a forward in time advection scheme and adaptive multilevel flow solver for nearly incompressible atmospheric flow this paper presents a new forward in time advection method for nearly incompressible flow mu and its application to an adaptive multilevel flow solver for atmospheric flows mu is a modification of leonardet al sutopia scheme mu like utopia is based on third order accurate semi lagrangian multidimensional upwinding for constant velocity flows for varying velocity fields mu is a second order conservative method mu has greater stability and accuracy than utopia and naturally decomposes into a monotone low order method and a higher order accurate correction for use with flux limiting its stability and accuracy make it a computationally efficient alternative to current finite difference advection methods we present a fully second order accurate flow solver for the anelastic equations a prototypical low mach number flow the flow solver is based on mu which is used for both momentum and scalar transport equations this flow solver can also be implemented with any forward in time advection scheme the multilevel flow solver conserves discrete global integrals of advected quantities and includes adaptive mesh refinement its second order accuracy is verified using a nonlinear energy conservation integral for the anelastic equations for a typical geophysical problem in which the flow is most rapidly varying in a small part of the domain the multilevel flow solver achieves global accuracy comparable to a uniform resolution simulation for 10 of the computational cost

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

a semi implicit numerical scheme for reacting flow an additive semi implicit projection scheme for the simulation of unsteady combustion in two dimensions is constructed the scheme relies on a zero mach number formulation of the compressible conservation equations with detailed chemistry the governing equations are discretized in space using second order differences and integrated in time using a semi implicit approach time integration of the evolution equations for species mass fraction thermodynamic pressure and density is performed using a semi implicit nonsplit scheme that combines a second order predictor corrector treatment of convection and diffusion terms and a stiff integrator for the reaction source terms meanwhile the momentum equations are integrated using a second order projection scheme the projection scheme is based on a predictor corrector approach that couples the evolution of the velocity and density fields in order to stabilize computations of reacting flows with large density variations a pressure poisson equation is inverted following both the predictor and corrector steps using a fast solver the advantages of the stiff integration of reaction source terms are analyzed by comparing the performance of the scheme to that of a predictor corrector scheme in which reaction and diffusion are integrated in a similar nonstiff fashion the comparison in based on both one dimensional 1d unsteady tests of a premixed methane air flame and unsteady two dimensional tests of the same flame interacting with a counterrotating vortex pair in both cases the grimech1 2 reaction mechanism with 32 species and 177 elementary reactions is used computed results show that the stiff reaction scheme enables selection of larger time steps and thus leads to substantial improvement in the performance of the computations for the present reaction mechanism and flame conditions speedup factors of about 10 are achieved in the 1d tests and about five in two dimensions possible extensions of the present scheme to further improve efficiency are also discussed

composite grid method for hybrid systems abstract a composite grid method is developed for solving symmetric indefinite linear systems arising from the three field hybrid variational principle which weakly enforces compatability and traction continuity conditions between independently modeled substructures the new solution approach decomposes the hybrid system into a hierarchical global local problem consisting of a positive definite global problem and an indefinite local system that are derived using minimization on the subspace and the stationarity principle respectively an innovative transformation of the indefinite local problem into an equivalent symmetric positive definite system makes the solution of the local problem possible either by a direct solver without pivoting or by a preconditioned iterative method alternative semi iterative solution procedures for the case when auxiliary coarse grid is not available are also developed this includes a multigrid like scheme that uses a mathematical model based on the collocation formulation as the auxiliary grid numerical performance studies for two dimensional plane stress and shell problems are carried out to test the usefulness of the proposed procedures

on the stability of godunov projection methods for incompressible flow an analysis of the stability of certain numerical methods for the linear advection diffusion equation in two dimensions is performed the advection diffusion equation is studied because it is a linearized version of the navier stokes equations the evolution equation for density in boussinesq flows and a simplified form of the equations for bulk thermodynamic temperature and mass fraction in reacting flows it is found that various methods currently in use which are based on a crank nicholson type temporal discretization utilizing second order godunov methods for explicitly calculating advective terms suffer from a time step restriction which depends on the coefficients of diffusive terms a simple modification in the computation of the advective derivatives results in a method with a stability condition that is independent of the magnitude of the coefficients of the diffusive terms

adaptive mesh refinement for coupled elliptic hyperbolic systems we present a modification to the berger and oliger adaptive mesh refinement algorithm designed to solve systems of coupled non linear hyperbolic and elliptic partial differential equations such systems typically arise during constrained evolution of the field equations of general relativity the novel aspect of this algorithm is a technique of extrapolation and delayed solution used to deal with the non local nature of the solution of the elliptic equations driven by dynamical sources within the usual berger and oliger time stepping framework we show empirical results demonstrating the effectiveness of this technique in axisymmetric gravitational collapse simulations and further demonstrate that the solution time scales approximately linearly with problem size we also describe several other details of the code including truncation error estimation using a self shadow hierarchy and the refinement boundary interpolation operators that are used to help suppress spurious high frequency solution components noise

