded in the open source finite element software freefem http www freefem org ff in particular we use the inverse of a generalized vandermonde matrix to build basis functions in duality with the degrees of freedom resulting in an easy to use but powerful interpolation operator we carefully address the problem of applying the same vandermonde matrix to possibly differently oriented tetrahedra of the mesh over the computational domain we investigate the preconditioning for maxwell s equations in the time harmonic regime which is an underdeveloped issue in the literature particularly for high order discretizations in the numerical experiments we study the effect of varying several parameters on the spectrum of the matrix preconditioned with overlapping schwarz methods both for 2d and 3d waveguide configurations

on oscillation of solutions of nth order delay differential equations abstract oscillatory behavior of the solutions of the nth order delay differential equation lnx t q t f x g t 0 is discussed the results obtained are extensions of some of the results by kim proc amer math soc 62 1977 77 82 for y n py 0

systems of multidimensional discrete inequalities on considere des systemes lineaires de m inegalites discretes comprenant des fonctions a n variables independantes

about some new integral inequalities of wendroff type for discontinuous functions abstract in this article we obtain some new nonlinear integral inequalities for discontinuous functions of two independent variables wendroff type by including also inequalities with delay we deduce new generalizations of earlier results given by r p agarwal r bellman i bihari b k bondge v lakshmikantham s leela b g pachpatte for continuous and discrete functions furthermore generalizations of some results for integro sum inequalities are obtained as well

goal oriented adaptivity using unconventional error representations for the multidimensional helmholtz equation in goal oriented adaptivity goa the error in the quantity of interest qoi is represented using the error functions of the direct and adjoint problems this error representation is subsequently bounded above by element wise error indicators that are used to drive optimal refinements in this work we propose to replace in the error representation the adjoint problem by an alternative operator the main advantage of the proposed approach is that when judiciously selecting such alternative operator the corresponding upper bound of the error representation becomes sharper leading to a more efficient goa while the method can be applied to a variety of problems we focus here on two and threedimensional 2d and 3d helmholtz problems we show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p adaptive process we also provide guidelines for finding operators delivering sharp error representation upper bounds we further extend the results to a convectiondominated diffusion problem as well as to problems with discontinuous material coefficients finally we consider a sonic logging while drilling lwd problem to illustrate the applicability of the proposed method

a simple evaluation of some principal value integrals for dyadic green s function using symmetry property for some special principal volume area the principal value integrals given by van bladel and yaghjian are evaluated using a reasoning of symmetry this approach is quite simple and the result of the two dimensional case is believed to be new

sharp conditions for existence of nontrivial invariant cones of nonnegative initial values of difference equations it is proved that an arbitrary scalar difference equation has a nontrivial invariant cone in the set of the nonnegative initial values if and only if the associated characteristic equation has a positive root by using this fact there is given a construction of the maximal invariant cone in the set of nonnegative initial values via a positive root of the associated characteristic equation finally the present results are compared with some earlier ones which were obtained for delay differential equations in the same direction

wkb based schemes for the oscillatory 1d schrodinger equation in the semiclassical limit an efficient and accurate numerical method is presented for the solution of highly oscillatory differential equations while standard methods would require a very fine grid to resolve the oscillations the presented approach uses first an analytic second order wkb type transformation which filters out the dominant oscillations the resulting ode is much smoother and can hence be discretized on a much coarser grid with significantly reduced numerical costs in many practically relevant examples the method is even asymptotically correct w r t the small parameter varepsilon that identifies the oscillation wavelength indeed in these cases the error then vanishes for varepsilon to0 even on a fixed spatial mesh applications to the stationary schrodinger equation are presented

oscillation theorems of nth order functional differential equations with forcing terms where a p q g h f r a co r are continuous b p t 0 q t 0 g r h t are nondecreasing c g t t and g t co as t co a function x t e cc 00 is called oscillatory if x t has arbitrarily large zeros in to co to o otherwise x t is called nonoscillatory we provide sufficient conditions for the above functional differential equations to be almost oscillatory in the sense that every solution x t of l 2 and 3 is either oscillatory or else it satisfies lim m x t 0 06i n 1

