time finite element methods a unified framework for numerical discretizations of odes abstract we present a unified framework for the numerical discretization of odes based on time finite element methods we relate time finite element methods to runge kutta methods with infinitely many stages by means of the corresponding numerical quadrature we establish the relation between time finite element methods and partitioned runge kutta methods we also provide order estimates and superconvergence of the corresponding numerical methods in use of the simplifying assumptions we apply time finite methods to hamiltonian systems and investigate the conservation of energy and symplectic structure for the resulting numerical discretizations for hamiltonian systems we also construct new classes of symplectic integrators by combining different time finite element methods and verify the results by performing some numerical experiments

on the space of orthonormal wavelets the space of orthonormal wavelets is described by a set of parameters for which a simple recurrence generates the coefficients for all orthonormal perfect reconstruction fir filters of arbitrary length the space splits into two halves each containing the time reverse of the other s filters a matlab implementation is given this paper considers the generation of two channel perfect reconstruction quadrature mirror filter banks corresponding to compactly supported orthonormal wavelets

analysis of a continuous finite element method for hyperbolic equations a finite element method for hyperbolic equations is analyzed in the context of a first order linear problem in r 2 the method is applicable over a triangulation of the domain and produces a continuous piecewise polynomial approximation which can be developed in an explicit fashion from triangle to triangle in a sense it extends the basic upwind difference scheme to higher order the method is shown to be stable and error estimates are obtained for nth degree approximation the errors in the approximate solution and its gradient are shown to be at least of order h n 1 4 and h n 1 2 respectively assuming sufficient regularity in the solution

numerical solution of a nonlinear wave equation in polar coordinates we study a finite element scheme and a difference scheme for the radial solutions of a nonlinear klein gordon equation

on a one dimensional nonlinear thermoviscoelastic model for structural phase transitions in shape memory alloys we shall prove the existence and the uniqueness of global classical solutions of the following one dimensional landau devonshire thermoviscoelastic model for structural phase transitions in shape memory alloys utt p ux x uxxt in 0 1 0 t 1 t xx 2 uxuxt u2xt g in 0 1 0 t u 0 t u 1 t 0 x 0 t x 1 t 0u x 0 u0 x ut x 0 u1 x x 0 0 x

global solutions to a model of structural phase transitions in shape memory alloys abstract global in time existence and uniqueness of solutions to a nonlinear system of coupled evolution equations is discussed the system represents a model of dynamical structural phase transitions martensitic transformations in shape memory alloys

quasilinear hyperbolic parabolic equations of one dimensional viscoelasticity abstract we study the global existence of solutions of initial boundary value problems for a quasilinear hyperbolic parabolic equation describing the longitudinal motion of a one dimensional viscoelastic rod we treat a variety of nonhomogeneous boundary conditions requiring separate analyses because they lead to distinctive physical effects we employ a constitutive equation giving the stress as a general nonlinear function of the strain and the strain rate all global analyses of this and related problems except that of dafermos j differential equations 1969 71 86 have employed a stress that is merely affine in the strain rate dafermos s assumptions are far more appropriate for shearing motions than for longitudinal motions our constitutive equation satisfies the physically natural requirement that an infinite amount of compressive stress is needed to produce a total compression at any point of the rod this requirement is the source of a severe singularity in the governing partial differential equations which is particularly acute when time dependent dirichlet data are prescribed the further novel yet physically reasonable restrictions we impose on the constitutive function yield estimates that preclude a total compression anywhere at any finite time the resulting estimates are crucial for the global existence theory we obtain

global attractor for a nonlinear thermoviscoelastic model with a non convex free energy density abstract this paper is concerned with the existence of a global attractor for a semiflow governed by the weak solutions to a nonlinear one dimensional thermoviscoelasticity with a non convex free energy density the constitutive assumptions for the helmholtz free energy include the model for the study of martensitic phase transitions in shape memory alloys to describe physically phase transitions between different configurations of crystal lattices we work in a framework in which the strain u belongs to l new approaches are introduced and more delicate estimates are derived to establish the crucial l estimate of strain u in deriving the compactness of the orbit of the semiflow and the existence of an absorbing set

