discontinuous galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity we consider a finite element in space and quadrature in time discretization of a compressible linear quasistatic viscoelasticity problem the spatial discretization uses a discontinous galerkin finite element method based on polynomials of degree r termed dg r and the time discretization uses a trapezoidal rectangle rule approximation to the volterra history integral both semi and fully discrete a priori error estimates are derived without recourse to gronwall s inequality and therefore the error bounds do not show exponential growth in time moreover the convergence rates are optimal in both h and r providing that the finite element space contains a globally continuous interpolant to the exact solution e g when using the standard k polynomial basis on simplicies or tensor product polynomials k on quadrilaterals when this is not the case e g using k on quadri laterals the convergence rate is suboptimal in r but remains optimal in h we also consider a reduction of the problem to standard linear elasticity where similarly optimal a priori error estimates are derived for the dg r approximation

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

models algorithms and error estimation for computational viscoelasticity this article reviews numerical algorithms for problems in solid polymer viscoelasticity in both small and large deformation for the linear small strain case we review both the quasistatic and the dynamic problem and give recent results on a posteriori error estimation for the large strain case we focus on the formulation and computational modelling of constrained membrane inflation the application of which is to the thermoforming process

numerical solution of linear volterra integral equations of the second kind with sharp gradients collocation methods are a well developed approach for the numerical solution of smooth and weakly singular volterra integral equations in this paper we extend these methods through the use of partitioned quadrature based on the qualocation framework to allow the efficient numerical solution of linear scalar volterra integral equations of the second kind with smooth kernels containing sharp gradients in this case the standard collocation methods may lose computational efficiency despite the smoothness of the kernel we illustrate how the qualocation framework can allow one to focus computational effort where necessary through improved quadrature approximations while keeping the solution approximation fixed the computational performance improvement introduced by our new method is examined through several test examples the final example we consider is the original problem that motivated this work the problem of calculating the probability density associated with a continuous time random walk in three dimensions that may be killed at a fixed lattice site to demonstrate how separating the solution approximation from quadrature approximation may improve computational performance we also compare our new method to several existing gregory sinc and global spectral methods where quadrature approximation and solution approximation are coupled

gaussian bounds for propagators perturbed by potentials abstract we develop the perturbation theory for propagators with the objective to prove gaussian bounds let u be a strongly continuous propagator i e a family of operators describing the solutions of a non autonomous evolution equation on an l p space and assume that u is positive and satisfies gaussian upper and lower bounds let v be a time dependent potential satisfying certain miyadera conditions with respect to u we show that then the perturbed propagator enjoys gaussian upper and lower bounds as well to prepare the necessary tools we extend the perturbation theory of strongly continuous propagators and the theory of absorption propagators

the galerkin continuous finite element method for delay differential equation with a variable term abstract in this paper we introduce and analyze the continuous finite element method for a class of delay differential equation with a variable term at first for the case of the constant delay and based on an orthogonal expansion in an element we derive optimal superconvergence u u o h p 2 p 2 at the p 1 order lobatto points and u u o h 2 p at the integer nodal points next we prove the stability of linear and quadratic finite element schemes for the case of constant delay for the case of variable delay by use of the approximative to the continuous delay in every subinterval we decompose the original problem into a series of subproblems with constant delay finally we give a recursive schemes which can be compute by every subproblem

optimal order a posteriori error estimates for a class of runge kutta and galerkin methods we derive a posteriori error estimates which exhibit optimal global order for a class of time stepping methods of any order that include runge kutta collocation rk c methods and the continuous galerkin cg method for linear and nonlinear stiff odes and parabolic pdes the key ingredients in deriving these bounds are appropriate one degree higher continuous reconstructions of the approximate solutions and pointwise error representations the reconstructions are based on rather general orthogonality properties and lead to upper and lower bounds for the error regardless of the time step they do not hinge on asymptotics

absorption semigroups their generators and schrodinger semigroups on presente des resultats de perturbations pour des c 0 semi groupes on considere le probleme aux valeurs initiales pour l equation de la chaleur avec absorption avec une solution dans l p r v avec une mauvaise fonction v r r on considere le meme probleme dans l 2 r pour l equation de schrodinger