an adaptive multigrid technique for three dimensional elasticity a program for finite element analysis of 3d linear elasticity problems is described the program uses quadratic hexahedral elements the solution process starts on an initial coarse mesh here error estimators are determined by the standard babuska rheinboldt method and local refinement is performed by partitioning of indicated elements each hexahedron into eight new elements then the discrete problem is solved on the second mesh and the refinement process proceeds in the following way on the ith mesh only the elements caused by refinement on the i 1 th mesh can be refined the control of refinement is the task of the user because the dimension of the discrete problem grows very rapidly in 3d the discrete problem is being solved by the frontal solution method on the initial mesh and by a newly developed and very efficient local multigrid method on the refined meshes the program can be successfully used for solving problems with structural singularities such as re entrant corners and moving boundary conditions a numerical example shows that such problems are solved with the same efficiency as regular problems

multidimensional upwind methods for hyperbolic conservation laws abstract we present a class of second order conservative finite difference algorithms for solving numerically time dependent problems for hyperbolic conservation laws in several space variables these methods are upwind and multidimensional in that the numerical fluxes are obtained by solving the characteristic form of the full multidimensional equations at the zone edge and that all fluxes are evaluated and differenced at the same time in particular operator splitting is not used correct behavior at discontinuities is obtained by the use of solutions to the riemann problem and by limiting some of the second order terms numerical results are presented which show that the methods described here yield the same high resolution as the corresponding operator split methods

supervisory control of a class of discrete event processes this paper studies the control of a class of discrete event processes i e processes that are discrete asynchronous and possibly nondeter ministic the controlled process is described as the generator of a formal language while the controller or supervisor is constructed from the grammar of a specified target language that incorporates the desired closed loop system behavior the existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language two examples are provided

block structured adaptive mesh refinement algorithms for vlasov simulation direct discretization of continuum kinetic equations like the vlasov equation are under utilized because the distribution function generally exists in a high dimensional 3d space and computational cost increases geometrically with dimension we propose to use high order finite volume techniques with block structured adaptive mesh refinement amr to reduce the computational cost the primary complication comes from a solution state comprised of variables of different dimensions we develop the algorithms required to extend standard single dimension block structured amr to the multi dimension case specifically algorithms for reduction and injection operations that transfer data between mesh hierarchies of different dimensions are explained in detail in addition modifications to the basic amr algorithm that enable the use of high order spatial and temporal discretizations are discussed preliminary results for a standard 1d 1v vlasov poisson test problem are presented results indicate that there is potential for significant savings for some classes of vlasov problems

solving the regularized strongly anisotropic cahn hilliard equation by an adaptive nonlinear multigrid method abstract we present efficient second order accurate and adaptive finite difference methods to solve the regularized strongly anisotropic cahn hilliard equation in 2d and 3d when the surface energy anisotropy is sufficiently strong there are missing orientations in the equilibrium level curves of the diffuse interface solutions corresponding to those missing from the sharp interface wulff shape and the anisotropic cahn hilliard equation becomes ill posed to regularize the equation a higher order derivative term is added to the energy this leads to a sixth order nonlinear parabolic equation for the order parameter an implicit time discretization is used to remove the high order time step stability constraints dynamic block structured cartesian mesh refinement is used to highly resolve narrow interfacial layers a multilevel nonlinear multigrid method is used to solve the nonlinear equations at the implicit time level one of the keys to the success of the method is the treatment of the anisotropic term this term is discretized in conservation form in space and is discretized fully implicitly in time numerical simulations are presented that confirm the accuracy efficiency and stability of the scheme we study the dynamics of interfaces under strong anisotropy and compare near equilibrium diffuse interface solutions to the sharp interface wulff shapes in 2d and 3d we also simulate large scale coarsening of a corrugated surface in 3d evolving by anisotropic surface diffusion we show the emergence of long range order during coarsening and an interesting mechanism of ordered coarsening

interface conditions for wave propagation through mesh refinement boundaries we study the propagation of waves across fixed mesh refinement boundaries in linear and nonlinear model equations in 1 d and 2 d and in the 3 d einstein equations of general relativity we demonstrate that using linear interpolation to set the data in guard cells leads to the production of reflected waves at the refinement boundaries implementing quadratic interpolation to fill the guard cells suppresses these spurious signals

multi grid method for periodic heterogeneous media part 2 multiscale modeling and quality control in multidimensional case abstract a multi grid method for a periodic heterogeneous medium in multidimensions is developed based on the homogenization theory special intergrid transfer operators have been developed to stimulate a low frequency response of the boundary value problem with oscillatory coefficients an adaptive strategy is developed to form a nearly optimal two scale computational model consisting of the finite element mesh entirely constructed on the microscale in the regions identified by the idealization error indicators while elsewhere the modeling level is only sufficient to capture the response of homogenized medium numerical experiments show the usefulness of the proposed adaptive multi level procedure for predicting a detailed response of composite specimens

