three dimensional element configurations for the discontinuous enrichment method for acoustics hexahedral and tetrahedral elements are proposed for the discontinuous enrichment method dem for boundary value problems governed by the helmholtz equation a procedure for obtaining sets of approximately uniform spherical enrichment directions with high flexibility in the choice of their number is constructed conditioning considerations indicate preferred representations of the oscillatory basis functions dispersion properties are used to rate the performance of different element configurations numerical tests assess accuracy indicating that high order dem elements exhibit little dispersion the dispersion and the numerical results are in a good agreement the proposed configurations of dem elements become more competitive as the number of enrichments and lagrange multipliers is increased copyright 2008 john wiley sons ltd

when coherent stochastic resonance appears we consider a system driven by a combination of gaussian white noise and a periodic signal it is demonstrated when the mean exit time for this system necessarily exhibits a minimum as a function of the frequency of the periodic bias and gives rise to the phenomenon known as coherent stochastic resonance this reasoning begins by calculating an exact expression for the mean exit time of a system driven by noise and a telegraph signal and showing that no resonancelike behavior arises this unexpected result lets us identify the sufficient conditions for resonance to appear the origin of coherent stochastic resonance lies in the different behavior of low frequency and high frequency periodic signals

a new eighth order implicit finite difference method to solve the three dimensional helmholtz equation abstract this paper introduces a novel method for deducing high order finite difference schemes for the three dimensional helmholtz equation p 2 p f the new method can reach an eighth order of accuracy it is based on an implicit formulation by computing simultaneously the unknown variable and its corresponding derivatives the scheme is obtained from taylor series expansion and plane wave theory analysis and it is constructed from a few modifications to the standard centered finite differences a classical dispersion analysis and the error between the numerical wavenumber and the exact wavenumber is given numerical experiments are presented to verify the feasibility and accuracy of the proposed methods for a broad range of values moreover the numerical method is capable to solve with great precision the monotone and oscillatory helmholtz problem with variable values thus the proposed formulation results in an attractive method easy to implement with high order of accuracy but nearly the same computational cost as those of an explicit formulation

an efficient high order nystrom scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface this text proposes a fast rapidly convergent nystrom method for the solution of the lippmann schwinger integral equation that mathematically models the scattering of time harmonic acoustic waves by inhomogeneous obstacles while allowing the material properties to jump across the interface the method works with overlapping coordinate charts as a description of the given scatterer in particular it employs partitions of unity to simplify the implementation of high order quadratures along with suitable changes of parametric variables to analytically resolve the singularities present in the integral operator to achieve desired accuracies in approximations to deal with the discontinuous material interface in a high order manner a specialized quadrature is used in the boundary region the approach further utilizes an fft based strategy that uses equivalent source approximations to accelerate the evaluation of large number of interactions that arise in the approximation of the volumetric integral operator and thus achieves a reduced computational complexity of o n log n for an n point discretization a detailed discussion on the solution methodology along with a variety of numerical experiments to exemplify its performance are presented in this paper

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

numerical dispersion of higher order nodal elements in the finite element method the discretization inherent in the finite element method results in numerical dispersion this dispersion is investigated for a time harmonic plane wave propagating through an infinite two dimensional finite element mesh composed of uniform quadrilateral and triangular elements the effects on the dispersion due to the propagation direction of the wave the order of the elements the node density and the mesh geometry are studied results are given which can serve as a guide in selecting the appropriate element order node density and mesh geometry when applying the finite element method

on the use of rational iterations and domain decomposition methods for the helmholtz problem an iterative algorithm for the numerical solution of the helmholtz problem is considered it is difficult to solve the problem numerically in particular when the imaginary part of the wave number is zero or small we develop a parallel iterative algorithm based on a rational iteration and a nonoverlapping domain decomposition method for such a non hermitian non coercive problem algorithm parameters artificial damping and relaxation are introduced to accelerate the convergence speed of the iteration convergence analysis and effective strategies for finding efficient algorithm parameters are presented numerical results carried out on an ncube2 are given to show the efficiency of the algorithm to reduce the boundary reflection we employ a hybrid absorbing boundary condition abc which combines the first order abc and the physical q abc computational results comparing the hybrid abc with non hybrid ones are presented

