marcinkiewicz zygmund inequalities and interpolation by spherical harmonics abstract we find necessary density conditions for marcinkiewicz zygmund inequalities and interpolation for spaces of spherical harmonics in s d with respect to the l p norm moreover we prove that there are no complete interpolation families for p 2

hilbert space frames containing a riesz basis and banach spaces which have no subspace isomorphic toc0 abstract we prove that a hilbert space frame fi i icontains a riesz basis if every subfamily fi i j j i is a frame for its closed span secondly we give a new characterization of banach spaces which do not have any subspace isomorphic toc0 this result immediately leads to an improvement of a recent theorem of holub concerning frames consisting of a riesz basis plus finitely many elements

the optimal symmetrical points for polynomial interpolation of real functions in the tetrahedron abstract the main result of this paper is the computation of the mean optimal symmetrical interpolation points in the tetrahedron up to degree 9 this interpoation set has the smallest lebesgue constant known today

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

landau s necessary density conditions for lca groups we derive necessary conditions for sampling and interpolation of bandlimited functions on a locally compact abelian group in line with the classical results of h landau for bandlimited functions on rd our conditions are phrased as comparison principles involving a certain canonical lattice

spectral theory of schrodinger operators with infinitely many point interactions and radial positive definite functions abstract a number of results on radial positive definite functions on r n related to schoenberg s integral representation theorem are obtained they are applied to the study of spectral properties of self adjoint realizations of two and three dimensional schrodinger operators with countably many point interactions in particular we find conditions on the configuration of point interactions such that any self adjoint realization has purely absolutely continuous non negative spectrum we also apply some results on schrodinger operators to obtain new results on completely monotone functions

bessel sequences of exponentials on fractal measures jorgensen and pedersen have proven that a certain fractal measure has no infinite set of complex exponentials which form an orthonormal set in l2 we prove that any fractal measure obtained from an affine iterated function system possesses a sequence of complex exponentials which forms a riesz basic sequence or more generally a bessel sequence in l2 such that the frequencies have positive beurling dimension

an unfitted finite element method based on nitsche s method for elliptic interface problems in this paper we propose a method for the finite element solution of elliptic interface problem using an approach due to nitsche the method allows for discontinuities internal to the elements in the approximation across the interface we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh further we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples

ingham type theorems for vector valued functions and observability of coupled linear systems we propose a new approach to study the observability of coupled linear distributed systems it is based on a generalization of some classical theorems of nonharmonic analysis to vector valued functions applying this method we answer some questions of j l lions controlabilite exacte et stabilisation de systems distribues vol 2 masson 1988 on the boundary observability of coupled linear distributed systems in particular cases

approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle the main results of this paper are the analysis of the quality of approximation of polynomial interpolation and the computation of the approximate optimal interpolation points in the triangle we introduce various norms for the interpolation operator computational results indicate that for a given polynomial degree the set that minimizes the mean l2 norm of the interpolation operator is close to the smallest lebesgue constant interpolation set in particular for the triangle this set gives the smallest lebesgue constant currently known

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

on the beurling dimension of exponential frames abstract we study fourier frames of exponentials on fractal measures associated with a class of affine iterated function systems we prove that under a mild technical condition the beurling dimension of a fourier frame coincides with the hausdorff dimension of the fractal

the finite cell method for three dimensional problems of solid mechanics this article presents a generalization of the recently proposed finite cell method to three dimensional problems of linear elasticity the finite cell method combines ideas from embedding or fictitious domain methods with the p version of the finite element method besides supporting a fast simple generation of meshes it also provides high convergence rates mesh generation for a boundary representation of solids and for voxel based data obtained from ct scans is addressed in detail in addition the implementation of non homogeneous neumann boundary conditions and the computation of cell matrices based on a composed integration is presented the performance of the proposed method is demonstrated by three numerical examples including the elastostatic computation of a human bone biopsy

on controllability of a rotating string we consider the problem of controllability of a rotating string we say that a set of initial data of the string is controllable if for any initial data of this set by suitable manipulation of the exterior force the string goes to rest the main result of the paper is a description of controllable sets of initial data the equation is not strongly hyperbolic to get exact controllability of such equation in the sharp time interval we use control functions from sobolev spaces with noninteger indices to prove our results we apply the method of moments that has been widely used in control theory of distributed parameter systems since the classical papers of h o fattorini and d l russell we use recent results about exponential bases in sobolev spaces with noninteger indices

