the h text p version of the finite element method for domains with curved boundaries this paper analyzes the convergence of the h text p version of the finite element method for elliptic problems on curved domains it proves that when the boundary of the domain is piecewise analytic boundary conditions are piecewise analytic and the sides of the elements are analytic the h text p version has an exponential rate of convergence numerical computations are included the paper generalizes the results previously obtained by the authors the h text p version of finite element method part 1 the basic approximation results comp mech 1 1986 pp 21 41 the h text p version of finite element method part 2 general results and applications comp mech 1 1986 pp 203 220

efficient arnoldi type algorithms for rational eigenvalue problems arising in fluid solid systems we develop and analyze efficient methods for computing damped vibration modes of an acoustic fluid confined in a cavity with absorbing walls capable of dissipating acoustic energy the discretization in terms of pressure nodal finite elements gives rise to a rational eigenvalue problem numerical evidence shows that there are no spurious eigenmodes for such discretization and also confirms that the discretization based on nodal pressures is much more efficient than that based on raviart thomas finite elements for the displacement field the trimmed linearization method is used to linearize the associated rational eigenvalue problem into a generalized eigenvalue problem gep of the form ax lbx for solving the gep we apply arnoldi algorithm to two different types of single matrices b 1a and ab 1 numerical accuracy shows that the application of arnoldi on ab 1 is better than that on b 1a

an a posteriori error estimator for hp adaptive discontinuous galerkin methods for computing band gaps in photonic crystals in this paper we propose and analyze an hp adaptive discontinuous finite element method for computing the band structure of 2d periodic photonic crystals the spectrum of a 2d photonic crystal is approximated by computing the discrete spectrum of members of a family of periodic hermitian eigenvalue problems on the primitive cell parametrized by a two dimensional parameter the quasimomentum we propose a residual based error estimator and show that it is reliable and efficient for all eigenvalue problems in the family in particular we prove that if the error estimator converges to zero then the distance of the computed eigenfunction from the true eigenspace also converges to zero and so the computed eigenvalue converges to a true eigenvalue the results hold for eigenvalues of any multiplicity we illustrate the benefits of the resulting hp adaptive method numerically both for fully periodic crystals and also for crystals with defects

an iterative adaptive finite element method for elliptic eigenvalue problems we consider the task of resolving accurately the nth eigenpair of a generalized eigenproblem rooted in some elliptic partial differential equation pde using an adaptive finite element method fem conventional adaptive fem algorithms call a generalized eigensolver after each mesh refinement step this is not practical in our situation since the generalized eigensolver needs to calculate n eigenpairs after each mesh refinement step it can switch the order of eigenpairs and for repeated eigenvalues it can return an arbitrary linear combination of eigenfunctions from the corresponding eigenspace in order to circumvent these problems we propose a novel adaptive algorithm that only calls a generalized eigensolver once at the beginning of the computation and then employs an iterative method to pursue a selected eigenvalue eigenfunction pair on a sequence of locally refined meshes both picard s and newton s variants of the iterative method are presented the underlying partial differential equation pde is discretized with higher order finite elements hp fem but the algorithm also works for standard low order fem the method is described and accompanied with theoretical analysis and numerical examples instructions on how to reproduce the results are provided

solving elliptic eigenvalue problems on polygonal meshes using discontinuous galerkin composite finite element methods in this paper we introduce a discontinuous galerkin method on polygonal meshes this method arises from the discontinuous galerkin composite finite element method dgfem for source problems on domains with micro structures in the context of the present paper the flexibility of dgfem is applied to handle polygonal meshes we prove the a priori convergence of the method for both eigenvalues and eigenfunctions for elliptic eigenvalue problems numerical experiments highlighting the performance of the proposed method for problems with discontinuous coefficients and on convex and non convex polygonal meshes are presented

numerical realization of dirichlet to neumann transparent boundary conditions for photonic crystal wave guides the computation of guided modes in photonic crystal wave guides is a key issue in the process of designing devices in photonic communications existing methods such as the super cell method provide an efficient computation of well confined modes however if the modes are not well confined the modelling error of the super cell method becomes prohibitive and advanced methods applying transparent boundary conditions for periodic media are needed in this work we demonstrate the numerical realization of a recently proposed dirichlet to neumann approach and compare the results with those of the super cell method for the resulting non linear eigenvalue problem we propose an iterative solution based on newton s method and a direct solution using chebyshev interpolation of the non linear operator based on the dirichlet to neumann approach we present a formula for the group velocity of guided modes that can serve as an objective function in the optimization of photonic crystal wave guides