arbitrary high order c0 tensor product galerkin finite element methods for the electromagnetic scattering from a large cavity the paper is concerned with the electromagnetic scattering from a large cavity embedded in an infinite ground plane the electromagnetic cavity problem is described by the helmholtz equation with a nonlocal boundary condition on the aperture of the cavity and dirichlet or neumann boundary conditions on the walls of the cavity a tensor product galerkin finite element method fem is proposed in which spaces of c 0 piecewise polynomials of degree k 1 are employed by the fast fourier transform and the toeplitz type structure of the approximation to the nonlocal operator in the nonlocal boundary condition a fast algorithm is designed for solving the linear system arising from the cavity problem with vertically layered media which requires o n 2logn operations on an nxn uniform partition numerical results for model problems illustrate the efficiency of the fast algorithm and exhibit the expected optimal global convergence rates of the finite element methods moreover our numerical results also show that the high order approximations are especially effective for problems with large wave numbers

preconditioning harmonic unsteady potential flow calculations this paper considers finite element discretizations of the helmholtz equation and its generalization arising from harmonic acoustic perturbations to a nonuniform steady potential flow a novel elliptic positive definite preconditioner with a multigrid implementation is used to accelerate the iterative convergence of krylov subspace solvers both theory and numerical results show that for a model 1 d helmholtz test problem the preconditioner clusters the discrete system s eigenvalues and lowers its condition number to a level independent of grid resolution for the 2 d helmholtz equation grid independent convergence is achieved using a quasi minimal residual krylov solver significantly outperforming the popular symmetric successive over relaxation preconditioner impressive results are also presented on more complex domains including an axisymmetric aircraft engine inlet with nonstagnant mean flow and modal boundary conditions

on the pollution effect in fe solutions of the 3d helmholtz equation we study convergence of the fem for a 3d problem of rigid scattering using separation of variables analytical results are presented for stability and error estimation for an h version galerkin fem with mesh refinement in radial direction we prove an upper bound that contains a pollution term similar to previous results in 1d and 2d this estimate is evaluated computationally using a 3d fem iem infinite elements implementation the tabulated and plotted errors show the theoretically predicted pollution effect in the fe error

a continuous localization and completion the main goal of this paper is to construct a localization and completion of bousfield kan type as a continuous functor for a virtually nilpotent cw complex then the localization and completion of an an space is given to be an an homomorphism between an spaces for any general compact lie group this gives a continuous equivariant localization and completion for a virtually nilpotent g cw complex more generally we have a continuous localization with respect to a system of core rings for a virtually nilpotent dcw complex for a polyhedral category d the simplicial construction of bousfield and kan 2 automatically generalizes to the homology localization of a d cw complex 6 for a discrete category d including the equivariant case for finite groups but this fails for general compact lie groups a elmendorf 8 constructs functorially an equivariant eilenberg mac lane space by using may s method and the iteration of a bar construction using this may et al 16 17 show the existence of an equivariant localization and completion using theorems 3 and 4 of 8 for nilpotent g cw complexes which is given with the arithmetic square theorem for nilpotent g spaces on the other hand t sumi 24 gives an equivariant localization of a 1 connected g cw complex with respect to a system of local rings by using the method of 20 but the functors are not continuous in this paper we construct a generalized eilenberg mac lane space r x by using the symmetric product 4 then by the methods of a triple an algebra functor 1 and a cosimplicial space we construct a nilpotent tower and a completion and a localization as continuous functors using this we show the arithmetic square theorem for a virtually nilpotent cw complex see drordwyer kan 5 as the localization is continuous the localization of an an space mapping is an an space mapping by the explicit definition of an an space in stasheff 22 and an an mapping between an spaces in 12 13 received by the editors february 19 1987 1980 mathematics subject classification 1985 revision primary 55p60 secondary 55n91 55p20 55u40