sharp discrete inequalities and applications to discrete variational problems some new discrete inequalities involving monotonic or convex functions are obtained while these are interesting inequalities in their own right they can be applied to solving certain types of discrete variational problems effectively

a symmetric trefftz dg formulation based on a local boundary element method for the solution of the helmholtz equation a general symmetric trefftz discontinuous galerkin method is built for solving the helmholtz equation with piecewise constant coefficients the construction of the corresponding local solutions to the helmholtz equation is based on a boundary element method a series of numerical experiments displays an excellent stability of the method relatively to the penalty parameters and more importantly its outstanding ability to reduce the instabilities known as the pollution effect in the literature on numerical simulations of long range wave propagation

adaptive finite element method for the sound wave problems in two kinds of media abstract in this paper we consider the adaptive finite element method for sound wave propagation problems in two kinds of media which are the linear anisotropic acoustic materials the cloak metamaterials and penetrable media and the nonlinear acoustic materials a posteriori error estimator based on the new flux recovery technique and a residual type posteriori error estimator are proposed based on our a posteriori error estimators the adaptive finite element algorithm is given for numerical simulations of the helmholtz equations in different media extensive numerical results demonstrate the effectiveness of the adaptive algorithm

modeling of electromagnetic absorption scattering problems using hp adaptive finite elements a model problem for the steady state form of maxwell s equations is considered the problem is formulated in a weak form using a lagrange multiplier laying down a foundation for a general class of novel hp adaptive fe approximations a convergence proof for affine elements is presented the proposed method is illustrated and verified with a series of 2d experiments including elements with curved boundaries and nonhomogeneous media

note on forced oscillation of nth order sublinear differential equations in the case of oscillatory potentials we establish an oscillation theorem for the forced sublinear differential equation x n q t x sgnx e t t t0 no restriction is imposed on the forcing term e t to be the nth derivative of an oscillatory function in particular we show that all solutions of the equation x t sint x sgnx mt cost t 0 0 2 1 this provides an analogue of a result of nasr proc amer math soc 126 1998 123 for the forced superlinear equation and answers a question raised in an earlier paper j s w wong siam j math anal 19 1988 673

error estimation and adaptivity for the finite element method in acoustics 2d and 3d applications this paper is dedicated to the control of accuracy and to the adaptivity of the finite element simulation of sound propagation assuming time harmonic behaviour the mathematical models are given as boundary value problems for the helmholtz equation two singularities inherent to the operator are demonstrated the k singularity related to the phase shift between the exact and the numerical waves and the singularity corresponding to the singularity at the eigenfrequencies two a posteriori error estimators are developed and the numerical tests show that due to these specific singularities error control cannot in general be accomplished by just transplanting methods that work well in static computations furthermore for low wave numbers it is necessary also to control the influence of the geometric or physical singularities an h adaptive version with refinement is applied to 2d and 3d real life problems

a two point boundary value problem with a rapidly oscillating solution some numerical methods are developed for a two point boundary value problem with a rapidly oscillating solution the two point boundary value problem is chosen to model some of the difficulties that may be expected to occur in solving the reduced wave equation at moderately high frequencies

oscillation criteria for nth order forced functional differential equations in this paper by using the method of general means some new oscillation criteria for forced functional differential equations of the form x n t i 1n 1aix i t q t fxg t e t t0 0 are established where ai are real constants q t f t e t and g t are real continuous functions xf x 0 whenever x 0 and limt g t

a galerkin least squares finite element method for the two dimensional helmholtz equation in this paper a galerkin least squares gls finite element method in which residuals in least squares form are added to the standard galerkin variational equation is developed to solve the helmholtz equation in two dimensions an important feature of gls methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes previous work has accomplished this for the one dimensional helmholtz equation using dispersion analysis in this paper the selection of the gls mesh parameter for two dimensions is considered and leads to elements that exhibit improved phase accuracy for any given direction of wave propagation an optimal gls mesh parameter is determined using two dimensional fourier analysis in general problems the direction of wave propagation will not be known a priori in this case an optimal gls parameter is found which reduces phase error for all possible wave vector orientations over elements the optimal gls parameters are derived for both consistent and lumped mass approximations several numerical examples are given and the results compared with those obtained from the galerkin method the extension of gls to higher order quadratic interpolations is also presented

