a finite element method for equations of one dimensional nonlinear thermoelasticity abstract this paper presents a new coupled scheme for the equations of one dimensional nonlinear thermoelasticity by using the finite element method in the space variable and the difference method in the time variable the existence and the uniqueness of the approximate solutions are proved and the bounds of the error are analyzed

some convergence estimates for semidiscrete type schemes for time dependent nonselfadjoint parabolic equations l2 norm error estimates are shown for semidiscrete continuous in time galerkin finite element type approximations to solutions of general time dependent nonselfadjoint second order parabolic equations under dirichlet boundary conditions the semidiscrete solutions are defined in terms of given methods for the corresponding elliptic problem such as the standard galerkin method in which the boundary conditions are satisfied exactly but also methods for which this is not necessary the results are proved by energy arguments and include estimates for the homogeneous equation with both smooth and nonsmooth initial data

on mixed finite element methods for linear elastodynamics we construct and analyze finite element methods for approximating the equations of linear elastodynamics using mixed elements for the discretization of the spatial variables we consider two different mixed formulations for the problem and analyze semidiscrete and up to fourth order in time fully discrete approximations l 2 optimal order error estimates are proved for the approximations of displacement and stress

convergence estimates for semidiscrete parabolic equation approximations semidiscrete methods for approximating the solutions of initial boundary value problems for parabolic equations are studied the construction of these semidiscrete methods is based upon the availability of several different galerkin type approximation methods for the associated elliptic steadystate problem the properties required of the spacial discretization methods are listed and estimates of the error made by the resulting semidiscrete approximations and of their time derivatives are given in particular estimates are given that require only weak smoothness assumptions on the initial data verifications of the required properties for various galerkin type methods are provided

cosine methods for second order hyperbolic equations with time dependent coefficients we analyze efficient high order accurate methods for the approximation of the solutions of linear second order hyperbolic equations with time dependent coefficients the methods are based on galerkin type discretizations in space and on a class of fourth order accurate two step cosine time stepping schemes preconditioned iterative techniques are used to solve linear systems with the same operator at each time step the schemes are supplemented by single step high order starting procedures and need no evaluations of derivatives of operators l2 optimal error estimates are proved throughout

on runge kutta methods for parabolic problems with time dependent coefficients analyse des approximations completement discretes de galerkin pour les equations paraboliques a coefficients dependant du temps utilisation des methodes de runge kutta implicites

parallel algorithms for initial value problems for difference and differential equations abstract let yn be a trajectory defined sequentially by a nonlinear vector difference equation yn 1 fn 1 yn with y0 known a steffensen like iterative method is proposed which starts from a guessed sequence un 0 for the approximation of yn and allows certain computations to be performed in parallel the sequence u k 1 n is obtained from u k n by means of a formula of the form u k 1 n 1 v k 1 n 1 k 1 n 1 u k 1 n u k n here the vectors v k 1 n 1 and the matrices k 1 n 1 each require a function evaluation but can be computed in parallel with respect to n in a suitable interval nk mk the length of which depends on the number of processors available therefore for the algorithm to serve effectively the steffensen iteration must converge quickly and the machine used must possess a large number of processors finally note that the theory includes the case that the sequence yn is given by a one step ode solver

a linear matrix inequality approach for robust control of systems with delayed states abstract this paper addresses the robust stabilization problem for uncertain systems with delayed states the parameter uncertainties are unknown but norm bounded and the delay is time varying in this study a linear matrix inequality lmi approach for robust stability analysis for the nominal unforced system and a method for robust stabilization for a class of uncertain delay systems via linear memoryless state feedback control are presented the results depend on the size of the delay and are given in terms of linear matrix inequalities

higher order single step fully discrete approximations for second order hyperbolic equations with time dependent coefficients error estimates are proved for finite element approximations to the solution of an initial boundary value problem for a second order hyperbolic partial differential equation with time dependent coefficients optimal order rates of convergence in l 2 are shown for single step fully discrete schemes the order nu of the single step schemes is greater than or equal to one and less than or equal to four a class of iterative methods for approximately solving the fully discrete equations is analyzed and optimal order rates of convergence are also obtained these methods avoid the repeated work of new matrix factorizations at each time level