hopf bifurcation and symmetry travelling and standing waves on the circle in this paper we consider hopf bifurcation in the presence of o 2 symmetry the system of reaction diffusion equations u t d u xx f u provided with periodic boundary conditions may serve as a model problem we prove the bifurcation of a torus of standing waves and two circles of travelling waves and we compute the stability with asymptotic phase of these periodic solutions giving explicit formulae finally we demonstrate how a small perturbation which breaks part of the symmetry leads to secondary bifurcation

on controllability of a non homogeneous elastic string with memory abstract we are motivated by the problem of control for a non homogeneous elastic string with memory we reduce the problem of controllability to a non standard moment problem the solution of the latter problem is based on an auxiliary riesz basis property result for a family of functions quadratically close to the nonharmonic exponentials this result requires the detailed analysis of an integro differential equation and is of interest in itself for function theory controllability of the string implies observability of a dual system

higher order accurate integration of implicit geometries summary a unified strategy for the higher order accurate integration of implicitly defined geometries is proposed the geometry is represented by a higher order level set function the task is to integrate either on the zero level set or in the sub domains defined by the sign of the level set function in three dimensions this is either an integration on a surface or inside a volume a starting point is the identification and meshing of the zero level set by means of higher order interface elements for the volume integration special sub elements are proposed where the element faces coincide with the identified interface elements on the zero level set standard gauss points are mapped onto the interface elements or into the volumetric sub elements the resulting integration points may for example be used in fictitious domain methods and extended finite element methods for the case of hexahedral meshes parts of the approach may also be seen as a higher order marching cubes algorithm copyright 2015 john wiley sons ltd

complex dynamics in the hopf bifurcation with broken translation symmetry abstract the dynamics of the normal form for the o 2 equivariant hopf bifurcation perturbed by linear translation symmetry breaking terms are investigated a variety of stable small amplitude complex behavior is uncovered even for parameters for which the unperturbed system has no stable small amplitude solutions in some cases this behavior can be attributed to the proximity to global bifurcations of sil nikov and sil nikov hopf type in other cases cascades of gluing bifurcations involving various two frequency states are present since the equations studied can be obtained by center manifold reduction of the coupled complex ginzburg landau ccgl equations in a finite container this behavior must also be present in the ccgl equations the theory may explain the origin of erratic reversals in traveling wave convection and in spiral vortex flow

sampling and interpolation problems for vector valued signals in the paley wiener spaces a new approach to sampling interpolation problem for vector valued signals is proposed for a function signal f from the paley wiener space pw pi bbc n of entire vector valued functions the sampling data are defined as the inner products langf lambdan alphan rang of f lambdan and vectors alphan it is proved that alphan lambdan is a sampling and interpolating sequence for pw pi bbc n if and only if the family alphanei lambdan t forms a riesz basis in l 2 pi pi bbc n this relation allows to characterize all sampling and interpolating sequences in terms of a generating matrix function to produce a wide class of matrix functions generating sampling and interpolating sequences we consider the matrix sturm liouville problems and apply the boundary control theory for hyperbolic dynamical systems this approach allows to construct both asymptotically uniform and nonuniform sampling and interpolating sequences

symmetric approximation of frames and bases in hilbert spaces we introduce the symmetric approximation of frames by normalized tight frames extending the concept of the symmetric orthogonalization of bases by orthonormal bases in hilbert spaces we prove existence and uniqueness results for the symmetric approximation of frames by normalized tight frames even in the case of the symmetric orthogonalization of bases our techniques and results are new a crucial role is played by whether or not a certain operator related to the initial frame or basis is hilbert schmidt

modeling holes and inclusions by level sets in the extended finite element method a methodology to model arbitrary holes and material interfaces inclusions without meshing the internal boundaries is proposed the numerical method couples the level set method s osher j a sethian j comput phys 79 1 1988 12 to the extended finite element method x fem n moes j dolbow t belytschko int j numer methods engrg 46 1 1999 131 in the x fem the finite element approximation is enriched by additional functions through the notion of partition of unity the level set method is used for representing the location of holes and material interfaces and in addition the level set function is used to develop the local enrichment for material interfaces numerical examples in two dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique

local properties of hilbert spaces of dirichlet series we show that the asymptotic behavior of the partial sums of a sequence of positive numbers determine the local behavior of the hilbert space of dirichlet series defined using these as weights this extends results recently obtained describing the local behavior of dirichlet series with square summable coefficients in terms of local integrability boundary behavior carleson measures and interpolating sequences as these spaces can be identified with functions spaces on the infinite dimensional polydisk this gives new results on the dirichlet and bergman spaces on the infinite dimensional polydisk as well as the scale of besov sobolev spaces containing the drury arveson space on the infinite dimensional unit ball we use both techniques from the theory of sampling in paley wiener spaces and classical results from analytic number theory c 2011 elsevier inc all rights reserved less