solution of 3d laplace and helmholtz equations in exterior domains using hp infinite elements abstract this work is devoted to a convergence study for infinite element discretizations for laplace and helmholtz equations in exterior domains the proposed approximation applies to separable geometries only combining an hp fe discretization on the boundary of the domain with a spectral like representation resulting from the separation of variables in the radial direction the presentation includes a convergence proof for the laplace equation and a stability analysis for the variational formulation of the helmholtz equation in weighted sobolev spaces the theoretical investigations are verified and illustrated with numerical examples for the exterior spherical domain

on the use of physical spline finite element method for acoustic scattering this paper presents the comparison of physical spline finite element method psfem in which differential equations are incorporated into interpolations of basic elements with least squares finite element method lsfem and mixed galerkin finite element method mgfem on the numerical solution of one dimensional helmholtz equation applied to an acoustic scattering problem firstly all three methods are explained in detail and then it is shown that psfem reaches higher precision in a shorter time with fewer nodes than the other methods it is also observed that this method is well suited for high frequency acoustic problems consequently the results of psfem point out better efficiency in terms of number of unknowns and accuracy level

forced oscillation of nth order functional differential equations abstract in this paper we investigate the oscillation of forced functional differential equations x n t a t f x q t e t when the forcing term e t is not required to be the nth derivative of an oscillatory function some new oscillatory criteria are given we also apply our technique to the forced neutral differential equation of the form x t cx t n a t x t b t x t e t c t f 1 x t d t f 2 x t where xf1 x 0 and xf2 x 0 for x 0 n 1 are nonnegative constants a t 0 b t 0 c t 0 d t 0 which includes the special case f1 x x sgn x f2 x x sgn x 1 and 1

a two dimensional self adaptive hp finite element method for the characterization of waveguide discontinuities part i energy norm based automatic hp adaptivity abstract the accurate analysis and characterization of waveguide discontinuities is an important issue in microwave engineering in some cases the discontinuities are an unavoidable result of mechanical or electric transitions which effects have to be minimized in other cases the discontinuities are deliberately introduced into the waveguide to perform a certain electric function this is the first of two papers analyzing different types of rectangular waveguide discontinuities by using a fully automatic hp adaptive finite element method in this paper a fully automatic energy norm based hp adaptive finite element fe strategy applied to a number of relevant waveguide structures is presented the methodology produces exponential convergence rates in terms of the energy norm error of the solution against the problem size number of degrees of freedom extensive numerical results demonstrate the suitability of the hp method for solving different rectangular waveguide discontinuities these results illustrate the flexibility reliability and high accuracy of the method the self adaptive hp fem provides similar sometimes more accurate results than those obtained with semi analytical techniques such as the mode matching method for problems where semi analytical methods can be applied at the same time the hp fem provides the flexibility of modeling more complex waveguide structures and including the effects of dielectrics metallic screws round corners etc which cannot be easily considered when using semi analytical techniques in the second part the work is extended by presenting a fully automatic goal oriented hp adaptive fe in terms of the scattering or s parameters which are widely used in microwave engineering for the characterization of microwave devices

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

transverse electric scattering on inhomogeneous objects spectrum of integral operator and preconditioning the domain integral equation method with its fft based matrix vector products is a viable alternative to local methods in free space scattering problems however it often suffers from the extremely slow convergence of iterative methods especially in the transverse electric te case with large or negative permittivity we identify very dense line segments in the spectrum as being partly responsible for this behavior and the main reason why a normally efficient deflating preconditioner does not work we solve this problem by applying an explicit multiplicative regularizing operator which on the operator level transforms the system to the form identity plus compact on the matrix level this regularization reduces the length of the dense spectral segments roughly by a factor of four while preserving the ability to calculate the matrix vector products using the fft algorithm such a regularized system is then further preconditioned by deflating an apparently stable set of eigenvalues with largest magnitudes which results in a robust acceleration of the restarted gmres under constraint memory conditions