conservative numerical methods for x f x abstract two distinctly different numerical methods are developed for solving the conservative initial value problem x f x x o dot x o both methods conserve exactly the same total energy as does the given differential equation computer examples are described and

finite difference solutions of a non linear schrodinger equation abstract this paper describes a finite difference method to approximate a schrodinger equation with a power non linearity a special case of this equation is currently used to model the propagation of a laser beam in a plasma the main feature of the method we present is that it satisfies a discrete analogue of some important conservation laws of the equations we present numerical results which show in particular the propagation and the formation of solitions in the one dimensional case

convergence of a second order scheme for the nonlinear dynamical equations of elastic rods a second order energy preserving finite difference scheme to approximate the partial differential equations governing planar twist free motions of inextensible unshearable elastic rods is presented and analyzed

a stable finite element method for initial boundary value problems for first order hyperbolic systems a nonstandard finite element method for initial boundary value problems for first order hyperbolic systems in one space dimension with general boundary conditions is analyzed the method can be considered as a generalization of the box scheme we first establish a stability result for the method and then derive several error estimates u x t a x ux x t b x u x t f x t 1 1 o 0 t g kxu l t u x 0 u0 x for x t e i x j 0 1 x 0 where t 0 here aix and 5 x are n x n matrices we assume that aix has n real eigenvalues xx x a2 x a x and that there exist a constant t0 0 and an integer m 0 t0 for all xg furthermore we assume that for each x e there is a nonsingular matrix qix such that

on thermodynamic active control of shape memory alloy wires we propose a model for the control of high frequency oscillations in shape memory alloy wires we introduce a notion of generalized solution for a generalized control and in this context prove a local exact controllability result effectively corresponding to an approximate controllability result for the nonconvex pseudoelastic material system

behaviour in the large on numerical solutions to one dimensional nonlinear viscoelasticity by continuous time galerkin methods abstract we analyze the long time behaviour of fully discrete solutions to a one dimensional nonlinear viscoelastic problem it is shown that these approximations which are found by a continuous time galerkin method converge to a steady state the possible numerical steady states are characterized and in particular their high degree of dependence on initial data and mesh design is explained computational results are included which show the above dependence and indicate that the numerical solutions will typically not converge to unstable states

wavelet transforms associated with finite cyclic groups multiresolution analysis via decomposition on wavelet bases has emerged as an important tool in the analysis of signals and images when these objects are viewed as sequences of complex or real numbers an important class of multiresolution decompositions are the laplacian pyramid schemes in which the resolution is successively halved by recursively low pass filtering the signal under analysis and decimating it by a factor of two in general the principal framework within which multiresolution techniques have been studied and applied is the same as that used in the discrete time fourier analysis of sequences of complex numbers an analogous framework is developed for the multiresolution analysis of finite length sequences of elements from arbitrary fields attention is restricted to sequences of length 2 sup n for n a positive integer so that the resolution may be recursively halved to completion as in finite length fourier analysis a cyclic group structure of the index set of such sequences is exploited to characterize the transforms of interest for the particular cases of complex and finite fields

a nonconforming finite element method for the cahn hilliard equation this paper reports a fully discretized scheme for the cahn hilliard equation the method uses a convexity splitting scheme to discretize in the temporal variable and a nonconforming finite element method to discretize in the spatial variable and the scheme can preserve the mass conservation and energy dissipation properties of the original problem some typical phase transition phenomena are also observed through the numerical examples

finite element approximation of a spatially structured metapopulation pde model abstract we present a new fully spatially structured pde metapopulation model for predator prey dynamics in d 3 space dimensions a nonlinear reaction diffusion system of rosenzweig macarthur form models predator prey dynamics in two high quality patches embedded in a low quality subdomain where species can diffuse convect and die our model substantially generalizes and improves earlier fully structured metapopulation models after a nondimensionalization procedure in order to approximate the metapopulation model we present a fully discrete galerkin finite element method in two space dimensions which is a generalization of the finite element method analyzed in a previous single patch predator prey model the numerical solutions are illustrated for some test cases using matlab numerical experiments demonstrate that the initial local extinction of predators in one patch leads to waves of recolonization from another patch in an appendix we also give an outline for the proof of the well posedness of the model

