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

self diffusion in a system of interacting langevin particles the behavior of the self diffusion constant of langevin particles interacting via a pairwise interaction is considered the diffusion constant is calculated approximately within a perturbation theory in the potential strength about the bare diffusion constant it is shown how this expansion leads to a systematic double expansion in the inverse temperature beta and the particle density rho the one loop diagrams in this expansion can be summed exactly and we show that this result is exact in the limit of small beta and rhobeta constants the one loop result can also be resummed using a semiphenomenological renormalization group method which has proved useful in the study of diffusion in random media in certain cases the renormalization group calculation predicts the existence of a diverging relaxation time signaled by the vanishing of the diffusion constant possible forms of divergence coming from this approximation are discussed finally at a more quantitative level the results are compared with numerical simulations in two dimensions of particles interacting via a soft potential recently used to model the interaction between coiled polymers

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

the generalized finite element method for helmholtz equation theory computation and open problems in this paper we address the generalized finite element method for the helmholtz equation we obtain our method by employing the finite element method on cartesian meshes which may overlap the boundaries of the problem domain and by enriching the approximation by plane waves pasted into the finite element basis at each mesh vertex by the partition of unity method here we address the q convergence of the method where q is the number of plane waves added at each vertex for the class of smooth analytic solutions for which we get better than exponential convergence for sufficiently small h depending on p an important observation is that we can monitor the accuracy in any computed solution quantity of interest at negligible cost by using q extrapolation our results assume exact integration of all the employed integrals further studies are needed to analyze the effects of the numerical integrations and also the effect of the roundoff errors

x fem in isogeometric analysis for linear fracture mechanics the extended finite element method x fem has proven to be an accurate robust method for solving problems in fracture mechanics x fem has typically been used with elements using linear basis functions although some work has been performed using quadratics in the current work the x fem formulation is incorporated into isogeometric analysis to obtain solutions with higher order convergence rates for problems in linear fracture mechanics in comparison with x fem with conventional finite elements of equal degree the nurbs based isogeometric analysis gives equal asymptotic convergence rates and equal accuracy with fewer degrees of freedom dof results for linear through quartic nurbs basis functions are presented for a multiplicity of one or a multiplicity equal the degree

the generalized finite element method for helmholtz equation part ii effect of choice of handbook functions error due to absorbing boundary conditions and its assessment abstract in part i t strouboulis i babuska r hidajat the generalized finite element method for helmholtz equation theory computation and open problems comput methods appl mech engrg 195 2006 4711 4731 we introduced the q version of the generalized finite element method gfem for the helmholtz equation and we addressed its a pollution error due to the wave number b exponential q convergence c robustness to perturbations of the mesh the roundoff and numerical quadrature errors and d a posteriori error estimation here we continue the development of the gfem for helmholtz and we address the effects of a alternative handbook functions and mesh types b the error due to the artificial truncation boundary conditions and its assessment the conclusions are 1 the employment of plane wave wave band and vekua handbook functions lead to equivalent results and 2 for high q the most significant component of error may be the one due to the artificial truncation boundary conditions a rather straightforward approach for assessing this error is proposed

extended isogeometric boundary element method xibem for three dimensional medium wave acoustic scattering problems a boundary element method bem based on non uniform rational b splines nurbs is used to find solutions to three dimensional wave scattering problems governed by the helmholtz equation the method is extended in a partition of unity sense multiplying the nurbs functions by families of plane waves this method is called the extended isogeometric boundary element method xibem in this paper the collocation xibem formulation is described and numerical results are given the numerical results are compared against closed form or converged solutions comparisons are made against the conventional boundary element method and the non enriched isogeometric bem igabem when compared to non enriched boundary element simulations using xibem significantly reduces the number of degrees of freedom required to obtain a solution of a given error thus with a fixed computational resource problems of a shorter wavelength can be solved

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

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

statistical description of interacting brownian particles abstract an alternative approach to the statistical description of interacting brownian particles based on the microscopic langevin equation is presented applying the formalism of microscopic phase space densities a stochastic form of the well known bbgky hierarchy for reduced distribution functions in phase space is produced special attention is paid to both the particle interaction and the hydrodynamic interaction between the brownons considered on the two particle level assuming fast momentum relaxation we find a hierarchy of diffusion like equations in configuration space on the smoluchowski time scale for reduced particle densities and corresponding particle flows these equations are an extension of the ebeling falkenhagen diffusion equations with respect to hydrodynamics finally we formulate the direct correlation force formalism for inhomogeneous non equilibrium systems in order to break off the hierarchy of coupled equations examples for closure relations are given

plane wave enriched partition of unity isogeometric analysis puiga for 2d helmholtz problems abstract in general numerical concerns in simulation of time harmonic acoustic waves are mainly due to the dispersion errors and the demands in modeling the problems with higher number of degrees of freedom per wavelength to obtain acceptable results as a remedy plane wave enriched partition of unity finite element method pufem was earlier proposed by melenk and babuska 1996 even though pufem gives accurate results with lesser degrees of freedom and shows exponential convergence it suffers from the approximate representation of geometry and leads to the boundary representation induced errors in this regard plane wave enriched pufem under isogeometric framework named partition of unity isogeometric analysis puiga is proposed for accurate geometry representation and smooth solution approximation which can lead to a better solution several benchmark problems are considered in the present study to illustrate the computational effectiveness of puiga

