the steklov eigenvalue problem in a cuspidal domain in this paper we analyze the approximation by piecewise linear finite elements of a steklov eigenvalue problem in a plane domain with an external cusp this problem is not covered by the literature and its analysis requires a special treatment indeed we develop new trace theorems and we also obtain regularity results for the source counterpart moreover under appropriate assumptions on the meshes we present interpolation error estimates for functions in fractional sobolev spaces these estimates allow us to obtain appropriate convergence results of the source counterpart which in the context of the theory of compact operator are a fundamental tool in order to prove the convergence of the eigenpairs at the end we prove the convergence of the eigenpairs by using graded meshes and present some numerical tests

an iterative method for the calculation of narrow solitary excitations on atomic chains abstract we discuss the quasicontinuum approximation of collins and give a general condition for its validity for the solution of the difference differential equations of the full discrete model we develop an iteration procedure in fourier space the method is tested for solitary excitations with widths of the order of the lattice constant by a comparison with both exact results and computer simulations an additional result of the new mwthod is a simple derivation of the collins approximation which allows several generalizations

discrete lattice solitons properties and stability the authors present a detailed numerical study of the dynamics of high energy kink excitations in monatomic chains with a simple description using a microscopic model of the narrow kink core they present velocity amplitude and energy momentum curves for the kink which fit well the numerical results kink kink collisions are slightly inelastic depending on the velocities and point of collision within a lattice spacing

stationary breather modes of generalized nonlinear klein gordon lattices this paper studies the form of stationary breather modes in discrete generalized nonlinear klein gordon equations with symmetric and non symmetric potential energy functions in the case of static breathers the discrete nature of the spatial dimension has a much more subtle effect on the breather than the moving breather mode this effect is analysed using a variety of approximating partial differential equations whose solutions are found by using an extended multiple scales asymptotic approach to reduce the equation to a nonlinear schrodinger equation at leading order and more complex equations at higher order where secularity conditions are required to fully specify the solution as well as the much studied discrete sine gordon equation the methods are demonstrated on a discrete nonlinear klein gordon equation with second neighbour interactions and non symmetric on site potential new partial differential equations which approximate these lattice systems are also proposed and analysed

optimal finite element interpolation on curved domains this paper is devoted to a general theory of approximation of functions in finite element spaces in particular the case is considered where the functions to be interpolated are on the one hand not very smooth and on the other are defined on curved domains thus a number of well known optimal interpolation results are generalized

existence theorem for solitary waves on lattices in this article we give an existence theorem for localized travelling wave solutions on one dimensional lattices with hamiltonian h sum limits n in mathbb z left tfrac 1 2 p n 2 v q n 1 q n right wherev is the potential energy due to nearest neighbour interactions until now apart from rare integrable lattices like the toda latticev ab 1 e b b 1 the only evidence for existence of such solutions has been numerical our result in particular recovers existence of solitary waves in the toda lattice establishes for the first time existence of solitary waves in the nonintegrable cubic and quartic latticesv 1 2 2 1 3a 3 v 1 2 2 1 4b 4 thereby confirming the numerical findings in 1 and shedding new light on the recurrence phenomena in these systems observed first by fermi pasta and ulam 2 and shows that contrary to widespread belief the presence of exact solitary waves is not a peculiarity of integrable systems but generic in this class of nonlinear lattices the approach presented here is new and quite general and should also be applicable to other forms of lattice equations the travelling waves are sought as minimisers of a naturally associated variational problem obtained via hamilton s principle and existence of minimisers is then established using modern methods in the calculus of variations the concentration compactness principle of p l lions 3

external finite element approximations of eigenfunctions in the case of multiple eigenvalues abstract the paper deals with the finite element analysis of second order elliptic eigenvalue problems when the approximate domains h are not subdomains of the original domain r 2 the considerations are restricted to piecewise linear approximations special attention is devoted to the convergence of approximate eigenfunctions in the case of multiple exact eigenvalues as yet the approximate solutions have been compared with linear combinations of exact eigenfunctions with coefficients depending on the mesh parameter h we avoid this disadvantage