unique global solvability for initial boundary value problems in one dimensional nonlinear thermoviscoelasticity the balance laws of mass momentum and energy are considered for a broad class of one dimensional nonlinear thermoviscoelastic materials for the initial boundary value problem corresponding to pinned endpoints held at constant temperature we establish existence and uniqueness of temporally global classical solutions for initial data of unrestricted size our approach also applies to all boundary conditions involving pinned or stress free endpoints which are either held at constant temperature or insulated an additional and novel feature of the theory is that solid like and gaseous materials are treated in a unified way

methods for the numerical solution of the nonlinear schroedinger equation optimal l2 rates of convergence are established for several fully discrete schemes for the numerical solution of the nonlinear schroedinger equation both finite differences and finite elements are considered for the discretization in space while the integration in time is treated either by the leap frog technique or by a modified crank nicolson procedure which generalizes the one suggested by delfour fortin and payne and possesses two useful conserved quantities

conservation properties of a time fe method part iv higher order energy and momentum conserving schemes in the present paper a systematic development of higher order accurate time stepping schemes which exactly conserve total energy as well as momentum maps of underlying finite dimensional hamiltonian systems with symmetry is shown the result of this development is the enhanced galerkin eg finite element method in time the conservation of the eg method is generally related to its collocation property total energy conservation in particular is obtained by a new projection technique the eg method is moreover based on objective time discretization of the used strain measure this paper is concerned with particle dynamics and semi discrete non linear elastodynamics the related numerical examples show good performance in presence of stiffness as well as for calculating large strain motions copyright 2005 john wiley sons ltd

su i 2 f z 1 z for f a subfield of c let f be a subfield of c closed under complex conjugation and denote by u 2 f z 1 z the multiplicative group of two by two unitary matrices over the ring f z 1 z where lzl 1 let su1 2 f z 1 z be the subgroup of such unitary matrices with determinant equal to the constant polynomial 1 and which are equal to the identity matrix upon evaluation at z 1 the main result is the unique factorization theorem for s u1 2 f z 1 z which expresses each element as an unique product of a minimal number of simple factors in su1 2 f z 1 z since these factors are free of relations it follows that su1 2 f z 1 z is a nonabelian free group generated by the factors besides having interesting algebraic and topological properties from the standpoint of pure mathematics these unitary groups have application to the theory of quadrature mirror filter banks and to the theory of wavelets with compact support especially in parameterizing various classes of qmf banks and wavelet families after this manuscript was completed the author s attention was drawn to reference v which derives nonunique factorizations of the groups u d r z 1 z and u d c z 1 z for d 2 reference v part of the extensive qmf literature concentrates mainly on the application of these factorizations to design qmf banks for particular digital signal processing problems the theory of wavelets with compact support provides both a generalization of and an analytical viewpoint of qmf theory wavelets with compact support were recently discovered by ingrid daubechies as described in di and d2

a continuous space time finite element method for the wave equation we consider a finite element method for the nonhomogeneous second order wave equation which is formulated in terms of continuous approximation functions in both space and time thereby giving a unified treatment of the spatial and temporal discretizations our analysis uses primarily energy arguments which are quite common for spatial discretizations but not for time we present a priori nodal in time superconvergence error estimates without any special time step restrictions our method is based on tensor product spaces for the full discretization

on the enforcing energy conservation of time finite elements for discrete elasto dynamics problems in the context of the time finite element method algorithmic stresses which enable the conservation of energy are designed for temporal integrators derived from the midpoint and trapezoidal schemes this is achieved through an appropriate modification of the standard midpoint and trapezoidal quadrature rules used for the numerical integration of time integrals either scalar scaling or vectorial adjustments can be employed for the modification and well designed simple tests allow to investigate the quality of these different strategies of energy conserving enforcements numerical examples with semi discrete elasto dynamics problems are presented to show the superior stability of energy conserving schemes copyright 2006 john wiley sons ltd

global solutions to the equations of a ginzburg landau theory for structural phase transitions in shape memory alloys abstract global existence of smooth solutions is proved for the system of partial differential equations governing the dynamics of martensic phase transitions in shape memory alloys the free energy is assumed in ginzburg landau form and nonconvex in the order parameter