matched interface and boundary mib for the implementation of boundary conditions in high order central finite differences high order central finite difference schemes encounter great difficulties in implementing complex boundary conditions this paper introduces the matched interface and boundary mib method as a novel boundary scheme to treat various general boundary conditions in arbitrarily high order central finite difference schemes to attain arbitrarily high order the mib method accurately extends the solution beyond the boundary by repeatedly enforcing only the original set of boundary conditions the proposed approach is extensively validated via boundary value problems initial boundary value problems eigenvalue problems and high order differential equations successful implementations are given to not only dirichlet neumann and robin boundary conditions but also more general ones such as multiple boundary conditions in high order differential equations and time dependent boundary conditions in evolution equations detailed stability analysis of the mib method is carried out the mib method is shown to be able to deliver high order accuracy while maintaining the same or similar stability conditions of the standard high order central difference approximations the application of the proposed mib method to the boundary treatment of other non standard high order methods is also considered

a generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution when using the galerkin fem for solving the helmholtz equation in two dimensions the error of the corresponding solution differs substantially from the error of the best approximation and this effect increases with higher wave number k in this paper we will design a generalized finite element method gfem for the helmholtz equation such that the pollution effect is minimal

adaptive numerical solution of a discontinuous galerkin method for a helmholtz problem in low frequency regime we develop an a posteriori error analysis for helmholtz problem using the local discontinuous galerkin ldg for short approach for the sake of completeness we give a description of the main a priori results of this method indeed under some assumptions on regularity of the solution of an adjoint problem we prove that a the corresponding indefinite discrete scheme is well posed b the approach is convergent with the expected convergence rates as long as the meshsize h is small enough we give precise information on how small h has to be in terms of the size of the wave number and its distance to the set of eigenvalues for the same boundary value problem for the laplacian after that we present a reliable and efficient a posteriori error estimator with detailed information on the dependence of the constants on the wave number we finish presenting extensive numerical experiments which illustrate the theoretical results proven in this paper and suggest that stability and convergence may occur under less restrictive assumptions than those taken in the present work we derive an a posteriori error estimator that is reliable and locally quasi efficient the efficiency and reliability constants have been characterized with respect to the physical parameter we include numerical examples that validate our theoretical results recognizing singularities and inner layers for moderate values of

stable fft jvie solvers for fast analysis of highly inhomogeneous dielectric objects abstract a stable volume integral equation formulation based on equivalent volumetric currents is presented for modeling electromagnetic scattering of highly inhomogeneous dielectric objects the proposed formulation is numerically solved by means of galerkin method of moments on uniform grids allowing for acceleration of the matrix vector products associated with the iterative solver with the help of fft in addition the pertinent volume volume galerkin inner products are reduced to purely surface surface integrals with smoother kernels allowing for highly accurate and fast computation by readily available sophisticated cubatures numerical results demonstrate the convergence properties of the algorithm for scatterers with high dielectric contrast in addition we describe a road map for magnetic resonance specific volume integral equations fast solvers based on the proposed algorithm

pollution studies for high order isogeometric analysis and finite element for acoustic problems abstract it is well known that galerkin finite element methods suffer from pollution error when solving wave problems to reduce the pollution impact on the solution different approaches were proposed to enrich the finite element method with wave like functions so that the exact wavenumber is incorporated into the finite element approximation space solving wave problems with isogeometric analysis was also investigated in the literature where the superior behaviour of isogeometric analysis due to higher continuity in the underlying basis has been studied recently a plane wave enriched isogeometric analysis was introduced for acoustic problems however it remains unquantified the impact of these different approaches on the pollution or how they perform compared to each other in this work we show that isogeometric analysis outperforms finite element method in dealing with pollution we observe similar behaviour when both the methods are enriched with plane waves using higher order polynomials with fewer enrichment functions seems to improve the pollution compared to lower order polynomials with more functions however the latter still leads to smaller errors using similar number of degrees of freedom in conclusion we propose that partition of unity isogeometric analysis can be an efficient tool for wave problems as enrichment eliminates the need for domain re meshing at higher frequencies and also due to its ability to capture the exact geometry even on coarse meshes as well as its improved pollution behaviour

