a new a posteriori error estimate for the morley element in this paper we present a posteriori error analysis for the nonconforming morley element of the fourth order elliptic equation we propose a new residual based a posteriori error estimator and prove its reliability and efficiency these results refine those of beirao da veiga et al numer math 106 165 179 2007 by dropping two edge jump terms in both the energy norm of the error and the estimator and those of wang and zhang local a priori and a posteriori error estimates of finite elements for biharmonic equation research report 13 2006 by showing the efficiency in the sense of verfurth a review of a posteriori error estimation and adaptive mesh refinement techniques wiley teubner new york 1996 moreover the normal component in the estimators of beirao da veiga et al numer math 106 165 179 2007 and wang and zhang local a priori and a posteriori error estimates of finite elements for biharmonic equation research report 13 2006 is dropped and therefore only the tangential component of the stress on each edge comes into the estimator in addition we generalize these results to three dimensional case

a posteriori error estimates for continuous discontinuous galerkin approximations of the kirchhoff love plate we present energy norm a posteriori error estimates for continuous discontinuous galerkin c dg approximations of the kirchhoff love plate problem the method is based on a continuous displacement field inserted into a symmetric discontinuous galerkin formulation of the fourth order partial differential equation governing the deflection of a thin plate we also give explicit formulas for the penalty parameter involved in the formulation

a semi implicit relaxation scheme for modeling two phase flow in a pipeline this work is devoted to the numerical approximation of the solutions to the system of conservation laws which arises in the modeling of two phase flows in pipelines the pdes are closed by two highly nonlinear algebraic relations namely a pressure law and a hydrodynamic law we have previously proposed an explicit relaxation scheme which allows us to cope with these nonlinearities but the system considered has eigenvalues which are of very different orders of magnitude which prevents the explicit scheme from being effective since the time step has to be very small in order to solve this effectiveness problem we now proceed to construct a scheme which is explicit with respect to the small eigenvalues and linearly implicit with respect to the large eigenvalues numerical evidences are provided

l infty l estimates on trajectories confined to a closed subset for control systems with bounded time variation the term distance estimate for state constrained control systems refers to an estimate on the distance of an arbitrary state trajectory from the subset of state trajectories that satisfy a given state constraint distance estimates have found widespread application in state constrained optimal control they have been used to establish regularity properties of the value function to establish the non degeneracy of first order conditions of optimality and to validate the characterization of the value function as a unique solution of the hjb equation the most extensively applied estimates of this nature are so called linear l infty distance estimates the earliest estimates of this nature were derived under hypotheses that required the multifunctions or controlled differential equations describing the dynamic constraint to be locally lipschitz continuous w r t the time variable recently it has been shown that the lipschitz continuity hypothesis can be weakened to a one sided absolute continuity hypothesis this paper provides new less restrictive hypotheses on the time dependence of the dynamic constraint under which linear l infty estimates are valid here one sided absolute continuity is replaced by the requirement of one sided bounded variation this refinement of hypotheses is significant because it makes possible the application of analytical techniques based on distance estimates to important new classes of discontinuous systems including some hybrid control systems a number of examples are investigated showing that for control systems that do not have bounded variation w r t time the desired estimates are not in general valid and thereby illustrating the important role of the bounded variation hypothesis in distance estimate analysis

a posteriori error estimates for nonconforming finite element methods computable a posteriori error bounds for a large class of nonconforming finite element methods are provided for a model poisson problem in two and three space dimensions besides a refined residual based a posteriori error estimate an averaging estimator is established and an l 2 estimate is included the a posteriori error estimates are reliable and efficient the proof of reliability relies on a helmholtz decomposition

global existence of weak solutions for a viscous two phase model abstract the purpose of this paper is to explore a viscous two phase liquid gas model relevant for well and pipe flow our approach relies on applying suitable modifications of techniques previously used for studying the single phase isothermal navier stokes equations a main issue is the introduction of a novel two phase variant of the potential energy function needed for obtaining fundamental a priori estimates we derive an existence result for weak solutions in a setting where transition to single phase flow is guaranteed not to occur when the initial state is a true mixture of both phases some numerical examples are also included in order to demonstrate characteristic behavior of solutions in particular we illustrate how two phase flow is genuinely different compared to single phase flow concerning the behavior of an initial mass discontinuity

