a finite axiomatization of conditional independence and inclusion dependencies we present a complete finite axiomatization of the unrestricted implication problem for inclusion and conditional independence atoms in the context of dependence logic for databases this result implies a finite axiomatization of the unrestricted implication problem for inclusion functional and embedded multivalued dependencies in the unirelational case we also indicate the generality of our approach by showing the analogous result for inclusion and embedded join dependencies

application of the perfectly matched layer pml absorbing boundary condition to elastic wave propagation a method is presented for application of the perfectly matched layer pml absorbing boundary condition abc to the p sv velocity stress finite difference method the pml consists of a nonphysical material containing both passive loss and dependent sources that provides active absorption of fields it has been used in electromagnetic applications where it has provided excellent results for a wide range of angles and frequencies in this work numerical simulations are used to compare the pml and an optimal second order elastic abc peng and toksoz j acoust soc am 95 733 745 1994 reflection factors are used to compare angular performance for continuous wave illumination snapshots of potentials are used to compare performance for broadband illumination these comparisons clearly demonstrate the superiority of the pml formulation within the pml there is a 60 increase in the number of unknowns per grid cell relative to the velocity stress formulation however the high quality of the pml abc allows the use of a smaller grid which can result in a lower overall computational cost

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

variational and projection methods for the volume constraint in finite deformation elasto plasticity this paper focuses on the treatment of volume constraints which in the context of elasto plasticity typically arise as a result of assuming volume preserving plastic flow projection methods based on the modification of the discrete gradient operator b often proposed on an ad hoc basis are systematically obtained in the variational context furnished by a three field hu washizu principle the fully nonlinear formulation proposed here is based on a local multiplicative split of the deformation gradient into volume preserving and dilatational parts without relying on rate forms of the weak form of momentum balance this approach fits naturally in a formulation of plasticity based on the multiplicative decomposition of the deformation gradient and enables one to exactly enforce the condition of volume preserving plastic flow within the framework proposed in this paper rate forms and incrementally objective algorithms are entirely bypassed

the implication problem for functional and inclusion dependencies is undecidable the implication problem for a class of dependencies is the following given a finite set of dependencies determine if they logically imply another dependency we show that the implication problem is undecidable for the class of functional and inclusion dependencies this holds true even if the inclusion dependencies are restricted to be binary it may be noted that the implication problem is known to be decidable for functional and unary inclusion dependencies and also for inclusion dependencies without functional dependencies

an optimal 9 point finite difference frequency space 2 d scalar wave extrapolator in this study a new finite difference technique is designed to reduce the number of grid points needed in frequency space domain modeling the new algorithm uses optimal nine point operators for the approximation of the laplacian and the mass acceleration terms the coefficients can be found by using the steepest descent method so that the best normalized phase curves can be obtained this method reduces the number of grid points per wavelength to 4 or less with consequent reductions of computer memory and cpu time that are factors of tens less than those involved in the conventional second order approximation formula when a band type solver is used on a scalar machine

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

the inverse medium problem in heterogeneous pml truncated domains using scalar probing waves abstract we discuss the inverse medium problem associated with semi infinite domains in particular we attempt to image the spatial variability of shear moduli or shear wave velocities from scant surficial measurements of an arbitrarily heterogeneous semi infinite domain s response to prescribed dynamic excitations we use a full waveform approach to drive the inversion process within a pde constrained optimization framework due to the semi infinite extent of the targeted domains we introduce perfectly matched layers pmls to arrive at finite computational domains the numerical implementation is based on a mixed finite element method that is used to resolve the ensuing state and adjoint boundary value problems both of which are pml endowed to alleviate the inherent solution multiplicity we use tikhonov and total variation tv regularization schemes in conjunction with a regularization factor continuation scheme to further improve the optimizer s chances to converge we also discuss a source frequency continuation scheme we report on two dimensional numerical experiments using synthetic data included are layered profiles and profiles involving inclined layers and inclusions we also report on our methodology s reconstruction of the highly heterogeneous marmousi benchmark velocity model