finite element approximation of multi scale elliptic problems using patches of elements in this paper we present a method for the numerical solution of elliptic problems with multi scale data using multiple levels of not necessarily nested grids the method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming this paper provides proofs of the results published earlier see c r acad sci paris ser i 337 2003 679 684 gives a generalization of the latter to more than two domains and contains extensive numerical illustrations new results including the spectral analysis of the iteration operator and a numerical method to evaluate the constant of the strengthened cauchy buniakowski schwarz inequality are presented

multidisciplinary interactions in energy and environmental modeling abstract mathematical models have been widely used to understand predict or optimize many complex physical processes from semiconductor or pharmaceutical design to large scale applications such as global weather models to astrophysics in particular simulation of environmental effects of air polution is extensive here we address the need for using similar models to understand the fate and transport of groundwater contaminants and to design in situ remediation strategies three basic problem areas must be addressed in the modeling and simulation of the flow of groundwater contamination one must first obtain an effective model to describe the complex fluid fluid and fluid rock interactions that control the transport of contaminants in groundwater this includes the problem of obtaining accurate reservoir descriptions at various length scales and modeling the effects of this heterogeneity in the reservoir simulators next one must develop accurate discretization techniques that retain the important physical properties of the continuous models finally one should develop efficient numerical solution algorithms that utilize the potential of the emerging computing architectures we will discuss recent advances in each of these three areas

a numerical method for the incompressible navier stokes equations based on an approximate projection in this method we present a fractional step discretization of the time dependent incompressible navier stokes equations the method is based on a projection formulation in which we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields our treatment of the diffusion convection step uses a specialized second order upwind method for differencing the nonlinear convective terms that provides a robust treatment of these terms at a high reynolds number in contrast to conventional projection type discretizations that impose a discrete form of the divergence free constraint we only approximately impose the constraint i e the velocity field we compute is not exactly divergence free the approximate projection is computed using a conventional discretization of the laplacian and the resulting linear system is solved using conventional multigrid methods numerical examples are presented to validate the second order convergence of the method for euler finite reynolds number and stokes flow a second example illustrating the behavior of the algorithm on an unstable shear layer is also presented

convergence to equilibrium in the classical model of population genetics abstract in the classical model of population genetics for continuous time fisher s equation for n alleles a stationary point is called regular if the viability of each of the absent alleles is distinct from the average viability of the population it is shown that if the limit set of a trajectory contains a regular point then it contains only one point

variable kinematic modelling of laminated composite plates a displacement based variable kinematic global local finite element model is developed using hierarchical multiple assumed displacement fields at two different levels 1 at the element level and 2 at the mesh level the displacement field hierarchy contains both a conventional plate expansion 2 d and a full layerwise 3 d expansion depending on the accuracy desired the variable kinematic element can use various terms from the composite displacement field thus creating a hierarchy of different elements having a wide range of kinematic complexity and representing a number of different mathematical models the vkfe is then combined with the mesh superposition technique to further increase the computational efficiency and robustness of the computational algorithm these models are used to analyse a number of laminated composite plate problems that contain localized subregions where significant 3 d stress fields exist e g free edge effects

the fast adaptive composite grid fac method for elliptic equation the fast adaptive composite grid fag method is a systematic process for solving differential boundary value problems fac uses global and local uniform grids both to define the composite grid problem and to interact for its fast solution it can with little added cost substantially improve accuracy of the coarse grid solution and is very suitable for vector and parallel computation this paper develops both the theoretical and practical aspects of fac as it applies to elliptic problems 1 introduction the need for local resolution in physical models occurs frequently in practice special local features of the forcing function operator coefficients boundary and boundary conditions can demand resolution in restricted regions of the domain that is much finer than the required global resolution it is important that the discretization and solution processes account for this locally that is that the local phenomena do not precipitate a dramatic increase in the overall computation unfortunately this objective of efficiently adapting to local features is often in conflict with the solution process equation solvers can degrade or even fail to apply in the presence of varying discretization scales data structures that account for irregular grids can be cumbersome the computer architecture may not be able to effectively account for grid irregularity e g vectorizability may be inhibited etc in fact even the discretization process itself may find difficulty with this objective for finite differences it is problematic to develop accurate difference formulae for irregular grids for finite elements this objective is reflected in the substantial overhead costs needed to automate the discretization the fast adaptive composite grid method fac 11 is a discretization and solution method designed to achieve efficient local resolution by constructing the

a stable and convergent scheme for viscoelastic flow in contraction channels we present a new algorithm to simulate unsteady viscoelastic flows in abrupt contraction channels in our approach we split the viscoelastic terms of the oldroyd b constitutive equation using duhamel s formula and discretize the resulting pdes using a semi implicit finite difference method based on a lax wendroff method for hyperbolic terms in particular we leave a small residual elastic term in the viscous limit by design to make the hyperbolic piece well posed a projection method is used to impose the incompressibility constraint we are able to compute the full range of unsteady elastic flows in an abrupt contraction channel from the viscous limit to the elastic limit in a stable and convergent manner we demonstrate the range of our method for unsteady flow of a maxwell fluid with and without viscosity in planar contraction channels we also demonstrate stable and convergent results for benchmark high weissenberg number problems at we 1 and we 10

