an alternative least squares formulation of the navier stokes equations with improved mass conservation the focus of this paper is on incompressible flows in three dimensions modeled by least squares finite element methods lsfem and using a novel reformulation of the navier stokes equations lsfem are attractive because the resulting discrete equations yield symmetric positive definite systems of algebraic equations and the functional provides both a local and global error measure on the other hand it has been documented for existing reformulations that certain types of boundary conditions and high aspect ratio domains can yield very poor mass conservation it has also been documented that improved mass conservation with lsfem can be achieved by strengthening the coupling between the pressure and velocity the new reformulation presented here is demonstrated to provide both improved multigrid convergence rates because it is differentially diagonally dominant and improved mass conservation over existing methods because it increases the pressure velocity coupling along the inflow and outflow boundaries

robust stabilization of a system with delays in control a robust stabilization problem is discussed for a system with delays in control the authors derive a robust stabilizing law against additive perturbations then motivated by the feature of predictive controllers they investigate the structure of robust stabilizing law and provide some interpretations on the relation between the value of time delay and the achievable margin of robust stabilizability

enhanced mass conservation in least squares methods for navier stokes equations there are many applications of the least squares finite element method for the numerical solution of partial differential equations because of a number of benefits that the least squares method has however one of the most well known drawbacks of the least squares finite element method is the lack of exact discrete mass conservation in some contexts due to the fact that the least squares method minimizes the continuity equation in l 2 norm in this paper we explore the reason for the mass loss and provide new approaches to retain the mass even in a severely underresolved grid

fosll method for the eddy current problem with three dimensional edge singularities in the case that the domain has reentrant edges the standard finite element method loses its global accuracy because of singularities on the boundary to overcome this difficulty foslla is applied in this paper foslla is a methodology for solving pdes using the dual operator here a modified foslla method is developed that employs a partially weighted functional and allows the use of a standard finite element scheme without losing global accuracy

a chain observer for nonlinear systems with multiple time varying measurement delays this paper presents a method for designing state observers with exponential error decay for nonlinear systems whose output measurements are affected by known time varying delays a modular approach is followed where subobservers are connected in cascade to achieve a desired exponential convergence rate chain observer when the delay is small a single step observer is sufficient to carry out the goal two or more subobservers are needed in the the presence of large delays the observer employs delay dependent time varying gains to achieve the desired exponential error decay the proposed approach allows to deal with vector output measurements where each output component can be affected by a different delay relationships among the error decay rate the bound on the measurement delays the observer gains and the lipschitz constants of the system are presented the method is illustrated on the synchronization problem of continuous time hyperchaotic systems with buffered measurements

analysis of first order system least squares fosls for elliptic problems with discontinuous coefficients part i first order system least squares fosls is a recently developed methodology for solving partial differential equations among its advantages are that the finite element spaces are not restricted by the inf sup condition imposed for example on mixed methods and that the least squares functional itself serves as an appropriate error measure this paper studies the fosls approach for scalar second order elliptic boundary value problems with discontinuous coefficients irregular boundaries and mixed boundary conditions a least squares functional is defined and ellipticity is established in a natural norm of an appropriately scaled least squares bilinear form for some geometries this ellipticity is independent of the size of the jumps in the coefficients the occurrence of singularities at interface corners cross points reentrant corners and irregular boundary points is discussed and a basis of singular functions with local support around singular points is established a companion paper shows that the singular basis functions can be added at little extra cost and lead to optimal performance of standard finite element discretization and multilevel solver techniques also independent of the size of coefficient jumps for some geometries

finite element methods of least squares type we consider the application of least squares variational principles to the numerical solution of partial differential equations our main focus is on the development of least squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows linear elasticity and convection diffusion for many of these problems least squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard galerkin discretizations most notably the use of least squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf sup condition arising in mixed methods for the stokes and navier stokes equations as a result application of least squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance

reduction model approach for linear time varying systems with delays we study stabilization problems for time varying linear systems with constant input delays our reduction method ensures input to state stability with respect to additive uncertainties under arbitrarily long delays it applies to rapidly time varying systems and gives a lower bound on the admissible rapidness parameters we also cover slowly time varying systems including upper bounds on the allowable slowness parameters we illustrate our work using a pendulum model

hard core theorems for complexity classes nancy lynch proved that if a decision problem a is not solvable in polynomial time then there exists an infinite recursive subset x of its domain on which the decision is almost everywhere complex in this paper general theorems of this kind that can be applied to several well known automata based complexity classes including a common class of randomized algorithms are proved

hybrid first order system least squares finite element methods with application to stokes equations this paper combines first order system least squares fosls with first order system ll fosll to create a hybrid method the fosls approach minimizes the error bf e h bf u h bf u over a finite element subspace cal v h in the operator norm min bf u h in cal v h l bf u h bf u the fosll method looks for an approximation in the range of l setting bf u h l bf w h and choosing bf w h in cal w h a standard finite element space fosll minimizes the l 2 norm of the error over l cal w h min bf w h in cal w h l bf w h bf u fosls enjoys a locally sharp globally reliable and easily computable a posteriori error estimate while fosll does not however fosll has the major advantage that it applies to problems that do not exhibit enough smoothness to enable the full advantages that the fosls approach otherwise provides the hybrid method attempts to retain the best properties of both

