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

near optimal linear decision trees for k sum and related problems we construct near optimal linear decision trees for a variety of decision problems in combinatorics and discrete geometry for example for any constant k we construct linear decision trees that solve the k sum problem on n elements using o n log2 n linear queries moreover the queries we use are comparison queries which compare the sums of two k subsets when viewed as linear queries comparison queries are 2k sparse and have only l 1 0 1r coefficients we give similar constructions for sorting sumsets a b and for solving the subset sum problem both with optimal number of queries up to poly logarithmic terms our constructions are based on the notion of inference dimension recently introduced by the authors in the context of active classification with comparison queries this can be viewed as another contribution to the fruitful link between machine learning and discrete geometry which goes back to the discovery of the vc dimension

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

the use of continuous boundary elements in the boundary elements method for domains with non smooth boundaries via finite difference approach a numerical method is presented in this article to deal with the drawback of boundary elements method bem at corner points the use of continuous elements instead of the discontinuous ones has been recommended in the bem literature widely because of the simplicity and accuracy however the continuous elements lead to certain difficulties for problems where their domains contain corners in this paper the finite difference method fdm has been applied to obtain some constraints for boundary points near the corners to deal with this drawback because of its simplicity and capability the new scheme is applicable on bem problems for all geometries all governing equations and general boundary conditions easily since the dirichlet boundary condition is more critical than the other ones we will focus on it in the numerical implementation the numerical results show that the new treatment improves the accuracy of bem significantly

a boundary element method for steady incompressible thermoviscous flow a boundary element formulation is presented for moderate reynolds number steady incompressible thermoviscous flows the governing integral equations are written exclusively in terms of velocities and temperatures thus eliminating the need for the computation of any gradients furthermore with the introduction of reference velocities and temperatures volume modelling can often be confined to only a small portion of the problem domain typically near obstacles or walls the numerical implementation includes higher order elements adaptive integration and multiregion capability both the integral formulation and implementation are discussed in detail several examples illustrate the high level of accuracy that is obtainable with the current method

linear decision trees subspace arrangements and mobius functions let p be a set in rn given a point x we want to test if x e p our model of computation is a linear decision tree a rooted ternary tree t where with each interior node v is associated a linear function lv x l j ajxj b and the three edges connecting v to its descendants are labelled starting from the root we move down the tree at each internal node v we check whether iv x 0 and follow the appropriately labelled edge leaves are labelled yes and no and arriving at a leaf we read off the answer to the question is x e p linear decision trees are sometimes surprisingly powerful devices we mention here the result that some np complete problems like the knapsack problem have polynomial depth linear decision trees mh lower bound results on the size or depth of linear decision trees usually depend on counting the number of connected components of p or r n p this method was introduced by dobkin and lipton dl and further developed by steele and yao sy ben or bo and others in this paper we describe topological methods that can be applied to obtain lower bounds also if these sets are connected the basic idea of this paper was first explored in bly the euler characteristic of p or r n p was used to estimate the number of leaves in a linear decision tree for p here we improve and extend the topological methods of bly in particular we show that the euler characteristic can be replaced by the sum of the betti numbers of r n p in bl y the method was applied to the case when p is a subspace arrangement i e a union of a finite number of affine subspaces the euler characteristic of the union is essentially the mobius function of the intersection lattice of these subspaces in this paper using a formula of goresky and macpherson gm we work with the betti numbers of the complement of this union and thereby obtain better bounds the motivating problem in bly was the k equal problem given n real numbers xl xn decide if some k of them are equal it was shown that ev ry linear decision tree for the k equal problem has size at least 2njk 0 n and consequently depth at least q nlog 2njk which is best possible we

three dimensional curved crack in an elastic body abstract the boundary integral equations which give the relation between the crack opening displacement and traction on the surface of a crack embedded in an infinite isotropic elastic body are formulated the integral equations are transformed into spherical and cylindrical coordinates in the cases of cracks curved in the shape of spherical and cylindrical surfaces respectively so that these boundary integral equations may be converted into a system of algebraic equations by the boundary element method the dependence of stress intensity factors on the curvature of crack has been numerically calculated for the spherical crack with circular contour under a constant load

two methods for the numerical solution of bueckner s singular integral equation for plane elasticity crack problems abstract the classical collocation and galerkin methods are used for the numerical solution of singular integral equations of the first kind involving a finite part integral with a double pole singularity such equations appear in plane elasticity crack problems where they were suggested by bueckner and the unknown function in them is proportional to the crack opening displacement function an application of the proposed methods to the problem of a straight crack under an exponential normal loading distribution is also made and shows the rapid convergence of the obtained numerical results for the stress intensity factors at the crack tips to their theoretical values

higher order extensions of a discontinuous galerkin method for mid frequency helmholtz problems recently a discontinuous galerkin method with plane wave basis functions and lagrange multiplier degrees of freedom was proposed for the efficient solution of helmholtz problems in the mid frequency regime in this paper this method is extended to higher order elements performance results obtained for various two dimensional problems highlight the advantages of these elements over classical higher order galerkin elements such as q2 and q4 for the discretization of interior and exterior helmholtz problems copyright 2004 john wiley sons ltd

non singular boundary integral equation implementation the basis of the boundary integral equation bie development is the singular field solution for the kelvin point load problem the presence of the mathematical singularity in an otherwise non singular problem has created over the years a variety of numerical coping strategies the current paper extends some of the earlier works in a simple manner that eliminates the mathematical singularity through analytical integration of the critical terms the resulting formulation is fully non singular and is shown to be amenable to simple low order gaussian integration the critical class of problems involving thin sections and or highly graded bie meshes now can be solved without previous problems and concerns the accuracy of the numerical integrations used in creating the linear bie equations extension of the formulation approach to remove hypersingular formulation problems in fracture mechanics is seen to be a future beneficiary of the semi analytical bie implementation reported herein