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

error estimates and automatic time step control for nonlinear parabolic problems i in this note we extend results by one of the authors on time discretization error estimates and related automatic time step control for stiff ordinary differential equations to the case of a nonlinear parabolic problem the method for time discretization is the so called discontinuous galerkin method based on using piecewise polynomials of degree q geqq 0 we consider in this note the case q 0 corresponding to a variant of the backward euler method we prove a new almost optimal error estimate and present a related new algorithm for automatic time step control this algorithm is very simple but yet is efficient and gives control of the global error

analysis of a space time continuous galerkin method for convection dominated sobolev equations abstract the convergence of space time continuous galerkin stcg method for the sobolev equations with convection dominated terms is studied in this article it allows variable time steps and the change of the spatial mesh from one time interval to the next which can make this method suitable for numerical simulations on unstructured grids we prove the existence and uniqueness of the approximate solution and get the optimal convergence rates in l h 1 norm which do not require any restriction assumptions on the space and time mesh size finally some numerical examples are designed to validate the high efficiency of the method showed herein and to confirm the correctness of the theoretical analysis

a posteriori error estimates for space time finite element approximation of quasistatic hereditary linear viscoelasticity problems abstract we give a space time galerkin finite element discretisation of the quasistatic compressible linear viscoelasticity problem as described by an elliptic partial differential equation with a fading memory volterra integral the numerical scheme consists of a continuous galerkin approximation in space based on piecewise polynomials of degree p 0 cg p with a discontinuous galerkin piecewise constant dg 0 or linear dg 1 approximation in time a posteriori galerkin error estimates are derived by exploiting the galerkin framework and optimal stability estimates for a related dual backward problem the a posteriori error estimates are quite flexible strong lp energy norms of the errors are estimated using time derivatives of the residual terms when the data are smooth while weak energy norms are used when the data are non smooth in time we also give upper bounds on the dg 0 cg 1 a posteriori error estimates which indicate optimality however a complete analysis is not given

error analysis for discretizations of parabolic problems using continuous finite elements in time and mixed finite elements in space variational time discretization schemes are getting of increasing importance for the accurate numerical approximation of transient phenomena the applicability and value of mixed finite element methods in space for simulating transport processes have been demonstrated in a wide class of works we consider a family of continuous galerkin petrov time discretization schemes that is combined with a mixed finite element approximation of the spatial variables the existence and uniqueness of the semidiscrete approximation and of the fully discrete solution are established for this the banach necas babuska theorem is applied in a non standard way error estimates with explicit rates of convergence are proved for the scalar and vector valued variable an optimal order estimate in space and time is proved by duality techniques for the scalar variable the convergence rates are analyzed and illustrated by numerical experiments also on stochastically perturbed meshes

continuous finite elements in space and time for the nonhomogeneous wave equation finite element approximation in space and time for the wave equation with a forcing term is considered the rates of convergence in h1 l2 and l2 h 1 topology reconstructing the best approximation properties of the subspaces are derived

an a posteriori error estimate and adaptive timestep control for a backward euler discretization of a parabolic problem a simple algorithm for adaptive timestep control is presented for a backward euler discretization of a linear parabolic problem the algorithm is based on an a posteriori error estimate involving the computed approximate solution it is proved that with only very rough a priori information on the exact solution the algorithm will choose a sequence of timesteps for which the error will be controlled up to a constant uniformly in time on a given tolerance level

adaptive space time finite element solution for volterra equations arising in viscoelasticity problems abstract we give a short overview of our recent efforts towards constructing adaptive space time finite element solvers for some partial differential volterra equations arising in viscoelasticity theory

