on the poincare friedrichs inequality for piecewise h 1 functions in anisotropic discontinuous galerkin finite element methods the purpose of this paper is to propose a proof for the poincare friedrichs inequality for piecewise h 1 functions on anisotropic meshes by verifying suitable assumptions involved in the newly proposed proof we show that the poincare friedrichs inequality for piecewise h 1 functions holds independently of the aspect ratio which characterizes the shape regular condition in finite element analysis in addition under the maximum angle condition we establish the poincare friedrichs inequality for the crouzeix raviart non conforming linear finite element counterexamples show that the maximum angle condition is only sufficient

spectral theory for the periodic sine gordon equation a concrete viewpoint a summary of the spectral theory for quasiperiodic sine and sinh gordon equations is given analogies with whole line solitons and scattering theory motivates the discussion the relation between the ingredients in the inverse spectral solution of the periodic sine gordon equation and physical characteristics of sine gordon waves is emphasized the explicit topics covered are summarized in the table of contents in the introduction

error estimates for the raviart thomas interpolation under the maximum angle condition the classical error analysis for the raviart thomas interpolation on triangular elements requires the so called regularity of the elements or equivalently the minimum angle condition however in the lowest order case optimal order error estimates have been obtained in g acosta and r g duran siam j numer anal 37 2000 pp 18 36 replacing the regularity hypothesis by the maximum angle condition which was known to be sufficient to prove estimates for the standard lagrange interpolation in this paper we prove error estimates on triangular elements for the raviart thomas interpolation of any order under the maximum angle condition also we show how our arguments can be extended to the three dimensional case to obtain error estimates for tetrahedral elements under the regular vertex property introduced in g acosta and r g duran siam j numer anal 37 2000 pp 18 36

piecewise polynomial interpolation in muckenhoupt weighted sobolev spaces and applications we develop a constructive piecewise polynomial approximation theory in weighted sobolev spaces with muckenhoupt weights for any polynomial degree the main ingredients to derive optimal error estimates for an averaged taylor polynomial are a suitable weighted poincare inequality a cancellation property and a simple induction argument we also construct a quasi interpolation operator built on local averages over stars which is well defined for functions in l 1 l1 we derive optimal error estimates for any polynomial degree on simplicial shape regular meshes on rectangular meshes these estimates are valid under the condition that neighboring elements have comparable size which yields optimal anisotropic error estimates over n n rectangular domains the interpolation theory extends to cases when the error and function regularity require different weights we conclude with three applications nonuniform elliptic boundary value problems elliptic problems with singular sources and fractional powers of elliptic operators

a singularly perturbed problem with two parameters in two dimensions on graded meshes abstract a numerical approximation of a convection reaction diffusion problem by standard bilinear finite elements is considered using duran lombardi and duran shishkin type meshes we prove first order error estimates in an energy norm numerical examples confirm our theoretical results and show smaller errors compared to the well known shishkin mesh

uniform error estimates for certain narrow lagrange finite elements error estimates of dupont and scott are used to derive uniform error estimates for lagrange finite elements in jrn n 2 under the following conditions 1 the elements can be arbitrarily narrow in any coordinate direction such that a sufficient number of interpolation points are grouped on lines parallel to that coordinate axis and 2 the space of approximating functions ft in each element t must include the space of polynomials of degree 1 n 2 if n is odd this does not cover elements of lowest degree that are normally considered with the shape regularity requirement that the ratio of their outer and inner diameters be bounded for example if n 3 the usual requirement with shape regularity is that each ft contain all first degree polynomials the result of this paper requires that each ft contain all quadratic polynomials and consequently does not apply to linear courant elements in tetrahedrons or trilinear tensor elements in rectangular boxes counterexamples in these two cases are included

interpolation error estimates in w 1 p for degenerate q 1 isoparametric elements optimal order error estimates in h 1 for the q 1 isoparametric interpolation were obtained in acosta and duran siam j numer anal37 18 36 1999 for a very general class of degenerate convex quadrilateral elements in this work we show that the same conlusions are valid in w 1 p for 1 p 3 and we give a counterexample for the case p 3 showing that the result cannot be generalized for more regular functions despite this fact we show that optimal order error estimates are valid for any p 1 keeping the interior angles of the element bounded away from 0 and independently of the aspect ratio we also show that the restriction on the maximum angle is sharp for p 3