counting complexity classes for numeric computations i semilinear sets we define a counting class rm p add in the blum shub smale setting of additive computations over the reals structural properties of this class are studied including a characterization in terms of the classical counting class sf p introduced by valiant we also establish transfer theorems for both directions between the real additive and the discrete setting then we characterize in terms of completeness results the complexity of computing basic topological invariants of semilinear sets given by additive circuits it turns out that the computation of the euler characteristic is rm fp rm add rm p rm add complete while for fixed k the computation of the k th betti number is rm fpar rm add complete thus the latter is more difficult under standard complexity theoretic assumptions we use all of the above to prove some analogous completeness results in the classical setting

a multiple node method to resolve the difficulties in the boundary integral equation method caused by corners and discontinuous boundary conditions in the boundary integral equation method biem use of lagrangian shape functions together with conforming boundary elements requires continuity of functions at the interelement boundary when the flux or the traction is discontinuous due to the presence of corners or discontinuous boundary conditions conforming elements can be a source of error in this paper we detail a multiple node method in which this error can either be eliminated or substantially reduced the paper is limited to the laplace problem and the problem of elastostatics in two dimensions however the method can be easily extended to problems of other types and to higher dimensions

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

coupling finite elements and auxiliary sources abstract we consider second order scalar elliptic boundary value problems on unbounded domains which model for instance electrostatic fields we propose a discretization that relies on a trefftz approximation by multipole auxiliary sources in some parts of the domain and on standard mesh based primal lagrangian finite elements in other parts several approaches are developed and based on variational saddle point theory rigorously analyzed to couple both discretizations across the common interface 1 least squares based coupling using techniques from pde constrained optimization 2 coupling through dirichlet to neumann operators 3 three field variational formulation in the spirit of mortar finite element methods we compare these approaches in a series of numerical experiments

fem with trefftz trial functions on polyhedral elements the goal of this paper is to generalize a variant of the bem based fem for second order elliptic boundary value problems to three space dimensions here the emphasis lies on polyhedral meshes with polygonal faces where even non convex elements are allowed due to an implicit definition of the trial functions in the spirit of trefftz the strategy yields conforming approximations and is very flexible with respect to the meshes thus it gets into the line of recent developments in several areas the arising local problems are treated by two dimensional galerkin schemes coming from finite and boundary element formulations with the help of a new interpolation operator and its properties convergence estimates are proven in the h 1 as well as in the l 2 norm numerical experiments confirm the theoretical results

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

point location in arrangements of hyperplanes we present a solution to the point location problem in arrangements of hyperplanes in ed with running time o d5 log n and space o nd for arbitrary 0 where n is the number of hyperplanes the main result is the d5 factor in the expression for the running time all previously known algorithms are exponential in d or log n this leads to nonuniform polynomial algorithms for np complete problems

comparisons of three kinds of plane wave methods for the helmholtz equation and time harmonic maxwell equations with complex wave numbers abstract in this paper we are concerned with some plane wave discretization methods of the helmholtz equation and time harmonic maxwell equations with complex wave numbers we design two new variants of the variational theory of complex rays and the ultra weak variational formulation for the discretization of these types of equations respectively the well posedness of the approximate solutions generated by the two methods is derived moreover based on the pwls lsfe method introduced in hu and yuan 2018 we extend these two methods vtcr method and uwvf method combined with local spectral element to discretize nonhomogeneous helmholtz equation and maxwell s equations the numerical results show that the resulting approximate solution generated by the uwvf method is clearly more accurate than that generated by the vtcr method

a bi cubic transformation for the numerical evaluation of the cauchy principal value integrals in boundary methods the numerical strategies employed in the evaluation of singular integrals existing in the cauchy principal value cpv sense are undoubtedly one of the key aspects which remarkably affect the performance and accuracy of the boundary element method bem thus a new procedure based upon a bi cubic co ordinate transformation and oriented towards the numerical evaluation of both the cpv integrals and some others which contain different types of singularity is developed both the ideas and some details involved in the proposed formulae are presented obtaining rather simple and attractive expressions for the numerical quadrature which are also easily embodied into existing bem codes some illustrative examples which assess the stability and accuracy of the new formulae are included

a new method for evaluating singular integrals in stress analysis of solids by the direct boundary element method the purpose of this paper is to report on a new and efficient method for the evaluation of singular integrals in stress analysis of elastic and elasto plastic solids respectively by the direct boundary element method bem triangle polar co ordinates are used to reduce the order of singularity of the boundary integrals by one degree and to carry out the integration over mappings of the boundary elements onto plane squares the method was subsequently extended to the cubature of singular integrals over three dimensional internal cells as occur in applications of the bem to three dimensional elasto plasticity for this purpose so called tetrahedron polar co ordinates were introduced singular boundary integrals stretching over either linear triangular or quadratic quadilateral isoparametric boundry elements and singular volume integrals extending over either linear tetrahedral or quadratic hexahedral isoparametric internal cells are treated in case of higher order isoparametric boundary elements and internal cells division into a number of subelements and subcells respectively is necessary the analytical investigation is followed by a numerical study restricted to the use of quadratic quadrilateral isoparametric boundary elements this is justified by the fact that such elements as opposed to linear elements yield singular boundary integrals which cannot be integrated analytically the results of the numerical investigation demonstrate the potential of the developed concept

a phase based interior penalty discontinuous galerkin method for the helmholtz equation with spatially varying wavenumber this paper is concerned with an interior penalty discontinuous galerkin ipdg method based on a flexible type of non polynomial local approximation space for the helmholtz equation with varying wavenumber the local approximation space consists of multiple polynomial modulated phase functions which can be chosen according to the phase information of the solution we obtain some approximation properties for this space and a priori l2l2 error estimates for the h convergence of the ipdg method using duality argument we also provide ample numerical examples to show that building phase information into the local spaces often gives more accurate results comparing to using the standard polynomial spaces

