stability properties of implicit runge kutta methods new stability concepts bs stability and bsi stability internal bs stability are introduced bs stability is a modification and extension of b stability and enables the derivation of order results for runge kutta methods applied to general nonlinear stiff initial value problems these order results b consistency b convergence are not affected by stiffness several classes of implicit runge kutta methods are shown to be bs stable gauss radau ia and radau iia schemes

sharp eigenvalue enclosures for the perturbed angular kerr newman dirac operator we examine a certified strategy for determining sharp intervals of enclosure for the eigenvalues of matrix differential operators with singular coefficients the strategy relies on computing the second order spectrum relative to subspaces of continuous piecewise linear functions for smooth perturbations of the angular kerr newman dirac operator explicit rates of convergence linked to regularity of the eigenfunctions are established numerical tests which validate and sharpen by several orders of magnitude the existing benchmarks are also included

physics conforming constraints oriented numerical method a general high order finite element method for solving partial differential equations called the constraints oriented library cool method is presented this hp approach takes into account the underlying nature of the corresponding physical problem and thus avoids the generation of nonphysical solutions in the cool method all terms in a variational form are represented by the same functional dependence and by the same regularity thus eliminating regularity constraints imposed by standard numerical methods external constraints such as the incompressibility condition appearing in the maxwell or navier stokes equations can then be satisfied identically and are eliminated algebraically this reduces the number of variables and leads to well conditioned matrix problems the consequence is that only physically relevant solutions remain the cool method also satisfies automatically internal constraints such as occur for the grad div and curl curl operators and this for any geometry this approach can be applied to a wide range of physical problems including fluid flows electromagnetics material sciences ideal linear magnetohydrodynamic stability analysis and alfven wave heating of fusion plasmas results obtained by applying the cool method to the grad div and curl curl operators the stokes problem and the steady and unsteady navier stokes equations are presented

a high order isoparametric finite element method for the computation of waveguide modes we have studied the approximation of optical waveguide eigenvalues by a high order isoparametric vector finite element method isoparametric mappings are used for the approximation of domains with curved boundaries or curved material interfaces eigenvalue convergence for curved elements is investigated numerical results verify the predicted order of convergence and show the remarkable accuracy of the method

order results for implicit runge kutta methods applied to stiff systems an extensive order analysis is given for the local and global errors of certain classes of runge kutta methods applied to general nonlinear stiff initial value problems it turns out that there are three order levels for linear problems y ay the classical orders apply for general nonlinear problems however two types of order reductions occur there a distinction has to be made between stiff problems where only f y is large and stiff problems where also some of the other derivatives f t f tt f yy cdots are large for these two classes of problems order results can be guaranteed which differ only by one power of the stepsize parameter h the order levels for both classes of problems are considerably lower than the respective conventional orders these order results are optimal in the sense that higher orders would contradict numerical observations made with particular initial value problems

eigenvalue approximation by mixed and hybrid methods note univ maryland dept math college pk md 20742 ecole polytech ctr math appl f 91128 palaiseau france univ paris 6 f 75230 paris 05 france mercier b cea limeil f 94190 villeneuve france isi document delivery no lq584times cited 29cited reference count 44 reference asn article 1981 005doi 10 2307 2007651 record created on 2006 08 24 modified on 2017 05 12

verification of the ordinary character table of the baby monster abstract we prove the correctness of the character table of the sporadic simple baby monster group that is shown in the atlas of finite groups

non consistent approximations of self adjoint eigenproblems application to the supercell method in this article we introduce a general theoretical framework to analyze non consistent approximations of the discrete eigenmodes of a self adjoint operator we focus in particular on the discrete eigenvalues laying in spectral gaps we first provide a priori error estimates on the eigenvalues and eigenvectors in the absence of spectral pollution we then show that the supercell method for perturbed periodic schrodinger operators falls into the scope of our study we prove that this method is spectral pollution free and we derive optimal convergence rates for the planewave discretization method taking numerical integration errors into account some numerical illustrations are provided