global weak solutions for the two dimensional motion of several rigid bodies in an incompressible viscous fluid we consider the two dimensional motion of several non homogeneous rigid bodies immersed in an incompressible non homogeneous viscous fluid the fluid and the rigid bodies are contained in a fixed open bounded set of r 2 the motion of the fluid is governed by the navier stokes equations for incompressible fluids and the standard conservation laws of linear and angular momentum rule the dynamics of the rigid bodies the time variation of the fluid domain due to the motion of the rigid bodies is not known a priori so we deal with a free boundary value problem the main novelty here is the demonstration of the global existence of weak solutions for this problem more precisely the global character of the solutions we obtain is due to the fact that we do not need any assumption concerning the lack of collisions between several rigid bodies or between a rigid body and the boundary we give estimates of the velocity of the bodies when their mutual distance or the distance to the boundary tends to zero

finite element approximation of spectral problems with neumann boundary conditions on curved domains this paper deals with the finite element approximation of the spectral problem for the laplace equation with neumann boundary conditions on a curved nonconvex domain convergence and optimal order error estimates are proved for standard piecewise linear continuous elements on a discrete polygonal domain h in tile framework of the abstract spectral approximation theory

finite element approximations in a non lipschitz domain part ii fil acosta gabriel universidad de buenos aires facultad de ciencias exactas y naturales departamento de matematica argentina

asymptotic analysis of combined breather kink modes in a fermi pasta ulam chain we find approximations to travelling breather solutions of the one dimensional fermi pasta ulam fpu lattice both bright breather and dark breather solutions are found we find that the existence of localised bright solutions depends upon the coefficients of cubic and quartic terms of the potential energy generalising an earlier inequality derived by james cr acad sci paris 332 581 2001 we use the method of multiple scales to reduce the equations of motion for the lattice to a nonlinear schr o dinger equation at leading order and hence construct an asymptotic form for the breather we show that in the absence of a cubic potential energy term the lattice supports combined breathing kink waveforms the amplitude of breathing kinks can be arbitrarily small as opposed to traditional monotone kinks which have a nonzero minimum amplitude in such systems we also present numerical simulations of the lattice verifying the shape and velocity of the travelling waveforms and confirming the long lived nature of all such modes

finite element analysis of the vibration problem of a plate coupled with a fluid we consider the approximation of the vibration modes of an elastic plate in contact with a compressible fluid the plate is modelled by reissner mindlin equations while the fluid is described in terms of displacement variables this formulation leads to a symmetric eigenvalue problem reissner mindlin equations are discretized by a mixed method the equations for the fluid with raviart thomas elements and a non conforming coupling is used on the interface in order to prove that the method is locking free we consider a family of problems one for each thickness t 0 and introduce appropriate scalings for the physical parameters so that these problems attain a limit when t to 0 we prove that spurious eigenvalues do not arise with this discretization and we obtain optimal order error estimates for the eigenvalues and eigenvectors valid uniformly on the thickness parameter t finally we present numerical results confirming the good performance of the method

quasi continuum approximations to lattice equations arising from the discrete nonlinear telegraph equation we show how quasi continuum methods can be used to construct approximate solutions of nonlinear differential delay equations derived from symmetry reductions of the discrete nonlinear telegraph equation travelling wave solutions are proven to exist and the existence of solutions to two other symmetry reductions are studied two of the less familiar reductions are studied the first supports both a one parameter family of single pulse solitary wave type solutions and a one parameter family of periodic waves the size and shape of these waves are examined using the quasi continuum technique this approximates the differential difference equation with a higher order differential equation which is integrated and analysed using phase plane techniques in the large amplitude limit the shape of the pulse approaches a limiting form which has a corner at its peak the manner of this approach is elucidated using matched asymptotic expansions the second reduction though differing only by the addition of a single term appears not to support the solitary wave type of solution even in the limit where the additional term is premultiplied by an asymptotically small constant