korn s inequalities for piecewise h 1 vector fields korn s inequalities for piecewise h 1 vector fields are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

uniform convergence and superconvergence of mixed finite element methods on anisotropically refined grids the lowest order raviart thomas rectangular element is considered for solving the singular perturbation problem mbox div a nabla p bp f where the diagonal tensor a varepsilon 2 1 or a varepsilon 2 varepsilon 2 global uniform convergence rates of o n 1 for both p and a1 2 nabla p in the l2 norm are obtained in both cases where n is the number of intervals in either direction the pointwise interior away from the boundary layers convergence rates of o n 1 for p are also proved superconvergence i e o n 2 at special points and o n 2 global l2 estimate for both p and a 1 2 nabla p are obtained by a local postprocessing numerical results support our theoretical analysis moreover numerical experiments show that an anisotropic mesh gives more accurate results than the standard global uniform mesh

on the interpolation error estimates for q_1 quadrilateral finite elements in this paper we study the relation between the error estimate of the bilinear interpolation on a general quadrilateral and the geometric characters of the quadrilateral some explicit bounds of the interpolation error are obtained based on some sharp estimates of the integral over frac 1 j p 1 for 1 leq p leq infty on the reference element where j is the jacobian of the nonaffine mapping this allows us to introduce weak geometric conditions depending on p leading to interpolation error estimates in the w 1 p norm for any p in 1 infty which can be regarded as a generalization of the regular decomposition property rdp condition introduced in g acosta and r g duran siam j numer anal 38 2000 pp 1073 1088 for p 2 and new rdp conditions nrdp for p neq2 we avoid the use of the reference family elements which allows us to extend the results to a larger class of elements and to introduce the nrdp condition in a more unified way as far as we know the mesh condition presented in this paper is weaker than any other mesh conditions proposed in the literature for any p with 1 leq p leq infty

error estimates for cq_1 isoparametric elements satisfying a weak angle condition we prove optimal order error estimates in the h1 norm for the cq 1 isoparametric interpolation on convex quadrilateral elements under rather weak hypotheses improving previously known results choose one diagonal and divide the element into two triangles we show that if the chosen diagonal is the longest one then the constant in the error estimate depends only on the maximum angle of the two triangles otherwise the constant depends on that maximum angle and on the ratio between the two diagonals in particular we obtain the optimal order error estimate under the maximum angle condition as in the case of triangular elements consequently the error estimate is uniformly valid for a rather general class of degenerate quadrilaterals

corrections to stochastic barbalat s lemma and its applications jun 12 1537 1543 the proof of theorem 1 stochastic barbalat s lemma in the paper by wu is incorrect this note provides a new statement of stochastic barbalat s lemma in addition some definitions propositions and proofs are given to replace those in the paper by wu

lagrange and average interpolation over 3d anisotropic elements abstract an average interpolation is introduced for 3 rectangles and tetrahedra and optimal order error estimates in the h1 norm are proved the constant in the estimate depends weakly improving the results given in duran math comp 68 1999 187 199 on the uniformity of the mesh in each direction for tetrahedra the constant also depends on the maximum angle of the element on the other hand merging several known results acosta and duran siam j numer anal 37 1999 18 36 duran math comp 68 1999 187 199 krizek siam j numer anal 29 1992 513 520 al shenk math comp 63 1994 105 119 we prove optimal order error for the p 1 lagrange interpolation in w1 p p 2 with a constant depending on p as well as the maximum angle of the element again under the maximum angle condition optimal order error estimates are obtained in the h1 norm for higher degree interpolations

supercloseness on graded meshes for q1 finite element approximation of a reaction diffusion equation in this paper we analyze the standard piece wise bilinear finite element approximation of a model reaction diffusion problem we prove supercloseness results when appropriate graded meshes are used the meshes are those introduced in duran and lombardi 2005 8 but with a stronger restriction on the graduation parameter as a consequence we obtain almost optimal error estimates in the l 2 norm thus completing the error analysis given in duran and lombardi 2005 8

a fem for an optimal control problem of fractional powers of elliptic operators we study solution techniques for a linear quadratic optimal control problem involving fractional powers of elliptic operators these fractional operators can be realized as the dirichlet to neumann map for a nonuniformly elliptic problem posed on a semi infinite cylinder in one more spatial dimension thus we consider an equivalent formulation with a nonuniformly elliptic operator as the state equation the rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation we discretize the proposed truncated state equation using first degree tensor product finite elements on anisotropic meshes for the control problem we analyze two approaches one that is semidiscrete based on the so called variational approach where the control is not discretized and the other one that is fully discrete via the discretization of the control by piecewise constant functions for both approaches we derive a priori error estimates with respect to degrees of freedom numerica

error estimates on anisotropic elements for functions in weighted sobolev spaces in this paper we prove error estimates for a piecewise q 1 average interpolation on anisotropic rectangular elements i e rectangles with sides of different orders in two and three dimensions our error estimates are valid under the condition that neighboring elements have comparable size this is a very mild assumption that includes more general meshes than those allowed in previous papers in particular strong anisotropic meshes arising naturally in the approximation of problems with boundary layers fall under our hypotheses moreover we generalize the error estimates allowing on the right hand side some weighted sobolev norms this extension is of interest in singularly perturbed problems finally we consider the approximation of functions vanishing on the boundary by finite element functions with the same property a point that was not considered in previous papers on average interpolations for anisotropic elements as an application we consider the approximation of a singularly perturbed reaction diffusion equation and show that as a consequence of our results almost optimal order error estimates in the energy norm valid uniformly in the perturbation parameter can be obtained

isospectral hamiltonian flows in finite and infinite dimensions the approach to isospectral hamiltonian flow introduced in part i is further developed to include integration of flows with singular spectral curves the flow on finite dimensional ad invariant poisson submanifolds of the dual widetilde gl r of the positive part of the loop algebra widetilde gl r is obtained through a generalization of the standard method of linearization on the jacobi variety of the invariant spectral curves these curves are embedded in the total space of a line bundlet p1 c allowing an explicit analysis of singularities arising from the structure of the image of a moment map tilde j r m n r times m n r to widetilde gl r from the space of rank r deformations of a fixedn n matrixa it is shown how the linear flow of line bundles e t to tilde s over a suitably desingularized curve tilde s may be used to determine both the flow of matricial polynomialsl and the hamiltonian flow in the spacemn r mn r in terms of functions the resulting flows are proved to be completely integrable the reductions to subalgebras developed in part i are shown to correspond to invariance of the spectral curves and line bundles e t to tilde s under certain linear or anti linear involutions the integration of two examples from part i is given to illustrate the method the rosochatius system and the cnls coupled non linear schrodinger equation

discussion on barbalat lemma extensions for conformable fractional integrals in this paper the barbalat type lemmas for conformable fractional order integrals which can be used to conclude the convergence of a function to zero is discussed

the maximum angle condition for mixed and nonconforming elements application to the stokes equations for the lagrange interpolation it is known that optimal order error estimates hold for elements satisfying the maximum angle condition the objective of this paper is to obtain similar results for the raviart thomas interpolation arising in the analysis of mixed methods we prove that optimal order error estimates hold under the maximum angle condition for this interpolation both in two and three dimensions and moreover that this condition is indeed necessary to have these estimates error estimates for the mixed approximation of second order elliptic problems and for the nonconforming piecewise linear approximation of the stokes equations are derived from our results

on the maximum angle condition for linear tetrahedral elements synge s maximum angle condition for triangular elements is generalized to tetrahedral elements for the generalized condition it is proved that tetrahedra may degenerate in a certain way and the error of the standard linear interpolation remains o h in the w p 1 omega norm for sufficiently smooth functions and p in 1 infty

a pde approach to fractional diffusion we derive a computable a posteriori error estimator for the harmonic extension problem which localizes the fractional powers of elliptic operators supplemented with dirichlet boundary conditions our a posteriori error estimator relies on the solution of small discrete problems on anisotropic cylindrical stars it exhibits built in flux equilibration and is equivalent to the energy error up to data oscillation under suitable assumptions we design a simple adaptive algorithm and present numerical experiments which reveal a competitive performance