discontinuous galerkin methods for the helmholtz equation with large wave number this paper develops and analyzes some interior penalty discontinuous galerkin ipdg methods using piecewise linear polynomials for the helmholtz equation with the first order absorbing boundary condition in two and three dimensions it is proved that the proposed discontinuous galerkin methods are stable hence well posed without any mesh constraint for each fixed wave number k optimal order with respect to h error estimate in the broken h 1 norm and suboptimal order estimate in the l 2 norm are derived without any mesh constraint the latter estimate improves to optimal order when the mesh size h is restricted to the preasymptotic regime i e k 2 h gtrsim 1 numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect with respect to k in the error bounds the novelties of the proposed ipdg methods include the following first the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives second the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters since the helmholtz problem is a non hermitian and indefinite linear problem as expected the crucial and the most difficult part of the whole analysis is to establish the stability estimates i e a priori estimates for the numerical solutions to this end the cruxes of our analysis are to establish and to make use of a local version of the rellich identity for the laplacian and to mimic the stability analysis for the pde solutions given in p cummings analysis of finite element based numerical methods for acoustic waves elastic waves and fluid solid interactions in the frequency domain ph d thesis the university of tennessee knoxville tn 2001 p cummings and x feng math models methods appl sci 16 2006 pp 139 160 u hetmaniuk commun math sci 5 2007 pp 665 678

a frequency space 2 d scalar wave extrapolator using extended 25 point finite difference operator finite difference frequency domain modeling for the generation of synthetic seismograms and crosshole tomography has been an active field of research since the 1980s the generation of synthetic seismograms with the time domain finite difference technique has achieved considerable success for waveform crosshole tomography and for wider applications in seismic reverse time migration this became possible with the rapid development of high performance computers however the space frequency x finite difference modeling technique is still beyond the capability of the modern supercomputer in terms of both cost and computer memory therefore finite difference time domain modeling is much more popular among exploration geophysicists a limitation of the space frequency domain is that the recently developed nine point scheme still requires that g the number of grid points per wavelength be 5 this value is greater than for most other numerical modeling techniques for example the pseudospectral scheme

polynomial time implication problems for unary inclusion dependencies unary inclusion dependencies are database constraints expressing subset relationships the decidability of implication for these dependencies together with embedded implicational dependencies such as functional dependencies are investigated as shown by casanova et al the unrestricted and finite implication problems are different for the class of functional and unary inclusion dependencies also for this class and for any fixed k finite implication has no k ary complete axiomatization for both of these problems complete axiomatizations and polynomial time decision procedures are provided linear time for unrestricted implication and cubic time for finite implication it follows that functional and unary inclusion dependencies form a semantically natural class of first order sentences with equality which although not finitely controllable is efficiently solvable and docile generalizing from these results it is shown that the interaction between functionaland inclusion dependencies characterizes 1 unrestricted implication of unary inclusion and all embedded implicational dependencies 2 finite implication of unary inclusion and all full implicational dependencies 3 finite implication of unary inclusion and all embedded tuple generating dependencies as a direct consequence of this analysis most of the applications of dependency implication are extended within polynomial time to database design problems involving unary inclusion dependencies such examples are tests for lossless joins and tests for complementarity of projective views finally if one additionally requires that

a multigrid based preconditioned krylov subspace method for the helmholtz equation with pml abstract in this paper we generalize the complex shifted laplacian preconditioner to the complex shifted laplacian pml preconditioner for the helmholtz equation with perfectly matched layer helmholtz pml equation the helmholtz pml equation is discretized by an optimal 9 point difference scheme and the preconditioned linear system is solved by the krylov subspace method especially by the biconjugate gradient stabilized method bi cgstab the spectral analysis of the linear system is given and a new matrix based interpolation operator is proposed for the multigrid method which is used to approximately invert the preconditioner the numerical experiments are presented to illustrate the efficiency of the preconditioned bi cgstab method with the multigrid based on the new interpolation operator also numerical results are given for comparing the performance of the new interpolation operator with that of classic bilinear interpolation operator and the one suggested in erlangga et al 2006 10

