some notes on the paper cone metric spaces and fixed point theorems of contractive mappings abstract huang and zhang reviewed cone metric spaces in 2007 huang long guang zhang xian cone metric spaces and fixed point theorems of contractive mappings j math anal appl 332 2007 1468 1476 we shall prove that there are no normal cones with normal constant m 1 and for each k 1 there are cones with normal constant m k also by providing non normal cones and omitting the assumption of normality in some results of huang long guang zhang xian cone metric spaces and fixed point theorems of contractive mappings j math anal appl 332 2007 1468 1476 we obtain generalizations of the results

the adaptive computation of far field patterns by a posteriori error estimation of linear functionals this paper is concerned with the derivation of a priori and a posteriori error bounds for a class of linear functionals arising in electromagnetics which represent the far field pattern of the scattered electromagnetic field the a posteriori error bound is implemented into an adaptive finite element algorithm and a series of numerical experiments is presented

a partition of unity finite element method for three dimensional transient diffusion problems with sharp gradients abstract an efficient partition of unity finite element method for three dimensional transient diffusion problems is presented a class of multiple exponential functions independent of time variable is proposed to enrich the finite element approximations as a consequence of this procedure the associated matrix for the linear system is evaluated once at the first time step and the solution is obtained at subsequent time step by only updating the right hand side of the linear system this results in an efficient numerical solver for transient diffusion equations in three space dimensions compared to the conventional finite element methods with h refinement the proposed approach is simple more efficient and more accurate the performance of the proposed method is assessed using several test examples for transient diffusion in three space dimensions we present numerical results for a transient diffusion equation with known analytical solution to quantify errors for the new method we also solve time dependent diffusion problems in complex geometries we compare the results obtained using the partition of unity finite element method to those obtained using the standard finite element method it is shown that the proposed method strongly reduces the necessary number of degrees of freedom to achieve a prescribed accuracy

the discontinuous enrichment method for medium frequency helmholtz problems with a spatially variable wavenumber numerical dispersion or what is often referred to as the pollution effect presents a challenge to an efficient finite element discretization of the helmholtz equation in the medium frequency regime to alleviate this effect and improve the unsatisfactory pre asymptotic convergence of the classical galerkin finite element method based on piecewise polynomial basis functions several discretization methods based on plane wave bases have been proposed among them is the discontinuous enrichment method that has been shown to offer superior performance to the classical galerkin finite element method for a number of constant wavenumber helmholtz problems and has also outperformed two representative methods that use plane waves the partition of unity and the ultra weak variation formulation methods in this paper the discontinuous enrichment method is extended to the variable wavenumber helmholtz equation to this effect the concept of enrichment functions based on free space solutions of the homogeneous form of the governing differential equation is enlarged to include free space solutions of approximations of this equation obtained in this case by successive taylor series expansions of the wavenumber around a reference point this leads to plane wave enrichment functions based on the piece wise constant approximation of the wavenumber and to airy wave enrichment functions several elements based on these enrichment functions are constructed and evaluated on benchmark problems modeling sound hard scattering by a disk submerged in an acoustic fluid where the speed of sound varies in space all these elements are shown to outperform by a substantial margin their continuous polynomial counterparts

residual based adaptivity and pwdg methods for the helmholtz equation we present a study of two residual a posteriori error indicators for the plane wave discontinuous galerkin pwdg method for the helmholtz equation in particular we study the h version of pwdg in which the number of plane wave directions per element is kept fixed first we use a slight modification of the appropriate a priori analysis to determine a residual indicator numerical tests show that this is reliable but pessimistic in that the ratio between the true error and the indicator increases as the mesh is refined we therefore introduce a new analysis based on the observation that sufficiently many plane waves can approximate piecewise linear functions as the mesh is refined numerical results demonstrate an improvement in the efficiency of the indicators

kkm mappings in cone metric spaces and some fixed point theorems abstract in this paper we define kkm mappings in cone metric spaces and define n cone metric spaces to obtain some fixed point theorems and hence generalize the results obtained in a amini m fakhar j zafarani kkm mapping in metric spaces nonlinear anal 60 2005 1045 1052