discretization error for the discrete kirchhoff plate finite element approximation abstract we provide in this work the discretization error estimates that can guide an adaptive mesh refinement for the discrete kirchhoff plate finite elements the proposed developments are built upon the concept of error estimates for classical elasticity and adapted to suit the kirchhoff plate finite elements we give a detailed illustration of the proposed procedures for the discrete kirchhoff triangular plate element along with several different possibilities for constructing the enhancement of test space needed for error estimates the first novelty concerns the consistent displacement field in terms of the third order polynomial for the discrete kirchhoff triangle whereas the second novelty is the use of the argyris triangle with fifth order polynomials for constructing the enhanced test for error estimates we compare the latter against several alternatives that can be used for kirchhoff plates the results of numerical examples are given to illustrate the effectiveness of proposed discretization error estimates

the morley element for fourth order elliptic equations in any dimensions in this paper the well known nonconforming morley element for biharmonic equations in two spatial dimensions is extended to any higher dimensions in a canonical fashion the general n dimensional morley element consists of all quadratic polynomials defined on each n simplex with degrees of freedom given by the integral average of the normal derivative on each n 1 subsimplex and the integral average of the function value on each n 2 subsimplex explicit expressions of nodal basis functions are also obtained for this element on general n simplicial grids convergence analysis is given for this element when it is applied as a nonconforming finite element discretization for the biharmonic equation

a family of c0 finite elements for kirchhoff plates ii numerical results abstract a new family of c0 kirchhoff plate elements has been introduced by the authors in the theoretical counterpart of the present paper a family of c0 finite elements for kirchhoff plates i error analysis the method presented is a displacement formulation with the deflection and the rotation as unknowns in the theoretical part an a priori and an a posteriori error analysis has been accomplished for the family in the present contribution the authors first give a supplementary derivation of the method and recall the main theoretical results then focus on the computational aspects of the method and finally present a set of numerical results on various benchmark computations these tests verify the optimal convergence rate of the method and illustrate the robustness of the reliable and efficient residual based a posteriori error estimator for adaptive mesh refinements

the adaptive ciarlet raviart mixed method for biharmonic problems with simply supported boundary condition abstract in this paper we study the adaptive fashion of the ciarlet raviart mixed method for biharmonic equation eigenvalue problem with simply supported boundary condition in r d we propose an a posteriori error indicator of the ciarlet raviart approximate solution for the biharmonic equation and an a posteriori error indicator of the ciarlet raviart approximate eigenfuction and prove the reliability and efficiency of the indicators we also give an a posteriori error indicator for the approximate eigenvalue and prove its reliability we design an adaptive ciarlet raviart mixed method with piecewise polynomials of degree less than or equal to m and numerical experiments show that numerical eigenvalues obtained by the method can achieve the optimal convergence order o d o f 2 m d d 2 m 2 3 d 3 m 3

a posteriori error estimates for mixed fem in elasticity a residue based reliable and efficient error estimator is established for finite element solutions of mixed boundary value problems in linear planar elasticity the proof of the reliability of the estimator is based on helmholtz type decompositions of the error in the stress variable and a duality argument for the error in the displacements the efficiency follows from inverse estimates the constants in both estimates are independent of the lame constant lambda and so locking phenomena for lambda to infty are properly indicated the analysis justifies a new adaptive algorithm for automatic mesh refinement

a gel fand type spectral radius formula and stability of linear constrained switching systems abstract using ergodic theory in this paper we present a gel fand type spectral radius formula which states that the joint spectral radius is equal to the generalized spectral radius for a matrix multiplicative semigroup s restricted to a subset that need not carry the algebraic structure of s this generalizes the berger wang formula using it as a tool we study the absolute exponential stability of a linear switched system driven by a compact subshift of the one sided markov shift associated to s

energy norm a posteriori error estimates for discontinuous galerkin approximations of the linear elasticity problem we present a residual based a posteriori error estimate in an energy norm of the error in a family of discontinuous galerkin approximations of linear elasticity problems the theory is developed in two and three spatial dimensions and general nonconvex polygonal domains are allowed we also present some illustrating numerical examples

a posteriori error estimates for the morley plate bending element a local a posteriori error indicator for the well known morley element for the kirchhoff plate bending problem is presented the error indicator is proven to be both reliable and efficient the technique applied is general and it is shown to have also other applications

quasi optimal convergence rate for an adaptive hybridizable c 0 discontinuous galerkin method for kirchhoff plates in this paper we present an adaptive hybridizable c 0 discontinuous galerkin hcdg method for kirchhoff plates a reliable and efficient a posteriori error estimator is produced for this hcdg method quasi orthogonality and discrete reliability are established with the help of a postprocessed bending moment and the discrete helmholtz decomposition based on these the contraction property between two consecutive loops and complexity of the adaptive hcdg method are studied thoroughly the key points in our analysis are a postprocessed normal normal continuous bending moment from the hcdg method solution and a lifting of jump residuals from inter element boundaries to element interiors