band structure of absorptive photonic crystals the band structure of absorptive dielectric photonic crystals is investigated provided the frequency dependent electric permeability e x satisfies certain analyticity requirements as a function of frequency we show that no bandgaps exist in frequency regions where absorption takes place i e where e x has a non zero imaginary part in this case real eigenvalues of the helmholtz operator in the bloch decomposed formalism are absent using a suitable analytic continuation procedure we find that the former change into resonances i e complex numbers depending on k the wavevector from the first brillouin zone thus leading to complex bands in the lower half plane this is confirmed numerically for a simple one dimensional example

adaptive finite element methods for elliptic equations with non smooth coefficients nous considerons une equation elliptique du second ordre a coefficients discontinus ou anisotropes dans un domaine borne en dimension 2 ou 3 et sa discretisation par elements finis le but de cet article est de demontrer des estimations d erreur a priori et a posteriori dans une norme appropriee qui soient independantes de la variation des coefficients

periodic schrodinger operators with local defects and spectral pollution this article deals with the numerical calculation of eigenvalues of perturbed periodic schrodinger operators located in spectral gaps such operators are encountered in the modeling of the electronic structure of crystals with local defects and of photonic crystals the usual finite element galerkin approximation is known to give rise to spectral pollution in this article we give a precise description of the corresponding spurious states we then prove that the supercell model does not produce spectral pollution finally we extend results by lewin and sere on some no pollution criteria in particular we prove that using approximate spectral projectors enables one to eliminate spectral pollution in a given spectral gap of the reference periodic schrodinger operator

a wider class of stable gyroscopic systems abstract it is well known that in a physically interesting sense n n gyroscopic systems of the form l 2 i b c where c 0 and b is indefinite and invertible are stable whenever b ki k 1 c for some real k 0 it is shown that stability is retained under a considerably weaker condition formulated in terms of the spectral radius of b 1 i 1 c

numerical algorithms for undamped gyroscopic systems abstract the solutions of a gyroscopic vibrating system oscillating about an equilibrium position with no external applied forces and no damping forces are completely determined by the quadratic eigenvalue problem 2 i m i g k x i 0 for i 1 2 n where m g and k are real n n matrices and m is symmetric positive definite denoted by m 0 g is skew symmetric and either k 0 or k 0 gyroscopic system in motion about a stable equilibrium position with k 0 are well understood two lanczos type algorithms the pseudo skew symmetric lanczos algorithm and the j lanczos algorithm are studied for computing some extreme eigenpairs for solving gyroscopic systems in motion about an unstable equilibrium position with k 0 shift and invert strategies error bounds implementation issues and numerical results for both algorithms are presented in details

indecomposable higher chow cycles on jacobians we construct some natural indecomposable elements of ch g j c 1 with trivial regulator and in particular prove that ch g ind j c 1 mb q is uncountable for c a generic curve or a generic hyperelliptic curve of genus g geq 3

a posteriori discontinuous galerkin error estimator for linear elasticity abstract this paper presents for the first time the derivation of an hp a posteriori error estimator for the symmetric interior penalty discontinuous galerkin finite element method for linear elastic analysis any combination of neumann and dirichlet boundary conditions are admissible in the formulation including applying neumann and dirichlet on different components on the same region of the boundary therefore the error estimator is applicable to a variety of physical problems the error estimator is incorporated into an hp adaptive finite element solver and verified against smooth and non smooth problems with closed form analytical solutions as well as being demonstrated on a non smooth problem with complex boundary conditions the hp adaptive finite element analyses achieve exponential rates of convergence the performances of the hp adaptive scheme are contrasted against uniform and adaptive h refinement this paper provides a complete framework for adaptivity in the symmetric interior penalty discontinuous galerkin finite element method for linear elastic analysis

finite element galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems abstract this paper establishes some refined estimates for the approximation of the eigenvalues and eigenvectors of selfadjoint eigenvalue problems by finite element or more generally galerkin methods suppose lambda is an eigenvalue of multiplicity q of a selfajoint problem and let m lambda denote the space of eigenvectors corresponding to lambda keywords hilbert space

spectral approximation of quadratic operator polynomials arising in photonic band structure calculations galerkin spectral approximation theory for non self adjoint quadratic operator polynomials with periodic coefficients is considered the main applications are complex band structure calculations in metallic photonic crystals periodic waveguides and metamaterials we show that the spectrum of the considered operator polynomials consists of isolated eigenvalues of finite multiplicity with a nonzero imaginary part the spectral problem is equivalent to a non compact block operator matrix and norm convergence is shown for a block operator matrix having the same generalized eigenvectors as the original operator convergence rates of finite element discretizations are considered and numerical experiments with the p p version and the h h version of the finite element method confirm the theoretical convergence rates