negative norm error control for second kind convolution volterra equations summary we consider a piecewise constant finite element approximation to the convolution volterra equation problem of the second kind find u such that u f phi u in a time interval 0 t an a posteriori estimate of the error measured in the w 1 p 0 t norm is developed and used to provide a time step selection criterion for an adaptive solution algorithm numerical examples are given for problems in which phi is of a form typical in viscoelasticity theory

korn s inequalities and their applications in continuum mechanics korn s inequalities have played a central role in the development of linear elasticity not only in connection with the basic theoretical issues such as existence and uniqueness but also in a variety of applications the korn inequalities and other related inequalities for integrals of quadratic functionals also arise in the analysis of viscous incompressible fluid flow the dimensionless optimal constants appearing in these inequalities the korn constants depend only on the shape of the domains of concern information on the geometric dependence of these constants is essential in applications in this review article we summarize the major results on korn s inequalities for bounded domains in two and three dimensions with emphasis on results concerning the korn constants some applications in continuum mechanics are also described

a crank nicolson type space time finite element method for computing on moving meshes in this paper a space time finite element method for evolution problems that is second order accurate in both space and time is proposed for convection dominated problems the elements may be aligned along the characteristics in space time which results in a crank nicolson method along the characteristics the method is also suitable as an alternative to other moving mesh methods for problems in deforming domains numerical examples dealing with diffusion and convection are given

numerical techniques for the treatment of quasistatic viscoelastic stress problems in linear isotropic solids abstract for quasistatic stress problems two alternative constitutive relationships expressing the stress in a linear isotropic viscoelastic solid body as a linear functional of the strain are available in conjunction with the equations of equilibrium these form the mathematical models for the stress problems these models are first discretized in the space domain using a finite element method and semi discrete error estimates are presented corresponding to each constitutive relationship through the use respectively of quadrature rules and finite difference replacements each semi discrete scheme is fully discretized into the time domain so that two practical algorithms suitable for the numerical stress analysis of linear viscoelastic solids are produced the semi discrete estimates are then also extended into the time domain to give spatially h 1 error estimates for each algorithm the numerical schemes are predicated on exact analytical solutions for a simple model problem and finally on design data for a real polymeric material

a galerkin method for the forward backward heat equation in this paper a new variational method is proposed for the numeri cal approximation of the solution of the forward backward heat equation the approach consists of first reducing the second order problem to an equivalent first order system and then using a finite element procedure with continuous elements in both space and time for the numerical approximation under suit able regularity assumptions error estimates and the results of some numerical experiments are presented

on the stability of the discontinuous galerkin method for the heat equation this paper analyzes stability properties of a class of discontinuous galerkin methods for the heat equation it is shown that the finite element projection associated with these methods is stable with respect to a mesh dependent norm a discrete analogue of the space time l 2 norm optimal order regularity error bounds in l 2 0 t l 2 omega are derived

numerical null controllability of the 1d linear schrodinger equation abstract this paper deals with the numerical approximation to boundary controls that drive the solution to the 1 d linear schrodinger equation to a prescribed state at a final time using ideas from fursikov and imanuvilov we consider the control that minimizes over the class of admissible controls a functional that involves weighted integrals with weights that blow up at t we will see that this extremal problem is equivalent to a differential problem that is fourth order in space and second order in time adapting some numerical techniques applied by the first author and munch to the heat equation we approximate the variational formulation by introducing appropriate space time finite elements that are c 1 in space and c 0 in time we present two approaches the second one relies on a change of variable which leads to a lower condition number for the stiffness matrix the results of some experiments show the efficiency of these methods