a space time trefftz discontinuous galerkin method for the acoustic wave equation in first order formulation we introduce a space time trefftz discontinuous galerkin method for the first order transient acoustic wave equations in arbitrary space dimensions extending the one dimensional scheme of kretzschmar et al ima j numer anal 36 1599 1635 2016 test and trial discrete functions are space time piecewise polynomial solutions of the wave equations we prove well posedness and a priori error bounds in both skeleton based and mesh independent norms the space time formulation corresponds to an implicit time stepping scheme if posed on meshes partitioned in time slabs or to an explicit scheme if posed on tent pitched meshes we describe two trefftz polynomial discrete spaces introduce bases for them and prove optimal high order h convergence bounds

dispersive and dissipative behaviour of high order discontinuous galerkin finite element methods the dispersive and dissipative properties of hp version discontinuous galerkin finite element approximation are studied in three different limits for the small wave number limit hk 0 we show the discontinuous galerkin gives a higher order of accuracy than the standard galerkin procedure thereby confirming the conjectures of hu and atkins j comput phys 182 2 2002 516 if the mesh is fixed and the order p is increased it is shown that the dissipation and dispersion errors decay at a super exponential rate when the order p is much larger than hk finally if the order is chosen so that 2p 1 hk for some fixed constant 1 then it is shown that an exponential rate of decay is obtained

exact integration scheme for planewave enriched partition of unity finite element method to solve the helmholtz problem in this paper we present an exact integration scheme to compute highly oscillatory integrals that appear in the solution of the two dimensional helmholtz problem using the planewave enriched partition of unity finite element method in the proposed scheme such oscillatory integrals are computed by a recursive application of the divergence theorem eventually expressing the integrals in terms of evaluations of the corresponding integrands at the nodes of the finite element mesh the number of such function evaluations is independent of the wave number kk which permits the scheme to be used for arbitrary high values of kk we consider finite element meshes with unstructured triangular and structured rectangular elements and present numerical results for three canonical benchmark helmholtz problems to demonstrate the accuracy and efficacy of the method

a polynomial linear search algorithm for the n dimensional knapsack problem presentation d un algorithme de recherche lineaire qui reconnait le probleme de sac a dos a n dimensions par etapes de 2n 4 logn o n 3

lower bounds for linear degeneracy testing in the late nineties erickson proved a remarkable lower bound on the decision tree complexity of one of the central problems of computational geometry given n numbers do any r of them add up to 0 his lower bound of n r 2 for any fixed r is optimal if the polynomials at the nodes are linear and at most r variate we generalize his bound to s variate polynomials for s r erickson s bound decays quickly as r grows and never reaches above pseudo polynomial we provide an exponential improvement our arguments are based on three ideas i a geometrization of erickson s proof technique ii the use of error correcting codes and iii a tensor product construction for permutation matrices

efficient dg like formulation equipped with curved boundary edges for solving elasto acoustic scattering problems summary a discontinuous galerkin dg based approach is proposed for computing the scattered field from an elastic bounded object immersed in an infinite homogeneous fluid medium the proposed method possesses two distinctive features first it employs higher order polynomial shape functions needed to address the high frequency propagation regime second it is equipped with curved boundary edges to provide an accurate representation of the fluid structure interface the most salient benefits resulting from the latter feature as demonstrated by the numerical investigation are the following i an improvement by at least two orders of magnitude on the relative error and ii the disappearance of spurious resonance frequencies in the surrounding fluid medium in addition the reported numerical results reveal that when using cubic polynomials with less than three elements per wavelength the proposed dg method computes the scattered field with a relative error below 1 for an elastic scatterer of about 30 wavelengths this observation highlights the potential of the proposed solution methodology for efficiently solving mid frequency to high frequency elasto acoustic scattering problems copyright 2014 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

a ray based ipdg method for high frequency time domain acoustic wave propagation in inhomogeneous media abstract the numerical approximation of high frequency wave propagation in inhomogeneous media is a challenging problem in particular computing high frequency solutions by direct simulations requires several points per wavelength for stability and usually requires many points per wavelength for a satisfactory accuracy in this paper we propose a new method for the acoustic wave equation in inhomogeneous media in the time domain to achieve superior accuracy and stability without using a large number of unknowns the method is based on a discontinuous galerkin discretization together with carefully chosen basis functions to obtain the basis functions we use the idea from geometrical optics and construct the basis functions by using the leading order term in the asymptotic expansion also we use a wavefront tracking method and a dimension reduction procedure to obtain dominant rays in each cell we show numerically that the accuracy of the numerical solutions computed by our method is significantly higher than that computed by the ipdg method using polynomials moreover the relative errors of our method grow only moderately as the frequency increases

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

a discontinuous galerkin method with plane waves for sound absorbing materials poro elastic materials are commonly used for passive control of noise and vibration and are key to reducing noise emissions in many engineering applications including the aerospace automotive and energy industries more efficient computational models are required to further optimise the use of such materials in this paper we present a discontinuous galerkin method dgm with plane waves for poro elastic materials using the biot theory solved in the frequency domain this approach offers significant gains in computational efficiency and is simple to implement costly numerical quadratures of highly oscillatory integrals are not needed it is shown that the biot equations can be easily cast as a set of conservation equations suitable for the formulation of the wave based dgm a key contribution is a general formulation of boundary conditions as well as coupling conditions between different propagation media this is particularly important when modelling porous materials as they are generally coupled with other media such as the surround fluid or an elastic structure the validation of the method is described first for a simple wave propagating through a porous material and then for the scattering of an acoustic wave by a porous cylinder the accuracy conditioning and computational cost of the method are assessed and comparison with the standard finite element method is included it is found that the benefits of the wave based dgm are fully realised for the biot equations and that the numerical model is able to accurately capture both the oscillations and the rapid attenuation of the waves in the porous material

