hermite interpolation of nonsmooth functions preserving boundary conditions this article is devoted to the construction of a hermite type regularization operator transforming functions that are not necessarily c 1 into globally c 1 finite element functions that are piecewise polynomials this regularization operator is a projection it preserves appropriate first and second order polynomial traces and it has approximation properties of optimal order as an illustration it is used to discretize a nonhomogeneous navier stokes problem with tangential boundary condition

noncommutative rational functions invariant under the action of a finite solvable group abstract this paper describes the structure of invariant skew fields for linear actions of finite solvable groups on free skew fields in d generators these invariant skew fields are always finitely generated which contrasts with the free algebra case for abelian groups or solvable groups g with a well behaved representation theory it is shown that the invariant skew fields are free on g d 1 1 generators finally positivity certificates for invariant rational functions in terms of sums of squares of invariants are presented

conservation of generalized momentum maps in mechanical optimal control problems with symmetry summary in this paper a structure preserving direct method for the optimal control of mechanical systems is developed the new method accommodates a large class of one step integrators for the underlying state equations the state equations under consideration govern the motion of affine hamiltonian control systems if the optimal control problem has symmetry associated generalized momentum maps are conserved along an optimal path this is in accordance with an extension of noether s theorem to the realm of optimal control problems in the present work we focus on optimal control problems with rotational symmetries the newly proposed direct approach is capable of exactly conserving generalized momentum maps associated with rotational symmetries of the optimal control problem this is true for a variety of one step integrators used for the discretization of the state equations examples are the one step theta method a partitioned variant of the theta method and energy momentum em consistent integrators numerical investigations confirm the theoretical findings copyright 2016 john wiley sons ltd

symmetry preserving observers this paper presents the theory of invariant observers i e symmetry preserving observers we consider an observer to consist of a copy of the system and a correction term and we propose a constructive method based on the cartan moving frame method to find all the symmetry preserving correction terms the construction relies on an invariant frame a classical notion and on an invariant output error a less standard notion precisely defined here using the theory we build three non linear observers for three examples of engineering interest a non holonomic car a chemical reactor and an inertial navigation system for each example the design is based on physical symmetries and the convergence analysis relies on the use of invariant state errors a symmetry preserving way to define the estimation error

reduction of sufficient conditions for optimal control problems with subgroup symmetry in this paper we apply symmetry reduction techniques from geometric mechanics to sufficient conditions for local optimality in optimal control problems after reinterpreting some previous results for left invariant problems on lie groups we focus on optimal control problems with subgroup symmetry for these problems the necessary conditions for optimality can be simplified by exploiting symmetries so as to reduce the number of variables needed to describe trajectories of the system we show that sufficient conditions for optimality based on the non existence of conjugate points can be simplified in an analogous way to the necessary conditions we demonstrate these simplifications by analyzing an optimal control problem that models a spinning top in a gravitational field and we give particular attention to the example of an axisymmetric sleeping top the results we derive in this paper allow us to determine which trajectories of a sleeping top are locally optimal solutions of the optimal control problem which is a new result that has not appeared in previous literature

error estimators for a mixed method we present two a posteriori error estimators for elliptic partial differential equations when we use a mixed method one is based on a adequate evaluation of the residual of the finite element solution and the other on the solution of a local problem we prove their equivalence with the norm of the error when the data is locally smooth

axioms of adaptivity with separate marking for data resolution mixed finite element methods with flux errors in h div norms and div least squares finite element methods require a separate marking strategy in obligatory adaptive mesh refining the refinement indicator sigma 2 mathcal t k eta 2 mathcal t k mu 2 k of a finite element domain k in an admissible triangulation mathcal t consists of some residual based error estimator eta mathcal t k with some reduction property under local mesh refining and some data approximation error mu k separate marking means either dorfler marking if mu 2 mathcal t leq kappa eta 2 mathcal t or otherwise an optimal data approximation algorithm with controlled accuracy the axioms are sufficient conditions on the estimators eta mathcal t k and data approximation errors mu k for optimal asymptotic convergence rates the enfolded set of axioms of this paper simplifies c carstensen m feischl m page and d praetorius comput math appl 67 2014 pp 1195 1253 for collective m