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

variational space time elements for large scale systems abstract in this paper we introduce a new galerkin based formulation for transient continuum problems governed by partial differential equations in space and time therefore we aim at a direct finite element discretization of the space time suitable for massive parallel analysis of the arising large scale problem the proposed formulation is applied to thermal mechanical and fluid systems as well as to a kuramoto sivashinsky problem representing the general class of higher order formulations in material science using nurbs based shape functions we verify whenever possible the conservation properties of the formulation finally a series of examples demonstrate the applicability to all systems presented in this paper

perturbation of dirichlet forms lower semiboundedness closability and form cores we study perturbations e e qu of dirichlet forms e on some l2 space l2 m given by quadratic forms q g g d with a signed borel measure whose positive and negative parts are smooth measures with respect to the given dirichlet form we give necessary and sufficient conditions for e to be a lower semibounded closed quadratic form with a unique associated continuous semigroup pt on l2 m we also study the associated resolvent semigroup as well as the relation between e and the largest closed quadratic form which is smaller than e in particular we exhibit form cores

discontinuous galerkin method with a posteriori l_p 0 t_i error estimate for second kind volterra problems we consider a discontinuous galerkin finite element method applied in time to a model volterra equation of the second kind a residual based computable l p 0 t i galerkin error estimate is derived for 1 le p le infty this estimate does not explicitly contain the time step and therefore the time step control must be based on a heuristic criterion the estimate can then be used to demonstrate the integrity or otherwise of the finite element solution after performing some numerical experiments we conclude that this approach is at least competetive with classical discretizations since it is computationally simple to implement but has the added advantage of reliable error feedback

sharp estimates in smoothing theorems for schrodinger semigroups the smoothing properties of schrodinger semigroups e th h 12 v on the scale of bessel potential spaces lp are studied we strengthen the lp lp smoothing theorem due to m a kon and the author the new version of this theorem contains a sharp time estimate for the norm of the semigroup e th we also get an estimate for the constants arising in the form boundedness inequality for the kato class potentials

convergence of a continuous galerkin method with mesh modification for nonlinear wave equations we consider space time continuous galerkin methods with mesh modification in time for semilinear second order hyperbolic equations we show a priori estimates in the energy norm without mesh conditions under reasonable assumptions on the choice of the spatial mesh in each time step we show optimal order convergence rates estimates of the jump in the riesz projection in two successive time steps are also derived

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

variational time discretization for mixed finite element approximations of nonstationary diffusion problems we develop and study numerically two families of variational time discretization schemes for mixed finite element approximations applied to nonstationary diffusion problems continuous and discontinuous approximations of the time variable are encountered the solution of the arising algebraic block system of equations by a schur complement technique is described and an efficient preconditioner for the iterative solution process is constructed the expected higher order rates of convergence are demonstrated in numerical experiments moreover superconvergence properties are verified further the efficiency and stability of the approaches are illustrated for a more sophisticated three dimensional application of practical interest with discontinuous and anisotropic material properties

on the smoothing property of the galerkin method for parabolic equations we analyze the galerkin approximation of the general second order parabolic initial boundary value problem for the second order continuous time method with initial data only in l 2 we prove an l 2 error estimate of order o h 2 t analogous results are shown to hold for the error in negative sobolev norms and for the time derivative of the error our analysis uses only elementary energy techniques and the same techniques are shown to give a simple analysis of the backward euler time discretization

stabilized shock hydrodynamics i a lagrangian method abstract a new supg stabilized formulation for lagrangian hydrodynamics of materials satisfying mie gruneisen equation of state is proposed it allows the use of simplex type triangular tetrahedral meshes as well as the more commonly used brick type quadrilateral hexahedral meshes the proposed method yields a globally conservative formulation in which equal order interpolation p1 or q1 isoparametric finite elements is applied to velocities displacements and pressure as a direct consequence and in contrast to traditional cell centered multidimensional hydrocode implementations the proposed formulation allows a natural representation of the pressure gradient on element interiors the supg stabilization involves additional design requirements specific to the lagrangian formulation a discontinuity capturing operator in the form of a noh type viscosity with artificial heat flux is used to preserve stability and smoothness of the solution in shock regions a set of challenging shock hydrodynamics benchmark tests for the euler equations of gas dynamics in one and two space dimensions is presented in the two dimensional case computations performed on quadrilateral and triangular grids are analyzed and compared these results indicate that the new formulation is a promising technology for hydrocode applications