calculs sur les structures de langage denombrable we develop two models of calculus over stuctures of countable signature and the main items of the structural complexity theory which relates to them in particular we prove the existence of universal machines complete problems and structures with the p np property in this frame we are starting the study of z group without order and the field of p adic numbers qp

a plane wave discontinuous galerkin method with a dirichlet to neumann boundary condition for the scattering problem in acoustics abstract we consider the numerical solution of an acoustic scattering problem by the plane wave discontinuous galerkin method pwdg in the exterior of a bounded domain in r 2 in order to apply the pwdg method we introduce an artificial boundary to truncate the domain and we impose a non local dirichlet to neumann dtn boundary condition on the artificial curve to define the method we introduce new consistent numerical fluxes that incorporate the truncated series of the dtn map error estimates with respect to the truncation order of the dtn map and with respect to mesh width are derived numerical results suggest that the accuracy of the pwdg method for the acoustic scattering problem can be significantly improved by using dtn boundary conditions

a time domain boundary element method for poroelasticity the development of a general boundary element method bem for two and three dimensional quasistatic poroelasticity is discussed in detail the new formulation for the complete biot consolidation theory operates directly in the time domain and requires only boundary discretization as a result the dimensionality of the problem is reduced by one and the method becomes quite attractive for geotechnical analyses particularly those which involve extensive or infinite domains the presentation includes the definition of the two key ingredients for the bem namely the fundamental solutions and a reciprocal theorem then once the boundary integral equations are derived the focus shifts to an overview of the general purpose numerical implementation this implementation includes higher order conforming elements self adaptive integration and multi region capability finally several detailed examples are presented to illustrate the accuracy and suitability of this boundary element approach for consolidation analysis

orthogonal collocation in the nonconforming boundary element method this paper outlines the use of non conforming discontinuous elements in the collocation boundary element method for solving two dimensional potential and poisson type problems the roots of an orthogonal polynomial shifted jacobi polynomial are used as the collocation points this results in increased accuracy due to the least square minimization property of the orthogonal polynomials the advantage of using non conforming elements is realized when the method is applied i to problems with singularities both due to geometry and boundary conditions and ii in conjunction with domain decomposition techniques also the collocation points can be relocated within an element by changing two user defined parameters in the shifted jacobi polynomial thus providing an error indicator which can be used for mesh refinement purposes this technique called the rh method is discussed and illustrated the results obtained by using non conforming boundary elements for standard test problems are shown to be accurate

lower bounds for sorting of sums this paper addresses the following question how much can sorting by comparisons be speeded up if some information about the possible ordertypes the input sequence x1 xn might have is given in advance we extend a lower bound due to fredman 3 concerning sorting of sums of the form yi zj 1 i j m to the problem of sorting all sums of up to d out of m numbers

interpolation properties of generalized plane waves this paper is aimed at developing new shape functions adapted to the scalar wave equation with smooth possibly vanishing coefficients and investigates the numerical analysis of their interpolation properties the interpolation is local but high order convergence is shown with respect to the size of the domain considered the new basis functions are then implemented in a numerical method to solve a scalar wave equation problem with a mixed boundary condition the main theoretical result states that any given order of approximation can be achieved by an appropriate choice of parameters for the design of the shape functions the convergence is studied with respect to the size of the domain which is referred to in the literature as h h convergence

a posteriori error estimates for the virtual element method an posteriori error analysis for the virtual element method vem applied to general elliptic problems is presented the resulting error estimator is of residual type and applies on very general polygonal polyhedral meshes the estimator is fully computable as it relies only on quantities available from the vem solution namely its degrees of freedom and element wise polynomial projection upper and lower bounds of the error estimator with respect to the vem approximation error are proven the error estimator is used to drive adaptive mesh refinement in a number of test problems mesh adaptation is particularly simple to implement since elements with consecutive co planar edges faces are allowed and therefore locally adapted meshes do not require any local mesh post processing

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

on boundary integral equations for crack problems a ubiquitous linear boundary value problem in mathematical physics involves solving a partial differential equation exterior to a thin obstacle one typical example is the scattering of scalar waves by a curved crack or rigid strip neumann boundary condition in two dimensions this problem can be reduced to an integrodifferential equation which is often regularized we adopt a more direct approach and prove that the problem can be reduced to a hypersingular boundary integral equation similar reductions will obtain in more complicated situations computational schemes for solving this equation are described with special emphasis on smoothness requirements extensions to three dimensional problems involving an arbitrary smooth bounded crack in an elastic solid are discussed

is the helmholtz equation really sign indefinite the usual variational or weak formulations of the helmholtz equation are sign indefinite in the sense that the bilinear forms cannot be bounded below by a positive multiple of the appropriate norm squared this is often for a good reason since in bounded domains under certain boundary conditions the solution of the helmholtz equation is not unique at wavenumbers that correspond to eigenvalues of the laplacian and thus the variational problem cannot be sign definite however even in cases where the solution is unique for all wavenumbers the standard variational formulations of the helmholtz equation are still indefinite when the wavenumber is large this indefiniteness has implications for both the analysis and the practical implementation of finite element methods in this paper we introduce new sign definite also called coercive or elliptic formulations of the helmholtz equation posed in either the interior of a star shaped domain with impedance boundary conditions or the exterior of a star shape