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

the maximum angle condition for mixed and nonconforming elements application to the stokes equations for the lagrange interpolation it is known that optimal order error estimates hold for elements satisfying the maximum angle condition the objective of this paper is to obtain similar results for the raviart thomas interpolation arising in the analysis of mixed methods we prove that optimal order error estimates hold under the maximum angle condition for this interpolation both in two and three dimensions and moreover that this condition is indeed necessary to have these estimates error estimates for the mixed approximation of second order elliptic problems and for the nonconforming piecewise linear approximation of the stokes equations are derived from our results

finite element approximations in a nonlipschitz domain in this paper we analyze the approximation by standard piecewise linear finite elements of a nonhomogeneous neumann problem in a cuspidal domain since the domain is not lipschitz many of the results on sobolev spaces which are fundamental in the usual error analysis do not apply therefore we need to work with weighted sobolev spaces and to develop some new theorems on traces and extensions we show that in the domain considered here suboptimal order can be obtained with quasi uniform meshes even when the exact solution is in h 2 and we prove that the optimal order with respect to the number of nodes can be recovered by using appropriate graded meshes

finite element solution of nonlinear elliptic problems the study of the finite element approximation to nonlinear second order elliptic boundary value problems with mixed dirichlet neumann boundary conditions is presented in the discretization variational crimes are commited approximation of the given domain by a polygonal one numerical integration with the assumption that the corresponding operator is strongly monotone and lipschitz continuous and that the exact solutionu h1 the convergence of the method is proved under the additional assumptionu h2 the rate of convergenceo h is derived without the use of green s theorem

improving the rate of convergence of high order finite elements on polygons and domains with cusps let u and uv v be the solution and respectively the discrete solution of the non homogeneous dirichlet problem u f on u 0 for any m and any bounded polygonal domain we provide a construction of a new sequence of finite dimensional subspaces vn such that where f hm 1 is arbitrary and c is a constant that depends only on and not on n we do not assume u hm 1 the existence of such a sequence of subspaces was first proved in a ground breaking paper by babuska 8 our method is different it is based on the homogeneity properties of sobolev spaces with weights and the well posedness of non homogeneous dirichlet problem in suitable sobolev spaces with weights for which we provide a new proof and which is a substitute of the usual shift theorems for boundary value problems in domains with smooth boundary our results extended right away to domains whose boundaries have conical points we also indicate some of the changes necessary to deal with domains with cusps our numerical computation are in agreement with our theoretical results

on the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form in the approximation of linear elliptic operators in mixed form it is well known that the so called inf sup and ellipticity in the kernel properties are sufficient and in a sense to be made precise necessary in order to have good approximation properties and optimal error bounds one might think in the spirit of mercier osborn rappaz raviart and in consideration of the good behavior of commonly used mixed elements like raviart thomas or brezzi douglas marini elements that these conditions are also sufficient to ensure good convergence properties for eigenvalues in this paper we show that this is not the case in particular we present examples of mixed finite element approximations that satisfy the above properties but exhibit spurious eigenvalues such bad behavior is proved analytically and demonstrated in numerical experiments we also present additional assumptions fulfilled by the commonly used mixed methods already mentioned which guarantee optimal error bounds for eigenvalue approximations as well

a robust finite element method for nonhomogeneous dirichlet problems in domains with curved boundaries in this paper the authors consider a simple finite element method on an approximately polygonal domain using linear elements the dirichlet data are transferred in a natural way and the resulting linear system can be solved using multigrid techniques their analysis takes into account the change in domain and data transfer and optimal error estimates are obtained that are robust in the regularity of the boundary data provided they are at least square integrable it is proved that the natural extension of this finite element approximation to the original domain is optimal order accurate