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

a performance study of nurbs based isogeometric analysis for interior two dimensional time harmonic acoustics abstract this work evaluates the performance of a nurbs based isogeometric finite element formulation for solving stationary acoustic problems in two dimensions an initial assessment is made by studying eigenvalue problems for a square and a circular domain the spectral approximation properties of nurbs functions of varying order are compared to those of conventional polynomials and are found to be superior yielding more accurate representations of eigenvalues as well as eigenmodes the higher smoothness of nurbs shape functions yields better approximations over an extended frequency range when compared to conventional polynomials two numerical case studies including a geometrically complex domain are used to benchmark the method versus the traditional finite element method a convergence analysis confirms the higher efficiency of the isogeometric method on a per degree of freedom basis simulations over a wider frequency range also illustrate that the method suffers less from the dispersion effects that deteriorate the acoustic response towards higher frequencies the tensor product structure of nurbs however also imposes practical considerations when modelling a complex geometry consisting of multiple patches

long time dynamics of charged colloidal suspensions hydrodynamic interaction effects abstract for charge stabilized suspensions we study the influence of hydrodynamic interactions hi and electrostatic interactions on the dynamic structure factor s q t and on the long time self diffusion coefficient d s l both types of interactions give rise to memory effects which lead to a slower and non exponential decay of s q t and which cause d s l to be smaller than the short time self diffusion coefficient d s s a global measure of the non exponential decay of s q t is the non exponentiality factor q which can be determined experimentally by dynamic light scattering we derive microscopic expressions for the one particle irreducible memory functions associated with q and d s l based on the generalized smoluchowski equation and on a projection operator method developed recently by kawasaki hi are accounted for by a far field expansion of the two body hydrodynamic mobility tensors by including the leading terms we calculate q and d s l by constructing a novel mode coupling approximation scheme which accounts for hi and we extend the theory to moderately polydisperse suspensions our results for the measurable non exponentiality factor compare well with available experimental data the effect of hi in charge stabilized colloids is to reduce the non exponentiality of s q t and to cause an unexpected enhancement of d s l our calculations clearly demonstrate the importance of hi at volume fractions even as low as 10 3

mixed enrichment for the finite element method in heterogeneous media summary problems of multiple scales of interest or of locally nonsmooth solutions may often involve heterogeneous media these problems are usually very demanding in terms of computations with the conventional finite element method on the other hand different enriched finite element methods such as the partition of unity which proved to be very successful in treating similar problems are developed and studied for homogeneous media in this work we present a new idea to extend the partition of unity finite element method to treat heterogeneous materials the idea is studied in applications to wave scattering and heat transfer problems where significant advantages are noted over the standard finite element method although presented within the partition of unity context the same enrichment idea can also be extended to other enriched methods to deal with heterogeneous materials copyright 2014 john wiley sons ltd

convergence analysis for finite element discretizations of the helmholtz equation with dirichlet to neumann boundary conditions a rigorous convergence theory for galerkin methods for a model helmholtz problem in bbb r d d in 1 2 3 is presented general conditions on the approximation properties of the approximation space are stated that ensure quasi optimality of the method as an application of the general theory a full error analysis of the classical hp version of the finite element method is presented for the model problem where the dependence on the mesh width h the approximation order p and the wave number k is given explicitly in particular 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 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

fuzzy quantities in networks one of possible applications of the fuzzy sets theory in network analysis is described and discussed it is shown that the uncertainty connected with the length of activities forming the network can be represented by fuzzy quantities summing these fuzzy quantities it is possible to derive the uncertainty connected with more complex paths and reserves an illustrative numerical example is presented at the end of the paper

modelling of short wave diffraction problems using approximating systems of plane waves this paper describes a finite element model for the solution of helmholtz problems at higher frequencies that offers the possibility of computing many wavelengths in a single finite element the approach is based on partition of unity isoparametric elements at each finite element node the potential is expanded in a discrete series of planar waves each propagating at a specified angle these angles can be uniformly distributed or may be carefully chosen they can also be the same for all nodes of the studied mesh or may vary from one node to another the implemented approach is used to solve a few practical problems such as the diffraction of plane waves by cylinders and spheres the wave number is increased and the mesh remains unchanged until a single finite element contains many wavelengths in each spatial direction and therefore the dimension of the whole problem is greatly reduced issues related to the integration and the conditioning are also discussed

discrete dispersion relation for hp version finite element approximation at high wave number the dispersive properties of high order finite element schemes are analyzed in the setting of the helmholtz equation and an explicit form of the discrete dispersion relation is obtained for elements of arbitrary order it is shown that the numerical dispersion displays three different types of behavior depending on the size of the order of the method relative to the mesh size and the wave number quantitative estimates are obtained for the behavior and rates of decay of the dispersion error in the differing regimes all estimates are fully explicit and are shown to be sharp limits are obtained where transitions between the different regimes occur and used to provide guidelines for the selection of the mesh size and the polynomial order in terms of the wave number so that the dispersion error is controlled