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

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

a first order system least squares method for the helmholtz equation we present a first order system least squares fosls method for the helmholtz equation at high wave number k which always leads to a hermitian positive definite algebraic system by utilizing a non trivial solution decomposition to the dual fosls problem which is quite different from that of the standard finite element methods we give an error analysis to the h p version of the fosls method where the dependence on the mesh size h the approximation order p and the wave number k is given explicitly in particular under some assumption of the boundary of the domain the l 2 norm error estimate of the scalar solution from the fosls method is shown to be quasi optimal under the condition that k h p is sufficiently small and the polynomial degree p is at least o log k numerical experiments are given to verify the theoretical results

decision trees old and new results abstract in this paper we prove two general lower bounds for algebraic decision trees which test membership in a set s r n which is defined by linear inequalities let rank s be the maximal dimension of a linear sub space contained in the closure of s in euclidean topology first we show that any decision tree for s which uses products of linear functions we call such functions mlf functions must have depth at least n rank s this solves an open question raised by a c yao and can be used to show that mlf functions are not really more powerful than simple comparisons between the input variables when computing the largest k out of n elements yao proved this result in the special case when products of at most two linear functions are allowed our proof also shows that any decision tree for this problem must have exponential size using the same methods we can give an alternative proof of rabin s theorem namely that the depth of any decision tree for s using arbitrary analytic functions is at least n rank s

lower bounds for solving linear diophantine equations on random access machines the problem of recognizing the language l n l n k of solvable diophantine linear equations with n variables and solutions from o k n is considered the languages nen l n nen l n l the knapsack problem are np complete the o n 2 lower bound for l n 1 on linear search algorithms due to dobkin and lipton is generalized to an o n 2 log k 1 lower bound for l n k the method of klein and meyer auf der heide is further improved to carry over the o n 2 lower bound for l n 1 to random access machines rams in such a way that it holds for a large class of problems and for very small input sets by this method lower bounds that depend on the input size as is necessary for l n are proved thereby an o n 2 log k 1 lower bound is obtained for rams recognizing l n or l n k for inputs from 0 nk 0 n 2 n

a semi analytical scheme for highly oscillatory integrals over tetrahedra this is the peer reviewed version of the following article hospital bravo r sarrate j and diez p 2017 a semi analytical scheme for highly oscillatory integrals over tetrahedra int j numer meth engng 111 703 723 doi 10 1002 nme 5474 which has been published in final form at http onlinelibrary wiley com doi 10 1002 nme 5474 full this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving

on trefftz and weak trefftz discontinuous galerkin approaches for medium frequency acoustics in this paper the wave approach called the variational theory of complex rays vtcr which was developed for medium frequency acoustics and vibrations is revisited as a discontinuous galerkin method extensions leading to a weak trefftz constraint are introduced this weak trefftz discontinuous galerkin approach enables hybrid fem vtcr strategies to be developed easily and paves the way for new computational techniques for the resolution of engineering problems this paper presents some of the fundamental properties of the approach which is illustrated by several numerical examples

a hybridizable discontinuous galerkin method for the helmholtz equation with high wave number this paper analyzes the error estimates of the hybridizable discontinuous galerkin hdg method for the helmholtz equation with high wave number in two and three dimensions the approximation piecewise polynomial spaces we deal with are of order p geq 1 through choosing a specific parameter and using the duality argument it is proved that the hdg method is stable without any mesh constraint for any wave number kappa by exploiting the stability estimates the dependence of convergence of the hdg method on kappa h and p is obtained numerical experiments are given to verify the theoretical results

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

direct computation of cauchy principal value integrals in advanced boundary elements this paper presents a new method for the direct computation of strongly singular integrals existing in the cauchy principal value sense it can be usefully applied in the solution by the direct boundary element method of many different problems initially some considerations are provided in order to summarize the state of the art on this issue then the features of the proposed method are reported the procedure allows the direct calculation of cauchy principal value integrals with first order singularity and it is applicable even in advanced boundary element methods employing high order elements it requires only the use of standard gaussian quadrature formulae plus the computation of a logarithmic term some examples show the effectiveness and efficiency of the procedure

a self adaptive co ordinate transformation for efficient numerical evaluation of general boundary element integrals almost all general purpose boundary element computer packages include a curved geometry modelling capability thus numerical quadrature schemes play an important role in the efficiency of programming the technique the present work discusses this problem in detail and introduces efficient means of computing singular or nearly singular integrals currently found in two dimensional axisymmetric and three dimensional applications emphasis is given to a new third degree polynomial transformation which was found greatly to improve the accuracy of gaussian quadrature scheme s within the near singularity range the procedure can easily be implemented into existing be codes and presents the important feature of being self adaptive i e it produces a variable lumping of the gauss stations toward the singularity depending on the minimum distance from the source point to the element the self adaptiveness of the scheme also makes it inactive when not useful large source distances which makes it very safe for general usage

some improvements of accuracy and efficiency in three dimensional direct boundary element method numerical analysis with the boundary element method bem has been used more and more in various engineering fields in recent years in numerical techniques however there are some problems which have not been fully solved even now the most essential one is the drop in the accuracy of results for internal points near the boundary of the structure where the singularity of integrands in the boundary integral equation is too strong to be evaluated with the normal numerical method for the boundary integral equation of stress this problem became more serious and the accuracy can be improved only partly even though very refined boundary elements are used in this paper the boundary integral equation is newly formulated using a relative quantity of displacement in this way the singularity of boundary integrals is reduced by the order of 1 r and the accuracy of solution is improved significantly furthermore in order to integrate it more accurately two kinds of numerical integral methods are newly developed by using these methods both displacement and stress can be obtained with excellent accuracy at almost any point in the structure without any numerical difficulty although the discretization may be comparatively coarse the generality and practicability of the present formulation and integral methods are confirmed through some examples of three dimensional elastic problems

a phase based hybridizable discontinuous galerkin method for the numerical solution of the helmholtz equation we introduce a new hybridizable discontinuous galerkin hdg method for the numerical solution of the helmholtz equation over a wide range of wave frequencies our approach combines the hdg methodology with geometrical optics in a fashion that allows us to take advantage of the strengths of these two methodologies the phase based hdg method is devised as follows first we enrich the local approximation spaces with precomputed phases which are solutions of the eikonal equation in geometrical optics second we propose a novel scheme that combines the hdg method with ray tracing to compute multivalued solution of the eikonal equation third we utilize the proper orthogonal decomposition to remove redundant modes and obtain locally orthogonal basis functions which are then used to construct the global approximation spaces of the phase based hdg method and fourth we propose an appropriate choice of the stabilization parameter to guarantee stability and accuracy for the proposed method numerical experiments presented show that optimal orders of convergence are achieved that the number of degrees of freedom to achieve a given accuracy is independent of the wave number and that the number of unknowns required to achieve a given accuracy with the proposed method is orders of magnitude smaller than that with the standard finite element method