spectral analysis of the discrete helmholtz operator preconditioned with a shifted laplacian shifted laplace preconditioners have attracted considerable attention as a technique to speed up convergence of iterative solution methods for the helmholtz equation in this paper we present a comprehensive spectral analysis of the helmholtz operator preconditioned with a shifted laplacian our analysis is valid under general conditions the propagating medium can be heterogeneous and the analysis also holds for different types of damping including a radiation condition for the boundary of the computational domain by combining the results of the spectral analysis of the preconditioned helmholtz operator with an upper bound on the gmres residual norm we are able to provide an optimal value for the shift and to explain the mesh dependency of the convergence of gmres preconditioned with a shifted laplacian we illustrate our results with a seismic test problem

numerical implementation of a shape memory alloy thermomechanical constitutive model using return mapping algorithms previous studies by the authors and their co workers show that the structure of equations representing shape memory alloy sma constitutive behaviour can be very similar to those of rate independent plasticity models for example the boyd lagoudas polynomial hardening model has a stress elastic strain constitutive relation that includes the transformation strain as an internal state variable a transformation function determining the onset of phase transformation and an evolution equation for the transformation strain such a structure allows techniques used in rate independent elastoplastic behaviour to be directly applicable to smas in this paper a comprehensive study on the numerical implementation of sma thermomechanical constitutive response using return mapping elastic predictor transformation corrector algorithms is presented the closest point projection return mapping algorithm which is an implicit scheme is given special attention together with the convex cutting plane return mapping algorithm an explicit scheme already presented in an earlier work the closest point algorithm involves relatively large number of tensorial operations than the cutting plane algorithm besides the evaluation of the gradient of the transformation tensor in the flow rule and the inversion of the algorithmic tangent tensor a unified thermomechanical constitutive model which does not take into account reorientation of martensitic variants but unifies several of the existing sma constitutive models is used for implementation remarks on numerical accuracy of both algorithms are given and it is concluded that both algorithms are applicable for this class of sma constitutive models and preference can only be given based on the computational cost copyright 2000 john wiley sons ltd

compact difference scheme for parabolic and schrodinger type equations with variable coefficients abstract we develop a new compact scheme for the second order pde parabolic and schrodinger type with a variable time independent coefficient it has a higher order and smaller error than classic implicit scheme the dirichlet and neumann boundary problems are considered the relative finite difference operator is almost self adjoint

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

formal systems for tuple and equality generating dependencies we develop several formal systems for tuple and equality generating dependencies there are three kinds of systems based upon substitution tuple elimination and transitivity we specialize our systems to several subclasses total dependencies template dependencies and binary dependencies we also show that finding a formal system for embedded multivalued dependencies is equivalent to solving the implication problem for that class

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

a return mapping algorithm for plane stress elastoplasticity an unconditionally stable algorithm for plane stress elastoplasticity is developed based upon the notion of elastic predictor return mapping plastic corrector enforcement of the consistency condition is shown to reduce to the solution of a simple nonlinear equation consistent elastoplastic tangent moduli are obtained by exact linearization of the algorithm use of these moduli is essential in order to preserve the asymptotic rate of quadratic convergence of newton methods the accuracy of the algorithm is assessed by means of iso error maps the excellent performance of the algorithm for large time steps is illustrated in numerical experiments

solving helmholtz equation at high wave numbers in exterior domains this paper presents effective difference schemes for solving the helmholtz equation at high wave numbers considering the problems in the polar and spherical coordinates we show that pollution free difference schemes can be constructed for annulus and hollow sphere domains but the pollution effect cannot be avoided in the neighborhood of the origin of the coordinate using the fact that the pollution effect appears in a small neighborhood of the origin of the coordinate we propose a local mesh refinement approach so that a fine grid is applied only for a small region near the origin of the coordinate and a much larger grid size is employed in the remaining domain the most attractive feature of this approach is that the computational storage can be significantly reduced while keeping almost the same numerical accuracy moreover by applying a two level local refinement technique we can further enhance the performance of the proposed scheme numerical simulations are reported to verify the effectiveness of the proposed schemes for the helmholtz problem in exterior domains