values of indefinite quadratic forms at integral points and flows on spaces of lattices this mostly expository paper centers on recently proved conjectures in two areas a a conjecture of a oppenheim on the values of real indefinite quadratic forms at integral points b conjectures of dani raghunathan and margulis on closures of orbits in spaces of lattices such as sl r sl z at first sight a belongs to analytic number theory and b belongs to ergodic and lie theory and they seem to be quite unrelated they are discussed together here because of a very interesting connection between the two pointed out by m s raghunathan namely a special case of b yields a proof of a the first main goal of this talk is to describe the oppenheim conjecture and various refinements and to derive them from one statement about closures of orbits in the space of unimodular lattices in r3 see proposition 2 in 2 3 in 3 and 4 we put this statement in context and describe more general conjectures and results on orbit closures and invariant probability measures on quotients of lie groups by discrete subgroups 5 gives some brief comments on the proofs and further developments 6 7 and 8 are devoted to the so called s arithmetic setting where we consider products of real and p adic groups 6 is concerned with a generalized oppenheim conjecture 7 with a generalization of the orbit closure theorem proved by m ratner r8 and 8 with applications to quadratic forms since the subject matter of that last section has not been so far discussed elsewhere we take this opportunity to present proofs obtained jointly with g prasad finally 9 gives the proof of a lemma on symmetric simple lie algebras a special case of which is used in 8 i am glad to thank m ratner and g prasad for a number of remarks on and corrections to a preliminary version of this paper thanks to which many typos and inaccuracies have been eliminated

translate of horospheres and counting problems let g be a semisimple lie group without compact factors gamma be an irreducible lattice in g in the first part of the article we give the necessary and sufficient condition under which a sequence of translates of probability horospherical measures is convergent and the limiting measure is also determined when it is convergent see theorems 1 and 2 for the precise statements in the second part two applications are presented the first one is of geometric nature and the second one gives an alternative way to count the number of rational points on a flag variety

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

high order space time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling we revisit a method originally introduced by werder et al in comput methods appl mech engrg 190 6685 6708 2001 for temporally discontinuous galerkin fems applied to a parabolic partial differential equation in that approach block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions if the spatial finite element space is of dimension d and polynomials of degree r are used in time the block system has dimension r 1 d and is usually regarded as being too large when r 1 werder et al found that the space time coupling matrices are diagonalizable over formula see text for r 100 and this means that the time coupled computations within a time step can actually be decoupled by using either continuous galerkin or spectral element methods in space we apply this dg in time methodology for the first time to second order wave equations including elastodynamics with and without kelvin voigt and maxwell zener viscoelasticity an example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high order up to degree 7 temporal and spatio temporal approximations and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease copyright 2014 the authors international journal for numerical methods in engineering published by john wiley sons ltd

adaptive finite element methods for parabolic problems ii optimal error estimates in l l 2 and l l optimal error estimates are derived for a complete discretization of linear parabolic problems using space time finite elements the discretization is done first in time using the discontinuous galerkin method and then in space using the standard galerkin method the underlying partitions in time and space need not be quasi uniform and the partition in space may be changed from time step to time step the error bounds show in particular that the error may be controlled globally in time on a given tolerance level by controlling the discretization error on each individual time step on the same given level i e without error accumulation effects the derivation of the estimates is based on the orthogonality of the galerkin procedure and the use of strong stability estimates the particular and precise form of these error estimates makes it possible to design efficient adaptive methods with reliable automatic error control for parabolic problems in the norms under consideration