a comparison of high order polynomial and wave based methods for helmholtz problems the application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation two common strategies to address this issue are to use high order polynomial shape functions e g hp fem or to use physics based or trefftz methods where the shape functions are local solutions of the problem typically plane waves both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite element methods but they have yet to be compared in this paper a high order polynomial method p fem with lobatto polynomials and the wave based discontinuous galerkin method are compared for two dimensional helmholtz problems a number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties performance and conditioning it is generally assumed that a wave based method naturally provides better accuracy compared to polynomial methods since the plane waves or bessel functions used in these methods are exact solutions of the helmholtz equation results indicate that this expectation does not necessarily translate into a clear benefit and that the differences in performance accuracy and conditioning are more nuanced than generally assumed the high order polynomial method can in fact deliver comparable and in some cases superior performance compared to the wave based dgm in addition to benchmarking the intrinsic computational performance of these methods a number of practical issues associated with realistic applications are also discussed

contractive type non self mappings on metric spaces of hyperbolic type abstract let x d be a metric space of hyperbolic type and k a nonempty closed subset of x in this paper we study a class of mappings from k into x not necessarily self mappings on k which are defined by the contractive condition 2 1 below and a class of pairs of mappings from k into x which satisfy the condition 2 28 below we present fixed point and common fixed point theorems which are generalizations of the corresponding fixed point theorems of ciric l b ciric quasi contraction non self mappings on banach spaces bull acad serbe sci arts 23 1998 25 31 l b ciric j s ume m s khan h k t pathak on some non self mappings math nachr 251 2003 28 33 rhoades b e rhoades a fixed point theorem for some non self mappings math japon 23 1978 457 459 and many other authors some examples are presented to show that our results are genuine generalizations of known results from this area

a common fixed point theorem for compatible quasi contractive self mappings in metric spaces a common fixed point theorem for weakly commuting quasi contractive self mappings with contracting orbital diameters in metric spaces v berinde a common fixed point theorem for quasi contractive type mappings ann univ sci budapest 46 2003 81 90 is extended to the more general class of compatible quasi contractive self mappings our result does extend and generalize numerous related results in literature

some common fixed point theorems for a family of mappings in metrically convex spaces abstract in the present paper some common fixed point theorems for a sequence and a pair of nonself mappings in complete metrically convex metric spaces are proved which generalize such results due to khan et al m s khan h k pathak m d khan some fixed point theorems in metrically convex spaces georgian math j 7 3 2000 523 530 assad n a assad on a fixed point theorem of kannan in banach spaces tamkang j math 7 1976 91 94 chatterjea s k chatterjea fixed point theorems c r acad bulgare sci 25 1972 727 730 and several others some related results are also discussed

an exponentially convergent nonpolynomial finite element method for time harmonic scattering from polygons in recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems as with their close cousin the method of particular solutions high efficiency comes from using solutions to the helmholtz equation as basis functions we present and analyze such a method for the scattering of two dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions the solution is obtained by minimizing a least squares functional which we discretize in such a way that a matrix least squares problem is obtained we give computable exponential bounds on the rate of convergence of the least squares functional that are in very good agreement with the observed numerical convergence challenging numerical examples including a nonconvex polygon with several corner singularities and a cavity domain are solved to around 10 digits of accuracy with a few seconds of cpu time the examples are implemented concisely with mpspack a matlab toolbox for wave computations with nonpolynomial basis functions developed by the authors a code example is included

cone metric spaces and fixed point theorems of contractive mappings in this paper we introduce cone metric spaces prove some fixed point theorems of contractive mappings on cone metric spaces

the ultra weak variational formulation for elastic wave problems the ultra weak variational formulation has been used effectively to solve time harmonic acoustic and electromagnetic wave propagation in inhomogeneous media we develop the ultra weak variational formulation for elastic wave propagation in two space dimensions in order to improve the accuracy and stability of the method we find it necessary to approximate the s and p wave components of the solution in a balanced way some preliminary analysis is provided and numerical evidence is presented for the efficiency of the scheme in comparison to piecewise linear finite elements