the combined effect of numerical integration and approximation of the boundary in the finite element method for eigenvalue problems the paper deals with the finite element analysis of second order elliptic eigenvalue problems when the approximate domains omega h are not subdomains of the original domain omega subset r 2 and when at the same time numerical integration is used for computing the involved bilinear forms the considerations are restricted to piecewise linear approximations the optimum rate of convergence o h 2 for approximate eigenvalues is obtained provided that a quadrature formula of first degree of precision is used in the case of a simple exact eigenvalue the optimum rate of convergence o h for approximate eigenfunctions in the h 1 omega h norm is proved while in the l 2 omega h norm an almost optimum rate of convergence i e near to o h 2 is achieved in both cases a quadrature formula of first degree of precision is used quadrature formulas with degree of precision equal to zero are also analyzed and in the case when the exact eigenfunctions belong only to h 1 omega the convergence without the rate of convergence is proved in the case of a multiple exact eigenvalue the approximate eigenfunctions are compard in contrast to standard considerations with linear combinations of exact eigenfunctions with coefficients not depending on the mesh parameter h

quasi continuum approximations for travelling kinks in diatomic lattices we extend the quasi continuum method of approximation to the case of a diatomic lattice this is illustrated by a lattice in which both small and large atoms interact with first and second nearest neighbours we show that highly accurate quasi continuum techniques may be generalized to determine the shape of nontopological kinks many previous analyses of such systems have found travelling wave solutions only for one particular speed using a second order continuum theory we show i how wave shapes can be found for arbitrary speeds and ii how solution profiles can be calculated from fourth order partial differential equations which approximate the lattice alongside the theoretical analysis we also present numerical simulations of the lattice which demonstrate propagation of the predicted waves through the lattice we show that the particular speed for which solutions have been found in previous studies is a special speed for waves in the lattice but waves can travel for long periods of time at faster or slower speeds whilst slowly relaxing to this critical speed

approximations to solitary waves on lattices iii the monatomic lattice with second neighbour interactions we find new behaviour in a lattice with second neighbour interactions first we find solitary waves with oscillatory spatial decay this is something that cannot occur in lattices with nearest neighbour interactions alone this is found by standard asymptotic analysis which leads to an exact curve in parameter space where this behaviour starts a number of highly accurate quasi continuum approximations are derived and solved one of these suggests a possible method for subsonic solitary waves to cease existing an alternative method of approximation elegantly reveals the differences in shape between subsonic and supersonic solitary waves this is based on the weak form of the differential delay equation for the travelling wave and is derived using the calculus of variations

immersed finite element method for eigenvalue problem abstract we consider the approximation of elliptic eigenvalue problem with an interface the main aim of this paper is to prove the stability and convergence of an immersed finite element method ifem for eigenvalues using crouzeix raviart p 1 nonconforming approximation we show that spectral analysis for the classical eigenvalue problem can be easily applied to our model problem we analyze the ifem for elliptic eigenvalue problems with an interface and derive the optimal convergence of eigenvalues numerical experiments demonstrate our theoretical results

finite element approximation of spectral acoustic problems on curved domains this paper deals with the finite element approximation of the displacement formulation of the spectral acoustic problem on a curved non convex two dimensional domain convergence and error estimates are proved for raviart thomas elements on a discrete polygonal domain h inline equation in the framework of the abstract spectral approximation theory similar results have been previously proved only for polygonal domains numerical tests confirming the theoretical results are reported

laplace eigenvalues on regular polygons a series in 1 n abstract for regular polygons p n inscribed in a circle the eigenvalues of the laplacian converge as n to the known eigenvalues on a circle we compute the leading terms of n in a series in powers of 1 n by applying the calculus of moving surfaces to a piecewise smooth evolution from the circle to the polygon the o 1 n 2 term comes from hadamard s formula and reflects the change in area this term disappears if we transcribe the polygon scaling it to have the same area as the circle

a nonconforming finite element method for a two dimensional curl curl and grad div problem a numerical method for a two dimensional curl curl and grad div problem is studied in this paper it is based on a discretization using weakly continuous p 1 vector fields and includes two consistency terms involving the jumps of the vector fields across element boundaries optimal convergence rates up to an arbitrary positive epsilon in both the energy norm and the l 2 norm are established on graded meshes the theoretical results are confirmed by numerical experiments