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 fourth order optimal finite difference scheme for the helmholtz equation with pml abstract in this paper 17 point and 25 point finite difference fd schemes for the helmholtz equation with perfectly matched layer pml in the two dimensional domain are presented it is shown that the 17 point fd scheme is inconsistent in the presence of pml however the 25 point fd scheme is pointwise consistent an error analysis for the numerical approximation of the exact wavenumber is also presented we present the global and refined 25 point finite difference schemes based on minimizing the numerical dispersion numerical experiments are given to illustrate the improvement of the accuracy and the reduction of the numerical dispersion

an optimal 25 point finite difference scheme for the helmholtz equation with pml in this paper we present an optimal 25 point finite difference scheme for solving the helmholtz equation with perfectly matched layer pml in two dimensional domain based on minimizing the numerical dispersion we propose the refined choice strategy for choosing optimal parameters of the 25 point finite difference scheme numerical experiments are given to illustrate the improvement of the accuracy and the reduction of the numerical dispersion

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

a dispersion minimizing finite difference scheme and preconditioned solver for the 3d helmholtz equation in this paper a new 27 point finite difference method is presented for solving the 3d helmholtz equation with perfectly matched layer pml which is a second order scheme and pointwise consistent with the equation an error analysis is made between the numerical wavenumber and the exact wavenumber and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters a full coarsening multigrid based preconditioned bi cgstab method is developed for solving the linear system stemming from the helmholtz equation with pml by the finite difference scheme the shifted laplacian is extended to precondition the 3d helmholtz equation and a spectral analysis is given the discrete preconditioned system is solved by the bi cgstab method with a multigrid method used to invert the preconditioner approximately full coarsening multigrid is employed and a new matrix based prolongation operator is constructed accordingly numerical results are presented to demonstrate the efficiency of both the new 27 point finite difference scheme with refined parameters and the preconditioned bi cgstab method with the 3d full coarsening multigrid

high order finite difference methods for the helmholtz equation high order finite difference methods for solving the helmholtz equation are developed and analyzed in one and two dimensions on uniform grids the standard pointwise representation has a second order accurate local truncation error we also study two schemes which have a fourth order accurate local truncation error one of the high order schemes is based on generalizations of the pade approximation the second scheme is based on high order approximation to the derivative calculated from the helmholtz equation itself a symmetric high order representation is developed for a neumann boundary condition numerical results are presented on model problems approximated with the developed schemes

group velocity in finite difference schemes the relevance of group velocity to the behavior of finite difference models of time dependent partial differential equations is surveyed and illustrated applications involve the propagation of wave packets in one and two dimensions numerical dispersion the behavior of parasitic waves and the stability analysis of initial boundary value problems

a posteriori estimation and adaptive control of the pollution error in the h version of the finite element method abstract we studied the pollution error in the h version of the finite element method and its effect on the quality of the local error indicators resp the quality of the derivatives recovered by local postprocessing in the interior of the mesh here we show that it is possible to construct a posteriori estimates of the pollution error in a patch of elements by employing the local error indicators over the entire mesh we also give an adaptive algorithm for the local control of the pollution error in a patch of elements of interest

guaranteeing no interaction between functional dependencies and tree like inclusion dependencies functional dependencies fds and inclusion dependencies inds are the most fundamental integrity constraints that arise in practice in relational databases a given set of fds does not interact with a given set of inds if logical implication of any fd can be determined solely by the given set of fds and logical implication of any ind can be determined solely by the given set of inds the set of tree like inds constitutes a useful subclass of inds whose implication problem is polynomial time decidable we exhibit a necessary and sufficient condition for a set of fds and tree like inds not to interact this condition can be tested in polynomial time

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

on the decidability and axiomatization of query finiteness in deductive databases a database query is finite if its result consists of a finite sets tuples for queries formulated as sets of pure horn rules the problem of determining finiteness is in general undecidable in this paper we consider superfiniteness a stronger kind of finiteness which applies to horn queries whose function symbols are replaced by the abstraction of infinite relations with finiteness constraints abbr fc s we show that superfiniteness is not only decidable but also axiomatizable and the axiomatization yields an effective decision procedure although there are finite queries that are not superfinite we demonstrate that superfinite queries represent an interesting and nontrivial subclass within the class of all finite queries the we turn to the issue of inference of finiteness constraints an important practical problem that is instrumental in deciding if a query is evaluable by a bottom up algorithm although it is not known whether fc entailment is decidable for sets of function free horn rules we show that super entailment a stronger form of entailment is decidable we also show how a decision procedure for super entailment can be used to enhance tests for query finiteness

a proof procedure for data dependencies definition d une classe de dependances dependances generant l equalite et les triples et generalisation du processus de poursuite pour traiter ces dependances on montre quelques cas pour lesquels il y a une procedure de decision redondance de l egalite pour decider de l implication des dependances generant les triples et presque redondance pour decider de l implication des dependances generant l egalite