error estimates for semidiscrete galerkin type approximations to semilinear evolution equations with nonsmooth initial data error estimates for the semidiserete galerkin method for abstract semilinear evolution equations with non smooth initial data are given in concrete cases almost optimal order of convergence for linear finite elements results

incomplete iterations in multistep backward difference methods for parabolic problems with smooth and nonsmooth data backward difference methods for the discretization of parabolic boundary value problems are considered in this paper in particular we analyze the case when the backward difference equations are only solved approximately by a preconditioned iteration we provide an analysis which shows that these methods remain stable and accurate if a suitable number of iterations often independent of the spatial discretization and time step size are used results are provided for the smooth as well as nonsmooth initial data cases finally the results of numerical experiments illustrating the algorithms performance on model problems are given

algebraically stable and implementable runge kutta methods of high order there are three interesting properties of methods for stiff ordinary differential equations order stability and efficiency of implementation this paper constructs runge kutta methods of orders 5 and 6 which possess these properties to a high extent we further classify all algebraically stable methods of an arbitrary order and give various relationships between contractivity and order of implicit methods

galerkin runge kutta discretizations for parabolic equations with time dependent coefficients a new class of fully discrete galerkin runge kutta methods is constructed and analyzed for linear parabolic initial boundary value problems with time dependent coefficients unlike any classical counterpart this class offers arbitrarily high order convergence while significantly avoiding what has been called order reduction in support of this claim error estimates are proved and computational results are presented additionally since the time stepping equations involve coefficient matrices changing at each time step a preconditioned iterative technique is used to solve the linear systems only approximately nevertheless the resulting algorithm is shown to preserve the original convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with time independent coefficients furthermore it is noted that special runge kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method

numerical approximations for a miscible displacement process in porous media a reservoir simulation problem describing a miscible displacement process is stated and analyzed and a semidiscrete numerical scheme based on a galerkin method with mesh refinements is presented for its solution the underlying physical problem involves two miscible incompressible fluids of equal viscosity flowing in a porous medium injection and production wells are rigorously modelled which leads to a formulation containing nearly singular source and sink terms the singularities arising in the solution are described in some detail and it is shown how knowledge of their forms motivate the choice of an appropriate mesh refinement for the numerical scheme optimal order quadratic convergence is shown for the resulting semidiscrete approximation using piecewise linear elements

a classification theorem for albert algebras let k be a field whose characteristic is different om 2 and 3 and let l k be a quadratic extension in this paper we prove that for a fixed degree 3 central simple algebra b over l with an involution ff of the second kind over k the jordan algebra j b a 11 u obtained through tits second co nstruction is determined up to isomorphism by the class of 11 u in h1 k su b a thus settling a question raised by petersson and racine as a consequence we derive a skolem noether type theorem for albert algebras we also show that the cohomological invariants determine the isomorphism class of j b a u u if b cr is fixed introduction let k be a field with characteristic different from 2 and 3 exceptional simple jordan algebras over k are called albert algebras there are rational constructions due to tits of all albert algebras over k referred to as the frst and the second constructions we begin by recalling the second construction briefly let l k be a quadratic extension bar denottng its nontrivial k automorphisme let b be a degree 3 central simple associative algebra over l with an involution a of the second kind over k let u be a unit of b with v u u and n u l i u e l where n denotes the reduced norm on b let h b x e blv x x let j b a u 1 xt ba d3 b be the jordan algebra with the multiplication given by bo b bo bt bo bo b l5 b btu5 b bob bob y ff b x o b u l where x y 2 xy yx z 2 t x x and x x y x y 2 t x y 2 t v x 2 t x t v t x tt t denoting the reduced trace of b the cubic norm of an element aoaa of j b c u is given by 81 n ao a n ao hn a hn ff a t aoau5 a for any unit w e b we have an isomorphism j b c u o j b c wu5 w n w h given by ao a l ao aw the following converse problem is posecs in p r question if j b c u j b c u y then does there e is 1 unit w e b such that u wuv w n w y received by the editors june 12 1996 1991 mathematics subject classification primary 17c40