an iterative method for the helmholtz equation an iterative algorithm for the solution of the helmholtz equation is developed the algorithm is based on a preconditioned conjugate gradient iteration for the normal equations the preconditioning is based on an ssor sweep for the discrete laplacian numerical results are presented for a wide variety of problems of physical interest and demonstrate the effectiveness of the algorithm

on the coupling of bem and fem for exterior problems for the helmholtz equation this paper deals with the coupled procedure of the boundary element method bem and the finite element method fem for the exterior boundary value problems for the helmholtz equation a circle is selected as the common boundary on which the integral equation is set up with fourier expansion as a result the exterior problems are transformed into nonlocal boundary value problems in a bounded domain which is treated with fem and the normal derivative of the unknown function at the common boundary does not appear the solvability of the variational equation and the error estimate are also discussed

finite element methods applied to nearly one way wave propagation abstract a numerical method is developed for approximately solving stationary wave propagation problems for which most of the energy propagates in an angle band of less than 90 about a fixed direction this property often occurs in realistic physical models the numerical method is based on factoring out the principal propagation direction in the solution and then applying the finite element method the method is applied to wave propagation in a duct or waveguide and to the exterior scattering problem it is shown both theoretically and by means of numerical examples that this method requires considerably less grid points and computational cost than standard discretization methods

coherent stochastic resonance in a sextic double well potential abstract we study the response of a stochastic system described by the one dimensional fokker planck equation with a sextic double well potential u 6 x c 4 x 4 c 6 x 6 to a weak sinusoidal periodic signal two types of boundary conditions have been investigated i absorbing boundary conditions and ii natural boundary conditions the unperturbed propagators are calculated by the eigenfunction expansion method we find that for case i the mean survival time shows the coherent stochastic resonance and for case ii the signal to noise ratio exhibits the stochastic resonance the results are compared with those of the bistable quartic double well potential u 4 x 1 2 x 2 1 4 x 4

compact 2d and 3d sixth order schemes for the helmholtz equation with variable wave number several studies have presented compact fourth order accurate finite difference approximation for the helmholtz equation in two or three dimensions several of these formulae allow for the wave number k to be variable other papers have extended this further to include variable coefficients within the laplacian which models non homogeneous materials in electromagnetism later papers considered more accurate compact sixth order methods but these were restricted to constant k in this paper we extend these compact sixth order schemes to variable k in both two and three dimensions results on 2d and 3d problems with known analytic solutions verify the sixth order accuracy we demonstrate that for large wave numbers the second order scheme cannot produce comparable results with reasonable grid sizes

errors due to spatial discretization and numerical precision in the finite element method the effects of the spatial discretization and the numerical precision on a plane wave propagating through a finite element mesh are investigated in this work the spatial discretization results in dispersion in the amplitude and the phase of the wave and in a non uniform rate of convergence within an element the finite precision in the calculations used in a finite element code results in degraded accuracy these errors are investigated as a function of the node density the order of the elements and the precision of the calculations used in the finite element code the errors for first through eighth order elements are investigated both analytically and numerically

moments of the first passage time under external driving a general theory is derived for the moments of the first passage time of a one dimensional markov process in the presence of a weak time dependent forcing the linear corrections to the moments can be expressed by quadratures of the potential and of the time dependent probability density of the unperturbed system or equivalently by its laplace transform if none of the latter functions is known the derived formulas may still be useful for specific cases including a slow driving or a driving with power at only small or large times in the second part of the paper explicit expressions for the mean and variance of the first passage time are derived for the cases of a linear or a parabolic potential and an exponentially decaying driving force the analytical results are found to be in excellent agreement with computer simulations of the respective first passage processes the particular examples furthermore demonstrate that already the effect of a simple exponential driving can be fairly involved implying a nontrivial nonmonotonic behavior of mean and variance as functions of the driving s time scale