a residual based finite element method for the helmholtz equation a new residual based finite element method for the scalar helmholtz equation is developed this method is obtained from the galerkin approximation by appending terms that are proportional to residuals on element interiors and inter element boundaries the inclusion of residuals on inter element boundaries distinguishes this method from the well known galerkin least squares method and is crucial to the resulting accuracy of this method in two dimensions and for regular bilinear quadrilateral finite elements it is shown via a dispersion analysis that this method has minimal phase error numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods it is found that even for unstructured meshes this method retains a high level of accuracy copyright 2000 john wiley sons ltd

on oscillation of solutions of higher order forced functional differential inequalities abstract oscillation criteria for the class of forced functional differential inequalities x t lnx t f t x t x g1 t x gm t h t 0 for n even and x t lnx t f t x t x g1 t x gm t h t 0 for n odd are established

dispersion analysis of a non conforming finite element method for the helmholtz and elastodynamic equations summary we investigate the dispersive properties of a non conformingnite element method to solve the two dimensional helmholtz and elastodynamics equations the study is performed by deriving and analysing the dispersion relations and by evaluating the derived quantities such as the dimensionless phase and group velocities also the phase dierence between exact and numerical solutions is investigated the studied method which yields a linear spatial approximation is shown to be less dispersive than a conforming bilinearnite element method in the two cases shown herein moreover it almost halves the number of points per wavelength necessary to reach a given accuracy when calculating the mentioned velocities in both cases here presented copyright 2003 john wiley sons ltd

convergence analysis of an adaptive continuous interior penalty finite element method for the helmholtz equation abstract we are concerned with an adaptive continuous interior penalty finite element method for the helmholtz equation a convergence result with explicit constraints on the wave number k and the initial mesh size h 0 is derived in particular it is shown that if k 1 s h 0 s is sufficiently small where 1 2 s 1 depends on the regularity of the weak solution of the model problem then the adaptive continuous interior penalty finite element method is a contraction with respect to the sum of the energy error and a scaled error estimator numerical experiments are provided to verify the theoretical findings and show advantages of the adaptive continuous interior penalty finite element method

a comparison theorem for difference inequalities in the study of growth and decay properties of nonlinear evolution equations in particular hyperbolic and parabolic partial differential equations as t the following difference inequality which is satisfied by an energy expression arises mathop sup limits t leq s leq t 1 u s 1 alpha leq c 1 t r u t u t 1 g t

oscillation theorems for nth order delay differential equations abstract oscillation criteria are obtained for a class of functional differential equations including the generalized emden fowler equation x n q t x g t sgn x g t 0 0 as a special case the purpose of this paper is to extend and improve known fundamental oscillation criteria for superlinear sublinear and linear differential equations with retarded arguments

mathematical modelling of 2d electromagnetic scattering abstract the singular integral equation method is used for solving the problem of diffraction by infinite nonhomogeneous cylinder a numerical method is developed and certain aspects of its practical application are considered some numerical results are described

systems of multidimensional volterra integral equations and inequalities on obtient des bornes superieures explicites pour les solutions u x d equations integrales de volterra lineaires ou non lineaires on demontre des theoremes d existence globale

on gronwall bellman bihari type integral inequalities in several variables with retardation the celebrated gronwall bellman bihari inequality and its various linear and nonlinear generalizations have been of vital importance in the study of existence uniqueness continuous dependence comparison perturbation boundedness and stability results of ordinar differential equations and integral equations 2 6 14 15 18 20 22 23 371 in recent years the more than one variable generalizations of this inequality have evoked considerable interest since they are handy tools in the theory of partial differential and integral equations specifically we refer to the results of bondge and pachpatte 7 11 131 bondge et 121 chu and metcalf 161 colan and diaz 171 headley 21 agarwal 11 pachpatte 24 301 rasmussen 31 shih and yeh 32 41 singare and pachpatte 33 snow 34 35 than dapani and agarwal 36 yeh 39 40 42 young 43 and zahariev and bainov 44 among others in this paper we establish some new nonlinear integral inequalities with delay in n independent variables which include and unify recent results on integral inequalities in several variables by shih and yeh 41 yeh 42 and zahariev and bainov 44 we also deduce from our results new generalizations of earlier results of bihari 6 bellman 4 dhongade and deo 18 191 and pachpatte 22 231 some applications of our results to a certain class of nonlinear hyperbolic functional partial differential equations are also given