analysis of quasi monte carlo methods for elliptic eigenvalue problems with stochastic coefficients we consider the forward problem of uncertainty quantification for the generalised dirichlet eigenvalue problem for a coercive second order partial differential operator with random coefficients motivated by problems in structural mechanics photonic crystals and neutron diffusion the pde coefficients are assumed to be uniformly bounded random fields represented as infinite series parametrised by uniformly distributed i i d random variables the expectation of the fundamental eigenvalue of this problem is computed by a truncating the infinite series which define the coefficients b approximating the resulting truncated problem using lowest order conforming finite elements and a sparse matrix eigenvalue solver and c approximating the resulting finite but high dimensional integral by a randomly shifted quasi monte carlo lattice rule with specially chosen generating vector we prove error estimates for the combined error which depend on the truncation dimension s the finite element mesh diameter h and the number of quasi monte carlo samples n under suitable regularity assumptions our bounds are of the particular form mathcal o h 2 n 1 delta where delta 0 is arbitrary and the hidden constant is independent of the truncation dimension which needs to grow as h rightarrow 0 and n rightarrow infty as for the analogous pde source problem the conditions under which our error bounds hold depend on a parameter p in 0 1 representing the summability of the terms in the series expansions of the coefficients although the eigenvalue problem is nonlinear which means it is generally considered harder than the source problem in almost all cases p ne 1 we obtain error bounds that converge at the same rate as the corresponding rate for the source problem the proof involves a detailed study of the regularity of the fundamental eigenvalue as a function of the random parameters as a key intermediate result in the analysis we prove that the spectral gap between the fundamental and the second eigenvalues is uniformly positive over all realisations of the random problem

on the spectrum of an operator pencil with applications to wave propagation in periodic and frequency dependent materials we study wave propagation in periodic and frequency dependent materials when the medium in a frequency interval is characterized by a real valued permittivity the spectral parameter relates to the quasi momentum which leads to spectral analysis of a quadratic operator pencil where frequency is a parameter we show that the underlying operator has a discrete spectrum where the eigenvalues are symmetrically placed with respect to the real and imaginary axis moreover we discretize the operator pencil with finite elements and use a krylov space method to compute eigenvalues of the resulting large sparse matrix pencil

finite element analysis of a quadratic eigenvalue problem arising in dissipative acoustics a quadratic eigenvalue problem arising in the determination of the vibration modes of an acoustic fluid contained in a cavity with absorbing walls is considered the problem is shown to be equivalent to the spectral problem for a noncompact operator and a thorough spectral characterization is given a numerical discretization based on raviart thomas finite elements is analyzed the method is proved to be free of spurious modes and to converge with optimal order implementation issues and numerical experiments confirming the theoretical results are reported

examples of smash nilpotent cycles on rationally connected varieties abstract voevodsky has conjectured that numerical and smash equivalence coincide on a smooth projective variety we prove this conjecture holds for uniruled 3 folds 4 folds whose mrcc quotient has dimension 2 and for smooth complete intersections with very small degree

a pade approximate linearization algorithm for solving the quadratic eigenvalue problem with low rank damping a pad e approximate linearization algorithm for solving the quadratic eigenvalue problem with low rank damping ding lu xin huang zhaojun bai yangfeng su june 30 2014 abstract the low rank damping term appears commonly in quadratic eigenvalue problems arising from physical simulations to exploit the low rank damping property we propose a pad e approximate linearization pal algorithm the advantage of the pal algorithm is that the dimension of the resulting linear eigenvalue problem is only n lm which is generally substantially smaller than the dimension 2n of the linear eigenvalue problem produced by a direct linearization approach where n is the dimension of the quadratic eigenvalue problem l and m are the rank of the damping matrix and the order of a pad e approximant respectively numerical examples show that by exploiting the low rank damping property the pal algorithm runs 33 47 faster than the direct linearization approach for solving modest size quadratic eigenvalue problems introduction we consider the quadratic eigenvalue problem qep q x 2 m c k x 0 where m c and k are n n matrices referred to as mass damping and stiffness matrices respectively in structural dynamics analysis the low rank damping property refers to the case where the damping matrix c is of rank l l n and admits the rank revealing decomposition c ef t where e and f are n l full column rank matrices the qep with the low rank damping arises frequently from analysis of structural dynam ics 10 18 and structural acoustic interaction 3 6 32 in these applications the damping school of mathematical sciences fudan university shanghai 200433 p r china dinglu fudan edu cn part of this work was done while this author was visiting at university of california davis supported by china scholarship council school of mathematical sciences fudan university shanghai 200433 china xinhuang fudan edu cn part of this work was done while this author was visiting at university of california davis supported by china scholarship council department of computer science and department of mathematics university of california davis ca 95616 usa bai cs ucdavis edu school of mathematical sciences fudan university shanghai 200433 china yfsu fudan edu cn