a survey of finite element methods for time harmonic acoustics many of the current issues and methodologies related to finite element methods for time harmonic acoustics are reviewed the effective treatment of unbounded domains is a major challenge most prominent among the approaches that have been developed for this purpose are absorbing boundary conditions infinite elements and absorbing layers standard computational methods are unable to cope with wave phenomena at short wave lengths due to resolutions required to control dispersion and pollution errors leading to prohibitive computational demands since computation naturally separates the scales of a problem according to the mesh size multiscale considerations provide a useful framework for viewing these difficulties and developing methods to counter them other issues addressed are related to the efficient solution of systems of specialized algebraic equations and inverse problems of acoustics the tremendous progress that has been made in all of the above areas in recent years will surely continue leading to many more exciting developments

finite element solution of the helmholtz equation with high wave number part ii the h p version of the fem in this paper which is part ii in a series of two the investigation of the galerkin finite element solution to the helmholtz equation is continued while part i contained results on the h version with piecewise linear approximation the present part deals with approximation spaces of order p ge 1 as in part i the results are presented on a one dimensional model problem with dirichlet robin boundary conditions in particular there are proven stability estimates both with respect to data of higher regularity and data that is bounded in lower norms the estimates are shown both for the continuous and the discrete spaces under consideration further there is proven a result on the phase difference between the exact and the galerkin finite element solutions for arbitrary p that had been previously conjectured from numerical experiments these results and further preparatory statements are then employed to show error estimates for the galerkin finite element method fem it becomes evident that the error estimate for higher approximation can with certain assumptions on the data be written in the same form as the piecewise linear case namely as the sum of the error of best approximation plus a pollution term that is of the order of the phase difference the paper is concluded with a numerical evaluation

domain decomposition iterative procedures for solving scalar waves in the frequency domain the propagation of dispersive waves can be modeled relevantly in the frequency domain a wave problem in the frequency domain is difficult to solve numerically in addition to having a complex valued solution the problem is neither hermitian symmetric nor coercive in a wide range of applications in geophysics or quantum mechanics in this paper we consider a parallel domain decomposition iterative procedure for solving the problem by finite differences or conforming finite element methods the analysis includes the decomposition of the domain into either the individual elements or larger subdomains blocks of finite elements to accelerate the speed of convergence we introduce relaxation parameters on the subdomain interfaces and an artificial damping iteration the convergence rate of the resulting algorithm turns out to be independent on the mesh size and the wave number numerical results carried out on an ncube2 parallel computer are presented to show the effectiveness of the method

a parallel fictitious domain method for the three dimensional helmholtz equation the application of a fictitious domain domain embedding method to the three dimensional helmholtz equation with absorbing boundary conditions is considered the finite element discretization is performed by using locally fitted meshes and an algebraic fictitious domain method with a separable preconditioner is used in the iterative solution of the resultant linear systems such a method is based on embedding the original domain into a larger one with a simple geometry with this approach it is possible to realize the gmres iterations in a low dimensional subspace and use the partial solution method to solve the linear systems with the preconditioner an efficient parallel implementation of the iterative algorithm is introduced results of numerical experiments demonstrate good scalability properties on distributed memory parallel computers and the ability to solve high frequency acoustic scattering problems

coherent stochastic resonance in one dimensional diffusion with one reflecting and one absorbing boundaries it is shown that the single step periodic signal periodic telegraph signal can not produce coherent stochastic resonance for diffusion on a segment with one absorbing and one reflecting end points while the multi step periodic signal does the general features of this process are exihibited the resonant frequency is found to decrease and the mean first passage time at resonant frequency increases linearly as we increase the length of the medium the cycle variable is shown to be the proper argument to express the first passage probability at resonance a formula for first passage probability at resonance is derived in terms of two universal functions which clearly isolates its dependence on the length of the medium