discontinuous galerkin methods with plane waves for time harmonic problems a general framework for discontinuous galerkin methods in the frequency domain with numerical flux is presented the main feature of the method is the use of plane waves instead of polynomials to approximate the solution in each element the method is formulated for a general system of linear hyperbolic equations and is applied to problems of aeroacoustic propagation by solving the two dimensional linearized euler equations it is found that the method requires only a small number of elements per wavelength to obtain accurate solutions and that it is more efficient than high order drp schemes in addition the conditioning of the method is found to be high but not critical in practice it is shown that the ultra weak variational formulation is in fact a subset of the present discontinuous galerkin method a special extension of the method is devised in order to deal with singular solutions generated by point sources like monopoles or dipoles aeroacoustic problems with non uniform flows are also considered and results are presented for the sound radiated from a two dimensional jet

remarks on some integrals and series involving the stirling numbers and from the perspective of the well known identity 2f a b c 1 rc cc a b we clarify the connections between the stirling numbers sk and the riemann zeta function c n as a consequence certain series and integrals can be evaluated in terms of c n and sn

approximation of point of coincidence and common fixed points of quasi contraction mappings using the jungck iteration scheme let x d be a cone metric space over a solid vector space y in this paper we prove a convergence theorem with error estimates and localization formula for jungck iteration process for approximating points of coincidence and common fixed points of two selfmappings t and f of x satisfying a quasi contraction condition of the type d t x t y co d f x f y d f x t x d f y t y d f x t y d f y t x for all x y x where 0 1 is a constant our result complements the recent result of ding et al 9 the main result is new even in the classical metric space setting moreover this result answers an open question posed by olaleru 17 concerning common fixed points in metric spaces

a discontinuous enrichment method for three dimensional multiscale harmonic wave propagation problems in multi fluid and fluid solid media an evanescent wave occurs when a propagating incident wave impinges on an interface between two fluid solid or fluid solid media at a subcritical angle mathematical properties of such a wave make it difficult to capture with standard finite element discretization schemes for this reason the discontinuous enrichment method dem developed in comput methods appl mech eng 2001 190 6455 6479 comput methods appl mech eng 2003 192 1389 1419 comput methods appl mech eng 2003 192 3195 3210 int j numer meth engng 2004 61 1938 1956 wave motion 2004 39 4 307 317 int j numer meth engng 2006 66 2086 2114 int j numer meth engng 2006 66 796 815 is extended here to the solution of a class of three dimensional evanescent wave problems in the frequency domain to this effect new dem elements for three dimensional elastodynamic problems are first proposed then these and other dem elements previously developed for the efficient solution of the helmholtz problem are further enriched with free space solutions of model evanescent wave problems in order to achieve high accuracy at practical mesh resolution for fluid fluid and fluid solid applications the performance of the extended dem elements is reported to be better than that of its basic helmholtz and navier counterparts and superior to that achieved by the classical high order polynomial finite element method copyright 2008 john wiley sons ltd

a least squares method for the helmholtz equation abstract we investigate the use of least squares methods to approximate the helmholtz equation the basis used in the discrete method consists of solutions of the helmholtz equation either consisting of plane waves or bessel functions on each element of a finite element grid unlike previous methods of this type we do not use polynomial based finite elements the use of small elements and relatively few basis functions per element allows us to prove convergence theorems for the method and to some extent control the conditioning of the resulting linear sy stem numerical results show the efficiency of the new method and suggest that it may be possible to obtain accurate results with a coarser grid than is usual for standard finite element methods

common coupled fixed point theorems in cone metric spaces for w compatible mappings in this paper we introduce the concept of a w compatible mappings to obtain coupled coincidence point and coupled point of coincidence for nonlinear contractive mappings in cone metric space with a cone having non empty interior coupled common fixed point theorems for such mappings are also proved our results generalize extend and unify several well known comparable results in the literature results are supported by three examples