on the use of xfem within the arlequin framework for the simulation of crack propagation abstract the arlequin method is a generic numerical method that allows by local superposition and coupling of models to address multimodel and multiscale mechanical problems in particular this method has already been used to super impose cracked patches on sound structures reducing this way the global simulation resources a classical finite element approach would have required in this paper one of the key features of the extended finite element method namely the heaviside enrichment function is used within the arlequin framework to further reduce the costs of crack propagation simulations the main goal of the paper is to describe the proposed methodology and to assess its performance through numerical experiments

on the finite volume element method the finite volume element method fve is a discretization technique for partial differential equations it uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations then restricts the admissible functions to a finite element space to discretize the solution this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with fve based on triangulations with linear finite element spaces and a general type of control volume we establisho h estimates of the error in a discreteh 1 semi norm under an additional assumption of local uniformity of the triangulation the estimate is improved too h 2 results on the effects of numerical integration are also included

h k trichotomies and asymptotics of nonautonomous difference systems abstract discrete nonautonomous nonlinear systems possessing h k trichotomies are considered we specialize to the case of solutions moving inside and in a neighborhood of an invariant manifold of a special kind which we call h k hyperbolic the aulbach coppel knobloch transformation is then developed and used to state conditions for granting asymptotic equivalence of solutions our results generalize some of aulbach s results in which the manifold considered was made of stationary points and the system taken was autonomous

behavior of solutions near manifolds of periodic solutions abstract consider an autonomous differential system x f x of dimension n that admits a k dimensional invariant manifold in x space represented by a k parameter family of periodic solutions then k characteristic multipliers of the corresponding variational equation are equal to 1 under the hypothesis that none of the remaining n k characteristic multipliers has modulus 1 the behavior of the solutions of x f x near is investigated the main results are the description of a neighborhood of in terms of stable and unstable manifolds and the statement that whenever a solution of x f x approaches the invariant manifold as t it tends to with asymptotic amplitude and phase

a fully implicit fully adaptive time and space discretisation method for phase field simulation of binary alloy solidification a fully implicit numerical method based upon adaptively refined meshes for the simulation of binary alloy solidification in 2d is presented in addition we combine a second order fully implicit time discretisation scheme with variable step size control to obtain an adaptive time and space discretisation method the superiority of this method compared to widely used fully explicit methods with respect to cpu time and accuracy is shown due to the high nonlinearity of the governing equations a robust and fast solver for systems of nonlinear algebraic equations is needed to solve the intermediate approximations per time step we use a nonlinear multigrid solver which shows almost h independent convergence behaviour

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

acceleration by aggregation of successive approximation methods abstract methods of successive approximation for solving linear systems or minimization problems are accelerated by aggregation disaggregation processes these processes which modify the iterates being produced are characterized by a two directional flow of information between the original higher dimensional problem and a lower dimensional aggregated version this technique is characterized by means of galerkin approximations and this in turn permits analysis of the method a deterministic as well as probabilistic analysis is given of a number of specific aggregation disaggregation examples numerical experiments have been performed and these confirm the analysis and demonstrate the acceleration

finite element approach for density functional theory calculations on locally refined meshes we present a quadratic finite element approach to discretize the kohn sham equations on structured non uniform meshes a multigrid fac preconditioner is proposed to iteratively solve the equations by an accelerated steepest descent scheme the method was implemented using samrai a parallel software infrastructure for general amr applications examples of applications to small nanoclusters calculations are presented

a second order projection method for variable density flows this paper describes a second order projection method for variabledensity incompressible flows the method is suitable for both finite amplitude density variations and for fluids that are modeled using a boussinesq approximation it is based on a second order fractional step scheme in which diffusion convection terms are advanced without enforcing the incompressibility condition and the resulting intermediate velocity field is then projected onto the space of discretely divergencefree vector fields the nonlinear convection terms are treated using a godunov type procedure that is second order for smooth flow and remain stable and non oscillatory for nonsmooth flows with low fluid viscosities the method is described for finite amplitude density variation and the simplifications for a boussinesq approximation are sketched numerical results are presented that validate the convergence properties of the method and demonstrate its performance on more realistic problems

nested grids ilu decomposition ngilu a preconditioning technique is described which shows in many cases grid independent convergence this technique only requires an ordering of the unknowns based on the different levels of multigrid and an incomplete lu decomposition based on a drop tolerance the method is demonstrated on a variety of well known elliptic test problems including strongly varying coefficients advective terms and grid refinement

asymptotic phase revisited abstract we discuss the classical problem on asymptotic phase of periodic orbits of planar systems the existence of asymptotic phase for non hyperbolic periodic orbits is completely determined with hypotheses on the derivatives of a poincare map and a return time map smoothness of the vector field turns out to be crucial for existence of asymptotic phase for hyperbolic periodic orbits a new proof for the existence of invariant foliations is provided