a multigrid method enhanced by krylov subspace iteration for discrete helmholtz equations standard multigrid algorithms have proven ineffective for the solution of discretizations of helmholtz equations in this work we modify the standard algorithm by adding gmres iterations at coarse levels and as an outer iteration we demonstrate the algorithm s effectiveness through theoretical analysis of a model problem and experimental results in particular we show that the combined use of gmres as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as 200 for fixed wave numbers it displays grid independent convergence rates and has costs proportional to the number of unknowns

application of hp adaptive be fe methods to elastic scattering abstract the investigation focuses on the development of hp adaptive methods for simulation of acoustical scattering problems the present formulation is based on steady state linear elasticity equations coupled with the burton miller integral equation representing the behavior of a surrounding acoustical medium the highlights of the proposed approach include a mathematically consistent variational formulation of the problem an hp adaptive fe be approximation based on standard galerkin methods an element by element solver a residual based a posteriori error estimate an hp adaptive strategy particular attention is given to asymptotic convergence and to a comparison of theory and numerical simulations

on accuracy conditions for the numerical computation of waves abstract the helmholtz equation k 2 n 2 u f with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems such problems can be solved numerically by first truncating the given unbounded domain imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization for example using a finite element method in practical applications the mesh size h and the wave number k are not independent but are constrained by the accuracy of the desired computation it will be shown that the number of points per wavelength measured by kh 1 is not sufficient to determine the accuracy of a given discretization for example the quantity k 3 h 2 is shown to determine the accuracy in the l 2 norm for a second order discretization method applied to several propagation models

asymptotic and oscillatory behavior of nth order forced functional differential equations abstract some new oscillation criteria for the forced functional differential equation x n t a t x q t e t are established

edged based smoothed point interpolation method for acoustic radiation with perfectly matched layer abstract it is known that for exterior acoustics the domain based finite element method always encounters the accuracy loss for high wavenumbers and the difficulty in dealing with the sommerfeld radiation condition in this work by using the polynomial and radial bases with specific node selection schemes the point interpolation method with the generalized gradient smoothing technique is employed to improve the accuracy for acoustic computations the appropriately softened acoustical stiffness of the discrete model is constructed based on the edge based smoothing domains via the gradient smoothing technique in addition the perfectly matched layer technique enables the smoothed point interpolation method to investigate exterior acoustics the complex coordinate transformation is utilized to determine the proper parameters for the perfectly matched layer through theoretical and numerical analyses several typical acoustic problems are analyzed revealing that better accuracy is obtained by the present method compared with the finite element method especially for high wavenumbers

enriched isogeometric collocation for two dimensional time harmonic acoustics abstract in this work the isogeometric collocation iga c is paired with two types of enrichment plane wave pw and generalized harmonic polynomial ghp functions to solve 2d problems for the helmholtz equation a parametric study is conducted and a detailed assessment of the performance of the method in a number of benchmark problems is provided three different collocation methods are tested for the non enriched formulation namely greville abscissae ga approximated cauchy galerkin acg and superconvergent sc points showing that the acg scheme is the best choice in terms of overall error convergence rate and ease of solving the linear system then the influence of the number of shape functions in the original and enriched basis location and number of collocation points and wave number over both the convergence rate as well as the condition number of the stiffness matrix are studied the numerical results show that 1 there is an improvement over the non enriched formulation 2 the improvement depends on the choice of the number and type of enrichment and 3 the pollution error is not completely alleviated with the enriched formulations

how large a shift is needed in the shifted helmholtz preconditioner for its effective inversion by multigrid the shifted helmholtz operator has received a lot of attention over the past decade as a preconditioner for the iterative solution of the helmholtz equation the idea is that if one uses a small complex shift the shifted helmholtz operator is still close to the original helmholtz operator and could thus be an effective preconditioner it was shown in m j gander i g graham and e a spence numer math 53 2015 pp 573 579 that the shift can be at most o k to prove rigorously wave number independent convergence of the preconditioned system solved with gmres provided the preconditioner is inverted exactly in practice however the preconditioner is inverted only approximately and if one shifts enough this can be done effectively by standard multigrid methods we show in this paper that for a finite element discretization the shift has to be at least o k 2 to be able to invert the shifted helmholtz preconditioner using multigrid there is therefore a gap between being a good precondi