some energy preserving finite element schemes based on the discrete variational derivative method abstract in this paper we discuss energy preserving finite element scheme for partial differential equations we firstly consider linear heat equation and linear wave equation as the typical example to derive energy preserving finite element scheme we employ the discrete variational method as the application of nonlinear partial differential equations we consider allen cahn equation and fujita problem

continuous finite element methods which preserve energy properties for nonlinear problems abstract continuous finite elements are used to develop numerical methods which preserve energy properties for nonlinear ordinary and partial differential equations schemes are constructed that give high orders of accuracy time discretization algorithms for the cahn hilliard klein gordon korteweg de vries and schrodinger equations are presented comparisons are made with existing schemes

combined dg cg time stepping for wave equations the continuous and discontinuous galerkin time stepping methodologies are combined to develop approximations of second order time derivatives of arbitrary order this eliminates the doubling of the number of variables that results when a second order problem is written as a first order system stability convergence and accuracy of these schemes is established in the context of the wave equation it is shown that natural interpolation of nonhomogeneous boundary data can degrade accuracy and that this problem can be circumvented using interpolants matched with the time stepping scheme

maximal attractor for the system of a landau ginzburg theory for structural phase transitions in shape memory alloys abstract in this paper a system of nonlinear partial differential equations modelling the dynamics of martensitic phase transitions in shape memory alloys is further investigated in a previous paper published in siam j math anal the global existence of a unique solution and the asymptotic behaviour of the solution as time goes to infinity have been established in the present paper the existence of a compact maximal attractor is proved

convergence of galerkin approximations for the korteweg de vries equation standard galerkin approximations using smooth splines on a uniform mesh to 1 periodic solutions of the korteweg de vries equation are analyzed optimal rate of convergence estimates are obtained for both semidiscrete and second order in time fully discrete schemes at each time level the resulting system of nonlinear equations can be solved by newton s method it is shown that if a proper extrapolation is used as a starting value then only one step of the newton iteration is required

exponential stability for a nonlinear one dimensional heat conductive viscous real gas abstract in this paper we prove the existence and the exponential stability in h i i 1 2 an incomplete metric subspace of hi hi hi i 1 2 of a nonlinear c0 semigroup s t for a nonlinear one dimensional heat conductive viscous real gas in bounded domain 0 1 the results in this paper improve those previously related results

finite element approximation of spatially extended predator prey interactions with the holling type ii functional response we study the numerical approximation of the solutions of a class of nonlinear reaction diffusion systems modelling predator prey interactions where the local growth of prey is logistic and the predator displays the holling type ii functional response the fully discrete scheme results from a finite element discretisation in space with lumped mass and a semi implicit discretisation in time we establish a priori estimates and error bounds for the semi discrete and fully discrete finite element approximations numerical results illustrating the theoretical results and spatiotemporal phenomena are presented in one and two space dimensions the class of problems studied in this paper are real experimental systems where the parameters are associated with real kinetics expressed in nondimensional form the theoretical techniques were adapted from a previous study of an idealised reaction diffusion system garvie and blowey in eur j appl math 16 5 621 646 2005

galerkin based mechanical integrators for finite elastodynamics formulated in principal stretches pitfalls and remedies in the present paper we deal with integrators relying on finite elements in time for general hyperelastic models which guarantee the conservation of the momentum maps as well as of the total energy within the calculation accuracy in the isotropic case eigenvalue based constitutive laws are on the one hand very significant to model the behaviour of various materials on the other hand it will be shown that the efficient algorithmic treatment of principal stretches based on a perturbation approach requires especially within the context of conserving schemes advanced techniques to circumvent numerical problems accordingly we propose an adequate solution strategy to avoid potential pitfalls which are related to numerical artefacts of the applied non standard quadrature rule finally the excellent numerical and mechanical performance of the resulting galerkin based time stepping schemes will be demonstrated by means of representative numerical examples