a mass conservative adaptive fas multigrid solver for cell centered finite difference methods on block structured locally cartesian grids abstract we present a mass conservative full approximation storage fas multigrid solver for cell centered finite difference methods on block structured locally cartesian grids the algorithm is essentially a standard adaptive fas afas scheme but with a simple modification that comes in the form of a mass conservative correction to the coarse level force this correction is facilitated by the creation of a zombie variable analogous to a ghost variable but defined on the coarse grid and lying under the fine grid refinement patch we show that a number of different types of fine level ghost cell interpolation strategies could be used in our framework including low order linear interpolation in our approach the smoother prolongation and restriction operations need never be aware of the mass conservation conditions at the coarse fine interface to maintain global mass conservation we need only modify the usual fas algorithm by correcting the coarse level force function at points adjacent to the coarse fine interface we demonstrate through simulations that the solver converges geometrically at a rate that is h independent and we show the generality of the solver applying it to several nonlinear time dependent and multi dimensional problems in several tests we show that second order asymptotic h 0 convergence is observed for the discretizations provided that 1 at least linear interpolation of the ghost variables is employed and 2 the mass conservation corrections are applied to the coarse level force term

structural transformations of probabilistic finite state machines probabilistic finite state machines have recently emerged as a viable tool for modelling and analysis of complex non linear dynamical systems this paper rigorously establishes such models as finite encodings of probability measure spaces defined over symbol strings the well known nerode equivalence relation is generalized in the probabilistic setting and pertinent results on existence and uniqueness of minimal representations of probabilistic finite state machines are presented the binary operations of probabilistic synchronous composition and projective composition which have applications in symbolic model based supervisory control and in symbolic pattern recognition problems are introduced the results are elucidated with numerical examples and are validated on experimental data for statistical pattern classification in a laboratory environment

invariant manifolds with asymptotic phase for nonautonomous difference equations abstract for autonomous difference equations with an invariant manifold conditions are known which guarantee that a solution approaching this manifold eventually behaves like a solution on this manifold in this paper we extend the fundamental result in this context to difference equations which are nonautonomous and whose solutions are guaranteed only in forward time

optimal supervisory control of finite state automata this paper presents optimal supervisory control of dynamical systems that can be represented by deterministic finite state automaton dfsa models the performance index for the optimal policy is obtained by combining a measure of the supervised plant language with possible penalty on disabling of controllable events the signed real measure quantifies the behaviour of controlled sublanguages based on a state transition cost matrix and a characteristic vector as reported in earlier publications synthesis of the optimal control policy requires at most n iterations where n is the number of states of the dfsa model generated from the unsupervised plant language the computational complexity of the optimal control synthesis is polynomial in n syntheses of the control algorithms are illustrated with two application examples

an adaptive mesh projection method for viscous incompressible flow many fluid flow problems of practical interest particularly at high reynolds number are characterized by small regions of complex and rapidly varying fluid motion surrounded by larger regions of relatively smooth flow efficient solution of such problems requires an adaptive mesh refinement capability to concentrate computational effort where it is most needed we present in this paper a fractional step version of chorin s projection method for incompressible flow with adaptive mesh refinement which is second order accurate in both space and time convection terms are handled by a high resolution upwind method which provides excellent resolution of small scale features of the flow while a multilevel iterative scheme efficiently solves the parabolic and elliptic equations associated with viscosity and the projection numerical examples demonstrate the performance of the method on two dimensional problems involving vortex spindown with viscosity and inviscid vortex merger

invariant manifolds with asymptotic phase this paper is concerned with invariant manifolds allowing a generic approach to solutions of the underlying autonomous differential system only this means that any solution approaching such an invariant manifold m ultimately behaves like a particular solution on j4 in the sense that the difference n t i l of the two solutions tends to 0 as t cq in other words only solutions on the stable manifold of m can approach m as f cc invariant manifolds with that property are said to have an asymptotic phase the simplest nontrivial example of an invariant manifold with asymptotic phase is the orbit of an isolated periodic solution whose floquet multipliers lie all but one in the open unit disc of the complex plane malkin 12 proved a manifold of stationary solutions to have an asymptotic phase if each of these solutions has all eigenvalues with nonpositive real parts and if in addition the number of eigenvalues on the imaginary axis equals the dimension of the manifold more generally a k dimensional manifold generated by a family of periodic solutions has an asymptotic phase if all but k floquet multipliers have modulus less than 1 see hale stokes 9 conditions for a torus to be asymptotically stable with asymptotic phase are given by samoilenko 13 and coppel 5 invariant manifolds without any specified structure are considered by fenichel 6 7 and in kirchgraber s note 111 moreover each center manifold has an asymptotic phase if the corresponding equilibrium is stable see carr al amood 4 and henry lo in any case the condition guaranteeing the existence of an asymptotic phase amounts to the fact that the decay rate of solutions toward the manifold is greater than the decay rate of the solutions within the manifold in each of the cases quoted so far the invariant manifold is asymptotically stable and thus the question arises whether an asymptotic phase can exist only in connection with asymptotic stability a negative answer to this question is given by the author in l 21 where hyperbolic manifolds of periodic solutions are proved to have an asymptotic phase in l the manifold is supposed to be compact while the noncompact case is treated in 2 under the restriction that the generating family of periodic solutions has an amplitude independent period recently hale massatt 8 extended the corresponding result for manifolds of stationary solutions to a certain class of partial differential equations in this paper we drop the periodicity assumption of the flow on the given manifold and prove the following result a compact invariant manifold has an asymptotic phase if the manifold is hyperbolic assumption hl and carries a flow parallel in the sense of our assumption h2 this includes most of the aforementioned results