a new approach to spectral approximation a new technique for approximating eigenvalues and eigenvectors of a self adjoint operator is presented the method does not incur spectral pollution uses trial spaces from the form domain has a self adjoint algorithm and exhibits superconvergence

analysis of finite element approximation of evolution problems in mixed form this paper deals with the finite element approximation of evolution problems in mixed form following d boffi f brezzi and l gastaldi math comp 69 2000 pp 121 140 we handle separately two types of problems a model for the first case is the heat equation in mixed form while the time dependent stokes problem fits within the second one for either case we give sufficient conditions for a good approximation in the natural functional spaces the results are not obvious in the first situation in this case the well known conditions for the well posedness and convergence of the corresponding steady problem are not sufficient for the good approximation of the time dependent problem this issue is demonstrated with a numerical counter example and justified analytically

short note a primal formulation for the helmholtz decomposition in 1999 jean paul caltagirone and jerome breil have developed in their paper caltagirone j breil sur une methode de projection vectorielle pour la resolution des equations de navier stokes c r acad sci paris 327 serie ii b 1999 1179 1184 a new method to compute a divergence free velocity they have used the grad div operator to extract the solenoidal part of a given vector field in this contribution we explain how this method can be considered as a real helmholtz decomposition and we present a stable approximation in the framework of spectral methods numerical results are presented to illustrate the efficiency of this approach

approximation of eigenvalues in mixed form discrete compactness property and application to hp mixed finite elements abstract in this paper we discuss the discrete compactness property dcp which is a well known tool for the analysis of finite element approximations of maxwell s eigenvalues we restrict our presentation to maxwell s eigenvalues but the theory applies to more general situations and in particular to mixed finite element schemes that can be written in the framework of de rham complex and which enjoy suitable compactness properties we investigate the relationships between dcp and standard mixed conditions for the good approximation of eigenvalues as a consequence of our theory the convergence analysis of the rectangular hp version of raviart thomas finite elements for the approximation of laplace eigenvalues is presented as a corollary of the analogous result for hp edge elements applied to the approximation of maxwell s eigenvalues d boffi m costabel m dauge l demkowicz discrete compactness for the hp version of rectangular edge finite elements siam j numer anal 44 3 2006 979 1004

guaranteed and robust a posteriori bounds for laplace eigenvalues and eigenvectors a unified framework this paper develops a general framework for a posteriori error estimates in numerical approximations of the laplace eigenvalue problem applicable to all standard numerical methods guaranteed and computable upper and lower bounds on an arbitrary simple eigenvalue are given as well as on the energy error in the approximation of the associated eigenvector the bounds are valid under the sole condition that the approximate i th eigenvalue lies between the exact i 1 th and i 1 th eigenvalue where the relative gaps are sufficiently large we give a practical way how to check this the precision of the resulting estimates depends on these relative gaps our bounds feature no unknown solution regularity or polynomial degree dependent constant are optimally convergent efficient and polynomial degree robust under a further explicit a posteriori minimal resolution condition the multiplicative constant in our estimates can be reduced by a fixed factor moreover when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants this multiplicative constant can be brought to the optimal value of 1 with mesh refinement applications of our framework to nonconforming discontinuous galerkin and mixed finite element approximations of arbitrary polynomial degree are provided along with numerical illustrations our key ingredient are equivalences between the i th eigenvalue error the associated eigenvector energy error and the dual norm of the residual we extend them in an appendix to the generic class of bounded below self adjoint operators with compact resolvent

spectral approximations by the hdg method we consider the numerical approximation of the spectrum of a secondorder elliptic eigenvalue problem by the hybridizable discontinuous galerkin hdg method we show for problems with smooth eigenfunctions that the approximate eigenvalues and eigenfunctions converge at the rate 2k 1 and k 1 respectively here k is the degree of the polynomials used to approximate the solution its flux and the numerical traces our numerical studies show that a rayleigh quotient like formula applied to certain locally postprocessed approximations can yield eigenvalues that converge faster at the rate 2k 2 for the hdg method as well as for the brezzidouglas marini bdm method we also derive and study a condensed nonlinear eigenproblem for the numerical traces obtained by eliminating all the other variables