simultaneous control problems for systems of elastic strings and beams abstract controllability problems for n strings or beams controlled from a common endpoint are studied we give necessary and sufficient conditions for simultaneous spectral and approximate controllability and describe the space of simultaneously reachable states as a function of the position of the endpoint for each type of controllability result the sharp controllability time is obtained the proofs are based on new results in nonharmonic fourier series which describe riesz bases of exponential divided differences

the hp d adaptive finite cell method for geometrically nonlinear problems of solid mechanics summary the finite cell method fcm combines the fictitious domain approach with the p version of the finite element method and adaptive integration for problems of linear elasticity it offers high convergence rates and simple mesh generation irrespective of the geometric complexity involved this article presents the integration of the fcm into the framework of nonlinear finite element technology however the penalty parameter of the fictitious domain is restricted to a few orders of magnitude in order to maintain local uniqueness of the deformation map as a consequence of the weak penalization nonlinear strain measures provoke excessive stress oscillations in the cells cut by geometric boundaries leading to a low algebraic rate of convergence therefore the fcm approach is complemented by a local overlay of linear hierarchical basis functions in the sense of the hp d method which synergetically uses the h adaptivity of the integration scheme numerical experiments show that the hp d overlay effectively reduces oscillations and permits stronger penalization of the fictitious domain by stabilizing the deformation map the hp d adaptive fcm is thus able to restore high convergence rates for the geometrically nonlinear case while preserving the easy meshing property of the original fcm accuracy and performance of the present scheme are demonstrated by several benchmark problems in one two and three dimensions and the nonlinear simulation of a complex foam sample copyright 2011 john wiley sons ltd

numerical simulation of a cylinder in uniform flow in this study a virtual boundary technique is applied to the numerical simulation of stationary and moving cylinders in uniform flow this approach readily allows the imposition of a no slip boundary within the flow field by a feedback forcing term added to the momentum equations in the present work this technique is used with a high order finite difference method effectively eliminating spurious oscillations caused by the feedback forcing when used with spectrally discretized flow solvers very good agreement is found between the present calculations and previous computational and experimental results for steady and time dependent flow at low reynolds numbers

strong and weak arbitrary discontinuities in spectral finite elements methods for constructing arbitrary discontinuities within spectral finite elements are described and studied we use the concept of the extended finite element method xfem which introduces the discontinuity through a local partition of unity so there is no requirement for the mesh to be aligned with the discontinuities a key aspect of the implementation of this method is the treatment of the blending elements adjacent to the local partition of unity we found that a partition constructed from spectral functions one order lower than the continuous approximation is optimal and no special treatment is needed for higher order elements for the quadrature of the galerkin weak form since the integrand is discontinuous we use a strategy of subdividing the discontinuous elements into 6 and 10 node triangles the order of the element depends on the order of the spectral method for curved discontinuities several numerical examples are solved to examine the accuracy of the methods for straight discontinuities we achieved the optimal convergence rate of the spectral element for the curved discontinuity the convergence rate in the energy norm error is suboptimal we attribute the suboptimality to the approximations in the quadrature scheme we also found that modification of the adjacent elements is only needed for lower order spectral elements copyright 2005 john wiley sons ltd

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

finite element analysis on implicitly defined domains an accurate representation based on arbitrary parametric surfaces in this paper we present some novel results and ideas for robust and accurate implicit representation of geometric surfaces in finite element analysis the novel contributions of this paper are threefold 1 describe and validate a method to represent arbitrary parametric surfaces implicitly 2 represent arbitrary solids implicitly including sharp features using level sets and boolean operations 3 impose arbitrary dirichlet and neumann boundary conditions on the resulting implicitly defined boundaries the methods proposed do not require local refinement of the finite element mesh in regions of high curvature ensure the independence of the domain s volume on the mesh do not rely on boundary regularization and are well suited to methods based on fixed grids such as the extended finite element method xfem numerical examples are presented to demonstrate the robustness and effectiveness of the proposed approach and show that it is possible to achieve optimal convergence rates using a fully implicit representation of object boundaries this approach is one step in the desired direction of tying numerical simulations to computer aided design cad similarly to the isogeometric analysis paradigm

cutfem discretizing geometry and partial differential equations we discuss recent advances on robust unfitted finite element methods on cut meshes these methods are designed to facilitate computations on complex geometries obtained for example from computer

marcinkiewicz zygmund inequalities and interpolation by spherical polynomials with respect to doubling weights abstract we find necessary density conditions for marcinkiewicz zygmund inequalities and interpolation for spaces of spherical polynomials in the weighted l p 0 p space with respect to a doubling weight moreover we show that these conditions are sharp