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 unifying theory of a posteriori finite element error control residual based a posteriori error estimates are derived within a unified setting for lowest order conforming nonconforming and mixed finite element schemes the various residuals are identified for all techniques and problems as the operator norm l of a linear functional of the form in the variable of a sobolev space v the main assumption is that the first order finite element space is included in the kernel ker l of l as a consequence any residual estimator that is a computable bound of l can be used within the proposed frame without further analysis for nonconforming or mixed fe schemes applications are given for the laplace stokes and navier lame equations

a unifying theory of a posteriori error control for nonconforming finite element methods residual based a posteriori error estimates were derived within one unifying framework for lowest order conforming nonconforming and mixed finite element schemes in carstensen numer math 100 617 637 2005 therein the key assumption is that the conforming first order finite element space v h c annulates the linear and bounded residual l written v h c subseteq rm ker ell that excludes particular nonconforming finite element methods ncfems on parallelograms in that v h c not subset rm ker ell the present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates also for mortar and discontinuous galerkin methods the key assumption is the existence of some bounded linear operator pi v h c rightarrow v h nc with some elementary properties it is conjectured that the more general hypothesis h1 h3 can be established for all known ncfems applications on various nonstandard finite element schemes for the laplace stokes and navier lame equations illustrate the presented unifying theory of a posteriori error control for ncfem

multi agent compositional stability exploiting system symmetries this paper considers nonlinear symmetric control systems by exploiting the symmetric structure of the system stability results are derived that are independent of the number of components in the system this work contributes to the fields of research directed toward compositionality and composability of large scale system in that a system can be built up by adding components while maintaining system stability the modeling framework developed in this paper is a generalization of many existing results which focus on interconnected systems with specific dynamics the main utility of the stability result is one of scalability or compositionality if the system is stable for a given number of components under appropriate conditions stability is then guaranteed for a larger system composed of the same type of components which are interconnected in a manner consistent with the smaller system the results are general and applicable to a wide class of problems the examples in this paper focus on the formation control problems for multi agent robotic systems

a posteriori error estimate for the mixed finite element method a computable error bound for mixed finite element methods is established in the model case of the poisson problem to control the error in the h div x l 2 norm the reliable and efficient a posteriori error estimate applies e g to raviart thomas brezzi douglas marini and brezzi douglas fortin marini elements

geometric approach to non linear optimal control problems minimum energy control problems are investigated for affine non linear systems evolving on manifolds in the framework of the geometric theory of hamiltonian systems some insights into the problems are gained via this machinery

symmetries in optimal control it is argued that the existence of symmetries may simplify as in classical mechanics the solution of optimal control problems a procedure for obtaining symmetries for the optimal hamiltonian resulting from the maximum principle is given this avoids the actual calculation of the optimal hamiltonian this procedure is based upon the notion of symmetry for the hamiltonian system with inputs and outputs associated with an optimal control problem

a multiplicative property characterizes quasinormal composition operators in l2 spaces abstract a densely defined composition operator in an l 2 space induced by a measurable transformation is shown to be quasinormal if and only if the radon nikodym derivatives h n attached to powers n of have the multiplicative property h n h n almost everywhere for n 0 1 2

a linear nonconforming finite element method for nearly incompressible elasticity and stokes flow abstract we introduce a new triangular element for nearly incompressible elasticity and incompressible fluid flow the method consists of conforming linear elements for one of the displacement or velocity for flows component and linear non conforming elements for the other component the element is proved to give an optimal approximation and this is also confirmed by several numerical examples

robust residual based a posteriori arnold winther mixed finite element analysis in elasticity this paper presents a residual based a posteriori error estimator for the arnold winther mixed finite element that utilises a post processing for the skew symmetric part of the strain numerical experiments verify the proven reliability and efficiency for suitable approximation of the skew symmetric deformation gradient numerical evidence supports that the l2l2 stress error estimator is robust in the poisson ratio and allows stable error control even in the incompressible limit