legendre spectral methods for the grad div operator this paper describes two legendre spectral methods for the grad div eigenvalue problem in r 2 the first method uses a single grid resulting from the p n circle times p n discretization in primal and dual variational formulations as is well known this method is unstable and exhibits spectral pollution effects increased number of singular eigenvalues and increased multiplicity of some eigenvalues belonging to the regular spectrum our study aims at the understanding of these effects the second spectral method is based on a staggered grid of the p n circle times pn 1 discretization this discretization leads to a stable algorithm free of spurious eigenmodes and with spectral convergence of the regular eigenvalues eigenvectors towards their analytical values in addition divergence free vector fields with sufficient regularity properties are spectrally projected onto the discrete kernel of grad div a clear indication of the robustness of this algorithm c 2007 elsevier b v all rights reserved

an algebraic characterization ofb convergent runge kutta methods in the analysis of discretization methods for stiff intial value problems stability questions have received most part of the attention in the past b stability and the equivalent criterion algebraic stability are well known concepts for runge kutta methods applied to dissipative problems however for the derivation ofb convergence results error bounds which are not affected by stiffness it is not sufficient in many cases to requireb stability alone in this paper necessary and sufficient conditions forb convergence are determined

on approximation of the eigenvalues of perturbed periodic schrodinger operators this paper addresses the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic schrodinger operator perturbed by a fast decreasing potential we use a recently developed technique the so called quadratic projection method in order to achieve convergence free from spectral pollution we describe the theoretical foundations of the method in detail and illustrate its effectiveness by several examples

on the relation between algebraic stability andb convergence for runge kutta methods this paper is concerned with the numerical solution of stiff initial value problems for systems of ordinary differential equations using runge kutta methods for these and other methods frank schneid and ueberhuber 7 introduced the important concept ofb convergence i e convergence with error bounds only depending on the stepsizes the smoothness of the exact solution and the so called one sided lipschitz constant spijker 19 proved for the case 0 thatb convergence follows from algebraic stability the well known criterion for contractivity cf 1 2 we show that the order ofb convergence in this case is generally equal to the stage order improving by one half the order obtained in 19 further it is proved that algebraic stability is not only sufficient but also necessary forb convergence

a schwarz generalized eigen oscillation spectral element method gesem for 2 d high frequency electromagnetic scattering in dispersive inhomogeneous media in this paper we propose a parallel schwarz generalized eigen oscillation spectral element method gesem for 2 d complex helmholtz equations in high frequency wave scattering in dispersive inhomogeneous media this method is based on the spectral expansion of complex generalized eigen oscillations for the electromagnetic fields and the schwarz non overlapping domain decomposition iteration method the gesem takes advantages of a special real orthogonality property of the complex eigen oscillations and a new radiation interface condition for the system of equations for the spectral expansion coefficients numerical results validate the high resolution and the flexibility of the method for various materials

on the linearization of regge calculus we study the linearization of three dimensional regge calculus around euclidean metric we provide an explicit formula for the corresponding quadratic form and relate it to the curl t curl operator which appears in the quadratic part of the einstein hilbert action and also in the linear elasticity complex we insert regge metrics in a discrete version of this complex equipped with densely defined and commuting interpolators we show that the eigenpairs of the curl t curl operator approximated using the quadratic part of the regge action on regge metrics converge to their continuous counterparts interpreting the computation as a non conforming finite element method

the second order spectrum and optimal convergence the method of second order relative spectra has been shown to reliably approximate the discrete spectrum for a self adjoint operator we extend the method to normal operators and find optimal convergence rates for eigenvalues and eigenspaces the convergence to eigenspaces is new while the convergence rate for eigenvalues improves on the previous estimate by an order of magnitude

discontinuous galerkin approximation of the laplace eigenproblem in this paper we analyse the problem of computing eigenvalues and eigenfunctions of the laplace operator by means of discontinuous galerkin dg methods it results that several dg methods actually provide a spectrally correct approximation of the laplace operator we present here the convergence theory which applies to a wide class of dg methods as well as numerical tests demonstrating the theoretical results