high order unfitted finite element methods on level set domains using isoparametric mappings abstract we introduce a new class of unfitted finite element methods with high order accurate numerical integration over curved surfaces and volumes which are only implicitly defined by level set functions an unfitted finite element method which is suitable for the case of piecewise planar interfaces is combined with a parametric mapping of the underlying mesh resulting in an isoparametric unfitted finite element method the parametric mapping is constructed in a way such that the quality of the piecewise planar interface reconstruction is significantly improved allowing for high order accurate computations of unfitted domain and surface integrals we present the method discuss implementational aspects and present numerical examples which demonstrate the quality and potential of this method

higher order conformal decomposition fem cdfem abstract a higher order accurate finite element method is proposed which uses automatically generated meshes based on implicit level set data for the description of boundaries and interfaces in two and three dimensions the method is an alternative for fictitious domain and extended finite element methods the domain of interest is immersed in a background mesh composed by higher order elements the zero level sets are identified and meshed followed by a decomposition of the cut background elements into conforming sub elements adaptivity is a crucial ingredient of the method to guarantee the success of the mesh generation it ensures the successful decomposition of cut elements and enables improved geometry descriptions and approximations it is confirmed that higher order accurate results with optimal convergence rates are achieved with the proposed conformal decomposition finite element method cdfem

decomposition of riesz frames and wavelets into a finite union of linearly independent sets we characterize riesz frames and prove that every riesz frame is the union of a finite number of riesz sequences furthermore it is shown that for piecewise continuous wavelets with compact support the associated regular wavelet systems can be decomposed into a finite number of linearly independent sets finally for finite sets an equivalent condition for decomposition into a given number of linearly independent sets is presented

frames of exponentials lower frame bounds for finite subfamilies and approximation of the inverse frame operator abstract we give lower frame bounds for finite subfamilies of a frame of exponentials e i k k z in l2 we also present a method for approximation of the inverse frame operator corresponding to e i k k z where knowledge of the frame bounds for finite subfamilies is crucial

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

regular article an accurate cartesian grid method for viscous incompressible flows with complex immersed boundaries a cartesian grid method has been developed for simulating two dimensional unsteady viscous incompressible flows with complex immersed boundaries a finite volume method based on a second order accurate central difference scheme is used in conjunction with a two step fractional step procedure the key aspects that need to be considered in developing such a solver are imposition of boundary conditions on the immersed boundaries and accurate discretization of the governing equation in cells that are cut by these boundaries a new interpolation procedure is presented which allows systematic development of a spatial discretization scheme that preserves the second order spatial accuracy of the underlying solver the presence of immersed boundaries alters the conditioning of the linear operators and this can slow down the iterative solution of these equations the convergence is accelerated by using a preconditioned conjugate gradient method where the preconditioner takes advantage of the structured nature of the underlying mesh the accuracy and fidelity of the solver is validated by simulating a number of canonical flows and the ability of the solver to simulate flows with very complicated immersed boundaries is demonstrated

on the connection between exponential bases and certain related sequences in l2 π π a sequence of vectors fn in a separable hilbert space h is a frame if there are positive constants a b such that a f 2 n f fn 2 b f 2 for all f h and fn is a riesz sequence if it is a riesz basis in the closure of the space spanned by the vectors fn the latter of the following two questions has been raised by khrushchev nikolskii and pavlov can every frame of complex exponentials ei nx in l2 be made into a riesz basis by removing from ei nx a suitable collection of the functions ei nx can every riesz sequence ei nx in l2 be made into a riesz basis by adjoining to ei nx a suitable collection of exponentials ei nx ei nx we show that this is indeed so for all frames and riesz sequences ei nx studied so far and we prove that we can always solve both problems in a certain weak sense however our main conclusion is that the answer to both questions is no

on fourier frame of absolutely continuous measures let be a compactly supported absolutely continuous probability measure on rn we show that l2 k d admits a fourier frame if and only if its radon nikodym derivative is bounded above and below almost everywhere on the support k as a consequence we prove that if is an equal weight absolutely continuous self similar measure on r1 and l2 k d admits a fourier frame then the density of must be a characteristic function of self similar tile in particular this shows for almost everywhere 1 2 1 the l2 space of the bernoulli convolutions cannot admit a fourier frame