sharp estimates for multigrid rates of convergence with general smoothing and acceleration in this paper we prove the convergence of the multilevel iterative method for solving linear equations that arise from elliptic partial differential equations our theory is presented entirely in terms of the generalized condition number kappa of the matrix a and the smoothing matrix b this leads to a completely algebraic analysis of the method as an iterative technique for solving linear equations the properties of the elliptic equation and discretization procedure enter only when we seek to estimate kappa just as in the case of most standard iterative methods here we consider the fundamental two level iteration and the v and w cycles of the j level iteration j 2 we prove that the v and w cycles converge even when only one smoothing iteration is used we present several examples of the computation of kappa using both fourier analysis and standard finite element techniques we compare the predictions of our theorems with the actual rate of convergence our analysis also shows that

a parallel fictitious domain multigrid preconditioner for the solution of poisson s equation in complex geometries abstract a parallel multilevel preconditioner based on domain decomposition and fictitious domain methods has been presented for the solution of the poisson equation in complicated geometries rectangular blocks with matching grids on interfaces on a structured rectangular mesh have been used for the decomposition of the problem domain sloping sides or curved boundary surfaces are approximated using stepwise surfaces formed by the grid cells a seven point stencil based on the central difference scheme has been used for the discretization of the laplacian for both interior and boundary grid points and this results in a symmetric linear algebraic system for any type of boundary condition the preconditioned conjugate gradient method has been used for the solution of this symmetric system the multilevel preconditioner for the cg is based on a v cycle multigrid applied to the poisson equation on a fictitious domain formed by the union of the rectangular blocks used for the domain decomposition numerical results are presented for two typical poisson problems in complicated geometries one related to heat conduction and the other one arising from the les dns of incompressible turbulent flow over a packed array of spheres these results clearly show the efficiency and robustness of the proposed approach

optimal control of infinite horizon partially observable decision processes modelled as generators of probabilistic regular languages decision processes with incomplete state feedback have been traditionally modelled as partially observable markov decision processes in this article we present an alternative formulation based on probabilistic regular languages the proposed approach generalises the recently reported work on language measure theoretic optimal control for perfectly observable situations and shows that such a framework is far more computationally tractable to the classical alternative in particular we show that the infinite horizon decision problem under partial observation modelled in the proposed framework is approximable and in general is not harder to solve compared to the fully observable case the approach is illustrated via two simple examples

signed real measure of regular languages for discrete event supervisory control this paper reviews expands and clarifies the underlying concepts of a signed real measure of regular languages which has been used as a novel tool for synthesis of discrete event supervisory control systems the language measure is constructed upon the principles of automata theory and real analysis it allows total ordering of a set of partially ordered sublanguages of a regular language for quantitative evaluation of the supervised behaviour of deterministic finite state automata dfsa under different supervisors in the setting of the language measure a supervisor s performance is superior if the supervised plant is more likely to terminate at a good marked state and or less likely to terminate at a bad marked state the computational complexity of the language measure algorithm is polynomial in the number of dfsa states

an algebraic study of a local multigrid method for variational problems a two level iterative method for solving systems of equations arising from local refinement of finite element meshes is proposed the smoothing steps consists of gauss seidel relaxations applied only to the local mesh a direct solver is required on the global coarse mesh we analyze this scheme by modifying the standard algebraic convergence theory the estimates of the convergence factor do not require any regularity assumption the estimates can be evaluated element by element the evaluated estimates are compared with the actual convergence factor and with another bound in several numerical examples the theory also applies to a method with a direct solver on the local fine mesh in this case we have obtained very sharp and in fact optimal estimates of the convergence factors

multi grid domain decomposition approach for solution of navier stokes equations in primitive variable form a new multi grid two grid pseudospectral element method has been carried out for solution of incompressible flow in terms of primitive variable formulation the main objective of the proposed method is to apply the multi grid technique solving the incompressible flow problems associated with three commonly encountered multi grid environments in domain decomposition terminology it includes i partially overlapped subdomains each of which has same types of grids ii partially overlapped subdomains each of which has different types of grids iii local adaptive subdomains fully overlapped with the original computational domain composite grids the approach for flow problems complex geometry or not is to first divide the computational domain into a number of subdomains with the inter overlapping area partially or fully overlapped in categories i and ii the fine grid or coarse grid subdomains can be defined by their representation while in category iii the fine grid or coarse grid subdomains are defined as usual next implement the schwarz alternating procedure sap to exchange the data among subdomains where the coarse grid correction is used to remove the high frequency error that occurs when the data interpolation from the fine grid subdomain to the coarse grid subdomain is conducted the strategy behind the coarse grid correction is to adopt the operator of the divergence of velocity field which intrinsically links the pressure equation into this process the solution of each subdomain can be efficiently solved by the direct or iterative eigenfunction expansion technique or preconditioned method with the least storage requirement i e o n2 in 2 d numerical results of i driven cavity flow re 100 400 with cartesian grids category i in each subdomain ii driven cavity flow re 3200 with local adaptive grids category iii in each subdomain and iii flow over a cylinder re 250 with o grids in one subdomain and cartesian grids in another category ii will be presented in the paper to account for the versatility of the proposed multi grid method