projection methods for discrete schrodinger operators let h be the discrete schrodinger operator acting on l2 z where the potential v is real valued and v n 0 as n let p be the orthogonal projection onto a closed linear subspace l2 z in a recent paper e b davies defines the second order spectrum spec2 h of h relative to as the set of z c such that the restriction to of the operator p h z 2p is not invertible within the space the purpose of this article is to investigate properties of spec2 h when is large but finite dimensional we explore in particular the connection between this set and the spectrum of h our main result provides sharp bounds in terms of the potential v for the asymptotic behaviour of spec2 h as increases towards l2 z 2000 mathematics subject classification 47b36 primary 47b39 81 08 secondary

on the convergence of runge kutta methods for stiff non linear differential equations this paper studies the convergence properties of general runge kutta methods when applied to the numerical solution of a special class of stiff non linear initial value problems it is proved that under weaker assumptions on the coefficients of a runge kutta method than in the standard theory of b convergence it is possible to ensure the convergence of the method for stiff non linear systems belonging to the above mentioned class thus it is shown that some methods which are not algebraically stable like the lobatto iiia or a stable sirk methods are convergent for the class of stiff problems under consideration finally some results on the existence and uniqueness of the runge kutta solution are also presented

a stabilized finite element method for the two field and three field stokes eigenvalue problems in this paper the stabilized finite element approximation of the stokes eigenvalue problems is considered for both the two field displacement pressure and the three field stress displacement pressure formulations the method presented is based on a subgrid scale concept and depends on the approximation of the unresolvable scales of the continuous solution in general subgrid scale techniques consist in the addition of a residual based term to the basic galerkin formulation the application of a standard residual based stabilization method to a linear eigenvalue problem leads to a quadratic eigenvalue problem in discrete form which is physically inconvenient as a distinguished feature of the present study we take the space of the unresolved subscales orthogonal to the finite element space which promises a remedy to the above mentioned complication in essence we put forward that only if the orthogonal projection is used the residual is simplified and the use of term by term stabilization is allowed thus we do not need to put the whole residual in the formulation and the linear eigenproblem form is recovered properly we prove that the method applied is convergent and present the error estimates for the eigenvalues and the eigenfunctions we report several numerical tests in order to illustrate that the theoretical results are validated

on mass conservation in least squares methods we compare three least squares finite element reformulations of the stokes equations paying particular attention to mass conservation the first problem we approximate has a simple analytical solution over a convex region even for this simple problem without special treatment of the conservation of mass term very poor numerical solutions may result sufficiently weighting this term leads to a dramatic improvement in the results over a range of test problems

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

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

mixed discontinuous galerkin approximation of the elasticity eigenproblem we introduce a discontinuous galerkin method for the mixed formulation of the elasticity eigenproblem with reduced symmetry the analysis of the resulting discrete eigenproblem does not fit in the standard spectral approximation framework since the underlying source operator is not compact and the scheme is nonconforming we show that the proposed scheme provides a correct approximation of the spectrum and prove asymptotic error estimates for the eigenvalues and the eigenfunctions finally we provide several numerical tests to illustrate the performance of the method and confirm the theoretical results

a div curl lemma for edge elements a variant of murat and tartar s div curl lemma is stated and proved for nedelec s edge elements given two sequences of vector fields of this type converging weakly in l2 as the mesh width tends to 0 we prove that their scalar product converges in the sense of distributions when one of the sequences consists of so called discrete divergence free fields whereas the other has relatively compact curl in h 1 the proof uses a uniform norm equivalence related to discrete compactness properties of vector finite element spaces and a super approximation property of scalar finite element spaces

stable and semistable low order finite elements for viscous flows it is shown that i the bilinear constant and linear constant box elements are not uniformly stable and ii certain element pairs derived from those above by restricting the pressure spaces are stable the notion of div semistability is developed to study the behavior of the bilinear constant and linear constant box elements in detail

mathematical analysis of a finite element method without spurious solutions for computation of dielectric waveguides in this paper we propose a finite element method to compute dielectric waveguides which does not present spurious solutions in addition to a theoretical analysis showing the good stability properties of this method we give numerical results for some classical test problems