advanced development of the boundary element method for steady state heat conduction considerable progress has been made in recent years toward advancing the state of the art in solid mechanics boundary element technology in the present work much of this new technology is applied in the development of a general purpose boundary element method bem for steady state heat conduction in particular the bem implementation involves the use of higher order conforming elements self adaptive integration and multi region capability two and three dimensional as well as axisymmetric analyses are incorporated within a unified framework in addition techniques are introduced for the calculation of boundary flux and for the inclusion of thermal resistance across interfaces as a final extension an efficient formulation is developed for the analysis of solid three dimensional bodies with embedded holes for this last class of problems the new bem formulation is particularly attractive since use of the alternatives i e finite element or finite difference methods is not practical a number of detailed examples illustrate the suitability and robustness of the present approach for steady state heat conduction

simulating probabilistic by deterministic algebraic computation trees a probabilistic algebraic computation tree probabilistic act which recognizes l rn in expected time t and which gives the wrong answer with probability 12 can be simulated by a deterministic act in o t2n steps the same result holds for linear search algorithms lsas the result for acts establishes a weaker version of results previously shown by the author for lsas namely that lsas can only be slightly sped up by their nondeterministic versions this paper shows that acts can only be slightly sped up by their probabilistic versions the result for lsas solves a problem posed by snir 1983 he found an example where probabilistic lsas are faster than deterministic ones and asked how large this gap can be

threesomes degenerates and love triangles the 3sum problem is to decide given a set of n real numbers whether any three sum to zero it is widely conjectured that a trivial o n2 time algorithm is optimal on the real ram and optimal even in the nonuniform linear decision tree model over the years the consequences of this conjecture have been revealed this 3sum conjecture implies n2 lower bounds on numerous problems in computational geometry and a variant of the conjecture for integer inputs implies strong lower bounds on triangle enumeration dynamic graph algorithms and string matching data structures in this article we refute the conjecture that 3sum requires n2 in the real ram and refute more forcefully the conjecture that its complexity is n2 in the linear decision tree model in particular we prove that the decision tree complexity of 3sum is o n3 2s log n and give two subquadratic 3sum algorithms a deterministic one running in o n2 log n log log n 2 3 time and a randomized one running in o n2 log log n 2 log n time with high probability our results lead directly to improved bounds on the decision tree complexity of k variate linear degeneracy testing for all odd k 3 finally we give a subcubic algorithm for a generalization of the min product over real valued matrices and apply it to the problem of finding zero weight triangles in edge weighted graphs we give a depth o n5 2s log n decision tree for this problem as well as a deterministic algorithm running in time o n3 log log n 2 log n

the 3 d thermoelastic boundary element method semi analytical integration for subparametric triangular elements in this paper a semi analytical integration scheme is described which is designed to reduce the errors that can result with the numerical evaluation of integrals with singular integrands the semi analytical scheme can be applied to quadratic subparametric triangular elements for use in thermoelastic problems established analytical solutions for linear isoparametric triangular elements are combined with standard quadrature techniques to provide an accurate integration scheme for quadratic subparametric triangular elements the use of subparametric elements provides an efficient means for coupling thermal and elastostatic analyses it is possible for the same mesh to be employed with linear isoparametric elements used for thermal analysis and quadratic subparametric elements used for deformation analysis numerical tests are performed on simple test problems to demonstrate the advantages of the semi analytical approach which is shown to be orders of magnitude more accurate than standard quadrature techniques moreover the expected increased accuracy with subparametric elements is also demonstrated 1998 john wiley sons ltd

well posedness and generalized plane waves simulations of a 2d mode conversion model certain types of electro magnetic waves propagating in a plasma can undergo a mode conversion process in magnetic confinement fusion this phenomenon is very useful to heat the plasma since it permits to transfer the heat at or near the plasma center this work focuses on a mathematical model of wave propagation around the mode conversion region from both theoretical and numerical points of view it aims at developing for a well posed equation specific basis functions to study a wave mode conversion process these basis functions called generalized plane waves are intrinsically based on variable coefficients as such they are particularly adapted to the mode conversion problem the design of generalized plane waves for the proposed model is described in detail their implementation within a discontinuous galerkin method then provides numerical simulations of the process these first 2d simulations for this model agree with qualitative aspects studied in previous works

wavenumber explicit analysis of a dpg method for the multidimensional helmholtz equation we study the properties of a novel discontinuous petrov galerkin dpg method for acoustic wave propagation the method yields hermitian positive definite matrices and has good pre asymptotic stability properties numerically we find that the method exhibits negligible phase errors otherwise known as pollution errors even in the lowest order case theoretically we are able to prove error estimates that explicitly show the dependencies with respect to the wavenumber the mesh size h and the polynomial degree p but the current state of the theory does not fully explain the remarkably good numerical phase errors theoretically comparisons are made with several other recent works that gave wave number explicit estimates numerically comparisons are made with the standard finite element method and its recent modification for wave propagation with clever quadratures the new dpg method is designed following the previously established principles of optimal test functions in addition to the nonstandard test functions in this work we also use a nonstandard wave number dependent norm on both the test and trial space to obtain our error estimates