regular article finite difference schemes for u t x α δg δu that inherit energy conservation or dissipation property we propose a new procedure for designing by rote finite difference schemes that inherit energy conservation or dissipation property from nonlinear partial differential equations such as the korteweg de vries kdv equation and the cahn hilliard equation the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly since the inherited properties are kept even if the time mesh size changes in the time evolution process we can use some appropriate time mesh adaptive methods to obtain numerical solutions through the derived schemes because of these properties the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions and sufficiently flexible to treat the inheritance of the energy conservation and dissipation properties are verified numerically for the kdv equation and the cahn hilliard equation

universal attractor in h4 for the nonlinear one dimensional compressible navier stokes equations abstract this paper is concerned with the regularity exponential stability of solutions and existence of universal attractor in h 4 for a nonlinear one dimensional compressible navier stokes equations describing a motion of heat conductive viscous real gas in a bounded domain 0 1 some new ideas and more delicated estimates are introduced to prove these results

improved technique for design of perfect reconstruction fir qmf banks with lossless polyphase matrices a technique is developed for the design of analysis filters in an m channel maximally decimated perfect reconstruction finite impulse response quadrature mirror filter fir qmf bank that has a lossless polyphase component matrix e z the aim is to optimize the parameters characterizing e z until the sum of the stopband energies of the analysis filters is minimized there are four novel elements in the procedure reported here the first is a technique for efficient initialization of one of the m analysis filters as a spectral factor of an mth band filter the factorization itself is done in an efficient manner using the eigenfilters approach without the need for root finding techniques the second element is the initialization of the internal parameters which characterize e z based on the above spectral factor the third element is a modified characterization mostly free from rotation angles of the fir e z the fourth is the incorporation of symmetry among the analysis filters so as to minimize the number of unknown parameters being optimized the resulting design procedure always gives better filter responses than earlier ones for a given filter length and converges much faster

energy momentum consistent finite element discretization of dynamic finite viscoelasticity this paper is concerned with energy momentum consistent time discretizations of dynamic finite viscoelasticity energy consistency means that the total energy is conserved or dissipated by the fully discretized system in agreement with the laws of thermodynamics the discretization is energy momentum consistent if also momentum maps are conserved when group motions are superimposed to deformations the performed approximation is based on a three field formulation in which the deformation field the velocity field and a strain like viscous internal variable field are treated as independent quantities the new non linear viscous evolution equation satisfies a non negative viscous dissipation not only in the continuous case but also in the fully discretized system the initial boundary value problem is discretized by using finite elements in space and time thereby the temporal approximation is performed prior to the spatial approximation in order to preserve the stress objectivity for finite rotation increments incremental objectivity although the present approach makes possible to design schemes of arbitrary order the focus is on finite elements relying on linear lagrange polynomials for the sake of clearness the discrete energy momentum consistency is based on the collocation property and an enhanced second piola kirchhoff stress tensor the obtained coupled non linear algebraic equations are consistently linearized the corresponding iterative solution procedure is associated with newly proposed convergence criteria which take the discrete energy consistency into account the iterative solution procedure is therefore not complicated by different scalings in the independent variables since the motion of the element is taken into account for solving the viscous evolution equation representative numerical simulations with various boundary conditions show the superior stability of the new time integration algorithm in comparison with the ordinary midpoint rule both the quasi rigid deformations during a free flight and large deformations arising in a dynamic tensile test are considered copyright 2009 john wiley sons ltd

computations on the cahn hilliard model of solidification we begin numerical studies of interface migration on the two dimensional cahn hilliard equation our results add evidence to support the connection between this equation and the well known quasistatic model of solidification finite elements coupled with a second order time discretization scheme are used the nonlinear systems of equations that arise on each time step are solved by fixed point iteration while the linear systems are solved by preconditioned conjugate gradients

global attractor for the ginzburg landau thermoviscoelastic systems with hinged boundary conditions this paper is concerned with the following one dimensional nonlinear system of equations 0 1 utt p e x uxxt ruxxxx f 0 2 cv t xx p et et2 g where both and r are positive constants the corresponding free energy density is assumed to be in ginzburg landau form and nonconvex as a function of the order parameter results concerning the existence and uniqueness of the global solution the asymptotic behavior of the solution as time tends to infinity and the compactness of the orbit are obtained furthermore we investigate dynamics of the system and prove the existence of global attractor