finite element vibration analysis of fluid solid systems without spurious modes this paper deals with the finite element approximation of the vibration modes of a problem with fluid structure interaction displacement variables are used for both the fluid and the solid to avoid the typical spurious modes of this formulation we introduce a nonconforming discretization error estimates for the approximation of eigenvalues and eigenvectors are given

murre s conjectures and explicit chow kunneth projectors for varieties with a nef tangent bundle in this paper we investigate murre s conjectures on the structure of rational chow groups and exhibit explicit chow kiinneth projectors for some examples more precisely the examples we study are the varieties which have a nef tangent bundle for surfaces and threefolds which have a nef tangent bundle explicit chow kunneth projectors are obtained which satisfy murre s conjectures and the motivic hard lefschetz theorem is verified

motives and filtrations on chow groups in this paper sequential to sa we discuss a philosophy of bloch and beilinson concerning filtrations on chow groups of algebraic vareties which provides us with a good understanding on the structure of at least certain portions of chow groups let k be a fixed ground field and let c be the category of smooth projective schemes over k with a fixed suitable weil cohomology theory cf notations x in mathcal c rightsquigarrow h x mathop oplus limits i in z h i x

hp adaptive composite discontinuous galerkin methods for elliptic eigenvalue problems on complicated domains in this paper we develop the a posteriori error estimation of hp adaptive discontinuous galerkin composite finite element methods dgfems for the discretization of second order elliptic eigenvalue problems dgfems allow for the approximation of problems posed on computational domains which may contain local geometric features the dimension of the composite finite element space is independent of the number of geometric features this is in contrast with standard finite element methods as the minimal number of elements needed to represent the underlying domain can be very large and so the dimension of the finite element space computable upper bounds on the error for both eigenvalues and eigenfunctions are derived numerical experiments highlighting the practical application of the proposed estimators within an automatic hp adaptive refinement procedure will be presented

linearization techniques for band structure calculations in absorbing photonic crystals band structure calculations for photonic crystals require the numerical solution of eigenvalue problems in this paper we consider crystals composed of lossy materials with frequency dependent per

modified edge finite elements for photonic crystals we consider maxwell s equations with periodic coefficients as it is usually done for the modeling of photonic crystals using bloch floquet theory the problem reduces in a standard way to a modification of the maxwell cavity eigenproblem with periodic boundary conditions following 8 a modification of edge finite elements is considered for the approximation of the band gap the method can be used with meshes of tetrahedrons or parallelepipeds a rigorous analysis of convergence is presented together with some preliminary numerical results in 2d which fully confirm the robustness of the method the analysis uses well established results on the discrete compactness for edge elements together with new sharper interpolation estimates

a convergent adaptive method for elliptic eigenvalue problems we prove the convergence of an adaptive linear finite element method for computing eigenvalues and eigenfunctions of second order symmetric elliptic partial differential operators the weak form is assumed to yield a bilinear form which is bounded and coercive in h 1 each step of the adaptive procedure refines elements in which a standard a posteriori error estimator is large and also refines elements in which the computed eigenfunction has high oscillation the error analysis extends the theory of convergence of adaptive methods for linear elliptic source problems to elliptic eigenvalue problems and in particular deals with various complications which arise essentially from the nonlinearity of the eigenvalue problem because of this nonlinearity the convergence result holds under the assumption that the initial finite element mesh is sufficiently fine

a multiscale hp fem for 2d photonic crystal bands a multiscale generalised hp finite element method msfem for time harmonic wave propagation in bands of locally periodic media of large but finite extent e g photonic crystal phc bands is presented the method distinguishes itself by its size robustness i e to achieve a prescribed error its computational effort does not depend on the number of periods the proposed method shows this property for general incident fields including plane waves incident at a certain angle to the infinite crystal surface and at frequencies in and outside of the bandgap of the phc the proposed msfem is based on a precomputed problem adapted multiscale basis this basis incorporates a set of complex bloch modes the eigenfunctions of the infinite phc which are modulated by macroscopic piecewise polynomials on a macroscopic fe mesh the multiscale basis is shown to be efficient for finite phc bands of any size provided that boundary effects are resolved with a simple macroscopic boundary layer mesh the msfem constructed by combing the multiscale basis inside the crystal with some exterior discretisation is a special case of the generalised finite element method g fem for the rapid evaluation of the matrix entries we introduce a size robust algorithm for integrals of quasi periodic micro functions and polynomial macro functions size robustness of the present msfem in both the number of basis functions and the computation time is verified in extensive numerical experiments