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

numerical modelling of elastic wave scattering in frequency domain by the partition of unity finite element method in this paper we investigate a numerical approach based on the partition of unity finite element method for the time harmonic elastic wave equations the aim of the proposed work is to accurately model two dimensional elastic wave problems with fewer elements capable of containing many wavelengths per nodal spacing and without refining the mesh at each frequency the approximation of the displacement field is performed via the standard finite element shape functions enriched by superimposing pressure and shear plane wave basis which incorporate knowledge of the wave propagation a variational framework able to handle mixed boundary conditions is described numerical examples dealing with the radiation and the scattering of elastic waves by a circular body are presented the results show the performance of the proposed method in both accuracy and efficiency copyright 2008 john wiley sons ltd

isogeometric analysis for geometric modelling and acoustic attenuation performances of reactive mufflers abstract in the present study the acoustic attenuation performances of mufflers are predicted by employing three dimensional isogeometric analysis iga and multi patch technique for the first time circular elliptical and conical reactive expansion chamber mufflers are exactly parameterized by non uniform rational b splines nurbs and the helmholtz equation governing the interior acoustic field is also solved by nurbs compared with traditional finite element method fem which is rooted in lagrange and hermite functions iga can effectively avoid the time consuming meshing and totally preserve exact geometrical information during the modelling analysis process the computational effectiveness of multi patch iga for calculating transmission loss is verified by comparing with experimental approach boundary element method and other numerical methods from available research articles the effects of geometry such as shapes of the expansion cavity and lengths of the extended inlet outlet are discussed in detail

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 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

self diffusion of spherical brownian particles with hard core interaction abstract the n particle smoluchowski equation is solved exactly for a system of spherical brownian particles interacting through hard core potentials to first order in the volume concentration the self diffusion coefficient is found to be given by d s d 0 1 2 where d 0 is the free diffusion constant the hydrodynamic interaction is then taken into account in a perturbative way using the oseen approximation d s d 0 1 0 09 is found whereas the improved form for the hydrodynamic interaction due to felderhof gives d s d 0 1 1 89

extended isogeometric boundary element method xibem for two dimensional helmholtz problems isogeometric analysis is a topic of considerable interest in the field of numerical analysis the boundary element method bem requires only the bounding surface of geometries to be described this makes non uniform rational b splines nurbs which commonly describe the bounding curve or surface of geometries in cad software appear to be a natural tool for the approach this isogeometric analysis bem igabem provides accuracy benefits over conventional bem schemes due to the analytical geometry provided by nurbs when applied to wave problems it has been shown that enriching bem approximations with a partition of unity basis in what has become known as the pu bem allows highly accurate solutions to be obtained with a much reduced number of degrees of freedom this paper combines these approaches and presents an extended isogeometric bem xibem which uses partition of unity enriched nurbs functions this new approach provides benefits which surpass those of both the igabem and the pu bem two numerical examples are given a single scattering cylinder and a multiple scatterer made up of two capsules and a cylinder

isogeometric analysis in electromagnetics b splines approximation abstract we introduce a new discretization scheme for maxwell equations in two space dimension inspired by the new paradigm of isogeometric analysis introduced in hughes et al 2005 16 we propose an algorithm based on the use of bivariate b splines spaces suitably adapted to electromagnetics we construct b splines spaces of variable interelement regularity on the parametric domain these spaces and their push forwards on the physical domain form a de rham diagram and we use them to solve the maxwell source and eigen problem our scheme has the following features i is adapted to treat complex geometries ii is spectral correct iii provides regular e g globally c 0 discrete solutions of maxwell equations

dynamic computer simulation of concentrated hard sphere suspensions ii re analysis of mean square displacement data in view of recent theoretical results by cichocki and felderhof we reevaluate previously published simulation data for the mean square displacement of a particle in a hard sphere suspension without hydrodynamic interactions this new evaluation is consistent with the theoretically predicted algebraic long time tails we observe better agreement for dense systems and obtain values for the relaxation time almost twice as large as previously

linear kinetic theory of a suspension of interacting brownian particles i enskog renormalization abstract time dependent equilibrium correlation functions for interacting brownian particles are analyzed to this end the kinetic theory method recently proposed by blawzdziewicz and cichocki is used the elementary processes i e single particle diffusion and many particle interactions contributing to the evolution of the time correlation functions are renormalized by the equilibrium correlations the final result which corresponds to the enskog renormalization in the kinetic theory of fluids is a convenient tool for an analysis of dynamical properties of dense suspensions

excluded volume effects in the diffusion of hard spheres excluded volume effects can play an important role in determining transport properties in diffusion of particles here the diffusion of finite sized hard core interacting particles in two or three dimensions is considered systematically using the method of matched asymptotic expansions the result is a nonlinear diffusion equation for the one particle distribution function with excluded volume effects enhancing the overall collective diffusion rate an expression for the effective collective diffusion coefficient is obtained stochastic simulations of the full particle system are shown to compare well with the solution of this equation for two examples

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

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