numerical solution of linear quasistatic hereditary viscoelasticity problems we give a space time galerkin finite element discretization of the linear quasistatic compressible viscoelasticity problem as described by an elliptic partial differential equation with a volterra memory term the discretization consists of a continuous piecewise linear approximation in space with a discontinuous piecewise constant or linear approximation in time we derive an a priori maximum energy error estimate by exploiting galerkin orthogonality and the data stability of a related discrete backward problem illustrative numerical experiments are also included as also is a brief description of our first results on a posteriori error estimation this allows for adaptive control of the space mesh but not of the time step

adaptive finite element methods in computational mechanics we present a general approach to adaptivity for finite element methods and give applications to linear elasticity non linear elasto plasticity and nonlinear conservation laws including numerical results

optimal long time l_p 0 t stability and semidiscrete error estimates for the volterra formulation of the linear quasistatic viscoelasticity problem the purpose of this article is to show how the solution of the linear quasistatic compressible viscoelasticity problem written in volterra form with fading memory may be sharply bounded in terms of the data if certain physically reasonable assumptions are satisfied the bounds are derived by making precise assumptions on the memory term which then make it possible to avoid the gronwall inequality and use instead a comparison theorem which is more sensitive to the physics of the problem once the data stability estimates are established we apply the technique also to deriving a priori error bounds for semidiscrete finite element approximations our bounds are derived for viscoelastic solids and fluids under the small strain assumption in terms of the eigenvalues of a certain matrix derived from the stress relaxation tensor for isotropic materials we can be explicit about the form of these bounds while for the general case we give a formula for their computation

on the heat equation with a time dependent singular potential abstract we introduce various classes of potentials on 0 t r n and study the corresponding nonautonomous heat equation given by u t 1 2 u v t x u we prove the existence of the feynman kac propagators for the nonautonomous heat equation and the l p l q smoothing theorem for the propagators

superconvergence of the continuous galerkin finite element method for delay differential equation with several terms he continuous galerkin finite element method for linear delay differential equation with several terms is studied adding some lower terms in the remainder of orthogonal expansion in an element so that the remainder satisfies more orthogonal condition in the element and obtain a desired superclose function to finite element solution thus the superconvergence of p degree finite element approximate solution on p 1 order lobatto points is derived

negative norm estimates for fully discrete finite element approximations to the wave equation with nonhomogeneous l 2 dirichlet boundary data this paper treats time and space finite element approximations of the solution to the nonhomogeneous wave equation with l 2 boundary terms and smooth right hand side for the case of l 2 boundary data the rates of convergence in negative norms are derived in the case of smooth forcing term and zero boundary data optimal rates of convergence in positive norms are provided

nonautonomous kato classes of measures and feynman kac propagators the behavior of the feynman kac propagator corresponding to a time dependent measure on r n is studied we prove the boundedness of the propagator in various function spaces on r n and obtain a uniqueness theorem for an exponentially bounded distributional solution to a nonautonomous heat equation

on a parabolic equation with a singular lower order term abstract we obtain the existence of the weak green s functions of parabolic equations with lower order coefficients in the so called parabolic kato class which is being proposed as a natural generalization of the kato class in the study of elliptic equations as a consequence we are able to prove the existence of solutions of some initial boundary value problems moreover based on a lower and an upper bound of the green s function we prove a harnack inequality for the non negative weak solutions

on direct methods for the discretization of a heat conduction equation using spline functions the finite element collocation method is sometimes equivalent to the galerkin method the two methods are therefore considered together here the nonhomogeneous heat conduction equation with nonhomogeneous boundary condition is chosen for consideration the equation is transformed into a variational form by continuous time galerkin method and expressed in terms of linear spline basis function on the other hand the collocation method is applied on the heat equation with cubic splines as the basis functions the two methods of discretization transform the heat equation to systems of ordinary differential equations in time respectively the corresponding stiffness matrices for the two systems of equations turn out to be the same these matrices are sparse banded and symmetric and are therefore convenient for computer application