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

a finite element method for solving helmholtz type equations in waveguides and other unbounded domains a finite element method is described for solving helmholtz type boundary value problems in unbounded regions including those with infinite boundaries typical examples include the propagation of acoustic or electromagnetic waves in waveguides the radiation condition at infinity is based on separation of variables and differs from the classical sommerfeld radiation condition it is shown that the problem may be replaced by a boundary value problem on a fixed bounded domain the behavior of the solution near infinity is incorporated in a nonlocal boundary condition this problem is given a weak or variational formulation and the finite element method is then applied it is proved that optimal error estimates hold

a fast method for the solution of the helmholtz equation in this paper we consider the numerical solution of the helmholtz equation arising from the study of the wave equation in the frequency domain the approach proposed here differs from those recently considered in the literature in that it is based on a decomposition that is exact when considered analytically so the only degradation in computational performance is due to discretization and roundoff errors in particular we make use of a multiplicative decomposition of the solution of the helmholtz equation into an analytical plane wave and a multiplier which is the solution of a complex valued advection diffusion reaction equation the use of fast multigrid methods for the solution of this equation is investigated numerical results show that this is an efficient solution algorithm for a reasonable range of frequencies

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

some response characteristics of finite element tidal models abstract small scale error can arise in finite element tidal models from different aspects of the spatial discretization one source is incompatibility in smoothness of interpolation of the pressure and velocity fields noise of this type can be eliminated completely by using stokes helmholtz potentials to represent the velocity and interpolating these potentials with the same smoothness as is given to the pressure another source is inhomogeneous triangulation in which the number of links is not the same at all nodes noise of this type can be suppressed by using a high resolution grid the resolution in question being that of the response rather than of the domain these points are illustrated by examining the response characteristics of idealized models it is found that in the models considered small scale noise can be ascribed to resonance anomalies associated with the method of spatial discretization in this analysis the time is continuous but some remarks are appended on the effect of time discretization and a formal proof is given of the time stepping stability of a general discretized form of laplace s tidal equations

a parallel multigrid based preconditioner for the 3d heterogeneous high frequency helmholtz equation we investigate the parallel performance of an iterative solver for 3d heterogeneous helmholtz problems related to applications in seismic wave propagation for large 3d problems the computation is no longer feasible on a single processor and the memory requirements increase rapidly therefore parallelization of the solver is needed we employ a complex shifted laplace preconditioner combined with the bi cgstab iterative method and use a multigrid method to approximate the inverse of the resulting preconditioning operator a 3d multigrid method with 2d semi coarsening is employed we show numerical results for large problems arising in geophysical applications

preasymptotic error analysis of cip fem and fem for helmholtz equation with high wave number part ii hp version in this paper which is the second in a series of two the preasymptotic error analysis of the continuous interior penalty finite element method cip fem and the fem for the helmholtz equation in two and three dimensions is continued while part i contained results on the linear cip fem and fem the present part deals with approximation spaces of order p ge 1 by using a modified duality argument preasymptotic error estimates are derived for both methods under the condition of frac kh p le c 0 frac p k frac 1 p 1 where k is the wave number h is the mesh size and c 0 is a constant independent of k h p and the penalty parameters it is shown that the pollution errors of both methods in h 1 norm are o k 2p 1 h 2p if p o 1 and are o frac k p 2 frac kh sigma p 2p if the exact solution u in h 2 omega which coincide with existent dispersion analyses for the fem on cartesian grids here sigma is a constant independent of k h p and the penalty par

the numerical solution of the helmholtz equation for wave propagation problems in underwater acoustics abstract the helmholtz equation k2n2 u 0 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 a numerical algorithm has been developed and a computer code implemented that can effectively solve this equation in the intermediate frequency range the equation is discretized by using the finite element method thus allowing for the modeling of complicated geometries including interfaces and complicated boundary conditions a global radiation boundary condition is imposed at the far field boundary that is exact for an arbitrary number of propagating modes the resulting large nonselfadjoint system of linear equations with indefinite symmetric part is solved by using the preconditioned conjugate gradient method applied to the normal equations a new preconditioner based on the multigrid method is developed this preconditioner is vectorizable and is extremely effective over a wide range of frequencies provided the number of grid levels is reduced for large frequencies a heuristic argument is given that indicates the superior convergence properties of this preconditioner the relevant limit to analyze convergence is for k increasing and a fixed prescribed accuracy level the efficiency and robustness of the numerical algorithm are confirmed for large acoustic models including interfaces with strong velocity contrasts