a noniterative method for the design of linear robust controllers a procedure for determining a linear control law which guarantees asymptotic stability for an uncertain system is derived using an elementary matrix identity an alternative proof of the existence of stabilizing control law for the matched case is provided bounds for the rates of decay are established and by using these bounds it is shown that every trajectory of the closed loop system can be made to decay at a prespecified exponential rate the technique is simpler to use than existing techniques since it does not require a numerical procedure but allows the control law to be obtained from some simple formulas

letter to the editor convergence of fem with interpolated coefficients for semilinear hyperbolic equation to solve spatially semidiscrete approximative solution of a class of semilinear hyperbolic equations the finite element method fem with interpolated coefficients is discussed by use of semidiscrete finite element for linear problem as comparative function the error estimate in l norm is derived by the nonlinear argument in chen structure theory of superconvergence of finite elements hunan press of science and technology changsha 2001 in chinese this indicates that convergence of fems with interpolated coefficients for a semilinear equation is similar to that of classical fems

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

convergence of runge kutta approximations for parabolic problems with neumann boundary conditions we analyze a class of algebraically stable runge kutta standard galerkin methods for inhomogeneous linear parabolic equations with time dependent coefficients under neumann boundary conditions and derive an l 2 error bound of o h r k min nu p 3 over2 provided frac k h is bounded

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

cyclic compositions and trisotopies abstract cyclic compositions in the sense of springer t a springer oktaven jordan algebren und ausnahmegruppen universitat gottingen 1963 and knus merkurjev rost and tignol m a knus a merkurjev m rost j p tignol the book of involutions amer math soc colloq publ vol 44 amer math soc providence ri 1998 with a preface in french by j tits are investigated by means of cyclic trisotopies a concept originally due to albert a a albert a construction of exceptional jordan division algebras ann of math 2 67 1958 1 28 using the quadrupling of composition algebras we enumerate cyclic trisotopies and compositions in a rational manner i e without extending the base field we relate cyclic trisotopies explicitly to simple associative algebras of degree 3 with involution and to the tits process of cubic jordan algebras

semidiscrete and single step fully discrete approximations for second order hyperbolic equations with time dependent coefficients l2 norm error estimates are proved for finite element approximations to the solutions of initial boundary value problems for second order hyperbolic partial differential equations with time dependent coefficients optimal order rates of convergence are shown for semidiscrete and single step fully discrete schemes using specially constructed initial data the initial data are designed so that the data used for the fully discrete equation is reasonable to compute and so that the optimal order estimates can be proved

galerkin runge kutta discretizations for semilinear parabolic equations a new class of fully discrete galerkin runge kutta methods is constructed and analyzed for semilinear parabolic initial boundary value problems unlike any classical counterpart this class offers arbitrarily high order convergence without suffering from what has been called order reduction in support of this claim error estimates are proved and computational results are presented furthermore it is noted that special runge kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method

delay dependent robust stabilization of uncertain systems with multiple state delays this paper deals with the problem of robust stabilization for uncertain systems with multiple state delays the parameter uncertainties are time varying and unknown but are norm bounded and the delays are time varying a new method for achieving robust stabilization is presented for a class of uncertain time delay systems via linear memoryless state feedback control the results depend on the size of the delays and are given in terms of several linear matrix inequalities

efficiently implementable algebraically stable runge kutta methods recent work by butcher bit 16 1976 pp 237 240 on the implementation of implicit runge kutta methods has shown that if his technique is used the most efficiently implementable methods are those whose runge kutta matrix has a single real eigenvalue in this paper therefore we analyze the algebraic stability of two families of methods with this property namely singly implicit and diagonally implicit methods