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

approximating thermo viscoelastic heating of largely strained solid rubber components abstract mechanically induced viscoelastic dissipation is difficult to compute when the constitutive model is defined by history integrals the computation of the viscous energy dissipated is in the form of a double convolution integral in this study we present a method to approximate the dissipation for constitutive models in history integral form that represent maxwell like materials the dissipation is obtained without directly computing the double convolution integral the approximation requires that the total stress can be separated into elastic and viscous components and that the relaxation form of the constitutive law is defined with a prony series a numerical approach often taken to approximate a history integral involves interpolating the history integral s kernel across a time step integration then yields finite difference equations for the evolution of the viscous stresses in time in the case when the material is modeled with a prony series the form of these finite difference equations is similar to the form of the finite difference equations for a maxwell solid since the dissipation rate in a maxwell solid can be easily computed from knowledge of its viscous stress and the prony series constants spring dashpot constants we computationally investigated employing a maxwell solid s dissipation function to couple thermal and large strain history integral based finite element models of solid rubber components numerical data is provided to support this analogy and to help understand its limitations a rubber cylinder with an imbedded steel disk is dynamically loaded and the non uniform heating within the cylinder is computed

exact smoothing properties of schrodinger semigroups we study schrodinger semigroups in the scale of sobolev spaces and show that for kato class potentials the range of such semigroups in lp has exactly two more derivatives than the potential this proves a conjecture of b simon we show that eigenfunctions of schrodinger operators are generically smoother by exactly two derivatives in given sobolev spaces than their potentials we give applications to the relation between the potential s smoothness and particle kinetic energy in the context of quantum mechanics and characterize kinetic energies in coulomb systems the techniques of proof involve leibniz and chain rules for fractional derivatives which are of independent interest as well as a new characterization of the kato class 1 introduction in this paper we attempt a precise study of the action of schrodinger semigroups in the scale of sobolev spaces work in this area was begun by b simon si4 in 1985 he proved a number of positive and negative results on such smoothing and also presented an open question in the form of a conjecture regarding necessary and sufficient conditions on the potential v for smoothing of order s by the semigroup of functions in l2 rd study of schrodinger semigroups is of interest for a number of reasons see

error estimates with sharp constants for a fading memory volterra problem in linear solid viscoelasticity the problem characterizing nonageing linear isothermal quasi static isotropic compressible solid viscoelasticity in the time interval 0 t is described this is essentially a volterra equation of the second kind arrived at by adding smooth fading memory to the elliptic linear elasticity equations we analyze the errors resulting from replacing the relaxation functions with practical approximations in a semidiscrete finite element approximation and in a fully discrete scheme derived by replacing the hereditary integral with the trapezoidal rule for numerical integration the error estimates are sharp in the sense that if certain bounds on the data are independent of t then so also are the constants involved in them this is a consequence of bypassing the usual gronwall lemmas with arguments that are more sensitive to the fading memory of the physical problem

a local projection stabilization continuous galerkin petrov method for incompressible flow problems abstract a local projection stabilization lps method in space is considered to approximate the evolutionary oseen equations optimal error bounds with constants independent of the viscosity parameter are obtained in the continuous in time case for both the velocity and pressure approximation in addition the fully discrete case in combination with higher order continuous galerkin petrov cgp methods is studied error estimates of order k 1 are proved where k denotes the polynomial degree in time assuming that the convective term is time independent numerical results show that the predicted order is also achieved in the general case of time dependent convective terms

classes of time dependent measures non homogeneous markov processes and feynman kac propagators we study the inheritance of properties of free backward propagators associated with transition probability functions by backward feynman kac propagators corresponding to functions and time dependent measures from non autonomous kato classes the inheritance of the following properties is discussed the strong continuity of backward propagators on the space l r the l r l q smoothing property of backward propagators and various generalizations of the feller property we also prove that a propagator on a banach space is strongly continuous if and only if it is separately strongly continuous and locally uniformly bounded

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis