on the p and hp extension of nedelec s curl conforming elements abstract we establish interpolation error estimates for the hp extension of nedelec s curl and divergence conforming three dimensional finite elements using these interpolation estimates we investigate the use of higher order edge finite elements for the linear three dimensional magnetostatic problem our estimates show that the standard mixed method for the magnetostatic problem can be approximated to optimal order in both h and p

wavelet solvers for hp fem discretizations in 3d using hexahedral elements in this paper we investigate the discretization of an elliptic boundary value problem in 3d by means of the hp version of the finite element method using a mesh of hexahedrons the corresponding linear system is solved by a preconditioned conjugate gradient method the construction of the preconditioner is based on an inexact additive overlapping schwarz method the remaining subproblems are treated by a tensor product based preconditioner this preconditioner uses a basis transformation into a basis which is stable in l 2 and h 1 several numerical examples show the efficiency of the proposed method

the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

the continuous heisenberg chain and constrained harmonic motion it is shown that the equation describing the evolution of the classical continuous heisenberg ferromagnet can be regarded as one aspect of quadratically constrained harmonic motion as it is also the case in a number of other integrable systems two ingredients are used a method to solve the inverse scattering problem for second order operators using ordinary differential rather than integral equations and the equivalence of the heisenberg chain with the nonlinear schrodinger equation

analysis of a finite element method for maxwell s equations the use of finite elements to discretize the time dependent maxwell equations on a bounded domain in three dimensional space is analyzed energy norm error estimates are provided when general finite element methods are used to discretize the equations in space in addition it is shown that if some curl conforming elements due to nedelec are used error estimates may also be proved in the l 2 norm

error estimates for the combinedh andp versions of the finite element method in theh version of the finite element method convergence is achieved by refining the mesh while keeping the degree of the elements fixed on the other hand thep version keeps the mesh fixed and increases the degree of the elements in this paper we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh in addition it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom even in the presence of corner singularities numerical results comparing theh version p version and combinedh p version for a one dimensional problem are presented

spectral element discretization of the maxwell equations on considere un probleme variationnel equivalent aux equations de l electromagnetisme avec conditions aux limites absorbantes sur une partie de la frontiere qu on prouve etre bien pose puis on propose une discretisation de ce probleme par schema aux differences finies en temps et elements spectraux en espace et on etablit des estimations d erreur entre solutions exacte et approchee

an energy preserving crank nicolson galerkin spectral element method for the two dimensional nonlinear schrodinger equation abstract a crank nicolson galerkin spectral element method for solving the nonlinear schrodinger nls equation in two dimensions is proposed in this paper our key idea is twofolds first the 2d nls equation is rewritten as an infinite dimensional hamiltonian pde and the hamiltonian pde is discreted by using the galerkin spectral element gse method in space second we cast the resulted odes into a finite dimensional canonical hamiltonian system and discrete the system by using the crank nicolson cn method the relay leads to a fully discretized and energy preserved scheme without grid ratio restrictions the order of convergence of our new method is o 2 h 2 if the discrete l 2 norm is employed the fast fourier transform and the matrix diagonalization method are applied to the new method to increase computing efficiency numerical examples are given to further illustrate the conservation properties and convergence of the energy preserving scheme constructed

mixed fem of higher order for time dependent contact problems abstract in this paper mixed finite element methods of higher order for time dependent contact problems are discussed the mixed methods are based on resolving the contact conditions by the introduction of lagrange multipliers dynamic signorini problems with and without friction are considered involving thermomechanical and rolling contact rothe s method is used to provide a suitable time and space discretization to discretize in time a stabilized newmark method is applied as an adequate time stepping scheme the space discretization relies on finite elements of higher order in each time step the resulting problems are solved by uzawa s method or alternatively by methods of quadratic programming via a suitable formulation in terms of the lagrange multipliers numerical results are presented towards an application in production engineering the results illustrate the performance of the presented techniques for a variety of problem formulations

efficient preconditioning for the p version finite element method in two dimensions parallel preconditioners are formulated and analyzed for systems of equations arising from the p version finite element method applied to second order self adjoint elliptic boundary value problems in two dimensions by using new theoretical results for polynomial spaces it is proved that the condition number of the preconditioned system grows as log 2 p where p is the degree of the polynomial space numerical results are presented showing that the condition number indeed grows very slowly with p