implementation of an interior point source in the ultra weak variational formulation through source extraction the ultra weak variational formulation uwvf is a powerful numerical method for the approximation of acoustic elastic and electromagnetic waves in the time harmonic regime the use of trefftz type basis functions incorporates the known wave like behaviour of the solution in the discrete space allowing large reductions in the required number of degrees of freedom for a given accuracy when compared to standard finite element methods however the uwvf is not well disposed to the accurate approximation of singular sources in the interior of the computational domain we propose an adjustment to the uwvf for seismic imaging applications which we call the source extraction uwvf differing fields are solved for in subdomains around the source and matched on the inter domain boundaries numerical results are presented for a domain of constant wavenumber and for a domain of varying sound speed in a model used for seismic imaging

common fixed point theorems on generalized distance in ordered cone metric spaces in this paper we introduce a concept of the c distance in a cone metric space and by using the concept of the c distance prove some fixed point theorems in ordered cone metric spaces

plane wave decomposition in the unit disc convergence estimates and computational aspects this paper deals with the numerical simulation of time harmonic wave fields using progressive plane waves it is shown that a plane wave travelling in arbitrary direction can be numerically recovered with an accuracy of the order of the machine precision with a collocation formulation and the square root of the machine precision with a least square formulation however strongly evanescent and nearly singular wave fields cannot be properly recovered with standard double precision floating point arithmetic some of the ideas are applied to the elastic wave equation and a simple optimization algorithm is proposed to find a good compromise between the accuracy and the number of plane waves

remarks on a paper by f browder about contraction throughout this note x denotes a complete metric space with distance function d and t is a map from x into itself powers of t are defined by tax x and t lx t t x n 2 0 occasionally we use the notation xk tkx in particular x0 x x1 tx for the sake of brevity the set x tx t x t x is called an orbit starting at x and denoted by o x the set 0 x y is the union of the two orbits starting at x and y 0 x y xk yk k 3 o the diameter diam a of a nonempty subset a of x is the supremum of the numbers d x y where x and y vary in a the letter 4 denotes a contractive gauge function i e a continuous increasing function from r the non negative reals into r which satisfies 4 s 0

new matrix bounds and iterative algorithms for the discrete coupled algebraic riccati equation abstractthe discrete coupled algebraic riccati equation dcare has wide applications in control theory and linear system in general for the dcare one discusses every term of the coupled term respectively in this paper we consider the coupled term as a whole which is different from the recent results when applying eigenvalue inequalities to discuss the coupled term our method has less error in terms of the properties of special matrices and eigenvalue inequalities we propose several upper and lower matrix bounds for the solution of dcare further we discuss the iterative algorithms for the solution of the dcare in the fixed point iterative algorithms the scope of lipschitz factor is wider than the recent results finally we offer corresponding numerical examples to illustrate the effectiveness of the derived results

on one dimensional digamma and polygamma series related to the evaluation of feynman diagrams we consider summations over digamma and polygamma functions often with summands of the form 1 n j n p q n r and 1 n j m n p q n r where m p q and r are positive integers we develop novel general integral representations and present explicit examples special cases of the sums reduce to known linear euler sums the sums of interest find application in quantum field theory including evaluation of feynman amplitudes

common fixed points for self mappings on compact metric spaces in this paper we obtain a result of existence of points of coincidence and of common fixed points for two self mappings on compact metric spaces satisfying a contractive condition of suzuki type we also present some examples to illustrate our results moreover using the scalarization method of du we deduce a result of common fixed point in compact cone metric spaces

on stirling numbers and euler sums abstract in this paper we propose another yet generalization of stirling numbers of the first kind for noninteger values of their arguments we discuss the analytic representations of stirling numbers through harmonic numbers the generalized hypergeometric function and the logarithmic beta integral we present then infinite series involving stirling numbers and demonstrate how they are related to euler sums finally we derive the closed form for the multiple zeta function p 1 1 for re p 1

application of an ultra weak variational formulation of elliptic pdes to the two dimensional helmholtz problem a new technique to solve elliptic linear pdes called ultra weak variational formulation uwvf in this paper is introduced in b despres c r acad sci paris 318 1994 pp 939 944 this paper is devoted to an evaluation of the potentialities of this technique it is applied to a model wave problem the two dimensional helmholtz problem the new method is presented in three parts following the same style of presentation as the classical one of the finite elements method even though they are definitely conceptually different methods the first part is committed to the variational formulation and to the continuous problem the second part defines the discretization process using a galerkin procedure the third part actually studies the efficiency of the technique from the order of convergence point of view this is achieved using theoretical proofs and a series of numerical experiments in particular it is proven and shown the order of convergence is lower bounded by a linear function of the number of degrees of freedom an application to scattering problems is presented in a fourth part

values of the polygamma functions at rational arguments gauss in 1812 in his celebrated memoir on the hypergeometric series presented a remarkable formula for the psi or digamma function z at rational arguments z which can be expressed in terms of elementary functions davis in 1935 extended gauss s result to the polygamma functions by using a known series representation of n z in an elementary yet technical way k lbig in 1996 in his cern technical report also gave two extensions to n z by using the series definition of polylogarithm function and the above known series representation here we aim at deriving general formulae expressing n z as rational arguments in terms of other functions which will be obtained in two ways in addition several special cases are also considered and as a by product of our main results we derive in a simple and unified manner all formulae given by gauss davis and k lbig finally it should be noted that all our results in view of the relationship between n z and the hurwitz zeta function s a could be rewritten in the representation of s a

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

common fixed point results for noncommuting mappings without continuity in cone metric spaces the existence of coincidence points and common fixed points for mappings satisfying certain contractive conditions without appealing to continuity in a cone metric space is established these results generalize several well known comparable results in the literature

common fixed points for four maps in cone metric spaces the existence of coincidence points and common fixed points for four mappings satisfying generalized contractive conditions without exploiting the notion of continuity of any map involved therein in a cone metric space is proved these results extend unify and generalize several well known comparable results in the existing literature

peter monk s contributions to numerical analysis and maxwell s equations abstract peter monk has greatly contributed to the numerical analysis of partial differential equations and integral equations his name is mainly connected to maxwell s equations still his work has been fundamental in the advancement of numerical methods also for several other applications in this article i will try to discuss some of peter s contributions to these fields

pair of non self mappings and common fixed points we study quasi contraction type non self mappings on takahashi convex metric spaces and common fixed point theorems for a pair of maps results generalizing and unifying fixed point theorems of imdad and kumar das and naik jungck ciric ume khan and pathak and ciric are established

on cone metric spaces a survey abstract using an old m krein s result and a result concerning symmetric spaces from s radenovic z kadelburg quasi contractions on symmetric and cone symmetric spaces banach j math anal 5 1 2011 38 50 we show in a very short way that all fixed point results in cone metric spaces obtained recently in which the assumption that the underlying cone is normal and solid is present can be reduced to the corresponding results in metric spaces on the other hand when we deal with non normal solid cones this is not possible in the recent paper m a khamsi remarks on cone metric spaces and fixed point theorems of contractive mappings fixed point theory appl 2010 7 pages article id 315398 doi 10 1115 2010 315398 the author claims that most of the cone fixed point results are merely copies of the classical ones and that any extension of known fixed point results to cone metric spaces is redundant also that underlying banach space and the associated cone subset are not necessary in fact khamsi s approach includes a small class of results and is very limited since it requires only normal cones so that all results with non normal cones which are proper extensions of the corresponding results for metric spaces cannot be dealt with by his approach