integration of hp adaptivity and a two grid solver for electromagnetic problems we present implementation details and analyze convergence of a two grid solver forming the core of a fully automatic hp adaptive strategy for electromagnetic problems the solver delivers a solution for a fine grid obtained from an arbitrary coarse hp grid by a global hp refinement the classical v cycle algorithm combines an overlapping block jacobi smoother with optimal relaxation and a direct solve on the coarse grid a theoretical analysis of the two grid solver is illustrated with numerical experiments several electromagnetic applications show the efficiency of combining the fully automatic hp adaptive strategy with the two grid solver

discrete inequalities of the gronwall bellman type in n independent variables ii soit r 0 n l ensemble des entiers non negatifs et n un entier positif soit u x k x n n r et f x s n 2n r avec s x si u x k x s 0 x 1 f x s u s pour x n n alors u x k x sd 1 0 xu 1 1 1 s 0 x 1 f x s 1 s ou k x sup k x 0 s x f x s sup f t s 0 t x

robustness and dispersion analysis of the partition of unity finite element method applied to the helmholtz equation abstract the partition of unity finite element method pufem has been widely used for the numerical simulation of the helmholtz equation in different physical settings in fact it is a numerical pollution free alternative method to the classical piecewise polynomial based finite element methods taking into account a plane wave enrichment of the piecewise linear finite element method the main goal of this work is focused on the derivation of the numerical dispersion relation and the robustness analysis of the pufem discretization when a spurious perturbation is presented in the wave number value used in the enrichment definition from the one dimensional helmholtz equation the discrete wave number is estimated based on a bloch s wave analysis and a priori error estimates are computed explicitly in terms of the mesh size the wave number and the perturbation value

a spectral finite element approach to modeling soft solids excited with high frequency harmonic loads abstract an approach for efficient and accurate finite element analysis of harmonically excited soft solids using high order spectral finite elements is presented and evaluated the helmholtz type equations used to model such systems suffer from additional numerical error known as pollution when excitation frequency becomes high relative to stiffness i e high wave number which is the case for example for soft tissues subject to ultrasound excitations the use of high order polynomial elements allows for a reduction in this pollution error but requires additional consideration to counteract runge s phenomenon and or poor linear system conditioning which has led to the use of spectral element approaches this work examines in detail the computational benefits and practical applicability of high order spectral elements for such problems the spectral elements examined are tensor product elements i e quad or brick elements of high order lagrangian polynomials with non uniformly distributed gauss lobatto legendre nodal points a shear plane wave example is presented to show the dependence of the accuracy and computational expense of high order elements on wave number then a convergence study for a viscoelastic acoustic structure interaction finite element model of an actual ultrasound driven vibroacoustic experiment is shown the number of degrees of freedom required for a given accuracy level was found to consistently decrease with increasing element order however the computationally optimal element order was found to strongly depend on the wave number

dispersion analysis of plane wave discontinuous galerkin methods summary the plane wave dg pwdg method for the helmholtz equation was introduced and analyzed in gittelson c hiptmair r and perugia i plane wave discontinuous galerkin methods analysis of the h version math model numer anal 43 2009 297 331 as a generalization of the so called ultra weak variational formulation see o cessenat and b despres application of an ultra weak variational formulation of elliptic pdes to the two dimensional helmholtz equation siam j numer anal 35 1998 pp 255 299 the method relies on trefftz type local trial spaces spanned by plane waves of different directions and links cells of the mesh through numerical fluxes in the spirit of dg methods we conduct a partly empirical dispersion analysis of the method in a discrete translation invariant setting by studying the mismatch of wave numbers of discrete and continuous plane waves traveling in the same direction we find agreement of the wave numbers for directions represented in the local trial spaces for other directions the pwdg methods turn out to incur both phase and amplitude errors this manifests itself as a pollution effect haunting the h version of the method our dispersion analysis allows a quantitative prediction of the strength of this effect and its dependence on the wave number and number of plane waves copyright 2014 john wiley sons ltd