an hp error analysis of mitc plate elements we give an hp error analysis of several rectangular families of finite elements for the reissner mindlin plate bending equations we consider both the original mitc families k j bathe f brezzi and m fortin internat j numer methods engrg 28 1989 pp 1787 1801 and some new ones introduced in this paper for the deflection and rotation we give error estimates which are optimal with respect to the mesh size h and optimal up to ordo p varepsilon varepsilon 0 arbitrary with respect to the polynomial degree p we also obtain estimates for the error in the shear force calculated via two different methods our analysis utilizes some recent results of ours for the mixed method for the stokes problem as well as hp interpolation estimates for mixed methods for second order elliptic equations in this regard we derive new hp results in this paper for the brezzi douglas fortin marini spaces and improve upon previous estimates for the brezzi douglas marini spaces

tensored polynomial models the theme of the present paper is the study of two different versions of tensor products of functional models and present some applications to various problems related to system theory among those are stability of higher order systems tangent spaces spaces of intertwining maps invariant factors of tensor products of linear transformations and the solvability of sylvester equations

the optimal convergence rate of the p version of the finite element method optimal error estimates for the p version of the finite element method in two dimensions are proven for the case when u in h k omega or u has singularities induced by the corners of the domain the case of nonhomogenous essential boundary conditions is also analyzed

mixed interpolated elements for reissner mindlin plates dcpartement de mathtmatiyue l nit ersitt 1 aral quthec canado summary we present in this paper a procedure to establish reissner mindlin plate bending elements the procedure is based on the idea to combine known resuits on the approximation of stokes problems with known results on the approximation of elliptic problems the proposed elements satisfy the mathematical conditions of stability and convergence and some of them promise to provide efficient elements for practical solutions

hp approximation theory for bdfm and rt finite elements on quadrilaterals we study approximation properties of hp finite element subspaces of boldsymbol mathsf h mathop rm div omega and boldsymbol mathsf h mathop rm rot omega on a polygonal domain omega using brezzi douglas fortin marini bdfm or raviart thomas rt elements approximation theoretic results are derived for the hp version finite element method on non quasi uniform meshes of quadrilateral elements with hanging nodes for functions belonging to weighted sobolev spaces boldsymbol mathsf h omega s ell omega and the countably normed spaces pmb cal b w ell omega these results culminate in a proof of the characteristic exponential convergence property of the hp version finite element method on suitably designed meshes under similar conditions needed for the analysis of the boldsymbol mathsf h 1 omega case by way of illustration exponential convergence rates are deduced for mixed hp approximation of flow in porous media

continuous galerkin methods for solving the time dependent maxwell equations in 3d geometries a few years ago costabel and dauge proposed a variational setting which allows one to solve numerically the time harmonic maxwell equations in 3d geometries with the help of a continuous approximation of the electromagnetic field in this paper we investigate how their framework can be adapted to compute the solution to the time dependent maxwell equations in addition we propose some extensions such as the introduction of a mixed variational setting and its discretization to handle the constraint on the divergence of the field 2007 elsevier inc all rights reserved

the p and hp versions of the finite element method for problems with boundary layers we study the uniform approximation of boundary layer functions exp x d for x 0 1 d 0 1 by the p and hp versions of the finite element method for the p version with fixed mesh we prove super exponential convergence in the range p 1 2 e 2d we also establish for this version an overall convergence rate of o p 1 ln p in the energy norm error which is uniform in d and show that this rate is sharp up to the ln p term when robust estimates uniform in d 0 1 are considered for the p version with variable mesh i e the hp version we show that exponential convergence uniform in d 0 1 is achieved by taking the first element at the boundary layer to be of size o pd numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates even when few degrees of freedom are used and when d is as small as e g 10 8 they also illustrate the superiority of the hp approach over other methods including a low order h version with optimal exponential mesh refinement the estimates established in this paper are also applicable in the context of corresponding spectral element methods

the h p and h p versions of the finite element methods in 1 dimension part iii the adaptive h p version abstract the paper is the third and final part in the series of three devoted to the detailed analysis of the three basic versions of the finite element method in one dimension the first part analyzed the p version and the second part concentrated on the h and h p version this paper analyzes a theoretical frame of the adaptive h p version and based on it the authors provide concrete algorithm for the one dimensional problem it is proven that in the case that the solution has x sub alpha type singularity the adaptive algorithm give an exponential rate of convergence very close to the optimal one analyzed in the second part of the paper additional keyword error analysis

on a finite element method for solving the three dimensional maxwell equations abstract the aim of this paper is to present a method for solving the time domain three dimensional maxwell equations which can be coupled with a particle solver for this purpose maxwell s equations are reformulated as a constrained wave equation system with lagrange multipliers associated to the conditions 0 and e e 0 we approximate both the fields and the lagrange multipliers with a finite element method using a taylor hood element

analysis for quadrilateral mitc elements for the reissner mindlin plate problem the present paper is made up of two parts in the first part we study the mathematical stability and convergence of the quadrilateral mitc elements for the reissner mindlin plate problem in an abstract setting we generalize the brezzi bathe fortin conditions to the quadrilateral mitc elements by weakening the second and fourth conditions under these conditions we show the well posedness of the discrete problem and establish an abstract error estimate in the energy norm the conclusion of this part is sparsity in the mathematical research of the quadrilateral mitc elements in the sense that one only needs to check these five conditions in the second part we extend four families of rectangular mitc elements of stenberg and suri to the quadrilateral meshes we prove that these quadrilateral elements satisfy the generalized brezzi bathe fortin conditions from the first part we develop the h p error estimates in both energy and l 2 norm for these quadrilateral elements for the first three families of quadrilateral elements the error estimates indicate that their convergent rates in both energy and l 2 norm depend on the mesh distortion parameter a we can get optimal error estimates for them provided that a 1 in addition we show the optimal convergence rates in energy norm uniformly in a for the fourth family of quadrilateral elements like their rectangular counterparts these quadrilateral elements are locking free

error analysis of variable degree mixed methods for elliptic problems via hybridization a new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree raviart thomas method for second order elliptic problems the approach gives error estimates for the lagrange multipliers without using error estimates for the other variables error estimates for the primal and flux variables then follow from those for the lagrange multipliers in contrast traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the lagrange multipliers the new approach not only gives new error estimates for the new variable degree raviart thomas method but also new error estimates for the classical uniform degree method with less stringent regularity requirements than previously known estimates the error analysis is achieved by using a variational characterization of the lagrange multipliers wherein the other unknowns do not appear this approach can be applied to other hybridized mixed methods as well

linear integral transformations and hierarchies of integrable nonlinear evolution equations abstract integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations these are riemann hilbert type of integral transformations which leave invariant an infinite sequence of ordinary differential matrix equations of increasing order in an indefinite parameter k the potential matrices in these equations obey a set of nonlinear recursion relations leading to a heirarchy of nonlinear partial differential equations in decreasing order the same equations give rise to a reciprocal hierarchy associated with heisenberg ferromagnet type of equations central in the treatment is an embedding of the hierarchy into an infinite matrix structure which is constructed on the basis of the integral equations in terms of this infinite matrix structure the equations governing the hierarchies become quite simple furthermore it leads in a straightforward way to various generalizations such as to other types of linear spectral problems multicomponent system and lattice equations generalizations to equations associated with noncommuting flows follow as a direct consequence of the treatment finally some results on conserved densities and the hamiltonian structure are briefly discussed

a combined mixed and discontinuous galerkin method for compressible miscible displacement problem in porous media in this paper we present a numerical scheme for solving the coupled system of compressible miscible displacement problem in porous media the flow equation is solved by the mixed finite element method and the transport equation is approximated by a discontinuous galerkin method the scheme is continuous in time and a priori hp error estimates is presented

a polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensions iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems a p version finite element method based on continuous piecewise q p functions is considered for second order elliptic problems in three dimensions this special method can also be viewed as a conforming spectral element method an iterative method is designed for which the condition number of the relevant operator grows only in proportion to 1 log p 2 this bound is independent of jumps in the coefficient of the elliptic problem across the interfaces between the subregions numerical results are also reported which support the theory

hp interpolation of nonsmooth functions and an application to hp a posteriori error estimation the quasi interpolation operators of clement and scott zhang type are generalized to the setting of the hp fem finite element method new polynomial lifting and inverse estimates are presented the classical residual based a posteriori error estimator is generalized to the hp fem

augmented formulations for solving maxwell equations we consider augmented variational formulations for solving the static or time harmonic maxwell equations for that a term is added to the usual h curl conforming formulations it consists of a weighted l2 scalar product between the divergence of the em and the divergence of test fields in this respect the methods we present are h curl div conforming we also build mixed augmented variational formulations with either one or two lagrange multipliers to dualize the equation on the divergence and when applicable the relation on the tangential or normal trace of the field it is proven that one can derive formulations which are equivalent to the original static or time harmonic maxwell equations in the latter case spurious modes are automatically excluded numerical analysis and experiments will be presented in the forthcoming paper augmented formulations for solving maxwell equations numerical analysis and experiments in preparation

two level domain decomposition preconditioning for the p version finite element method in three dimensions this paper presents a flexible approach to domain decomposition preconditioning for problems expressed in terms of local stiffness matrices of subdomains evaluation of the preconditioner requires fully parallelizable local computations and the solution of a global auxiliary problem with few variables per subdomain convergence factors can be bounded using properties of the local stiffness matrices only the method is applied to the p version finite element method for three dimensional elasticity we treat each element as a subdomain and compute explicit convergence bounds

a uniformly accurate finite element method for the reissner mindlin plate a simple finite element method for the reissner mindlin plate model in the prim itive variables is presented and analyzed the method uses nonconforming linear finite elements for the transverse displacement and conforming linear finite elements enriched by bubbles for the rotation with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise averaging it is proved that the method converges with optimal order uniformly with respect to thickness

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

dual primal feti algorithms for edge element approximations two dimensional h and p finite elements on shape regular meshes a family of dual primal finite element tearing and interconnecting feti methods for edge element approximations in two dimensions is proposed and analyzed the primal constraints here are averages over subdomain edges it is shown that the condition number of the corresponding method is independent of the number of substructures and grows only polylogarithmically with the number of unknowns associated with individual substructures the estimate is also independent of the jumps of both of the coefficients of the original problem numerical results validating our theoretical bounds are given

the treatment of nonhomogeneous dirichlet boundary conditions by thep version of the finite element method the paper addresses the problem of the implementation of nonhomogeneous essential dirichlet type boundary conditions in thep version of the finite element method

an implementation of the hp version of the finite element method for reissner mindlin plate problems reissner mindlin plate theory is still a topic of research in finite element analysis one reason for the continuous development of new plate elements is that it is still difficult to construct elements which are accurate and stable against the well known shear locking effect in this paper we suggest an approach which allows high order polynomial degrees of the shape functions for deflection and rotations a balanced adaptive mesh refinement and increase of the polynomial degree in an hp version finite element program is presented and it is shown in numerical examples that the results are highly accurate and that high order elements show virtually no shear locking even for very small plate thickness

spectral discretization of the vorticity velocity and pressure formulation of the stokes problem we consider the stokes problem in a square or a cube provided with nonstandard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity we write a variational formulation of this problem with three independent unknowns the vorticity the velocity and the pressure next we propose a discretization by spectral methods which relies on this formulation and since it leads to an inf sup condition on the pressure in a natural way we prove optimal error estimates for the three unknowns we present numerical experiments which are in perfect coherence with the analysis

mixed finite element methods for quasilinear second order elliptic problems a mixed finite element method is developed to approximate the solution of a quasilinear second order elliptic partial differential equation the existence and uniqueness of the approximation are demonstrated and optimal rate error estimates are derived

a hierarchy of approximate models for the maxwell equations in this paper we perform an asymptotic study of the maxwell equations with respect to the small parameter eps v c where v is the characteristic velocity associated with the physical problem and c is the speed of light this enables us to derive the quasistatic and darwin models as respectively first and second order approximations of the maxwell equations moreover an interpretation of the obtained variational formulations gives us the appropriate boundary conditions for these models

the problem of selecting the shape functions for a p type finite element abstract the paper addresses the question of the optimal selection of the shape functions for p type finite elements and discusses the effectivity of the conjugate gradient and multilevel iteration method for solving the corresponding linear system the selection of the shape functions is of major importance for the performance of the solver based on iterative methods neither the theory nor practice of the optimal selection of the shape functions is available yet we have seen that the condensation approach which has obvious advantages from the point of parallel computations is a very effective tool for keeping the condition number under the control and is especially advantageous for the conjugate gradient method

the hp mitc finite element method for the reissner mindlin plate problem the popular mitc finite elements used for the approximation of the reissner mindlin plate are extended to the case where elements of non uniform degree p distribution are used on locally refined meshes such an extension is of particular interest to the hp version and hp adaptive finite element methods a priori error bounds are provided showing that the method is locking free the analysis is based on new approximation theoretic results for non uniform brezzi douglas fortin marini spaces and extends the results obtained in the case of uniform order approximation on globally quasi uniform meshes presented by stenberg and suri siam j numer anal 34 1997 544 numerical examples illustrating the theoretical results and comparing the performance with alternative standard galerkin approaches are presented for two new benchmark problems with known analytic solution including the case where the shear stress exhibits a boundary layer the new method is observed to be locking free and able to provide exponential rates of convergence even in the presence of boundary layers

discretization by finite elements of a model parameter dependent problem the discretization by finite elements of a model variational problem for a clamped loaded beam is studied with emphasis on the effect of the beam thickness which appears as a parameter in the problem on the accuracy it is shown that the approximation achieved by a standard finite element method degenerates for thin beams in contrast a large family of mixed finite element methods are shown to yield quasioptimal approximation independent of the thickness parameter the most useful of these methods may be realized by replacing the integrals appearing in the stiffness matrix of the standard method by gauss quadratures

the optimal convergence of the h p version of the finite element method with quasi uniform meshes in the framework of the jacobi weighted besov spaces we analyze the convergence of the h p version of finite element solutions on quasi uniform meshes and the lower and upper bounds of errors for elliptic problems on polygons both lower and upper bounds are proved to be optimal in h and p which leads to the optimal convergence of the h p version of the finite element method with quasi uniform meshes for elliptic problems on polygons the results proved for the h p version include the h version with quasi uniform meshes and the p version with quasi uniform degrees as two special cases

on tangential matrix interpolation abstract the paper presents an algebraic approach using polynomial and rational models over an arbitrary field to tangential interpolation problems both by polynomial as well as rational matrix functions appropriate extensions of scalar problems associated with the names of lagrange first order hermite high order and newton recursive are derived the relation of interpolation problems to the matrix chinese remainder theorem are clarified some two sided interpolation problems are discussed using the theory of tensored functional models

on computing the pressure by thep version of the finite element method for stokes problem this paper introduces and analyzes two ways of extracting the hydrostatic pressure when solving stokes problem using thep version of the finite element method when one uses a localh 1 projection we show that optimal rates of convergence for the pressure approximation is achieved when the pressure is not inh 1 or the value of the pressure is only needed at a few points one may extract the pressure pointwise using e g a single layer potential recovery negative zero and higher norm estimates for the stokes velocity are derived within the framework of thep version of the f e m

mixed finite element methods for nonlinear elliptic problems the h p version mixed finite element methods for strongly nonlinear second order elliptic problems are proposed and analyzed existence and uniqueness of the approximate solution are demonstrated using a fixed point argument convergence and stability of the method are proved both with respect to mesh refinement and increase of the degree of the approximating polynomials the analysis is carried out in detail using raviart thomas nedelec spaces as an example numerical results for minimal surface problems are obtained using brezzi douglas marini spaces graphs of the approximate solutions are presented for two different problems

an efficient implicit spectral element method for time dependent nonlinear diffusion equations by evaluating integrals at one quadrature point we present an implicit spectral element method to approximate the solution of time dependent nonlinear diffusion equations in complex geometries we propose a nodal expansion to approximate derivatives of an unknown function so integrals involving nonlinearities are evaluated at one quadrature point and discrete form of nonlinear diffusion equations is obtained without forming local and global stiffness matrices using the method of lines mol the nonlinear partial differential equation is reduced to a nonlinear system of ordinary differential equations such that the mol formulation is derived in terms of the differentiation matrix instead of the stiffness matrix since the arising system is stiff an implicit method is employed to solve it also with a simple algorithm we obtain a closed analytical form for the jacobian matrix of the system of nonlinear algebraic equations in addition the differentiation matrices for triangular elements and deformed quadrilateral and hexahedral elements are obtained for deformed elements an isoparametric mapping is used where the expansion coefficients are obtained with the aid of the gordon hall mapping several benchmarks including the scalar and system of these equations are carried out as physical applications the cell motion model over long time the tumour angiogenesis model on a mixed quadrangle triangle mesh and two and three dimensional p laplace equations for various choices of the parameter p are considered

polynomial approximation of divergence free functions etude de la meilleure approximation d une fonction sans divergence par un polynome algebrique ou trigonometrique sans divergence etablissement d une estimation optimale

a priori error analysis of stochastic galerkin mixed approximations of elliptic pdes with random data we construct stochastic galerkin approximations to the solution of a first order system of pdes with random coefficients under the standard finite dimensional noise assumption we transform the variational saddle point problem to a parametric deterministic one approximations are constructed by combining mixed finite elements on the computational domain with m variate tensor product polynomials we study the inf sup stability and well posedness of the continuous and finite dimensional problems the regularity of solutions with respect to the m parameters describing the random coefficients and establish a priori error estimates for stochastic galerkin finite element approximations

a hierarchical iterative procedure for the analysis of composite laminates a multilevel recursively defined preconditioner for use with the preconditioned conjugate gradient pcg algorithm is developed and applied to solve thin and thick laminated plate problems the preconditioner is constructed from a sequence of hierarchical vector spaces arising from the p version of the finite element fe method numerical studies have been conducted using quadrilateral elements for two dimensional problems up to p 8 and serendipity brick elements for three dimensional problems up to p 5 the efficiency of the iterative procedure is illustrated using bending response of laminated plates the effects of element span ratios convergence tolerance polynomial order material properties and plate span ratios on the number of iterations required for convergence has been investigated results indicate that the preconditioner developed herein can be used to produce an efficient iterative solver for two and three dimensional problems in structural mechanics

the h p and h p versions of the finite element method in 1 dimension this paper is the first one in the series of three which are addressing in detail the properties of the three basic versions of the finite element method in the one dimensional setting the main emphasis is placed on the analysis when the exact solution has singularity of x type the first part analyzes thep version the second theh version and generalh p version and the final third part addresses the problems of the adaptiveh p version

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

the h p and h p versions of the finite element method in 1 dimension part ii the error analysis of the h and h p versions the paper is the second in the series of three devoted to the detailed analysis of the three basic versions of the finite element method in one dimensional setting the first part 1 analyzed thep version the present one concentrates on theh andh p versions

tangent spaces of rational matrix functions we consider the task of determining tangent spaces for classes of rational matrix valued functions our analysis is based on methods from control theory and in particular the theory of polynomial models explicit descriptions of tangent spaces of rational transfer functions stable rational transfer functions rational inner functions and symmetric rational transfer functions are obtained moreover a new proof of delchamps s decomposition formula for the tangent bundle of rational transfer functions is given a riemannian metric as well as a symplectic structure is defined 0 1998 elsevier science inc

analytic feedback control and the algebraic riccati equation the central result of this correspondence is lemma 1 1 which states that the stabilizing solution p to the algebraic riccati equation in a b c depends analytically on a b c in the remainder of the correspondence various control and system theoretic ramifications of the analyticity lemma are considered an important consequence of lemma 1 1 is that any smooth state feedback control law for a finite dimensional plant may be implemented using dynamic input output compensators whose transfer functions depend smoothly on the plant transfer function

duality in polynomial models with some applications to geometric control theory duality is studied in the context of polynomial models for linear systems the output injection group the dual of the feedback group is studied and a polynomial characterization of c a invariant subspaces as well as of the maximal reachability subspaces contained in ker c is given

on the accuracy of p version elements for the reissner mindlin plate problem this paper addresses the question of accuracy of p version finite element formulations for reissner mindlin plate problems three model problems a circular arc a rhombic plate and a geometrically complex structure are investigated whereas displacements and bending moments turn out to be very accurate without any post processing even for very coarse meshes the quality of shear forces computed from constitutive equations is poor it is shown that significantly improved results can be obtained if shear forces are computed from equilibrium equations instead a consistent computation of second derivatives of the shape functions is derived 1998 john wiley sons ltd