approximation of common fixed points of weakly compatible pairs using the jungck iteration abstract we show that the jungck iteration scheme can be used to approximate the common fixed points of some weakly compatible pairs of generalized quasicontractive operators defined on metric spaces the existence of coincidence points are also discussed for those pair of maps the results are generalizations of well known results of the convergence of picard iterations for single self maps of banach spaces in particular the results improve generalize and extend the recent results of berinde v berinde a common fixed point theorem for compatible quasi contractive self mappings in metric spaces applied mathematics and computation 213 2009 348 354 and answers the open question posed in the paper

a discontinuous galerkin method with lagrange multipliers for the solution of helmholtz problems in the mid frequency regime we present a discontinuous galerkin method dgm for the solution of the helmholtz equation in the mid frequency regime our approach is based on the discontinuous enrichment method in which the standard polynomial field is enriched within each finite element by a non conforming field that contains free space solutions of the homogeneous partial differential equation to be solved hence for the helmholtz equation the enrichment field is chosen here as the superposition of plane waves we enforce a weak continuity of these plane waves across the element interfaces by suitable lagrange multipliers preliminary results obtained for two dimensional model problems discretized by uniform meshes reveal that the proposed dgm enables the development of elements that are far more competitive than both the standard linear and the standard quadratic galerkin elements for the discretization of helmholtz problems

a domain decomposition method for the helmholtz equation and related optimal control problems we present an iterative domain decomposition method to solve the helmholtz equation and related optimal control problems the proof of convergence of this method relies on energy techniques this method leads to efficient algorithms for the numerical resolution of harmonic wave propagation problems in homogeneous and heterogeneous media

a discontinuous enrichment method for capturing evanescent waves in multiscale fluid and fluid solid problems abstract an evanescent wave is produced when a propagating incident wave impinges on an interface between two media or materials at a sub critical angle the sub scale nature of such a wave makes it difficult to be captured computationally in this paper the discontinuous enrichment method dem developed in c farhat i harari l p franca the discontinuous enrichment method comput methods appl mech engrg 190 2001 6455 6479 c farhat i harari u hetmaniuk a discontinuous galerkin method with lagrange multipliers for the solution of helmholtz problems in the mid frequency regime comput methods appl mech engrg 192 2003 1389 1419 c farhat i harari u hetmaniuk the discontinuous enrichment method for multiscale analysis comput methods appl mech engrg 192 2003 3195 3210 c farhat r tezaur p weidemann goiran higher order extensions of a discontinuous galerkin method for mid frequency helmholtz problems int j numer methods engrg 61 2004 1938 1956 c farhat p wiedemann goiran r tezaur a discontinuous galerkin method with plane waves and lagrange multipliers for the solution of short wave exterior helmholtz problems on unstructured meshes new computational methods for wave propagation wave motion 39 4 2004 307 317 l zhang r tezaur c farhat the discontinuous enrichment method for elastic wave propagation in the medium frequency regime int j numer methods engrg 66 2006 2086 2114 r tezaur c farhat three dimensional discontinuous galerkin elements with plane waves and lagrange multipliers for the solution of mid frequency helmholtz problems int j numer methods engrg 66 2006 796 815 is extended to a class of evanescent wave problems new dem elements are constructed by enriching polynomial approximations with free space evanescent solutions in order to achieve high accuracy for problems with fluid fluid or fluid solid interfaces on which evanescent waves may occur the new dem elements are applied to the solution of various two dimensional model problems with evanescent waves their performance is observed to be better than that of the basic helmholtz type dem elements and superior to that of the classical higher order polynomial finite element method

overview of the discontinuous enrichment method the ultra weak variational formulation and the partition of unity method for acoustic scattering in the medium frequency regime and performance comparisons

fixed point theorem for two non self mappings in cone metric spaces in this paper we extend a fixed point theorem of imdad and kumar for a pair of non self maps to non normal cone spaces