filippov s and filippov wazewski s theorems on closed domains abstract the celebrated filippov s theorem implies that given a trajectory x1 0 r n of a differential inclusion x f t x with the set valued map f measurable in t and k lipschitz in x for any initial condition x2 0 r n there exists a trajectory x2 starting from x2 0 such that x1 t x2 t ekt x1 0 x2 0 filippov wazewski s theorem establishes the possibility of approximating any trajectory of the convexified differential inclusion x co f t x by a trajectory of the original inclusion x f t x starting from the same initial condition in the present paper we extend both theorems to the case when the state variable x is constrained to the closure of an open subset r n the latter is allowed to be non smooth we impose a generalized soner type condition on f and yielding extensions of the above classical results to infinite horizon constrained problems applications to the study of regularity of value functions of optimal control problems with state constraints are discussed as well

a posteriori error analysis for the morley plate element with general boundary conditions this paper introduces and analyses a local residual based a posteriori error indicator for the morley finite element method of the biharmonic kirchhoff plate bending problem in the theoretical part of the paper a recent approach presented by the authors for clamped boundaries is extended to general boundary conditions the error indicator is proven to be both reliable and efficient the numerical part of the paper presents a set of results on various benchmark computations with different kinds of domains and boundary conditions these tests verify the reliability and efficiency of the error estimator and illustrate the robustness of the method for adaptive mesh refinements copyright 2009 john wiley sons ltd

linear estimates for trajectories of state constrained differential inclusions and normality conditions in optimal control abstract we consider solutions to a differential inclusion x f x constrained to a compact convex set here f is a compact possibly non convex valued lipschitz continuous multifunction whose convex closure co f satisfies a strict inward pointing condition at every boundary point x given a reference trajectory x taking values in an e neighborhood of we prove the existence of a second approximating trajectory x 0 t which satisfies the linear estimate x x ac 0 t k e if one of the following two cases occurs i the initial datum x 0 x 0 is given but lies in a compact set containing only points where the boundary is smooth ii the initial point x 0 of the approximating trajectory x can be chosen arbitrarily subsequently we employ these linear ac estimates to establish conditions for normality of the generalized euler lagrange condition for optimal control problems with state constraints in which we have an integral term in the cost we finally provide an illustrative example which underlines the fact that if conditions i and ii above are not satisfied then we can find a degenerate minimizer

a posteriori error estimates for nonconforming finite element methods for fourth order problems on rectangles the a posteriori error analysis of conforming finite element discretisations of the biharmonic problem for plates is well established but nonconforming discretisations are more easy to implement in practice the a posteriori error analysis for the morley plate element appears very particular because two edge contributions from an integration by parts vanish simultaneously this crucial property is lacking for popular rectangular nonconforming finite element schemes like the nonconforming rectangular morley finite element the incomplete biquadratic finite element and the adini finite element this paper introduces a novel methodology and utilises some conforming discrete space on macro elements to prove reliability and efficiency of an explicit residual based a posteriori error estimator an application to the morley triangular finite element shows the surprising result that all averaging techniques yield reliable error bounds numerical experiments confirm the reliability and efficiency for the established a posteriori error control on uniform and graded tensor product meshes

global solutions of the navier stokes equations for multidimensional compressible flow with discontinuous initial data abstract we prove the global existence of weak solutions of the navier stokes equations for compressible isothermal flow in two and three space dimensions when the initial density is close to a constant in l 2 and l and the initial velocity is small in l 2 and bounded in l 2 n in two dimensions the l 2 norms must be weighted slightly a great deal of qualitative information about the solution is obtained for example we show that the velocity and vorticity are relatively smooth in positive time as is the effective viscous flux f which is the divergence of the velocity minus a certain multiple of the pressure we find that f plays a crucial role in the entire analysis particularly in closing the required energy estimates understanding rates of regularization near the initial layer and most important obtaining time independent pointwise bounds for the density

the lower bound property of the morley element eigenvalues in this paper we prove that the morley element eigenvalues approximate the exact ones from below on regular meshes including adaptive local refined meshes for the fourth order elliptic eigenvalue problems with the clamped boundary condition in any dimension and we implement the adaptive computation with the morley element to obtain lower bounds of eigenvalues for the vibration problem of clamped plate under tension and a fourth order elliptic eigenvalue problem with variable coefficients