on bogovskiĭ and regularized poincare integral operators for de rham complexes on lipschitz domains we study integral operators related to a regularized version of the classical poincare path integral and the adjoint class generalizing bogovski s integral operator acting on differential forms in mathbb r n we prove that these operators are pseudodifferential operators of order 1 the poincare type operators map polynomials to polynomials and can have applications in finite element analysis for a domain starlike with respect to a ball the special support properties of the operators imply regularity for the de rham complex without boundary conditions using poincare type operators and with full dirichlet boundary conditions using bogovski type operators for bounded lipschitz domains the same regularity results hold and in addition we show that the cohomology spaces can always be represented by fancyscript c infty functions

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

a p version mitc finite element method for reissner mindlin plates with curved boundaries we consider the approximation of reissner mindlin plates with curved boundaries using a p version mitc finite element method we describe in detail the formulation and implementation of the method and emphasize the need for a piola type map in order to handle the curved geometry of the elements the results of our numerical computations demonstrate the robustness of the method and suggest that it gives near exponential convergence when the error is measured in the energy norm for the robust computation of quantities of engineering interest such as the shear force the proposed method yields very satisfactory results without the need for any additional post processing comparisons are made with the standard finite element formulation with and without post processing

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

on the stability and convergence of higher order mixed finite element methods for second order elliptic problems we investigate the use of higher order mixed methods for second order elliptic problems by establishing refined stability and convergence estimates which take into account both the mesh size h and polynomial degree p our estimates yield asymptotic convergence rates for the p and h p versions of the finite element method they also describe more accurately than previously proved estimates the increased rate of convergence expected when the h version is used with higher order polynomials

optimal error estimation for bfh rmcurl conforming p interpolation in two dimensions in this paper we prove an optimal error estimate for the bf h rm curl conforming projection based p interpolation operator introduced in l demkowicz and i babuska siam j numer anal 41 2003 pp 1195 1208 this result is proved on the reference element either triangle or square k for regular vector fields in bf h r rm curl k with arbitrary r 0 the formulation of the result in the bf h rm div conforming setting which is relevant for the analysis of high order boundary element approximations for maxwell s equations is provided as well

a meshless local petrov galerkin method for the time dependent maxwell equations abstract in this paper the meshless local petrov galerkin mlpg method is employed to solve the 2 d time dependent maxwell equations the mlpg method is a truly meshless method in which the trial and test functions are chosen from totally different functional spaces in the current work the moving least square reproducing kernel mlsrk scheme is chosen to be the trial function the method is applied for the unsteady maxwell equations in different media in the local weak form by employing the difference operator for evolution in time and simultaneously in time and space the semi discrete and fully discrete schemes are obtained respectively the error estimation is discussed for both the semi discrete and fully discrete numerical schemes for modelling the time dependent maxwell equations we show that provided that the time step size is sufficiently small the proposed scheme yields an error of o 2 m 1 2 in the l 2 norm for the square of error the new scheme is implemented and the numerical results are provided to justify our theoretical analysis

toda flows inverse spectral transform and realization theory abstract in this paper we establish some new relationships between the solution techniques for completely integrable nonlinear differential equations of the toda type and topics in realization theory

use of potentials in the study of magnet models of the heisenberg type abstract we show that a pair of coupled nonlinear evolution equations can be associated with an extended form of the continuous heisenberg magnet model applying a prolongation technique to these equations we derive a potential function in terms of which the original system can be expressed using the invariance property under the mobius group of a certain differential equation involving the potential function we outline a method to get new solutions of the spin system in the extended form starting from trivial ones as a special application of our machinery a few examples of solutions of the su 1 1 magnet model are presented

locking effects in the finite element approximation of plate models we analyze the robustness of various standard finite element schemes for a hierarchy of plate models and obtain asymptotic convergence estimates that are uniform in terms of the thickness d we identify h version schemes that show locking i e for which the asymptotic convergence rate deteriorates as d 0 and also show that the p version is free of locking in order to isolate locking erects from boundary layer erects which also arise as d 0 our analysis is carried out for the periodic case which is free of boundary layers we analyze in detail the lowest model of the hierarchy the well known reissner mindlin model and show that the locking and robustness of finite element schemes for higher models of the hierarchy are essentially identical to the riessner mindlin case