efficiency based adaptive local refinement for first order system least squares formulations in this paper we propose new adaptive local refinement alr strategies for first order system least squares finite elements in conjunction with algebraic multigrid methods in the context of nested iteration the goal is to reach a certain error tolerance with the least amount of computational cost and nearly uniform distribution of the error over all elements to accomplish this the refinement decision at each refinement level is determined based on optimizing efficiency measures that take into account both error reduction and computational cost two efficiency measures are discussed predicted error reduction and predicted computational cost these methods are first applied to a two dimensional 2d poisson problem with steep gradients and the results are compared with the threshold based methods described in w dorfler siam j numer anal 33 1996 pp 1106 1124 next these methods are applied to a 2d reduced model of the incompressible resistive magnetohydrodynamic equations these equations are used to simulate instabilities in a large aspect ratio tokamak we show that by using the new alr strategies on this system we are able to resolve the physics using only 10 percent of the computational cost used to approximate the solutions on a uniformly refined mesh within the same error tolerance

a least squares approach based on a discrete minus one inner product for first order systems the purpose of this paper is to develop and analyze a least squares approximation to a first order system the first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and or nonsymmetric the approach taken here is novel in that the least squares functional employed involves a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions in addition the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned thus providing an efficient method for solving the algebraic equations the preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems a task which is well understood

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

first order system least squares for the stokes equations with application to linear elasticity following our earlier work on general second order scalar equations here we develop a least squares functional for the two and three dimensional stokes equations generalized slightly by allowing a pressure term in the continuity equation by introducing a velocity flux variable and associated curl and trace equations we are able to establish ellipticity in an h1 product norm appropriately weighted by the reynolds number this immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems both estimates are naturally uniform in the reynolds number moreover our pressure perturbed form of the generalized stokes equations allows us to develop an analogous result for the dirichlet problem for linear elasticity where we obtain the more substantive result that the estimates are uniform in the poisson ratio

input delay compensation of linear systems with both state and input delays by adding integrators abstract this paper studies stabilization of linear systems with both state and input delays a dynamic input delay compensator obtained by adding integrators is established to compensate the input delays that can be arbitrarily large with the input delay compensator the original stabilization problem reduces to the problem of stabilizing an augmented linear time delay system without input delay three methods are also proposed to design stabilizing controllers for the augmented linear time delay system the first method is based on linear matrix inequalities lmis and the second method is based on model reduction the third method is based on pole placement and is built for the particular case that the original time delay system has only a pure delayed state vector on its right hand side for this method the optimal gain such that the decay rate of the closed loop system is maximized is also proposed the effectiveness of the proposed approaches is illustrated by three linear time delay systems that are open loop unstable

least squares methods for incompressible newtonian fluid flow linear stationary problems this paper develops and analyzes two least squares methods for the numerical so lution of linear stationary incompressible newtonian fluid flow in two and three dimensions both approaches use the l 2 norm to define least squares functionals one is based on the stress velocity formulation see section 3 2 and it applies to general boundary conditions the other is based on an equivalent formulation for the pseudostress and velocity see section 4 2 and it applies to pure velocity dirichlet boundary conditions the velocity gradient and vorticity can be obtained alge braically from this new tensor variable it is shown that the homogeneous least squares functionals are elliptic and continuous in the h div d h 1 d norm this immediately implies optimal error estimates for conforming finite element approximations as well it admits optimal multigrid solution methods if raviart thomas finite element spaces are used to approximate the stress or the pseudostress tensor

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

first order system cl cl fosll scalar elliptic partial differential equations the l2 norm version of first order system least squares fosls attempts to reformulate a given system of partial differential equations so that applying a least squares principle yields a functional whose bilinear part is h1 elliptic this ellipticity means that the minimization process amounts to solving a weakly coupled system of poisson like scalar equations an unfortunate limitation of the l2 norm fosls approach is that this product h1 equivalence generally requires sufficient smoothness of the original problem inverse norm fosls overcomes this limitation but at a substantial loss of real efficiency the fosll approach introduced here is a promising alternative that is based on recasting the original problem as a minimization principle involving the adjoint equations this paper provides a theoretical foundation for the fosll methodology and illustrates its performance by applying it numerically to several examples results for the so called two stage approach applied to discontinuous coefficient problems show promising robustness and optimality indeed fosll appears to exhibit the generality of the inverse norm fosls approach while retaining the full efficiency of the l2 norm approach

brief paper exponential stability of nonlinear time delay systems with delayed impulse effects the problem of exponential stability for nonlinear time delay systems with delayed impulses is addressed in this paper lyapunov based sufficient conditions for exponential stability are derived respectively for two kinds of delayed impulses i e destabilizing delayed impulses and stabilizing delayed impulses it is shown that if a nonlinear impulsive time delay system without impulse input delays is exponentially stable then under some conditions its stability is robust with respect to small impulse input delays moreover it is also shown that for a stable nonlinear impulsive time delay system if the magnitude of the delayed impulses is sufficiently small then under some conditions the delayed impulses do not destroy the stability irrespective of the sizes of the impulse input delays the efficiency of the proposed results is illustrated by three numerical examples

stabilization of linear systems with both input and state delays by observer predictors this paper studies the stabilization of linear systems with both state and input delays where the input delay can be arbitrarily large yet exactly known observer predictor based controllers are designed to predict the future states so that the input delay can be properly compensated necessary and sufficient conditions guaranteeing the stability of the closed loop system are provided in terms of the stability of some simple linear time delay systems refereed to as observer error systems by which the separation principle is discovered moreover approaches in terms of linear matrix inequalities are also provided to design both the state feedback gains and observer gains finally a numerical example illustrates that the proposed approaches are more effective and safe to implement than the existing methods