on the solvability complexity index the n pseudospectrum and approximations of spectra of operators we show that it is possible to compute spectra and pseudospectra of linear operators on separable hilbert spaces given their matrix elements the core in the theory is pseudospectral analysis and in particular the n pseudospectrum and the residual pseudospectrum we also introduce a new classification tool for spectral problems namely the solvability complexity index this index is an indicator of the difficultness of different computational spectral problems

on the unique solvability of the runge kutta equations we consider the existence of a unique solution to the systems of equations that arise when we apply a runge kutta method to a stiff nonlinear system of differential equationsu f t u withf satisfying a one sided lipschitz condition with constant for any given product h whereh denotes the step size we present algebraic conditions on the runge kutta matrixa which are necessary and sufficient for unique solvability of the equations as a second topic we consider the question whether the solution to the system of equations is stable with respect to perturbations known as bsi stability for this property also necessary and sufficient algebraic conditions ona are presented

hybridization and postprocessing techniques for mixed eigenfunctions we introduce hybridization and postprocessing techniques for the raviart thomas approximation of second order elliptic eigenvalue problems hybridization reduces the raviart thomas approximation to a condensed eigenproblem the condensed eigenproblem is nonlinear but smaller than the original mixed approximation we derive multiple iterative algorithms for solving the condensed eigenproblem and examine their interrelationships and convergence rates an element by element postprocessing technique to improve accuracy of computed eigenfunctions is also presented we prove that a projection of the error in the eigenspace approximation by the mixed method of any order superconverges and that the postprocessed eigenfunction approximations converge faster for smooth eigenfunctions numerical experiments using a square and an l shaped domain illustrate the theoretical results

limiting set of second order spectra let m be a self adjoint operator acting on a hilbert space h a complex number z is in the second order spectrum of m relative to a finite dimensional subspace l c dom m 2 iff the truncation to l of m z 2 is not invertible this definition was first introduced in davies 1998 and according to the results of levin and shargorodsky in 2004 these sets provide a method for estimating eigenvalues free from the problems of spectral pollution in this paper we investigate various aspects related to the issue of approximation using second order spectra our main result shows that under fairly mild hypothesis on m the uniform limit of these sets as l increases towards h contain the isolated eigenvalues of m of finite multiplicity therefore unlike the majority of the standard methods second order spectra combine nonpollution and approximation at a very high level of generality

on the convergence of second order spectra and multiplicity the notion of second order relative spectrum of a self adjoint operator acting on a hilbert space has been studied recently in connection with the phenomenon of spectral pollution in the galerkin method in this paper we examine how the second order spectrum encodes precise information about the multiplicity of the isolated eigenvalues of the underlying operator our theoretical findings are supported by various numerical experiments on the computation of guaranteed eigenvalue inclusions via finite element bases

on least squares finite element methods for the poisson equation and their connection to the dirichlet and kelvin principles least squares finite element methods for first order formulations of the poisson equation are not subject to the inf sup condition and lead to stable solutions even when all variables are approximated by equal order continuous finite element spaces for such elements one can also prove optimal convergence in the energy norm equivalent to a norm on h 1 omega times h omega mbox rm div for all variables and optimal l2 convergence for the scalar variable however showing optimal l2 convergence for the flux has proven to be impossible without adding the redundant curl equation to the first order system of partial differential equations in fact numerical evidence strongly suggests that nodal continuous flux approximations do not posses optimal l2 accuracy in this paper we show that optimal l2 error rates for the flux can be achieved without the curl constraint provided that one uses the div conforming family of brezzi douglas marini or brezzi douglas duran fortin elements then we proceed to reveal an interesting connection between a least squares finite element method involving h omega mbox rm div conforming flux approximations and mixed finite element methods based on the classical dirichlet and kelvin principles we show that such least squares finite element methods can be obtained by approximating through an l2 projection the hodge operator that connects the kelvin and dirichlet principles our principal conclusion is that when implemented in this way a least squares finite element method combines the best computational properties of finite element methods based on each of the classical principles