three dimensional biorthogonal multiresolution time domain method and its application to electromagnetic scattering problems a three dimensional 3 d multiresolution time domain mrtd analysis is presented based on a biorthogonal wavelet expansion with application to electromagnetic scattering problems we employ the cohen daubechies feauveau cdf biorthogonal wavelet basis characterized by the maximum number of vanishing moments for a given support we utilize wavelets and scaling functions of compact support yielding update equations involving a small number of proximate field components a detailed analysis is presented on algorithm implementation with example numerical results compared to data computed via the conventional finite difference time domain fdtd method it is demonstrated that for 3 d scattering problems the cdf based mrtd often provides significant computational savings in computer memory and run time relative to fdtd while retaining numerical accuracy

accurate finite difference methods for time harmonic wave propagation finite difference methods for solving problems of time harmonic acoustics are developed and analyzed multi dimensional inhomogeneous problems with variable possibly discontinuous coefficients are considered accounting for the effects of employing non uniform grids a weighted average representation is less sensitive to transition in wave resolution due to variable wave numbers or non uniform grids than the standard pointwise representation further enhancement in method performance is obtained by basing the stencils on generalizations of pade approximation or generalized definitions of the derivative reducing spurious dispersion anisotropy and reflection and by improving the representation of source terms the resulting schemes have fourth order accurate local truncation error on uniform grids and third order in the non uniform case guidelines for discretization pertaining to grid orientation and resolution are presented

local solutions to high frequency 2d scattering problems we consider the solution of high frequency scattering problems in two dimensions modeled by an integral equation on the boundary of a smooth scattering object we devise a numerical method to obtain solutions on only parts of the boundary with little computational effort the method incorporates asymptotic properties of the solution and can therefore attain particularly good results for high frequencies we show that the integral equation in this approach reduces to an ordinary differential equation

algebraic multilevel preconditioner for the helmholtz equation in heterogeneous media an algebraic multilevel ml preconditioner is presented for the helmholtz equation in heterogeneous media it is based on a multilevel incomplete ldl t factorization and preserves the inherent complex symmetry of the helmholtz equation the ml preconditioner incorporates two key components for efficiency and numerical stability symmetric maximum weight matchings and an inverse based pivoting strategy the former increases the block diagonal dominance of the system whereas the latter controls l 1 for numerical stability when applied recursively their combined effect yields an algebraic coarsening strategy similar to algebraic multigrid methods even for highly indefinite matrices the ml preconditioner is combined with a krylov subspace method and applied as a black box solver to a series of challenging two and three dimensional test problems mainly from geophysical seismic imaging the numerical results demonstrate the robustness and efficiency of the ml preconditioner even at higher frequency regimes

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

finite element solution of maxwell s equations for hyperthermia treatment planning abstract a general galerkin formulation is given for the periodic form of maxwell s equations on a bounded inhomogeneous medium the formulation conserves energy exactly and its dispersion relation is studied solutions for test problems representative of hyperthermia applications show good agreement with analytic solutions a representative clinical example is shown

fast alternating bidirectional preconditioner for the 2d high frequency lippmann schwinger equation this paper presents a fast iterative solver for the lippmann schwinger equation for high frequency waves scattered by a smooth medium with a compactly supported inhomogeneity the solver is based on the sparsifying preconditioner l ying multiscale model simul 13 2015 pp 644 660 and a domain decomposition approach similar to the method of polarized traces l zepeda nunez and l demanet j comput phys 308 2016 pp 347 388 the iterative solver has two levels the outer level in which a sparsifying preconditioner for the lippmann schwinger equation is constructed and the inner level in which the resulting sparsified system is solved quickly using an iterative solver preconditioned with a bidirectional matrix free variant of the method of polarized traces the complexity of the construction and application of the preconditioner is cal o n and cal o n log n respectively where n is the number of degrees of freedom numerical experiments in 2 dimensions indicate that t