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

boundedness and convergence on fractional order systems we establish conditions to guarantee boundedness and convergence of signals described by non integer order equations using caputo derivatives the case of linear time varying unforced equations is first studied and later results for linear time varying forced equations and time varying unforced non linear equations are presented and discussed

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

existence of ℋ matrix approximants to the inverse fe matrix of elliptic operators with l coefficients this article deals with the existence of blockwise low rank approximants so called matrices to inverses of fem matrices in the case of uniformly elliptic operators with l coefficients unlike operators arising from boundary element methods for which the matrix theory has been extensively developed the inverses of these operators do not benefit from the smoothness of the kernel function however it will be shown that the corresponding green functions can be approximated by degenerate functions giving rise to the existence of blockwise low rank approximants of fem inverses numerical examples confirm the correctness of our estimates as a side product we analyse the matrix property of the inverse of the fe mass matrix

error estimates of triangular finite elements under a weak angle condition in this note by analyzing the interpolation operator of girault and raviart given in v girault p a raviart finite element methods for navier stokes equations theory and algorithms in springer series in computational mathematics springer verlag berlin 1986 over triangular meshes we prove optimal interpolation error estimates for lagrange triangular finite elements of arbitrary order under the maximal angle condition in a unified and simple way the key estimate is only an application of the bramble hilbert lemma

stabilization mechanisms in discontinuous galerkin finite element methods in this paper we propose a new general framework for dg methods which allows to uncover a basic mechanism that ensures suitable stability properties of the methods we show that such a mechanism is essentially the same for apparently unrelated stabilizations such as jump penalty upwinding and hughes franca type residual based stabilizations

error estimates for 3 d narrow finite elements we obtain error estimates for finite element approximations of the lowest degree valid uniformly for a class of three dimensional narrow elements first for the lagrange interpolation we prove optimal error estimates both in order and regularity in l p for p 2 for p 2 it is known that this result is not true applying extrapolation results we obtain an optimal order error estimate for functions sligthly more regular than h 2 these results are valid both for tetrahedral and rectangular elements second for the case of rectangular elements we obtain optimal in order and regularity error estimates for an average interpolation valid for functions in w 1 s p with 1 p and 0 s 1

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

on the equivalence of regularity criteria for triangular and tetrahedral finite element partitions in this note we examine several regularity criteria for families of simplicial finite element partitions in r d email protected 2 3 these are usually required in numerical analysis and computer implementations we prove the equivalence of four different definitions of regularity often proposed in the literature the first one uses the volume of simplices the others involve the inscribed and circumscribed ball conditions and the minimal angle condition

separation of variables for soliton equations via their binary constrained flows binary constrained flows of soliton equations admitting 2 2 lax matrices have 2n degrees of freedom which is twice as many degrees of freedom than in the case of monoconstrained flows by using the normal method their lax matrices directly give rise to first n pairs of canonical separated variables for their separation of variables we propose a new method to introduce the other n pairs of canonical separated variables and additional separated equations the jacobi inversion problems for binary constrained flows are established finally the factorization of soliton equations by two commuting binary constrained flows and the separability of binary constrained flows enable us to construct the jacobi inversion problems for some soliton hierarchies

isospectral hamiltonian flows in finite and infinite dimensions i generalized moser systems and moment maps into loop algebras a moment map jr jia gl r is constructed from the poisson manifold j a of rank r perturbations of a fixed n xn matrix a to the dual r f e positive part of the formal loop algebra i r g r c 1 the adler kostant symes theorem is used to give hamiltonians which generate commutative isospectral flows on g r the pull back of these hamiltonians by the moment map gives rise to commutative isospectral hamiltonian flows in j a the latter may be identified with flows on finite dimensional coadjoint orbits in g and linearized on the jacobi variety of an invariant spectral curve xr which generically is an r sheeted riemann surface reductions of j a are derived corresponding to subalgebras of gl r c and sl r c determined as the fixed point set of automorphism groupes generated by involutions i e jdl the classical algebras as well as reductions to twisted subalgebras of s r c the theory is illustrated by a number of examples of finite dimensional isospectral flows defining integrable hamiltonian systems and their embeddings as finite gap solutions to integrable systems of pde s