on the penalty stabilization mechanism for upwind discontinuous galerkin formulations of first order hyperbolic systems abstract penalty fluxes are dissipative numerical fluxes for high order discontinuous galerkin dg methods which depend on a penalization parameter warburton 2013 ye et al 2016 we investigate the dependence of the spectra of high order dg discretizations on this parameter and show that as its value increases the spectrum of the dg discretization splits into two disjoint sets of eigenvalues one set converges to the eigenvalues of a conforming discretization while the other set corresponds to spurious eigenvalues which are damped proportionally to numerical experiments also demonstrate that undamped spurious modes present in both in the limit of zero and large penalization parameters are damped for moderate values of the upwind parameter

the concept of b convergence a useful concept of convergence for nonlinear stiff ode s will be developed which permits the derivation of uniform global error bounds independent of the stiffness of the considered problem this concept will be discussed for three simple methods for stiff systems the implicit euler scheme the implicit midpoint rule and the implicit trapezoidal rule

the p and hp versions of the virtual element method for elliptic eigenvalue problems abstract we discuss the p and h p versions of the virtual element method for the approximation of eigenpairs of elliptic operators with a potential term on polygonal meshes an application of this model is provided by the schrodinger equation with a pseudo potential term as an interesting byproduct we present for the first time in literature an explicit construction of the stabilization of the mass matrix we present in detail the analysis of the p version of the method proving exponential convergence in the case of analytic eigenfunctions the theoretical results are supplied with a wide set of experiments we also show numerically that in the case of eigenfunctions with finite sobolev regularity an exponential approximation of the eigenvalues in terms of the cubic root of the number of degrees of freedom can be obtained by employing h p refinements importantly the geometric flexibility of polygonal meshes is exploited in the construction of the h p spaces

on the spurious solutions in the high order finite difference methods for eigenvalue problems in this paper the origins of spurious solutions occurring in the high order finite difference methods are studied based on a uniform mesh spurious modes are found in the high order one sided finite difference discretizations of many eigenvalue problems spurious modes are classified as spectral pollution and non spectral pollution the latter can be partially avoided by mesh refinement while the former persists when the mesh is refined through numerical studies of some prototype eigenvalue problems such as those of the helmholtz and beam equations we show that perfect central differentiation schemes do not produce any spurious modes nevertheless high order central difference schemes encounter difficulties in implementing complex boundary conditions we further show that the central difference schemes will produce spurious modes as well if asymmetric approximation is involved in boundary treatments in general central difference schemes are less likely to produce spurious modes than one sided difference schemes 2007 elsevier b v all rights reserved

on spectral pollution in the finite element approximation of thin elastic membrane shells the bending terms in a shell are small with respect to membrane ones as the thickness tends to zero consequently the membrane approximation gives a good description of vibration properties of a thin shell this vibration problem is associated with a non compact resolvent operator and spectral pollution could appear when computing galerkin approximations that is to say there could exist sequences of eigenvalues of the approximate problems that converge to points of the resolvent set of the exact problem we give an account of the state of the art of this problem in shell theory a description of the phenomenon and its interpretation in terms of spectral families are given a theorem of localization of the region where pollution may appear is stated and its complete proof is published for the first time recipes are given for avoiding the pollution as well as indications on the possibility of a posteriori elimination

analysis of some mixed finite element methods related to reduced integration we prove error estimates for the following two mixed finite element methods related to reduced integration a method for stokes problem using rectangular elements with piecewise bilinear approximations for the velocities and piecewise constants for the pressure and one method for a plate problem using bilinear approximations for transversal displacement and rotations and piecewise constants for the shear stress the main idea of the proof in the case of stokes problem is to combine a weak babuska brezzi type stability estimate for the pressure with a superapproximability property for the velocities a similar technique is used in the case of the plate problem

the character table of 2 1 22 co2 abstract the second largest sporadic simple group the baby monster has a maximal subgroup 2 1 22 co 2 which is a non split extension of an extraspecial group of order 223 by the second sporadic simple conway group co 2 the construction of the character table of this group is described