efficient and reliable hp fem estimates for quadratic eigenvalue problems and photonic crystal applications we present a posteriori analysis of higher order finite element approximations hp fem for quadratic fredholm valued operator functions residual estimates for approximations of the algebraic eigenspaces are derived and we reduce the analysis of the estimator to the analysis of an associated boundary value problem for the reasons of robustness we also consider approximations of the associated invariant pairs we show that our estimator inherits the efficiency and reliability properties of the underlying boundary value estimator as a model problem we consider spectral problems arising in analysis of photonic crystals in particular we present an example where a targeted family of eigenvalues cannot be guaranteed to be semisimple numerical experiments with hp fem show the predicted convergence rates the measured effectivities of the estimator compare favorably with the performance of the same estimator on the associated boundary value problem we also present a benchmark estimator based on the dual weighted residual dwr approach which is more expensive to compute but whose measured effectivities are close to one

the zero section of the universal semiabelian variety and the double ramification cycle we study the chow ring of the boundary of the partial compactification of the universal family of principally polarized abelian varieties ppav we describe the subring generated by divisor classes and we compute the class of the partial compactification of the universal zero section which turns out to lie in this subring our formula extends the results for the zero section of the universal uncompactified family the partial compactification of the universal family of ppav can be thought of as the first two boundary strata in any toroidal compactification of ag our formula provides a first step in a program to understand the chow groups of a g especially of the perfect cone compactification by induction on genus by restricting to the image of mg under the torelli map our results extend the results of hain on the double ramification cycle answering eliashberg s question

convergence analysis of planewave expansion methods for 2d schrodinger operators with discontinuous periodic potentials in this paper we consider the problem of computing the spectrum of a two dimensional schrodinger operator with discontinuous periodic potential in two dimensions using fourier or planewave expansion methods problems of this kind are currently of great interest in the design of new optical devices to determine band gaps and to compute localized modes in photonic crystal materials although fourier methods may not be every applied mathematician s first choice for this problem because of the discontinuities in the potential we will show here that even though as expected the convergence is not exponential the method has several desirable features that make it competitive with other discretization techniques such as finite element methods both with respect to implementation and convergence properties in particular we will prove that simple preconditioners for the system matrix are optimal leading to a computational complexity of mathcal o n log n in the number of planewaves n using the fast fourier transform moreover we derive sharp error estimates that show that the method is essentially third order in the eigenvalues and of order frac 3 2 in the eigenfunctions in the h 1 norm and frac 5 2 in the l 2 norm to improve the planewave expansion method in the case of discontinuous potentials it has been proposed in the physics literature to replace the discontinuous potential with an effective potential that is smooth despite the additional error this incurs we will here answer the question whether this smoothing is worth it in fact our convergence analysis of the modified method provides an optimal choice for the smoothing parameter but it also shows that the overall rate of convergence is no faster than before and so smoothing does not seem to be worth it all the theoretical results are confirmed in our numerical experiments

on spurious solutions in finite element approximations of resonances in open systems in this paper we discuss problems arising when computing resonances with a finite element method in the pre asymptotic regime we detect for the one dimensional case spurious solutions in finite

spectral approximation for polynomial eigenvalue problems abstract in this paper we extent the classical spectral approximation theory for compact and bounded operators to general linear operators and then apply it to polynomial eigenvalue problems pep we also study the essential spectrum in peps and prove that this spectrum is stable under relatively compact perturbations based on this analysis we give some suggestions to make algorithms for solving peps more efficient

band gap calculations of photonic crystals by singular boundary method this paper applies the singular boundary method sbm to compute band gap structures of two dimensional 2d photonic crystals by using the fundamental solutions a linear eigenvalue equation is established to describe the relationship between the block wave and frequency for given frequencies the solution of the equation yields the related block wave vectors the convergence accuracy and efficiency are carefully examined through several benchmark cases

accurate pressure post process of a finite element method for elastoacoustics this paper deals with a post process to obtain a more accurate approximation of the fluid pressure from a finite element computation of the vibration modes of a fluid structure coupled system the underlying finite element method based on a displacement formulation for both media consists of using raviart thomas elements for the fluid combined with standard continuous elements for the solid an easy to compute post process of the pressure is derived the relation between this post process and an alternative finite element approximation of the problem based on discretizing the fluid pressure by enriched crouzeix raviart elements is studied higher order estimates for the l2 norm of the post processed pressure are proved by exploiting this relation as a by product higher order l2 estimates for the solid displacements obtained with the original method are also proved