composite overlapping meshes for the solution of partial differential equations abstract we discuss the generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them a composite overlapping grid consists of a set of curvilinear component grids that cover a region and overlap where they meet continuity conditions interpolation are imposed at the overlapping boundaries the principal advantage of composite grids is in the generation of grids for regions with complicated geometries the grid construction program cmpgrd is used to create composite grids with any number of component grids we describe some techniques for the solution of elliptic and time dependent pdes on composite meshes applications to the solution of the compressible navier stokes equations are presented

asymptotic stability regions via extensions of zubov s method i on considere un systeme differentiel x f t x satisfaisant les hypotheses suivantes f r n 1 r n est analytique f t x est presque periodique en t pour chaque x r n f t 0 0 t r la matrice principale fondamentale t s de y f x t 0 y satisfait t s exp t s t s avec et des constantes positives f t x f x t o x 0 x uniformement en t quand x 0

schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems part i algorithms and numerical results we describe sequential and parallel algorithms based on the schwarz alternating method for the solution of mixed finite element discretizations of elliptic problems using the raviart thomas finite element spaces these lead to symmetric indefinite linear systems and the algorithms have some similarities with the traditional block gauss seidel or block jacobi methods with overlapping blocks the indefiniteness requires special treatment the sub blocks used in the algorithm correspond to problems on a coarse grid and some overlapping subdomains and is based on a similar partition used in an algorithm of dryja and widlund for standard elliptic problems if there is sufficient overlap between the subdomains the algorithm converges with a rate independent of the mesh size the number of subdomains and discontinuities of the coefficients extensions of the above algorithms to the case of local grid refinement is also described convergence theory for these algorithms will be presented in a subsequent paper

adaptive global local refinement strategy based on the interior error estimates of the h method an adaptive global local refinement strategy based on the interior error estimates of the h method is proposed adaptive global local refinement strategy is aimed at constructing nearly optimal finite element meshes where the force transfer to the local region of interest is sufficiently accurate so that the local phenomena of interest is resolved with a user specified accuracy numerical examples for linear elasticity problems in two dimensions together with a comparison to the classical adaptive h refinement strategy based on the equidistribution of errors are presented to validate the present formulation

gerris a tree based adaptive solver for the incompressible euler equations in complex geometries an adaptive mesh projection method for the time dependent incompressible euler equations is presented the domain is spatially discretised using quad octrees and a multilevel poisson solver is used to obtain the pressure complex solid boundaries are represented using a volume of fluid approach second order convergence in space and time is demonstrated on regular statically and dynamically refined grids the quad octree discretisation proves to be very flexible and allows accurate and efficient tracking of flow features the source code of the method implementation is freely available

domain decomposition type iterative techniques for parabolic problems on locally refined grids based on an extension of the discontinuous galerkin finite element method discretization schemes for solving parabolic problems on grids with local refinement both in space and in time are constructed the stability of schemes constructed in this way is automatically ensured by the method the construction of two level preconditioners utilizing local timestepping and a global coarse grid solver both on standard rectangular and uniform grids is the main objective of the paper the optimal convergence properties of such two level preconditioners are studied the theory is illustrated by a set of numerical examples

a high order projection method for tracking fluid interfaces in variable density incompressible flows we present a numerical method for computing solutions of the incompressible euler or navier stokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties the method is based on a second order projection method for variable density flows using an approximate projection formulation the boundary between the fluids is tracked with a second order volume of fluid interface tracking algorithm we present results for viscious rayleigh taylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm we also present computational results for the rayleigh taylor instability in air helium and for bubbles and drops in an air water system without surface tension to demonstrate the behavior of the algorithm on problems with large density and viscosity contrasts