on the modeling and computations of nonlinear thermodynamics in sma wires we present a mathematical model describing the thermodynamic behavior of shape memory alloy wires as well as a computational technique to solve the resulting system of partial differential equations the model consists of continuum level conservation equations based on a new construction of the helmholtz free energy potential this construction allows specification of material constants for different shape memory wires the computational technique introduces a viscosity based continuation method which allows the model to handle dynamic applications where the temporally local behavior of solutions is desired computational experiments document that this combination of modeling and solution techniques appropriately predicts the thermally and stress induced martensitic phase transitions as well as the hysteretic behavior and production of latent heat associated with such materials

continuous finite elements in space and time for the heat equation on analyse une nouvelle methode variationnelle pour l approximation de l equation de la chaleur a l aide des elements finis continus dans l espace et le temps on presente les estimations d erreurs et les resultats d experiences numeriques

global existence and asymptotic behavior in nonlinear thermoviscoelasticity abstract we study global existence uniqueness and asymptotic behavior as time tends to infinity of weak solutions to the system of nonlinear thermoviscoelasticity various boundary conditions are considered it is shown that for any initial data u 0 v 0 0 l w 1 h 1 there is a unique global solution u v deformation gradient velocity temperature such that u c 0 l v c 0 w 1 l 0 w 1 c 0 h 1 the constitutive assumptions for the helmholtz free energy include the models for the study of phase transition problems in shape memory alloys

on the numerical approximation of an evolution problem in nonlinear viscoelasticity abstract an error analysis is furnished for the spatial approximation via the finite element method of certain nonlinear time dependent problems emanating in the theory of viscoelastic solids numerical examples are given first for a model equation and then for numerical equilibrium solutions in cases of a nonconvex double well potential and a convex potential the latter derived from the nonconvex one via relaxation by an equal area construction

energy dissipative time finite elements for classical mechanics abstract based upon the temporal discretisation of hamilton s canonical equations by means of the continuous galerkin method a petrov galerkin time finite element formulation is presented to analyse non linear mechanical systems for long time integration with the presence of dissipation the time stepping scheme features a correct estimation of physical dissipative energy for mechanical systems with weak dissipation this enables to introduce a moderate numerical dissipation to damp out undesired spurious high frequency modes in a controllable manner numerical examples are performed to demonstrate that the proposed scheme allows us to obtain a correct amount by which the energy changes over the integration run

numerical solution of the sine gordon equation two new difference schemes are proposed for the sine gordon equation they have the advantage that there is a discrete energy which is conserved their stability and convergence is proved also we show some numerical computations using one of these schemes

a second order accurate non linear difference scheme for the n component cahn hilliard system a b s t r a c t we consider a second order conservative nonlinear numerical scheme for the ncomponent cahn hilliard system modeling the phase separation of a n component mixture the scheme is based on a crank nicolson finite difference method and is solved by an efficient and accurate nonlinear multigrid method we numerically demonstrate the second order accuracy of the numerical scheme we observe that our numerical solutions are consistent with the exact solutions of linear stability analysis results we also describe numerical experiments such as the evolution of triple junctions and the spinodal decomposition in a quaternary mixture we investigate the effects of a concentration dependent mobility on phase separation

asymptotic behaviour and changes of phase in one dimensional nonlinear viscoelasticity in this paper we study the asymptotic behaviour as t co of solutions u x t to the nonlinear partial differential equation ut m4 a o o 1 1 with initial conditions u 0 q x u x u x o 0 1 3a

numerical simulation for a nonlinear partial differential equation with variable coefficients by means of the discrete variational derivative method partial differential equations with possibly discontinuous coefficients play an important part in engineering physics and ecology in this paper we will study nonlinear partial differential equations with variable coefficients arising from population models generally speaking it is difficult to analyze the behavior of nonlinear partial differential equations therefore we usually rely on the numerical approximation currently there is an increasing interest in designing numerical schemes that preserve energy properties for differential equations we will design the numerical schemes that preserve discrete energy property and show numerical experiments for a nonlinear partial differential equation with variable coefficients