coupling of the ultra weak variational formulation and an integral representation using a fast multipole method in electromagnetism many different methods have been developed for the solution of the time harmonic maxwell equations in exterior domains at high frequency volume based methods have the drawback of needing an artificial boundary far from the obstacle integral formulations enable one to avoid this difficulty by solving a problem on the surface of the obstacle however integral operators imply dense systems with bad condition numbers the ultra weak variational formulation uwvf is a volume based method using plane wave basis functions that allows the use of a coarser mesh in comparison with more classical low order finite element methods however the uwvf still involves the problem of the artificial boundary in this paper we suggest the use of an integral representation of the unknown field to obtain an exact artificial boundary condition in this way the distance between the obstacle and the artificial boundary can be reduced the use of the fast multipole method ensures a low cost for the calculation of various integral operators used in the representation in this paper we describe the combined algorithm demonstrate its accuracy on a model problem and discuss the complexity of the algorithm

computational aspects of the ultra weak variational formulation the ultra weak variational formulation uwvf approach has been proposed as an effective method for solving helmholtz problems with high wave numbers however for coarse meshes the method can suffer from instability in this paper we consider computational aspects of the ultra weak variational formulation for the inhomogeneous helmholtz problem we introduce a method to improve the uwvf scheme and we compare iterative solvers for the resulting linear system computations for the acoustic transmission problem in 2d show that the new approach enables helmholtz problems to be solved on a relatively coarse mesh for a wide range of wave numbers

the perfectly matched layer for the ultra weak variational formulation of the 3d helmholtz equation we investigate the feasibility of using the perfectly matched layer pml as an absorbing boundary condition for the ultra weak variational formulation uwvf of the 3d helmholtz equation the pml is derived using complex stretching of the spatial variables this leads to a modified helmholtz equation for which the uwvf can be derived in the standard discrete uwvf the approximating subspace is constructed from local solutions of the helmholtz equation in previous studies plane wave basis functions have been advocated because they simplify the building of the uwvf matrices for the pml domain we propose a special set of plane wave basis functions which allow fast computations and efficiently reduce spurious numerical reflections the method is validated by numerical experiments in comparison to a low order absorbing boundary condition the pml shows superior performance copyright 2004 john wiley sons ltd

plane wave discontinuous galerkin methods for the 2d helmholtz equation analysis of the p version plane wave discontinuous galerkin pwdg methods are a class of trefftz type methods for the spatial discretization of boundary value problems for the helmholtz operator delta omega 2 omega 0 they include the so called ultra weak variational formulation from o cessenat and b despres siam j numer anal 35 1998 pp 255 299 this paper is concerned with the a priori convergence analysis of pwdg in the case of p refinement that is the study of the asymptotic behavior of relevant error norms as the number of plane wave directions in the local trial spaces is increased for convex domains in two space dimensions we derive convergence rates employing mesh skeleton based norms duality techniques from p monk and d wang comput methods appl mech engrg 175 1999 pp 121 136 and plane wave approximation theory

a coupled coincidence point result in partially ordered metric spaces for compatible mappings abstract in this paper we introduce the notion of compatibility of mappings in a partially ordered metric space and use this notion to establish a coupled coincidence point result our work extends the work of bhaskar and lakshmikantham t gnana bhaskar v lakshmikantham fixed point theorems in partially ordered metric spaces and applications nonlinear anal tma 65 2006 1379 1393 an example is also given

the discontinuous enrichment method we propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it the enrichment contains free space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field continuity of the enrichment across element interfaces is enforced weakly by lagrange multipliers in this manner we expect to attain high coarse mesh accuracy without significant degradation of conditioning assuring good performance of the computation at any mesh resolution examples of application to acoustics and advection diffusion are presented

one integral in three ways moments of a quantum distribution very recently an exponential probability distribution with parameter has been used to calculate the decoherence factors for quantum states we derive all moments of this distribution systematically in three different ways presenting the results in terms of binomial coefficients or pochhammer symbols and stirling numbers of the first and second kinds we show how generalized harmonic numbers or polygamma functions provide another representation for the moments extensions of the approach are briefly mentioned