adaptive finite element methods for computing band gaps in photonic crystals in this paper we propose and analyse adaptive finite element methods for computing the band structure of 2d periodic photonic crystals the problem can be reduced to the computation of the discrete spectra of each member of a family of periodic hermitian eigenvalue problems on a unit cell parametrised by a two dimensional parameter the quasimomentum these eigenvalue problems involve non coercive elliptic operators with generally discontinuous coefficients and are solved by adaptive finite elements we propose an error estimator of residual type and show it is reliable and efficient for each eigenvalue problem in the family in particular we prove that if the error estimator converges to zero then the distance of the computed eigenfunction from the true eigenspace also converges to zero and the computed eigenvalue converges to a true eigenvalue with double the rate we also prove that if the distance of a computed sequence of approximate eigenfunctions from the true eigenspace approaches zero then so must the error estimator the results hold for eigenvalues of any multiplicity we illustrate the benefits of the resulting adaptive method in practice both for fully periodic structures and also for the computation of eigenvalues in the band gap of structures with defect using the supercell method

determination of electromagnetic bloch variety in a medium with frequency dependent coefficients abstract we provide a functional framework and a numerical algorithm to compute the bloch variety for maxwell s equations when the electric permittivity is frequency dependent we incorporate the idea of a mixed formulation for maxwell s equations to obtain a quadratic eigenvalue for the wave vector in terms of the frequency we reformulate this problem as a larger linear eigenvalue problem and prove that this results in the need to compute eigenvalues of a compact operator using finite elements we provide preliminary numerical examples of the scheme for both frequency independent and frequency dependent permittivity

mixed finite element methods in fluid structure systems the eigenvalue problem describing the frequencies of a fluid vibrating in a rigid cavity or within moving boundaries is considered based on the method of lagrange multipliers a three field mixed formulation is introduced in order to avoid the spurious circulating modes stability and optimal error bounds are proved for two choices of finite element spaces

universal algebraic equivalences between tautological cycles on jacobians of curves we present a collection of algebraic equivalences between tautological cycles on the jacobian j of a curve i e cycles in the subring of the chow ring of j generated by the classes of certain standard subvarieties of j these equivalences are universal in the sense that they hold for all curves of given genus we show also that they are compatible with the action of the fourier transform on tautological cycles and compute this action explicitly

the waveguide eigenvalue problem and the tensor infinite arnoldi method we present a new computational approach for a class of large scale nonlinear eigenvalue problems neps that are nonlinear in the eigenvalue the contribution of this paper is twofold we derive a new iterative algorithm for neps the tensor infinite arnoldi method tiar which is applicable to a general class of neps and we show how to specialize the algorithm to a specific nep the waveguide eigenvalue problem the waveguide eigenvalue problem arises from a finite element discretization of a partial differential equation used in the study waves propagating in a periodic medium the algorithm is successfully applied to accurately solve benchmark problems as well as complicated waveguides we study the complexity of the specialized algorithm with respect to the number of iterations m and the size of the problem n both from a theoretical perspective and in practice for the waveguide eigenvalue problem we establish that the computationally dominating part of the algorithm has complexity mathcal o

a new finite element formulation for internal acoustic problems with dissipative walls this work concerns the variational formulation and the numerical computation of internal acoustic problems with absorbing walls the originality of the proposed approach compared to other existing methods is the introduction of the normal fluid displacement field on the damped walls this additional variable allows to transpose formulations in frequency domain to time domain when the fluid is described by a scalar field pressure or fluid displacement potential with this new scalar unknown various absorbing wall models can be introduced in the variational formulation moreover the associated finite element matrix system in symmetric form can be solved in frequency and time domain

higher picard varieties and the height pairing let x be a smooth projective variety which is defined over a number field beilinson and bloch have defined under suitable asssumptions height pairings between chow groups of homologically trivial cycles on x beilinson has also formulated a hard lefschetz and a hodge index conjecture for these chow groups we show that the restriction of the height pairing to cycles algebraically equivalent to zero can be computed via abel jacobi maps in terms of the neron tate height pairing on the higher picard varieties of x this description is used in the case where x is an abelian variety to prove a consequence of beilinson conjectures namely we prove a hard lefschetz and a hodge index theorem for the groups of cycles algebraically equivalent to zero modulo incidence equivalence

a dirichlet to neumann approach for the exact computation of guided modes in photonic crystal waveguides this work deals with one dimensional infinite perturbation namely line defects in periodic media in optics such defects are created to construct an open waveguide that concentrates light the existence and computation of the eigenmodes is a crucial issue this is related to a self adjoint eigenvalue problem associated to a pde in an unbounded domain in the directions orthogonal to the line defect which makes both the analysis and the computations more complex using a dirichlet to neumann approach we show that this problem is equivalent to one set on a small neighborhood of the defect contrary to existing methods this one is exact but there is a price to be paid the reduction of the problem leads to a nonlinear eigenvalue problem of a fixed point nature 2013 society for industrial and applied mathematics