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

on the undecidability of implications between embedded multivalued database dependencies the implication and finite implication problems for embedded multivalued database dependencies are both shown to be algorithmically undecidable the proof is by an interpretation of semigroup word problems via systems of permuting equivalence relations into database dependencies in contrast it is shown that for each fixed premise h one has a decision procedure for implications h f

the implication problem for functional and inclusion dependencies there are two implication problems for functional dependencies and inclusion dependencies general implication and finite implication given a set of dependencies the problems are to determine whether holds in all databases satisfying or all finite databases satisfying contrary to the possibility suggested in casanova fagin and papadimitriou proceedings 1st acm conf on principles of database system pp 171 176 1982 there is a natural complete axiom system for general implication however a simple observation shows that both implication problems are recursively unsolvable it follows that there is no recursively enumerable set of axioms for finite implication

on the computational significance of the strain space formulation of plasticity theory the strain space and traditional stress space formulations of plasticity theory are compared we show how plasticity theory has been implemented numerically to solve static and dynamic problems and conclude that the numerical loading criteria that have been used for two decades are identical to the strain space loading criteria proposed by naghdi and co workers

a dispersion minimizing subgridding finite difference scheme for the helmholtz equation with pml in this paper we present a dispersion minimizing subgridding finite difference scheme for solving the helmholtz equation with perfectly matched layer pml in the two dimensional domain which is a second order scheme and pointwise consistent with the equation subgrids are used to discretize the computational domain and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters for transitional nodes numerical experiments are given to illustrate that the newly proposed schemes can produce highly accurate seismic modeling results with enhanced efficiency compared to uniform grids

axioms and algorithms for inferences involving probabilistic independence abstract this paper offers an axiomatic characterization of the probabilistic relation x is independent of y written x y where x and y are two disjoint sets of variables four axioms for x y are presented and shown to be complete based on these axioms a polynomial membership algorithm is developed to decide whether any given independence statement x y logically follows from a set of such statements i e whether x y holds in every probability distribution that satisfies the complexity of the algorithm is o k2 n where is the number of given statements n is the number of variables in x y and k is the number of variables in x y

dispersion and pollution of the fem solution for the helmholtz equation in one two and three dimensions for high wave numbers the helmholtz equation suffers the so called pollution effect this effect is directly related to the dispersion a method to measure the dispersion on any numerical method related to the classical galerkin fem is presented this method does not require to compute the numerical solution of the problem and is extremely fast numerical results on the classical galerkin fem p method is compared to modified methods presented in the literature a study of the influence of the topology triangles is also carried out the efficiency of the different methods is compared the numerical results in two of the mesh and for square elements show that the high order elements control the dispersion well the most effective modified method is the qsfem 1 2 but it is also very complicated in the general setting the residual free bubble 3 4 is effective in one dimension but not in higher dimensions the least square method 1 5 approach lowers the dispersion but relatively little the results for triangular meshes show that the best topology is the criss cross pattern copyright 1999 john wiley sons ltd

a dispersion minimizing finite difference scheme for the helmholtz equation based on point weighting abstract in this paper we develop a new dispersion minimizing finite difference scheme for the helmholtz equation with perfectly matched layer pml in two dimensional domain which is a second order 9 point scheme to discretize the second derivative operator we employ a linear combination of a point and its neighboring grid points to replace each of the five points in the traditional central difference scheme based on minimizing the numerical dispersion the combination weights are determined by minimizing the numerical dispersion with a flexible selection strategy the new scheme is simple rotation free and pointwise consistent with the equation which is different from the classical rotated 9 point difference scheme obtained by combining the cartesian coordinate system and the rotated system moreover it is a robust scheme even if the step sizes of different directions are not equal convergence analysis and dispersion analysis are given several numerical examples are presented to illustrate the numerical convergence and effectiveness of the new scheme