some new error estimates of a semidiscrete finite volume element method for a parabolic integro differential equation with nonsmooth initial data a semidiscrete finite volume element fve approximation to a parabolic integro differential equation pide is analyzed in a two dimensional convex polygonal domain an optimal order l 2 error estimate for smooth initial data and nearly the same optimal order l 2 error estimate for nonsmooth initial data are obtained more precisely for homogeneous equations an elementary energy technique and a duality argument are used to derive an error estimate of order o left t 1 h 2 ln h right in the l 2 norm for positive time when the given initial function is only in l 2

cosine methods for nonlinear second order hyperbolic equations construction et analyse des methodes efficaces avec une precision d ordre eleve pour l approximation des solutions regulieres d une classe d equations hyperboliques non lineaires du second ordre application des discretisations de type galerkin et des schemas a deux pas estimations d erreurs d ordre optimal

robust linear controllers using observers the possibility of determining a linear robust control law when the full state cannot be measured and observers are implemented to estimate the state is considered the focus is on systems where the uncertainty satisfies the matching condition the control law and the observer are designed using two riccati equations the first result establishes that if certain scalar parameters are chosen so that a matrix inequality is satisfied the closed loop system is stable for the uncertainty levels of interest it is then shown that if there are at least as many sensors as there are actuators and the transfer function of the nominal system or a squared down form of it does not have zeros in the closed right half plane the closed loop system can be stabilized by the technique regardless of the size of the uncertainty bounding set

a riccati equation approach to the design of linear robust controllers abstract in this paper by the lyapunov stability criterion and the riccati equation we derive a new procedure for determining a linear control law to stabilize an uncertain system the main features of this approach are that no precompensator is needed the required feedback gains are small and the uncertain system can be exponentially stabilized with any prescribed exponential rate in addition the robust control law can be made linear quadratic optimal with respect to an index function for the nominal system two examples are given to illustrate the results

fully discrete galerkin approximations of parabolic boundary value problems with nonsmooth boundary data in this paper we study the convergence properties of a fully discrete galerkin approximation with a backwark euler time discretization scheme an approach based on semigroup theory is used to deal with the nonsmooth dirichlet boundary data which cannot be handled by standard techniques this approach gives rise to optimal rates of convergence inl p o t l 2 norms for boundary conditions inl p o t l 2 1 p

discrete maximal parabolic regularity for galerkin finite element methods for nonautonomous parabolic problems the main goal of the paper is to establish time semidiscrete and space time fully discrete maximal parabolic regularity for the lowest order time discontinuous galerkin solution of linear parabolic

adaptive local overlapping grid methods for parabolic systems in two space dimensions abstract adaptive mesh refinement techniques are described for two dimensional systems of parabolic partial differential equations solutions are calculated using galerkin s method with a piecewise bilinear basis in space and backward euler integration in time a posteriori estimates of the local discretization error of piecewise bilinear finite element solutions are obtained by a p refinement technique these error estimates are used to control a local h refinement strategy where finer grids are recursively introduced in regions where a prescribed tolerance is exceeded fine grids at a given level of refinement may overlap each other and independent solutions are generated on each of them a version of the schwarz alternating principle is used to coordinate solutions between overlapping fine grids computational results demonstrating the performance of the adaptive procedure on linear and nonlinear problems and apparent convergence of the error estimate for linear heat conduction problem and uniform global refinement is presented

higher order single step fully discrete approximations for nonlinear second order hyperbolic equations abstract error estimates are proved for finite element approximations to the solution of an initial boundary value problem for nonlinear second order hyperbolic partial differential equations optimal order rates of convergence in l 2 are shown for single step fully discrete schemes which are linearized by extrapolations the order of accuracy in time is 2 3 or 4 a class of iterative methods for approximately solving the fully discrete equations is analyzed and optimal order rates of convergence are also obtained