displacement pressure based mixed finite element formulations for acoustic fluid structure interaction problems summary we present reliable finite element discretizations based on displacement pressure interpolations for the analysis of acoustic fluid structure interaction problems the finite element interpolations are selected using the inf sup condition and emphasis is given to the fact that the boundary conditions must satisfy the mass and momentum conservation we show that with our analysis procedure no spurious non zero frequencies are encountered as heretofore calculated with other displacement based discretizations 1997 by john wiley sons ltd

a virtual element method for the acoustic vibration problem we analyze in this paper a virtual element approximation for the acoustic vibration problem we consider a variational formulation relying only on the fluid displacement and propose a discretization by means of mathrm h mathrm div h div virtual elements with vanishing rotor under standard assumptions on the meshes we show that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates with this end we prove approximation properties of the proposed virtual elements we also report some numerical tests supporting our theoretical results

vibration analysis of fluid solid systems using a finite element displacement formulation this report presents a finite element solution for the vibration interaction between an inviscid fluid and a solid the equation of motion governing the inviscid fluid is expressed in terms of the displacements reduced integration is used when computing the element stiffness matrix contributed by the fluid in addition a projection is used on the element mass matrix in order to remove the spurious modes which result from the use of reduced integration

finite element computation of three dimensional elastoacoustic vibrations abstract in this paper the interior elastoacoustic problem in a 3d domain is solved displacement variables are used for both the fluid and the solid to avoid the typical spurious modes of this formulation a non standard discretization is used consisting of classical linear tetrahedral finite element for the solid and raviart thomas elements of lowest order for the fluid a new unknown is introduced on the interface between solid and fluid to impose the transmission conditions

computation of the band structure of two dimensional photonic crystals with hp finite elements the band structure of 2d photonic crystals a periodic material with discontinuous dielectrical properties and their eigenmodes can be efficiently computed with the finite element method fem for second order elliptic boundary value problems with piecewise analytic coefficients it is known that the solution converges extremly fast i e exponentially when using em p fem for smooth and em hp fem for polygonal interfaces and boundaries in this article we discretise the variational eigenvalue problems for photonic crystals with smooth and polygonal interfaces in scalar variables with quasi periodic boundary conditions by means of em p and em hp fem this for the transverse electric te and transverse magnetic tm modes our computations show exponential convergence of the numerical eigenvalues for smooth and polygonal lines of discontinuity of dielectric material properties

the order of convergence of eigenfrequencies in finite element approximations of fluid structure interaction problems in this paper we prove a double order for the convergence of eigenfrequencies in fluid structure vibration problems we improve estimates given recently for compressible and incompressible fluids to do this we extend classical results on finite element spectral approximation to nonconforming methods for noncompact operators

convergence of the supercell method for defect modes calculations in photonic crystals we present a rigorous study of the convergence of the supercell method used for determing defect modes in photonic crystals with compactly supported perturbations transverse electric and transverse magnetic polarized waves are investigated in 2 d structures we prove an exponential convergence of the defect frequencies with the supercell size and give a justification of the quasi independence of the corresponding eigenfunctions on the wave vector we also give a characterization of the supercell eigenvalues corresponding to the background photonic crystal

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

on occurrence of spectral edges for periodic operators inside the brillouin zone the paper discusses the following frequently arising question on the spectral structure of periodic operators of mathematical physics e g schrodinger maxwell waveguide operators etc is it true that one can obtain the correct spectrum by using the values of the quasimomentum running over the boundary of the reduced brillouin zone only rather than the whole zone or do the edges of the spectrum occur necessarily at the set of corner high symmetry points this is known to be true in 1d while no apparent reasons exist for this to be happening in higher dimensions in many practical cases though this appears to be correct which sometimes leads to the claims that this is always true there seems to be no definite answer in the literature and one encounters different opinions about this problem in the community in this paper starting with simple discrete graph operators we construct a variety of convincing multiply periodic examples showing that the spectral edges might occur deeply inside the brillouin zone on the other hand it is also shown that in a generic case the situation of spectral edges appearing at high symmetry points is stable under small perturbations this explains to some degree why in many maybe even most practical cases the statement still holds

mixed 3 manifolds are virtually special letm be a compact oriented irreducible 3 manifold which is neither a graph manifold nor a hyperbolic manifold we prove that 1m is virtually special

discrete perturbation estimates for eigenpairs of fredholm operator valued functions we present perturbation estimates for eigenvalue and eigenvector approximations for a class of fredholm operator valued functions our approach is based on perturbation estimates for the generalized resolvents and the exponential convergence of the contour integration by the trapezoidal rule we use discrete residual functions to estimate the resolvents a posteriori numerical experiments are also presented