residual based a posteriori error analysis for symmetric mixed arnold winther fem this paper introduces an explicit residual based a posteriori error analysis for the symmetric mixed finite element method in linear elasticity after arnold winther with pointwise symmetric and h text div conforming stress approximation the residual based a posteriori error estimator of this paper is reliable and efficient and truly explicit in that it solely depends on the symmetric stress and does neither need any additional information of some skew symmetric part of the gradient nor any efficient approximation thereof hence it is straightforward to implement an adaptive mesh refining algorithm numerical experiments verify the proven reliability and efficiency of the new a posteriori error estimator and illustrate the improved convergence rate in comparison to uniform mesh refining a higher convergence rate for piecewise affine data is observed in the l 2 stress error and reproduced in non smooth situations by the adaptive mesh refining strategy

computational competition of symmetric mixed fem in linear elasticity the navier lame equation for linear elasticity has evoked the design of various non standard finite element methods fem in order to overcome the locking phenomenon recent developments of arnold and winther in 2002 involve a stable mixed method which strongly fulfils the symmetry constraint subsequently two h div non conforming symmetric mixed methods arose this paper comments on the implementation of all those mixed fem and provides a numerical comparison of the different symmetric mixed schemes for linear elasticity the computational survey also includes the low order elements of weak symmetry peers the non conforming kouhia and stenberg ks elements plus the conforming displacement pk fem for k 1 2 3 4 numerical experiments confirm the theoretical convergence rates for sufficiently smooth solutions and illustrate the superiority of the symmetric mfem amongst the methods of second or third order

the arnold winther mixed fem in linear elasticity part i implementation and numerical verification abstract following some de rham complex arnold and winther have recently proposed a symmetric mixed finite element method mfem in linear elasticity this paper describes the implementation of the symmetric mfem and its 30 30 local stress stiffness matrices and studies the implementation of the lowest order scheme for general boundary conditions numerical experiments in model examples in computational mechanics illustrate the robust locking free convergence behavior and support the theoretically predicted experimental convergence rates

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

the norm of a discretized gradient in h div for a posteriori finite element error analysis this paper characterizes the norm of the residual of mixed schemes in their natural functional framework with fluxes or stresses in h mathrm div h div and displacements in l 2 l2 under some natural conditions on an associated fortin interpolation operator reliable and efficient error estimates are introduced that circumvent the duality technique and so do not suffer from reduced elliptic regularity for non convex domains for the laplace stokes and lame equations this generalizes known estimators to non convex domains and introduces new a posteriori error estimators

axioms of adaptivity this paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of discrete lower bounds inexact solvers inhomogeneous boundary data or the use of equivalent error estimators solely four axioms guarantee the optimality in terms of the error estimators compared to the state of the art in the temporary literature the improvements of this article can be summarized as follows first a general framework is presented which covers the existing literature on optimality of adaptive schemes the abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method second efficiency of the error estimator is neither needed to prove convergence nor quasi optimal convergence behavior of the error estimator in this paper efficiency exclusively characterizes the approximation classes involved in terms of the best approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant third some general quasi galerkin orthogonality is not only sufficient but also necessary for the r linear convergence of the error estimator which is a fundamental ingredient in the current quasi optimality analysis due to stevenson 2007 finally the general analysis allows for equivalent error estimators and inexact solvers as well as different non homogeneous and mixed boundary conditions

mixed finite elements for elasticity there have been many efforts dating back four decades to develop stable mixed finite elements for the stress displacement formulation of the plane elasticity system this requires the development of a compatible pair of finite element spaces one to discretize the space of symmetric tensors in which the stress field is sought and one to discretize the space of vector fields in which the displacement is sought although there are number of well known mixed finite element pairs known for the analogous problem involving vector fields and scalar fields the symmetry of the stress field is a substantial additional difficulty and the elements presented here are the first ones using polynomial shape functions which are known to be stable we present a family of such pairs of finite element spaces one for each polynomial degree beginning with degree two for the stress and degree one for the displacement and show stability and optimal order approximation we also analyze some obstructions to the construction of such finite element spaces which account for the paucity of elements available