robust stabilization of uncertain systems with time varying multistate delay the authors deal with the problem of stabilizing a class of uncertain linear systems with time varying multi state delay and subject to norm bounded parameter uncertainty via memoryless linear state feedback some sufficient conditions for the robust stabilizability are derived fur this class of uncertain systems if there exists a positive definite symmetric solution satisfying the algebraic riccati equation or inequality a suitable memoryless state feedback law can be derived also moreover all such parametric algebraic riccati inequalities have been transformed into some linear matrix inequality problems so there is no tuning of the parameters to gain a stabilizing solution

state feedback for uncertain dynamical systems we consider a linear dynamical system in which the system and input matrices as well as the input are uncertain we present a control system consisting of a linear control to stabilize the nominal system a nonlinear control to cope with the uncertainties and an insensitive observer for the state estimation practical stability is guaranteed for uncertainties with known bounds furthermore the control system is designed to achieve insensitivity against parameter variations the theoretical results are illustrated by an application to the suspension control of a maglev vehicle

galerkin approximations of abstract parabolic boundary value problems with rough boundary data l p theory etude des approximations de galerkin d un probleme aux valeurs limites parabolique abstrait avec des donnees aux limites grossieres determination des vitesses de convergence optimales dans les normes lp

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

cosine methods for a class of semilinear second order wave equations abstract we analyze fully discrete methods of fourth or second order temporal accuracy for the approximation of the solutions of a class of semilinear second order hyperbolic equations with a nonlinear term that depends on x t u u t u the methods are based on galerkin type discretizations in space and on a class of two step cosine time stepping schemes by suitably extrapolating from previous values in the nonlinear term we compute the approximations by solving linear systems of equations at each time step l 2 optimal error estimates are proved

mixed finite element approximations of parabolic integro differential equations with nonsmooth initial data we analyze the semidiscrete mixed finite element methods for parabolic integro differential equations that arise in the modeling of nonlocal reactive flows in porous media a priori l 2 error estimates for pressure and velocity are obtained with both smooth and nonsmooth initial data more precisely a mixed ritz volterra projection introduced earlier by ewing et al in siam j numer anal 40 2002 pp 1538 1560 is used to derive optimal l 2 error estimates for problems with initial data in h 2 cap h 0 1 in addition for homogeneous equations we derive optimal l 2 error estimates for initial data just in l 2 here we use an elementary energy technique and duality argument

error estimates for semidiscrete finite element approximations of linear and semilinear parabolic equations under minimal regularity assumptions semidiscrete finite element error estimates for linear parabolic equations are derived under minimal regularity with the help of l2 projectors then analogous minimal regularity semidiscrete error estimates for semilinear parabolic equations are derived

on some high order accurate fully discrete galerkin methods for the korteweg de vries equation we construct and analyze fully discrete galerkin finite element methods of high order of accuracy for the numerical solution of the periodic initial value problem for the korteweg de vries equation the methods are based on a standard space discretization using smooth periodic splines on a uniform mesh for the time stepping we use two schemes of third resp fourth order of accuracy which are modifications of well known diagonally implicit runge kutta methods and require the solution of two resp three nonlinear systems of equations at each time step these systems are solved approximately by newton s method provided the initial iterates are chosen in a specific accurate way we show that only one newton iteration per system is needed to preserve the stability and order of accuracy of the scheme under certain mild restrictions on the space mesh length and the time step we prove l2 error estimates of optimal rate of convergence for both schemes

runge kutta methods for parabolic equations and convolution quadrature we study the approximation properties of runge kutta time discretizations of linear and semilinear parabolic equations including incompressible navier stokes equations we derive asymptotically sharp error bounds and relate the temporal order of convergence which is generally noninteger to spatial regularity and the type of boundary conditions the analysis relies on an interpretation of runge kutta methods as convolution quadratures in a different context these can be used as efficient computational methods for the approximation of convolution integrals and integral equations they use the laplace transform of the confolution kernal via a discrete operational calculus 23 refs 2 tabs