nonlinear coupled coincidence and coupled fixed point theorems for not necessary commutative contractive mappings in partially ordered probabilistic metric spaces in this paper we introduce the notion of compatibility of mappings in partially ordered probabilistic metric spaces and use this notion to establish a coupled coincidence point result very recently hu and ma xin qi hu xiao yan ma coupled coincidence point theorems under contractive conditions in partially ordered probabilistic metric spaces nonlinear anal 74 2011 6451 6458 proved coupled coincidence point theorems for commuting mappings in partially ordered probabilistic metric spaces in this paper we proved results of hu and ma under a different set of conditions precisely we establish our results by assuming that two mappings on a partially ordered probabilistic metric spaces are compatible not necessary commutative and satisfy a more general contractive condition than the contractive condition in the main theorem of hu and ma our results improve and extend a coupled coincidence point theorem due to hu and ma as well as a coupled fixed point theorem due to bhaskar and lakshmikantham t g bhaskar v lakshmikantham fixed point theorems in partially ordered metric spaces and applications nonlinear anal 65 2006 1379 1393 an example is given to support our result

identifying the wavenumber for the inverse helmholtz problem using an enriched finite element formulation abstract we investigate the inverse problem of identifying the wavenumber for the helmholtz equation the problem solution is based on measurements taken at few points from inside the computational domain or on its boundary a novel iterative approach is proposed based on coupling the secant and the descent methods with the partition of unity method starting from an initial guess for the unknown wavenumber the forward problem is solved using the partition of unity method then the secant descent methods are used to improve the initial guess by minimizing a predefined objective function based on the difference between the solution and a set of data points in the next round of iterations the improved wavenumber estimate is used for the forward problem solution and the partition of unity approximation is improved by adding more enrichment functions the iterative process is terminated when the objective function has converged and a set of two predefined tolerances are met to evaluate the estimate accuracy we propose to utilize extra data points to validate the approach and test its efficiency two wave applications with known analytical solutions are studied the results show that the proposed approach can achieve high accuracy for the studied applications even when the considered data is contaminated with noise despite the clear advantages that were previously shown in the literature for solving the forward helmholtz problem this work presents a first attempt to solve the inverse helmholtz problem with an enriched finite element approach

the existence uniqueness and the fixed iterative algorithm of the solution for the discrete coupled algebraic riccati equation in this article applying the properties of m matrix and non negative matrix utilising eigenvalue inequalities of matrix s sum and product we firstly develop new upper and lower matrix bounds of the solution for discrete coupled algebraic riccati equation dcare secondly we discuss the solution existence uniqueness condition of the dcare using the developed upper and lower matrix bounds and a fixed point theorem thirdly a new fixed iterative algorithm of the solution for the dcare is shown finally the corresponding numerical examples are given to illustrate the effectiveness of the developed results

improvements for the ultra weak variational formulation summary in this paper we investigate strategies to improve the accuracy and efficiency of the ultra weak variational formulation uwvf of the helmholtz equation the uwvf is a trefftz type nonpolynomial method using basis functions derived from solutions of the adjoint helmholtz equation we shall consider three choices of basis function propagating plane waves original choice bessel basis functions and evanescent wave basis functions traditionally two dimensional triangular elements are used to discretize the computational domain however the element shapes affect the conditioning of the uwvf hence we investigate the use of different element shapes aiming to lower the condition number and number of degrees of freedom our results include the first tests of a plane wave method on meshes of mixed element types in many modeling problems evanescent waves occur naturally and are challenging to model therefore we introduce evanescent wave basis functions for the first time in the uwvf to tackle rapidly decaying wave modes the advantages of an evanescent wave basis are verified by numerical simulations on domains including curved interfaces copyright 2013 john wiley sons ltd

the ultra weak variational formulation of thin clamped plate problems we develop a new numerical scheme for a fourth order elliptic partial differential equation based on kirchhoff s thin plate theory in particular we extend the ultra weak variational formulation uwvf to thin plate problems with clamped plate boundary conditions the uwvf uses a finite element mesh and non polynomial basis functions after deriving the new method we then prove l 2 norm convergence on the boundary finally we investigate numerically the feasibility of the uwvf for both homogeneous and inhomogeneous problems and show examples of p and h convergence

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