finite element approximation of the elasticity spectral problem on curved domains we analyze the finite element approximation of the spectral problem for the linear elasticity equation with mixed boundary conditions on a curved non convex domain in the framework of the abstract spectral approximation theory we obtain optimal order error estimates for the approximation of eigenvalues and eigenvectors two kinds of problems are considered the discrete domain does not coincide with the real one and mixed boundary conditions are imposed some numerical results are presented

nonhomogeneous neumann problem for the poisson equation in domains with an external cusp abstract in this work we analyze the existence and regularity of the solution of a nonhomogeneous neumann problem for the poisson equation in a plane domain with an external cusp in order to prove that there exists a unique solution in h 1 using the lax milgram theorem we need to apply a trace theorem since is not a lipschitz domain the standard trace theorem for h 1 does not apply in fact the restriction of h 1 functions is not necessarily in l 2 so we introduce a trace theorem by using weighted sobolev norms in under appropriate assumptions we prove that the solution of our problem is in h 2 and we obtain an a priori estimate for the second derivatives of the solution

using voxel specific hemodynamic response function in eeg fmri data analysis most existing analytical techniques for eeg fmri data need specific assumptions about the hemodynamic response function hrf these assumptions may not be appropriate when the hrf varies from subject to subject or from region to region in this article we introduce a deconvolution method for eeg fmri activation detection which can be implemented with voxel specific hrfs a comparison of performance is made between three fixed hrfs and the deconvolution method under the framework of the general linear model the main results are as follows 1 the volume of detected regions from the deconvolved hrfs is larger 2 in some subjects the deconvolution technique can find areas of activation that have not been detected with the three fixed hrfs at our threshold of significance 3 deconvolution obtained higher adjusted coefficients of multiple determination compared to those obtained with the three fixed hrfs the results suggest that the fixed hrf methods may not be the most appropriate for the analysis of epileptic activity with eeg fmri and the deconvolution method may be a better choice

error estimate in an isoparametric finite element eigenvalue problem the aim of this paper is to obtain an eigenvalue approximation for elliptic operators defined on a domain i2 with the help of isoparametric finite elements of degree k we prove that i xh 0 h2k provided the boundary of ii is well approximated which is the same estimate as the one obtained in the case of conforming finite elements

a general statistical analysis for fmri data we propose a method for the statistical analysis of fmri data that seeks a compromise between efficiency generality validity simplicity and execution speed the main differences between this analysis and previous ones are a simple bias reduction and regularization for voxel wise autoregressive model parameters the combination of effects and their estimated standard deviations across different runs sessions subjects via a hierarchical random effects analysis using the em algorithm overcoming the problem of a small number of runs session subjects using a regularized variance ratio to increase the degrees of freedom

characterizing the hemodynamic response effects of presentation rate sampling procedure and the possibility of ordering brain activity based on relative timing rapid presentation event related functional mri er fmri allows neuroimaging methods based on hemodynamics to employ behavioral task paradigms typical of cognitive settings however the sluggishness of the hemodynamic response and its variance provide constraints on how er fmri can be applied in a series of two studies estimates of the hemodynamic response in or near the primary visual and motor cortices were compared across various paradigms and sampling procedures to determine the limits of er fmri procedures and more generally to describe the behavior of the hemodynamic response the temporal profile of the hemodynamic response was estimated across overlapping events by solving a set of linear equations within the general linear model no assumptions about the shape were made in solving the equations following estimation of the temporal profile the amplitude and timing were modeled using a gamma function results indicated that 1 within a region for a given subject estimation of the hemodynamic response is extremely stable for both amplitude r 2 0 98 and time to peak r 2 0 95 from one series of measurements to the next and slightly less stable for estimation of time to onset r 2 0 60 2 as the trial presentation rate changed from those spaced 20 s apart to temporally overlapping trials the hemodynamic response amplitude showed a small but significant decrease trial onsets spaced on average 5 s apart showed a 17 25 reduction in amplitude compared to those spaced 20 s apart power analysis indicated that the increased number of trials at fast rates outweighs this decrease in amplitude if statistically reliable response detection is the goal 3 knowledge of the amplitude and timing of the hemodynamic response in one region failed to predict those properties in another region even for within subject comparisons 4 across subjects the amplitude of the response showed no significant correlation with timing of the response for either time to onset or time to peak estimates 5 the within region stability of the response was sufficient to allow offsets in the timing of the response to be detected that were under a second placing event related fmri methods in a position to answer questions about the change in relative timing between regions