parallel solution of high frequency helmholtz equations using high order finite difference schemes abstract high frequency helmholtz equations pose a challenging computational task especially at high frequencies this work examines the parallel solution of such problems using 2nd 4th and 6th order finite difference schemes the examples include problems with known analytic solutions enabling error evaluation of the different schemes on various grids including problems on l shaped domains the resulting linear systems are solved with the block parallel carp cg algorithm which succeeded in lowering the relative residual indicating that it is a robust and reliable parallel solver of the resulting linear systems however lowering the error of the solution to reasonable levels with practical mesh sizes is obtained only with the higher order schemes these results corroborate the known limitations of the 2nd order scheme at modeling the helmholtz equation at high frequencies and they indicate that carp cg can also be used effectively with high order finite difference schemes furthermore the parallel scalability of carp cg improves when the wave number is increased on a fixed grid or when for a fixed wave number the grid size is decreased

convergence analysis of a discontinuous galerkin method with plane waves and lagrange multipliers for the solution of helmholtz problems we analyze the convergence of a discontinuous galerkin method dgm with plane waves and lagrange multipliers that was recently proposed by farhat harari and hetmaniuk comput methods appl mech engrg 192 2003 pp 1389 1419 for solving two dimensional helmholtz problems at relatively high wavenumbers we prove that the underlying hybrid variational formulation is well posed we also present various a priori error estimates that establish the convergence and order of accuracy of the simplest 1 5pt element associated with this method we prove that for k k h frac 2 3 sufficiently 1 5pt small the relative error in the l 2 norm resp in the h 1 seminorm is of order k k h frac 4 3 resp of order k h frac 2 3 for a solution being in h frac 5 3 omega in addition we establish an a posteriori error estimate that can be used as a practical error indicator when refining the partition of the computational domain

a study of discretization error in the finite element approximation of wave solutions a dispersion analysis is used to study the errors caused by the spatial discretization of the finite element method for the two dimensional scalar helmholtz equation it is shown that the error can be determined analytically for a uniform mesh of infinite extent numerical results are presented to show the effects of several parameters on the error these parameters are the nodal density the electrical size of the mesh the direction of propagation of the incident wave the type of element and the type of boundary condition

fast solution methods in electromagnetics various methods for efficiently solving electromagnetic problems are presented electromagnetic scattering problems can be roughly classified into surface and volume problems while fast methods are either differential or integral equation based the resultant systems of linear equations are either solved directly or iteratively a review of various differential equation solvers their complexities and memory requirements is given the issues of grid dispersion and hybridization with integral equation solvers are discussed several fast integral equation solvers for surface and volume scatterers are presented these solvers have reduced computational complexities and memory requirements

an analysis of discretizations of the helmholtz equation in l2 and in negative norms for a model helmholtz problem at high wavenumber k we present a wavenumber explicit error analysis in the l 2 and h 1 norms for the galerkin fem for the convergence in l 2 we show that the lowest order case p 1 is special in that the relative error in l 2 scales at best with k whereas it does not for higher order discretizations an alternative to the galerkin method with better dispersion properties is the optimally blended spectral finite element scheme of ainsworth and wajid 2010 for this method we present an error analysis in l 2 for the lowest order case p 1 in one dimension showing that the l 2 error is improved by a factor k compared to the lowest order galerkin fem

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

compressed absorbing boundary conditions via matrix probing absorbing layers are sometimes required to be impractically thick in order to offer an accurate approximation of an absorbing boundary condition for the helmholtz equation in a heterogeneous medium it is always possible to reduce an absorbing layer to an operator at the boundary by layer stripping elimination of the exterior unknowns but the linear algebra involved is costly we propose bypassing the elimination procedure and directly fitting the surface to surface operator in compressed form from a few exterior helmholtz solves with random dirichlet data the result is a concise description of the absorbing boundary condition with a complexity that grows slowly often logarithmically in the frequency parameter