on fractional extensions of barbalat lemma a b s t r a c t this paper presents barbalat like lemmas for fractional order integrals which can be used to conclude about the convergence of a function to zero based on some conditions upon its fractional integral some examples in the context of asymptotic behaviour of solutions of fractional order differential equations indicate the potential application of these lemmas in control theory

on the lyapunov theory for fractional order systems abstract we provide the main features of lyapunov theory when it is formulated for fractional order systems we give consistent extensions of lyapunov lasalle and chetaev classical theorems to the case of fractional order systems we give examples to illustrate the applications of the concepts and propositions introduced

isospectral flow in loop algebras and quasiperiodic solutions of the sine gordon equation the sine gordon equation is considered in the hamiltonian framework provided by the adler kostant symes theorem the phase space a finite dimensional coadjoint orbit in the dual space g of a loop algebra g is parameterized by a finite dimensional symplectic vector space w embedded into g by a moment map real quasiperiodic solutions are computed in terms of theta functions using a liouville generating function which generates a canonical transformation to linear coordinates on the jacobi variety of a suitable hyperelliptic curve

quantization of solitons and the restricted sine gordon model we show how to compute form factors matrix elements of local fields in the restricted sine gordon model at the reflectionless points by quantizing solitons we introduce quantum separated variables in which the hamiltonians are expressed in terms of quantum functions we explicitly describe the soliton wave functions and we explain how the restriction is related to an unusual hermitian structure we also present a semi classical analysis which enlightens the fact that the restricted sine gordon model corresponds to an analytical continuation of the sine gordon model intermediate between sine gordon and kdv

stochastic barbalat s lemma and its applications in the deterministic case a significant improvement on stability analysis of nonlinear systems is caused by introducing barbalat s lemma into control area after lyapunov s second method and lasalle s theorem were established this note considers the extension of barbalat s lemma to the stochastic case to this end the uniform continuity and the absolute integrability are firstly described in stochastic forms it is nevertheless a small generalization upon the existing references since our result can be used to adapted processes which are not necessarily ito diffusions when it is applied to ito diffusion processes many classical results on stochastic stability are covered as special cases

a pde approach to space time fractional parabolic problems we study solution techniques for parabolic equations with fractional diffusion and caputo fractional time derivative the latter being discretized and analyzed in a general hilbert space setting the spatial fractional diffusion is realized as the dirichlet to neumann map for a nonuniformly elliptic problem posed on a semi infinite cylinder in one more spatial dimension we write our evolution problem as a quasi stationary elliptic problem with a dynamic boundary condition we propose and analyze an implicit fully discrete scheme first degree tensor product finite elements in space and an implicit finite difference discretization in time we prove stability and error estimates for this scheme

numerical inverse spectral transform for the periodic sine gordon equation theta function solutions and their linearized stability abstract the inverse spectral transform for integrable nonlinear ordinary and partial differential equations such as the toda lattice korteweg de vries sine gordon and nonlinear schrodinger equations provides explicit algorithms to generate exact solutions under periodic or quasiperiodic boundary conditions these oscillatory wavetrains may be prescribed a priori to consist of a nonlinear superposition of n phases j x t j x j t 0 j j 1 n where the wave is 2 periodic independently in each phase this paper exhibits the numerical implementation of the inverse spectral solution of the sine gordon equation the general construction is outlined and then implemented for n 1 2 and 3 we compute 1 the exact theta function solutions 2 the floquet spectrum of x periodic solutions 3 the labelling of linearized instabilities of n phase solutions in terms of spectral data and 4 the linearized growth rate in each unstable mode the associated surfaces q n x t are displayed to illustrate a variety of spatial and dynamical phenomena in the oscillatory solution space of this integrable system