avoiding the order reduction of runge kutta methods for linear initial boundary value problems a new strategy to avoid the order reduction of runge kutta methods when integrating linear autonomous nonhomogeneous initial boundary value problems is presented the solution is decomposed into two parts one of them can be computed directly in terms of the data and the other satisfies an initial value problem without any order reduction a numerical illustration is given this idea applies to practical problems where spatial discretization is also required leading to the full order both in space and time

the potential for parallelism in runge kutta methods part 1 rk formulas in standard form the authors examine the potential for parallelism in runge kutta rk methods based on formulas in standard one step form both negative and positive results are presented many of the negative results are based on a theorem that bounds the order of an rk formula in terms of the minimal polynomial associated with its coefficient matrix the positive results are largely examples of prototypical formulas that offer a potential for effective coarse grain parallelism on machines with a few processors

stability of runge kutta methods for stiff ordinary differential equations this work analyzes the integration of initial value problems for stiff systems of ordinary differential equations by runge kutta methods the author uses the characterization of stiff initial value problems due to kreiss the jacobian matrix is essentially negative dominant and satisfies a relative lipschitz condition the existence and regularity of the numerical solution are established and conditions under which the runge kutta formula is stable are given

runge kutta nystrom methods for hyperbolic problems with time dependent coefficients abstract galerkin fully discrete approximations for hyperbolic equations with time dependent coefficients are analyzed the schemes are based on implicit runge kutta nystrom methods and are coupled with preconditioned iterative methods for the efficient albeit approximate solution of the resulting linear systems arbitrarily high order both conditionally and unconditionally stable methods are considered in addition it is shown that a specific class of methods exhibit parallel features that can be exploited to reduce the final execution time to that of a low order method

error estimates for the semidiscrete finite element approximation of a damped wave equation with a smoothing property error estimates are given for the semidiscrete finite element approximation of an inhomogeneous wave equation damped by a constant times the positive square root of the elliptic operator in the wave equation times the time derivative of the solution the damping term smooths the solution the smoothing property is used to obtain higher order error estimates for positive time even when the initial data are non smooth

error estimates for single step fully discrete approximations for nonlinear second order hyperbolic equations abstract optimal order rates of convergence are proved for fully discrete approximations for nonlinear second order hyperbolic equations which are based on a finite element approximation in the space variables and a single step method in the time variable the r h s of the differential equation can depend on u u 1 or u and the coefficients can depend on u this paper generalizes the work of bales where the r h s and coefficients were allowed to depend on u

maximum norm analysis of completely discrete finite element methods for parabolic problems we consider full discretizations of linear parabolic initial value problems in any space dimension finite elements are used for the discretization in space the resulting semidiscrete problem is integrated in time by means of a strongly a theta acceptable rational method stability estimates are derived and error bounds are established in the maximum norm for both smooth and rough initial data

differentiability and convergence rates of approximating semigroups for retarded functional differential equations in this paper the averaging approximations i e semidiscrete finite difference scheme for linear retarded functional differential equations are considered in the context of the semigroups on the hilbert space r n times l 2 it is shown that the approximating semigroups are all differentiable uniformly with respect to the index n determining the mesh size this fact makes it possible to prove the existence of a uniform exponential decay or growth rate for the approximating semigroups which can be made arbitrarily close to the decay respectively growth rate of the original semigroup more importantly the uniform differentiability makes it possible to establish convergence rates that improve with time and to prove the convergence of the approximating semigroups in the uniform operator topology the dependence of the convergence rate on the initial conditions and system parameters is also established

the etale tits process of jordan algebras revisited the etale tits process which was called the toral tits process by petersson racine 12 may be viewed as a jordan theoretical method to construct associative algebras with involution more specifically starting from a cubic etale algebra e and a quadratic etale algebra l both over an arbitrary base field as well as from invertible elements u e b l having the same norms the etale tits process produces an absolutely simple jordan algebra j j e l u b of degree 3 and dimension 9 which by structure theory must be the symmetric elements of a central simple associative algebra of degree 3 with involution of the second kind in addition e identifies canonically with a subalgebra of j