systems of fixed point sets let g be a compact lie group a canonical method is given for constructing a c space from homotopy theoretic information about its fixed point sets the construction is a special case of the categorical bar construction applications include easy constructions of certain classifying spaces as well as c eilenbergmac lane spaces and postnikov towers 0 introduction let g be a compact lie group and x a g space the equivariant homotopy theory of x is reflected to a remarkable extent in its system of fixed point sets defined as a functor from a certain category 0g to top the category of topological spaces our spaces will be compactly generated weak hausdorff they may or may not be equipped with a basepoint depending on the context these functors or systems have considerable technical advantages over g spaces it is easy to apply most homotopy theoretic constructions to them whereas in many cases it is unclear how to proceed for g spaces it is the purpose of this paper to present a canonical way of recovering from any system a g space which preserves all the homotopy theoretic structure of the system this allows us to give easy equivariant versions of some standard topological constructions such as eilenberg mac lane spaces and postnikov towers and to simplify other equivariant constructions 1 1 statements of the main theorems throughout g is a fixed compact lie group definitions the category of canonical orbits written 0g is a topological category with discrete object space 0g g 77 77 a closed subgroup of g and morphisms the g maps topologized by requiring the natural bijection hoxn0cig h g k g k h to be a homeomorphism by an oc space we shall mean a continuous contravariant functor from 0g to top these functors form the objects of a topological category in the usual manner we will also consider gg rings oc groups etc defined similarly definition let a1 be a g space the fixed point set system of x written x is an gg space defined as follows g 77 x received by the editors october 30 1981 and in revised form april 26 1982 1980 mathematics subject classification primary 57s10 55n25

computational aeroacoustics as applied to the diffraction of sound by cylindrical bodies the problem of the diffraction of sound by two dimensional rigid cylindrical objects has been formulated in generalized curvilinear coordinates the governing equations for the aeroacoustic field are derived by perturbing the unsteady euler equations and then extracting the steady mean flow equations from them the governing equations are then solved by an explicit four stage runge kutta time marching finite volume numerical scheme the scheme has been implemented with far field radiation and time accurate wall boundary conditions the cases of diffraction by a knife edge and a cylinder have been computed and compared against their corresponding analytical solutions in order to demonstrate the prediction capability of the numerical scheme the acoustic field resulting from the interaction of the incident reflected and diffracted fields has been predicted very well in both the shadow and the illuminated zones the scheme has also been applied to compute the diffraction of plane waves as well as that of a monopole acoustic field by a ga w 1 airfoil i introduction w hen sound waves impinge upon large rigid obstacles in their path the resultant aeroacoustic field comprises of the incident waves the reflected waves and the diffracted waves the diffraction of sound waves in the shadow zone plays a key role in assessing the acoustic shielding capability of aerodynamic bodies and forms a problem of fundamental importance in the field of aeroacoustics the various fluctuating quantities that describe the aeroacoustic field are governed by the euler equations which are of course nonlinear even if the governing equations are linearized it is extremely difficult to obtain analytical solutions in the cases of the realistic twodimensional obstacles such as airfoils without some unacceptable drastic simplifications the difficulty can be essentially attributed to the complexities associated with the treatment of the boundary conditions these difficulties pertaining to the analytical solutions have been appreciated by many investigators1 4 and have led to the emergence of a new field of computational aeroacoustics caa progress in the methods of caa has now advanced because of the considerable advances in the fields of computational fluid dynamics cfd and computer technology the aim of both cfd and caa is to find solutions to the practical problems numerically but the main differences between the two lie in the nature of the problems one deals with and in the type of the information one wishes to obtain cfd treats steady as well as unsteady problems whereas all the caa problems are by definition unsteady in current cfd problems interests are focused primarily on the flowfield in the immediate vicinity of an aerodynamic body and the objectives are to determine aerodynamic quantities such as pressure distributions skin friction lift and drag etc on the other hand in caa problems attention is directed to both the near field acoustic vibrations and the far field noise radiation away from the body and the objectives are to find the directivity and the spectral characteristics of the acoustic field therefore while the well established techniques of cfd can be utilized in order to develop the

an adaptive multilevel method for time harmonic maxwell equations with singularities we develop an adaptive edge finite element method based on reliable and efficient residual based a posteriori error estimates for low frequency time harmonic maxwell equations with singularities the resulting discrete problem is solved by the multigrid preconditioned minres minimum residual iteration algorithm we demonstrate the efficiency and robustness of the proposed method by extensive numerical experiments for cavity problems with singular solutions which include in particular scattering over screens