the h p version of the finite element method for problems with interfaces regularities of the solutions of interface problems in two dimensions are described in the frame of the weighted sobolev spaces and countably normed spaces based upon the regularity of solutions the geometric meshes and the distribution of polynomial degrees are properly designed so that the h p version of the finite element method for interface problems can lead to the exponential rate of convergence numerical results on an elliptic equation with interfaces are presented the optimal mesh factor optimal degree factors and optimal layer factors of the geometric mesh in neighbourhoods of singular points having varied intensities are discussed from both theoretical and practical point of view

spectral element method for band structures of three dimensional anisotropic photonic crystals a spectral element method sem is introduced for accurate calculation of band structures of three dimensional anisotropic photonic crystals the method is based on the finite element framework with curvilinear hexahedral elements gauss lobatto legendre polynomials are used to construct the basis functions in order to suppress spurious modes mixed order vector basis functions are employed and the bloch periodic boundary condition is imposed into the basis functions with tangential components at the boundary by multiplying a bloch phase factor the fields and coordinates in the curvilinear hexahedral elements are mapped to the reference domain by covariant mapping which preserves the continuity of tangential components of the field numerical results show that the sem has exponential convergence for both square lattice and triangular lattice photonic crystals the sampling density as small as 3 4 points per wavelength can achieve accuracy as high as 99 9 the band structures of several modified woodpile photonic crystals are calculated by using the sem

finite element computation of the vibration modes of a fluid solid system abstract in this paper we solve the interior elastoacoustic problem by a finite element method which does not present spurious or circulation modes for nonzero frequencies it consists of classical triangular lagrangian elements for the solid and lowest order triangular raviart thomas elements for the fluid transmission conditions at the fluid solid interface are taken into account in a weak sense numerical results for some test examples are presented which show the good performance of the proposed methodology

resolvent method for computations of localized defect modes of h polarization in two dimensional photonic crystals we have developed and tested a version of the resolvent or green s function method based on the shift inverse of the maxwellian operator that ensures stable convergence of iterative computations of localized defect modes of h polarization in two dimensional 2d photonic crystals the defect states are obtained by solving the eigenvalue problem for an associated compact operator with the expansion in bloch eigenfunctions of the unperturbed maxwellian operator this method can be extended to 3d photonic crystals we apply the method to a 2d square lattice of square dielectric rods in a dielectric background and compute with controlled precision of 1 the defect modes induced by the replacement of one rod the defect we investigate the rise and variation of the defect frequencies in a photonic band gap caused by the increase of the dielectric strength of the defect for four branches of localized modes of various symmetries

estimates for the errors in eigenvalue and eigenvector approximation by galerkinmethods with particular attention to the case of multiple eigenvalues refined estimates for the errors in eigenvalue and eigenvector approximation by finite element or more generally galerkin methods as they apply to selfadjoint problems are presented particular attention is given to the case of multiple eigenvalues the results are new in this case the proof is based on a novel approach which yields the known results for simple eigenvalues in a simple way numerical computations are presented and analyzed in light of the theoretical results

an oscillation free adaptive fem for symmetric eigenvalue problems a refined a posteriori error analysis for symmetric eigenvalue problems and the convergence of the first order adaptive finite element method afem is presented the h 1 stability of the l 2 projection provides reliability and efficiency of the edge contribution of standard residual based error estimators for p 1 finite element methods in fact the volume contributions and even oscillations can be omitted for courant finite element methods this allows for a refined averaging scheme and so improves mao et al in adv comput math 25 1 3 135 160 2006 the proposed afem monitors the edge contributions in a bulk criterion and so enables a contraction property up to higher order terms and global convergence numerical experiments exploit the remaining l 2 error contributions and confirm our theoretical findings the averaging schemes show a high accuracy and the afem leads to optimal empirical convergence rates

on the chow motive of an abelian scheme with non trivial endomorphisms let x be an abelian scheme over a base variety s with endomorphism algebra d we prove that the relative chow motive r x s has a canonical decomposition as a direct sum of motives r whereruns over an explicitly determined finite set of irreducible representations of the group d op such that r seen as a functor from chow motives to d op representations is isotypic our decomposition refines the motivic decomposition of deninger and murre as well as beauville s decomposition of the chow group the second main result is that we construct a canonical generalized motivic lefschetz decomposition inspired by work of looijenga and lunts in 17 the role of sl2 in the classical theory is here replaced by a larger lie algebra generated by all lefschetz and lambda operators associated to non degenerate line bundles we construct an action of this larger lie algebra on r x s and deduce from this a generalized lefschetz decomposition we also give a precise structure result for the lefschetz components that arise as an application of our techniques we provide a positive answer to a question of claire voisin concerning the analogue for abelian varieties of a conjecture of beauville ams 2010 mathematics subject classification 14c15 14k05