conforming versus non conforming boundary elements in three dimensional elastostatics in this work we present a critical comparison of two basic ways of implementing the boundary element method in three dimensional elastostatics the first way is by using non conforming elements i e elements which have the collocation nodes removed from their perimeter the number of nodes used for the collocation of the boundary tractions and displacements need not coincide with the number of nodes placed along the perimeter of the element for the purpose of describing the geometry the second way is by placing the collocation nodes along the perimeter of the element usually in coincidence with the geometry nodes thus interelement continuity of the displacements is obtained the basic reasons for the use of non conforming elements in the boundary element method are twofold first simplification in the assembly and solution of the system equations and secondly easy computation of the free and cauchy principal value terms appearing in the integral equations the state of the art in the boundary element method however has advanced in the last decade to the point where both of the aforementioned reasons are no longer problematic as will be shown in what follows conforming elements are able to produce more accurate results than non conforming ones with substantial economy in the final size of the system equations

a local wave tracking strategy for efficiently solving mid and high frequency helmholtz problems abstract we propose a procedure for selecting basis function orientation to improve the efficiency of solution methodologies that employ local plane wave approximations the proposed adaptive approach consists of a local wave tracking strategy each plane wave basis set within considered elements of the mesh partition is individually or collectively rotated to best align one function of the set with the local propagation direction of the field systematic determination of the direction of the field inside the computational domain is formulated as a minimization problem as the resultant system is nonlinear with respect to the directions of propagation the newton method is employed with exact characterization of the jacobian and hessian to illustrate the salient features and evaluate the performance of the proposed wave tracking approach we present error estimates as well as numerical results obtained by incorporating the procedure into a prototypical plane wave based approach the least squares method lsm developed by monk and wang 1999 1 the numerical results obtained for the case of a two dimensional rigid scattering problem indicate that a convergence was achievable to a prescribed level of accuracy even upon initial application of the tracking wave strategy outside the pre asymptotic convergence region and b the proposed approach reduced the size of the resulting system by up to two orders of magnitude depending on the frequency range with respect to the size of the standard lsm system

fast algorithms for n dimensional restrictions of hard problems let m be a parallel ram with p processors and arithmetic operations addition and subtraction recognizing l n n in t steps inputs for m are given integer by integer not bit by bit then l can be recognized by a sequential linear search algorithm lsa in o n 4 log n t log p steps thus many n dimensional restrictions of np complete problems binary programming traveling salesman problem etc and even that of the uniquely optimum traveling salesman problem which is p 2 complete can be solved in polynomial time by an lsa this result generalizes the construction of a polynomial lsa for the n dimensional restriction of the knapsack problem previously shown by the author and destroys the hope of proving nonpolynomial lower bounds on lsas for any problem that can be recognized by a pram as above with 2 poly n processors in poly n time

the 3 d elastodynamic boundary element method semi analytical integration for linear isoparametric triangular elements a semi analytical integration scheme is described in this paper which is designed to reduce the errors incurred when integrals with singular integrands are evaluated numerically this new scheme can be applied to linear triangular elements for use in steady state elastodynamic bem problems and is particularly useful for predicting displacement to high accuracy close to surfaces for a spectrum of frequencies the scheme involves the application of taylor expansions to formulate the integrals into two parts one part is regular and is evaluated numerically and the other part is singular but sufficiently simple to enable its transforma tion into a line integral the line integral is solved numerically using gauss legendre quadrature this approach caters for all the integral types that appear in steady state elastodynamic boundary elements but in particular no special treatment is required for the evaluation of the cauchy principal value singular integrals numerical tests are performed on a simple test problem for which a known analytical solution exists the results obtained using the semi analytical approach are shown to be considerably more accurate than those obtained using standard quadrature methods copyright 1999 john wiley sons ltd

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

wavenumber explicit convergence analysis for galerkin discretizations of the helmholtz equation we develop a stability and convergence theory for a class of highly indefinite elliptic boundary value problems bvps by considering the helmholtz equation at high wavenumber k as our model problem the key element in this theory is a novel k explicit regularity theory for helmholtz bvps that is based on decomposing the solution into two parts the first part has the sobolev regularity properties expected of second order elliptic pdes but features k independent regularity constants the second part is an analytic function for which k explicit bounds for all derivatives are given this decomposition is worked out in detail for several types of bvps namely the helmholtz equation in bounded smooth domains or convex polygonal domains with robin boundary conditions and in exterior domains with dirichlet boundary conditions we present an error analysis for the classical hp version of the finite element method hp fem where the dependence on the mesh width h the approximation order p and the wavenumber k is given explicitly in particular under the assumption that the solution operator for helmholtz problems is polynomially bounded in k it is shown that quasi optimality is obtained under the conditions that kh p is sufficiently small and the polynomial degree p is at least o log k

a class of discontinuous petrov galerkin methods part iv the optimal test norm and time harmonic wave propagation in 1d the phase error or the pollution effect in the finite element solution of wave propagation problems is a well known phenomenon that must be confronted when solving problems in the high frequency range this paper presents a new method with no phase errors for one dimensional 1d time harmonic wave propagation problems using new ideas that hold promise for the multidimensional case the method is constructed within the framework of the discontinuous petrov galerkin dpg method with optimal test functions we have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm in this paper we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm this is answered in the specific case of the helmholtz equation with l 2 norm as the energy norm we obtain uniform stability with respect to the wave number we illustrate the method with a number of 1d numerical experiments using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally a 1d theoretical stability analysis is also developed

conforming boundary elements in plane elasticity for shape design sensitivity the structural design sensitivity analysis of a two dimensional continuum using conforming continuous boundary elements is investigated implicit differentiation of the discretized boundary integral equations is performed to obtain design sensitivities in an efficient manner by avoiding the factorization of the perturbed matrices a singular formulation of the boundary element method is used implicit differentiation of the boundary integral equations produces terms that contain derivatives of the fundamental solutions employed in the analysis the behaviour of the singularity of these derivatives of the boundary element kernel functions with respect to the design variables is investigated a rigid body motion technique is presented to obtain the singular terms in the resulting sensitivity matrices thus avoiding the problems associated with their numerical integration a formulation for obtaining the design sensitivities of the continua under body forces of the gravitational and centrifugal types is also presented the design sensitivity results are seen to be of the same order of accuracy as the boundary element analysis results numerical data comparing the performance of conforming and non conforming formulations in the calculation of design sensitivities are also presented the accuracy of the present results is demonstrated through comparisons with existing analytical results