isogeometric analysis for sixth order boundary value problems of gradient elastic kirchhoff plates abstract sixth order boundary value problems of a one parameter gradient elastic kirchhoff plate model are formulated in a weak form within an h 3 sobolev space setting with the corresponding equilibrium equations and general boundary conditions the corresponding conforming galerkin method is proposed with error estimates for discretizations satisfying c 2 continuity requirements continuity coercivity and consistency of the corresponding bilinear form are utilized for proving the theoretical results numerical computations with conforming isogeometric discretizations of c p 1 continuous nurbs basis functions of order p 3 confirm the theoretical results and illustrate the features of the problem for both statics and free vibrations in particular the effects of the additional boundary conditions and parameter dependent boundary layers corresponding to the gradient elasticity theory are addressed by the numerical examples

optimal control with state space constraint i optimal control of piecewise deterministic processes with state space constraint is studied under appropriate assumptions it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain finally the uniform continuity of the value function is obtained under a condition on the deterministic drift

convergence and optimality of the adaptive morley element method this paper is devoted to the convergence and optimality analysis of the adaptive morley element method for the fourth order elliptic problem a new technique is developed to establish a quasi orthogonality which is crucial for the convergence analysis of the adaptive nonconforming method by introducing a new parameter dependent error estimator and further establishing a discrete reliability property sharp convergence and optimality estimates are then fully proved for the fourth order elliptic problem

framework for the a posteriori error analysis of nonconforming finite elements this paper establishes a unified framework for the a posteriori error analysis of a large class of nonconforming finite element methods the theory assures reliability and efficiency of explicit residual error estimates up to data oscillations under the conditions h1 h2 and applies to several nonconforming finite elements the crouzeix raviart triangle element the han parallelogram element the nonconforming rotated nr parallelogram element of rannacher and turek the constrained nr parallelogram element of hu and shi the p 1 element on parallelograms due to park and sheen and the dssy parallelogram element the theory is extended to include 1 irregular meshes with at most one hanging node per edge

new error estimates of the morley element for the plate bending problems in this paper we derive some new error estimates of a nonconforming morley element for the plate bending problems our aim is fourfold first it shows that the consistency error of the morley element can be controlled by its corresponding approximation error up to an arbitrarily high order term especially which can be of arbitrary order provided that f on the right hand side is piecewise smooth enough thus the total energy norm is dominated alone by the approximation error second as a byproduct we derive the error estimate under the regularity assumption u h 2 s with s 0 1 which fills the gap in the a priori error estimate of the morley element with low regularity for s 0 1 2 third based on this new error bound a robust convergence is proved even if the solution is exactly in h 2 finally by a slight modification of the variational formulation a new error estimate is also derived for the case f h 1 the key role is played by some quasi interpolation operators and the weak continuous properties of morley finite element space the main ideas and results presented here are motivated by mao and shi 2010 and can be viewed as its extension to fourth order problems

estimates for trajectories confined to a cone in mathbb r lowercase n we develop estimates on the distance of a trajectory associated with a differential inclusion dot x in f in mathbb r n to a given set of feasible f trajectories where feasible means with values confined to a given cone when the cone is a half space it is known that a linear k h x estimate of the mathbf w 1 1 distance is valid where k is a constant independent of the initial choice of trajectory and h x is a measure of the constraint violation a recent counterexample has unexpectedly demonstrated that linear estimates of the distance are no longer valid when the state constraint set is the intersection of two half spaces this paper addresses fundamental questions concerning the approximation of general f trajectories by f trajectories confined to a cone which is the intersection of two half spaces in this context we establish the validity of a superlinear k h x ln h x estimate of the distance we demonstrate by means of an example that the structure of this estimate is optimal we show furthermore that a linear estimate can be recovered in the case when the velocity set f is strictly convex

convergence of an adaptive mixed finite element method for kirchhoff plate bending problems some reliable and efficient a posteriori error estimators are produced for a mixed finite element method the hellan herrmann johnson h h j method for kirchhoff plate bending problems cf k hellan acta polytech scand civil engrg ser 46 1967 pp 1 28 l herrmann j eng mech div asce 93 1967 pp 13 26 c johnson numer math 21 1973 pp 43 62 based on these results with k 0 1 where k denotes the polynomial order of the discrete moment field space an adaptive mixed finite element method amfem is set up and its convergence and complexity are studied thoroughly the key points of the theoretical analysis include achieving a discrete helmholtz decomposition and a discrete inf sup condition which serve as the main tools in deducing the quasi orthogonality of the moment field and the discrete reliability of the estimator it is shown that the amfem is a contraction for the sum of the moment field error in an energy norm and the scaled error estimator between two consecutive adaptive loops moreover an estimate for the amfem s complexity via the number of elements is developed