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

the impact of temporal regularization on estimates of the bold hemodynamic response function a comparative analysis in fmri data analysis it has been shown that for a wide range of situations the hemodynamic response function hrf can be reasonably characterized as the impulse response function of a linear and time invariant system an accurate and robust extraction of the hrf is essential to infer quantitative information about the relative timing of the neuronal events in different brain regions when no assumptions are made about the hrf shape it is most commonly estimated using time windowed averaging or a least squares estimated general linear model based on either fourier or delta basis functions recently regularization methods have been employed to increase the estimation efficiency of the hrf typically these methods produce more accurate hrf estimates than the least squares approach goutte c nielsen f a hansen l k 2000 modeling the haemodynamic response in fmri using smooth fir filters ieee trans med imag 19 12 1188 1201 here we use simulations to clarify the relative merit of temporal regularization based methods compared to the least squares methods with respect to the accuracy of estimating certain characteristics of the hrf such as time to peak ttp height hr and width w of the response we implemented a bayesian approach proposed by marrelec et al marrelec g benali h ciuciu p pelegrini issac m poline j b 2003 robust estimation of the hemodynamic response function in event related bold fmri using basic physiological information hum brain mapp 19 1 17 marrelec g benali h ciuciu p poline j b bayesian estimation of the hemodynamic of the hemodynamic response function in functional mri in r f editor 2001 melville p 229 247 and its deterministic counterpart based on a combination of tikhonov regularization tikhonov a n arsenin v y 1977 solution of ill posed problems washington dc w h winston and generalized cross validation gcv wahba g 1990 spline models for observational data philadelphia siam for selecting the regularization parameter the performance of both methods is compared with least square estimates as a function of temporal resolution color and strength of the noise and the type of stimulus sequences used in almost all situations under the considered assumptions e g linearity time invariance and smooth hrf the regularization based techniques more accurately characterize the hrf compared to the least squares method our results clarify the effects of temporal resolution noise color and experimental design on the accuracy of hrf estimation

approximations to solitary waves on lattices ii quasi continuum methods for fast and slow waves a number of highly accurate analytic approximations to solitary waves on lattices are detailed and the results of numerical tests presented two types of approximation are considered the first set of approximations are generalizations of the continuum approximation based around various pade expansions of the operator these are convergent to the exact solution in the limit of wavespeed c to 1 or a dense lattice the author also considers ways of constructing approximations to fast solitary waves from a continuum type theory the two types are closely related in both cases significant improvements are made on existing approximations a symplectic hamiltonian integration scheme is used to perform computer simulations a new method of measuring the accuracy of a predicted waveform is also used on new and old approximations alike to compare the various methods considered

existence of weak solutions for the motion of rigid bodies in a viscous fluid we study the evolution of a finite number of rigid bodies within a viscous incompressible fluid in a bounded domain of r d d 2 or 3 with dirichlet boundary conditions by introducing an appropriate weak formulation for the complete problem we prove existence of solutions for initial velocities in h 1 0 omega in the absence of collisions solutions exist for all time in dimension 2 whereas in dimension 3 the lifespan of solutions is infinite only for small enough data

approximations to large amplitude solitary waves on nonlinear electrical lattices abstract in this paper we describe an approximate method to characterise solitary wave solutions of nonlinear lattice equations it is based upon one and two point pade approximations to a series of the real exponential travelling wave solutions of the underlying dispersive system the theory is applied to an example of a lattice system which models an experimental nonlinear transmission line and the results obtained are consistent with numerical simulations even for relatively large amplitude solitary waves the speed amplitude relation is investigated and compared with the derived using quasi continuum methods