finite difference discretizations of some initial and boundary value problems with interface we analyze the discretization of initial and boundary value problems with a stationary interface in one space dimension for the heat equation the schrodinger equation and the wave equation by finite difference methods extending the concept of the elliptic projection well known from the analysis of galerkin finite element methods to our finite difference case we prove second order error estimates in space and time in the 12 norm

finite element solution of diffusion problems with irregular data diffusion problems occuring in practice often involve irregularities in the initial or boundary data resulting in a local break down of the solution s regularity this may drastically reduce the accuracy of discretization schemes over the whole interval of integration unless certain precautions are taken the diagonal pade schemes of order 2 combined with a standard finite element discretization usually require an unnatural step size restriction in order to achieve even locally optimal accuracy it is shown here that this restriction can be avoided by means of a sample damping procedure which preserves the order of the discretization and in the case 1 does not increase the costs

a converse lyapunov theorem for a class of dynamical systems which undergo switching the authors investigate the stability of a system in which the dynamics at any instant in time will follow one of a fixed set of vector fields they allow switching between members of the family of vector fields to be completely random the focus of this paper is to prove a converse lyapunov theorem for this class of systems

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

spectral fractional step runge kutta discretizations for initial boundary value problems with time dependent boundary conditions in this paper we develop a technique for avoiding the order reduction caused by nonconstant boundary conditions in the methods called splitting alternating direction or more generally fractional step methods such methods can be viewed as the combination of a semidiscrete in time procedure with a special type of additive runge kutta method which is called the fractional step runge kutta method and a standard space discretization which can be of type finite differences finite elements or spectral methods among others spectral methods have been chosen here to complete the analysis of convergence of a totally discrete scheme of this type of improved fractionary steps the numerical experiences performed also show the increase of accuracy that this technique provides

finite element approximations of nonlinear elastic waves in this paper we study finite element methods for a class of problems of nonlinear elastodynamics we discretize the equations in space using galerkin methods for the temporal discretization the construction of our schemes is based on rational approximations of cos x and ex we analyze semidiscrete as well as secondand fourth order accurate in time fully discrete methods for the approximation of the solution of the problem and prove optimal order l2 error estimates for some schemes a taylor type technique is used so that only linear systems of equations need be solved at each time step in the proofs we need various estimates for a nonlinear elliptic projection the proofs of which are also established in the paper

invariants mod 2 and subgroups of g2 and f4 abstract let k be a field of characteristic different from 2 and 3 let g be a simple group of type f 4 or g 2 defined over k in this paper we discuss embeddings of connected simple algebraic groups of type a 1 and a 2 in g in terms of the mod 2 galois cohomological invariants attached to these groups we prove that k groups of type f 4 resp g 2 arising from division algebras are generated by k subgroups of type a 2 resp a 1 see theorem 3 11 theorem 4 1 we derive a necessary and sufficient condition for an albert algebra to have zero f 5 invariant theorem 3 4 further for a k group g of type f 4 we derive a condition necessary for a k group h of type a 1 or a 2 to embed in g over k we prove that in order to embed h in g over k the mod 2 invariant of h must divide f 5 g see section 2 and remark 2 8 remark 2 3 for definition of invariants along similar lines we derive a condition for a k group h of type a 2 to embed in a k group g of type g 2 we prove that h embeds in g over k if and only if f 3 h f 3 g theorem 4 4 next we derive a necessary and sufficient condition for a k group h of type a 1 to embed in a k group g of type g 2 if g aut c is a group of type g 2 over k for an octonion algebra c over k then this condition provides a natural bijection between k conjugacy classes of involutions in g k and isometry classes of 2 fold pfister divisors of the pfister form n c the norm form of the octonion algebra c section 4 proposition 4 2

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