fast isogeometric boundary element method based on independent field approximation abstract an isogeometric boundary element method for problems in elasticity is presented which is based on an independent approximation for the geometry traction and displacement field this enables a flexible choice of refinement strategies permits an efficient evaluation of geometry related information a mixed collocation scheme which deals with discontinuous tractions along non smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions all these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations the system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis for the required geometrical bisection of the domain a strategy for the evaluation of bounding boxes containing the supports of nurbs basis functions is presented the versatility and accuracy of the proposed methodology are demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions

studied x fem enrichment to handle material interfaces with higher order finite element an approach to improve the geometrical representation of surfaces with the extended finite element method is proposed surfaces are implicitly represented using the level set method the finite element approximation is enriched by additional functions through the notion of partition of unity to track material interfaces optimal rate of convergence is achieved with curved geometries using linear elements and linear level set in elements in order to accelerate the convergence the order of approximation shape functions is increased while keeping the same computational mesh the level set is represented on a finer sub mesh than the finite element mesh a special attention to integration procedure is necessary a new enrichment function is introduced to represent the behavior of curved material interfaces numerical examples including free surfaces and material interfaces in 2 d linear elasticity are presented to study convergence rates

irregular sampling toeplitz matrices and the approximation of entire functions of exponential type in many applications one seeks to recover an entire function of exponential type from its non uniformly spaced samples whereas the mathematical theory usually addresses the question of when such a function in l 2 r can be recovered numerical methods operate with a finite dimensional model the numerical reconstruction or approximation of the original function amounts to the solution of a large linear system we show that the solutions of a particularly efficient discrete model in which the data are fit by trigonometric polynomials converge to the solution of the original infinite dimensional reconstruction problem this legitimatizes the numerical computations and explains why the algorithms employed produce reasonable results the main mathematical result is a new type of approximation theorem for entire functions of exponential type from a finite number of values from another point of view our approach provides a new method for proving sampling theorems

a new 3d finite element for adaptive h refinement in 1 irregular meshes a new finite element viable for use in the three dimensional simulation of transient physical processes with sharply varying solutions is presented the element is intended to function in adaptive h refinement schemes as a versatile transition between regions of different refinement levels ensuring interelement continuity by constructing a piecewise linear solution at the element boundaries and retaining all degrees of freedom in the solution phase construction of the element shape functions is described and a numerical example is presented which illustrates the advantages of using such an element in an adaptive refinement problem the new element can be used in moving front problems such as those found in reservoir engineering and groundwater flow applications

stabilized finite element approximation of the mean curvature vector on closed surfaces the mean curvature vector of a surface is obtained by letting the laplace beltrami operator act on the embedding of the surface in bf r 3 in this contribution we develop a stabilized finite element approximation of the mean curvature vector of certain piecewise linear surfaces which enjoys first order convergence in l 2 the stabilization involves the jump in the tangent gradient in the direction of the outer co normals at each edge in the surface mesh we consider both standard meshed surfaces and so called cut surfaces that are level sets of piecewise linear distance functions we prove a priori error estimates and verify the theoretical results numerically

an embedding of domains approach in free boundary problems andoptimal design both free boundary problems and optimal design problems involve unknown or variable domains we describe a direct approach extending the formulation of the problem to a larger fixed domain without modifying the differential operator this argument is based on an approximate controllability type property

the generalized finite element method an example of its implementation and illustration of its performance the generalized finite element method gfem was introduced in reference 1 as a combination of the standard fem and the partition of unity method the standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data the geometry of the domain the loads and the boundary conditions the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation the essential boundary conditions can be imposed exactly as in the standard fem adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation this paper gives an example of how the gfem can be developed for the laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the gfem versus the standard fem copyright 2000 john wiley sons ltd

aspects of an adaptive hp finite element method adaptive strategy conforming approximation and efficient solvers the main components needed for an adaptive hp version finite element algorithm are discussed an adaptive hp refinement strategy effective methods for constructing conforming hp approximations and efficient solvers for the large ill conditioned systems of linear equations together these provide the methodology for an effective adaptive hp version algorithm the presentation emphasizes the links between the differing components showing how the algorithms may be implemented efficiently in practice the main principles are illustrated by use of concrete examples so that a non expert may develop their own adaptive hp code the performance of the whole algorithm is illustrated for some representative problems taken from linear elasticity

a perturbed hopf bifurcation with reflection symmetry we study the effects of a small symmetry breaking perturbation on a system of differential equations at a coupled hopf bifurcation with o 2 symmetry where the perturbation breaks the continuous rotation symmetry but retains a reflection z 2 symmetry it is shown that for a large range of parameter values the invariant manifolds of the unperturbed bifurcation persist and that for some values of normal form coefficients there are secondary bifurcations of nonsymmetric periodic standing wave solutions