staggered discontinuous galerkin methods for the helmholtz equation with large wave number abstract in this paper we investigate staggered discontinuous galerkin method for the helmholtz equation with large wave number on general polygonal meshes the method is highly flexible by allowing rough grids such as the trapezoidal grids and highly distorted grids and at the same time is numerical flux free furthermore it allows hanging nodes which can be simply treated as additional vertices by exploiting a modified duality argument the stability and convergence can be proved under the condition that h c 0 1 1 m 1 where m is the polynomial order is the wave number h is the mesh size and c 0 is a positive constant independent of h error estimates for both the scalar and vector variables in l 2 norm are established several numerical experiments are tested to verify our theoretical results and to present the capability of our method for capturing singular solutions

a general and effective way for evaluating the integrals with various orders of singularity in the direct boundary element method a general and efficient technique is developed for the evaluation of the integrals with various orders of singularity such as occur in the three dimensional boundary element method bem generalized extended triangle tetrahedron polar co ordinate mappings together with two conditions are used to remove the singularity of the integrals and to evaluate the corresponding non singular ones in a new numerical space triangle and tetrahedron polar co ordinates in reference 1 are proved to be a special case of the generalized ones in this paper with the developed idea boundary element results converge rapidly towards the analytical solutions for the strongly singular integrals evaluated directly and the analytical solutions can be gained in principle even when employing higher order triangular boundary elements and tetrahedral cells the generality and practicability of the method are demonstrated in the case of higher order elements discontinuous elements and large engineering problems

a method for direct evaluation of singular integral in direct boundary element method a new method is presented for evaluating the cauchy principal value integral and the free term in the three dimensional elastic boundary element method this method expresses the singular integral as the sum of a weakly singular integral and an integrable cauchy principal value integral the former can be computed by means of the usual standard gaussian quadrature after a transformation suggested by lachat and watson 10 who correctly limited its use only to weakly singular integral the latter can be calculated by direct integration in the cauchy principal value sense this paper also points out some limitations in the use of triangle polar co ordinate suggested by li et al 11 and later extended by zhang and xu 16 the general expression of the free term for a boundary point with arbitrary characteristic surface is also presented simple numerical examples which verify the efficiency and accuracy of the present method are given

accurate numerical integration of singular boundary element kernels over boundaries with curvature use of the green function for the solution of boundary value problems frequently results in singular integral equations algorithms are presented for the accurate and efficient treatment of singular kernels frequently encountered in the boundary element method bem they are based upon the use of appropriately weighted gaussian quadrature formulae together with numerical geometrical transformations of the region of integration the use of high order subdomain expansion functions for interpolation over nonplanar elements allows boundary curvature to be accommodated in particular the handling of green functions with logarithmic and r 1 behaviour are detailed volume integrals with r 2 singularity are outlined operations are performed on a simplex thus resulting in generality and ease of automation this scheme has been incorporated into boundary element method software and successfully applied to a variety of problems

three dimensional discontinuous galerkin elements with plane waves and lagrange multipliers for the solution of mid frequency helmholtz problems recently a discontinuous galerkin finite element method with plane wave basis functions and lagrange multiplier degrees of freedom was proposed for the efficient solution in two dimensions of helmholtz problems in the mid frequency regime in this paper this method is extended to three dimensions and several new elements are proposed computational results obtained for several wave guide and acoustic scattering model problems demonstrate one to two orders of magnitude solution time improvement over the higher order galerkin method

some notes on singular integral techniques in boundary element analysis the paper presented here reviews the numerical techniques used currently to calculate the singular integrals and nearly singular integrals in the boundary element analysis some incorrect algorithms published before are discussed and a new numerical technique to calculate the nearly singular integral is developed the numerical results show a significant improvement in both accuracy and efficiency compared to the traditional adaptive gaussian quadrature and subdivision techniques

trefftz approximations in complex media accuracy and applications abstract approximations by trefftz functions are rapidly gaining popularity in the numerical solution of boundary value problems of mathematical physics by definition these functions satisfy locally in weak form the underlying differential equations of the problem which often results in high order or even exponential accuracy with respect to the size of the basis set we highlight two separate examples in applied electromagnetics and photonics i homogenization of periodic structures and ii numerical simulation of electromagnetic waves in slab geometries extensive numerical evidence and theoretical considerations show that trefftz approximations can be applied much more broadly than is traditionally done they are effective not only in physically homogeneous regions but also in complex inhomogeneous ones two mechanisms underlying the high accuracy of trefftz approximations in such complex cases are pointed out the first one is related to trigonometric interpolation and the second one somewhat surprisingly to well posedness of random matrices

hypersingular integrals in boundary element fracture analysis a new general purpose boundary element method for domains with cracks has been recently developed this technique avoids the use of a multi domain decomposition by including an additional integral equation expressing the boundary condition on the crack the principal requirement of this technique is the analytic determination of certain hypersingular integrals of the green s function which arise from this equation in order to establish the applicability of this method for fracture these integrals are evaluated herein for the kelvin solution of the three dimensional navier equations of linear elasticity numerical results for fracture problems using the single domain boundary element analysis are also presented

discontinuous galerkin methods with trefftz approximations abstract we present a novel discontinuous galerkin finite element method for wave propagation problems the method employs space time trefftz basis functions that satisfy the underlying partial differential equations and the respective interface boundary conditions exactly in an element wise fashion the basis functions can be of arbitrarily high order and we demonstrate spectral convergence in the space time l 2 norm therefore high order time integration is an inherent property of the method and clearly sets it apart from methods that employ a high order approximation in space only

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