a discrete helmholtz decomposition with morley finite element functions and the optimality of adaptive finite element schemes the discrete reliability of a finite element method is a key ingredient to prove optimal convergence of an adaptive mesh refinement strategy and requires the interchange of a coarse triangulation and some arbitrary refinement of it one approach for this is the careful design of an intermediate triangulation with one level refinements and with the remaining difficulty to design some interpolation operator which maps a possibly nonconforming approximation into the finite element space based on the finer triangulation this paper enfolds the second possibility of some novel discrete helmholtz decomposition for the nonconforming morley finite element method this guarantees the optimality of a standard adaptive mesh refining algorithm for the biharmonic equation numerical examples illustrate the crucial dependence of the bulk parameter and the surprisingly short pre asymptotic range of the adaptive morley finite element method

trajectories of differential inclusions with state constraints abstract the paper deals with solutions of a differential inclusion x f x constrained to a compact convex set here f is a compact possibly non convex valued lipschitz continuous multifunction whose convex closure co f satisfies a strict inward pointing condition at every boundary point x given a reference trajectory x taking values in an e neighborhood of we prove the existence of a second trajectory x 0 t which satisfies x x w 1 1 c e 1 ln e as shown by an earlier counterexample this bound is sharp

l estimates on trajectories confined to a closed subset this paper concerns the validity of estimates on the distance of an arbitrary state trajectory from the set of state trajectories which lie in a given state constraint set these so called distance estimates have wide spread application in state constrained optimal control including justifying the use of the maximum principle in normal form and establishing regularity properties of value functions we focus on linear l distance estimates which of all the available estimates have so far been the most widely used such estimates are known to be valid for general closed state constraint sets provided the functions defining the dynamic constraint are lipschitz continuous with respect to the time and state variables we ask whether linear l distance estimates remain valid when the lipschitz continuity hypothesis governing t dependence of the data is relaxed we show by counter example that these distance estimates are not valid in general if the hypothesis of lipschitz continuity is replaced by continuity we also provide a new hypothesis absolute continuity from the left for the validity of linear l estimates the new hypothesis is less restrictive than lipschitz continuity and even allows discontinuous time dependence in certain cases it is satisfied in particular by differential inclusions exhibiting non lipschitz t dependence at isolated points governed for example by a fractional power modulus of continuity the relevance of distance estimates for state constrained differential inclusions permitting fractional power time dependence is illustrated by an example in engineering design where we encounter an isolated square root type singularity concerning the t dependence of the data

on trajectories satisfying a state constraint w 1 1 estimates and counterexamples this paper concerns properties of solutions to a differential inclusion f trajectories satisfying a state constraint estimates on the w 1 1 distance of a given f trajectory to the set of f trajectories satisfying the constraint have an important role in state constrained optimal control theory regarding the derivation of nondegenerate necessary conditions sensitivity analysis characterization of the value function in terms of the hamilton jacobi equation and other applications according to some of the earlier literature estimates in which the w 1 1 distance is related linearly to the degree of constraint violation of the original f trajectory are valid for state constraints defined by a collection of one or more inequality constraint functionals we show by counterexample that linear w 1 1 estimates are not in general valid for multiple state constraints we also identify cases involving several state constraints where not even a weaker linear l infty estimate holds we further show that it is possible to justify linear w 1 1 estimates by means of a modification of earlier constructive techniques when there is only one state constraint in a future companion paper estimates for trajectories confined to a cone in mathbb r n siam j control optim submitted we develop weaker estimates for multiple state constraints and identify additional hypotheses under which w 1 1 estimates are valid in this broader setting

on the qualitative theory of lower semicontinuous differential inclusions on etudie les proprietes qualitatives de l ensemble des solutions de caratheodory pour une inclusion differentielle de la forme x t f t x t t 0 t x 0 r n

on the wave structure of two phase flow models we explore the relationship between two common two phase flow models usually denoted as the two fluid and drift flux models they differ in their mathematical description of momentum transfer between the phases in this paper we provide a framework in which these two model formulations are unified the drift flux model employs a mixture momentum equation and treats interphasic momentum exchange indirectly through the slip relation which gives the relative velocity between the phases as a function of the flow parameters this closure law is in general highly complex which makes it difficult to analyze the model algebraically to facilitate the analysis we express the quasi linear formulation of the drift flux model as a function of three parameters the derivatives of the slip with respect to the vector of unknown variables the wave structure of this model is investigated using a perturbation technique then we rewrite the drift flux model with a general slip relation such that it is expressed in term