a robust incompressible navier stokes solver for high density ratio multiphase flows abstract this paper presents a robust adaptive numerical scheme for simulating high density ratio and high shear multiphase flows on locally refined staggered cartesian grids that adapt to the evolving interfaces and track regions of high vorticity the algorithm combines the interface capturing level set method with a variable coefficient incompressible navier stokes solver that is demonstrated to stably resolve material contrast ratios of up to six orders of magnitude the discretization approach ensures second order pointwise accuracy for both velocity and pressure with several physical boundary treatments including velocity and traction boundary conditions the paper includes several test cases that demonstrate the order of accuracy and algorithmic scalability of the flow solver to ensure the stability of the numerical scheme in the presence of high density and viscosity ratios we employ a consistent treatment of mass and momentum transport in the conservative form of discrete equations this consistency is achieved by solving an additional mass balance equation which we approximate via a strong stability preserving runga kutta time integrator and by employing the same mass flux obtained from the mass equation in the discrete momentum equation the scheme uses higher order total variation diminishing tvd and convection boundedness criterion cbc satisfying limiter to avoid numerical fluctuations in the transported density field the high order bounded convective transport is done on a dimension by dimension basis which makes the scheme simple to implement we also demonstrate through several test cases that the lack of consistent mass and momentum transport in non conservative formulations which are commonly used in practice or the use of non cbc satisfying limiters can yield very large numerical error and very poor accuracy for convection dominant high density ratio flows our numerical scheme also uses well balanced surface tension and gravity force discretizations in the hydrostatic limit we show that the well balanced formulation mitigates spurious flow currents and achieves discrete force balance between the pressure gradient and surface tension or gravity

optimal multilevel methods for graded bisection grids we design and analyze optimal additive and multiplicative multilevel methods for solving h 1 problems on graded grids obtained by bisection we deal with economical local smoothers after a global smoothing in the finest mesh local smoothing for each added node during the refinement needs to be performed only for three vertices the new vertex and its two parent vertices we show that our methods lead to optimal complexity for any dimensions and polynomial degree the theory hinges on a new decomposition of bisection grids in any dimension which is of independent interest and yields a corresponding decomposition of spaces we use the latter to bridge the gap between graded and quasi uniform grids for which the multilevel theory is well established

three dimensional adaptive mesh refinement for hyperbolic conservation laws a local adaptive mesh refinement algorithm for solving hyperbolic systems of conservation laws in three space dimensions is described the method is based on the use of local grid patches superimpo

an iterative method for adaptive finite element computation abstract an iterative method designed for solving systems of equations arising from adaptive finite element computation is presented the method is based on the block gauss seidel algorithm the convergence theory presented in the paper gives very sharp convergence bounds which can be easily evaluated the method applies for h p extensions and various types of elements

a multilevel variational method for au bu on composite grids abstract there are two existing general approaches to the numerical solution of differential eigen problems by multigrid one which uses a linear multigrid solver for the inner loop of a linearization technique e g inverse iteration and the other which integrates multigrid directly into the problem solution e g fas the present paper developes a third multigridlike approach rqmg that appeals directly to the variational problem of minimizing the rayleigh quotient applying easily to both global and local grid implementations numerical results are presented for a simplified single group diffusion problem

ν a robot path planning algorithm based on renormalised measure of probabilistic regular languages this article introduces a novel path planning algorithm called that reduces the problem of robot path planning to optimisation of a probabilistic finite state automaton the algorithm makes use of renormalised measure of regular languages to plan the optimal path for a specified goal although the underlying navigation model is probabilistic the algorithm yields path plans that can be executed in a deterministic setting with automated optimal trade off between path length and robustness under dynamic uncertainties the algorithm has been experimentally validated on segway robotic mobility platforms in a laboratory environment

adaptive s method for linear elastostatics abstract a methodology has been developed to simultaneously treat phenomena at several scales the key features of the method are i a detailed error diagnosis which identifies critical regions and ii a feedback procedure which overlays portions of the finite element mesh where unacceptable errors have been identified by local finite element mesh es and then subsequently improves the quality of the solution in the global and local response until the desired level of accuracy is achieved the method can be routinely used as a global local method and or as an adaptive technique which provides a measure of reliability of global and local computations numerical experiments for problems in homogeneous and heterogeneous laminated composites media are presented to validate the present formulation

multigrid elliptic equation solver with adaptive mesh refinement in this paper we describe in detail the computational algorithm used by our parallel multigrid elliptic equation solver with adaptive mesh refinement our code uses truncation error estimates to adaptively refine the grid as part of the solution process the presentation includes a discussion of the orders of accuracy that we use for prolongation and restriction operators to ensure second order accurate results and to minimize computational work code tests are presented that confirm the overall second order accuracy and demonstrate the savings in computational resources provided by adaptive mesh refinement

evaluation of language measure parameters for discrete event manufacturing systems with multiproduct machines the concept of language measure provides a mathematical framework for quantitative analysis and synthesis of discrete event supervisory control systems the language measure is assigned two parameters namely the state transition cost matrix and the state cost vector in a recent paper khatab and nourelfath 2006 an analytical approach has been formulated to evaluate the language measure parameters for discrete event manufacturing systems composed of several monoproduct machines i e each machine is able to perform only one kind of product this paper generalizes these results to the multiproduct case i e to the case where each machine in the manufacturing system is able to perform different kinds of products the proposed approach is based on the theory of markov stochastic processes and kronecker algebra the main advantage of the proposed approach is that the mathematical expressions of the system language measure parameters are derived from data of individual elementary machines without generat

a conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer

a moving mesh finite difference method for equilibrium radiation diffusion equations an efficient moving mesh finite difference method is developed for the numerical solution of equilibrium radiation diffusion equations in two dimensions the method is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of meshing partial differential equations the mesh adaptation is controlled through a hessian based monitor function and the so called equidistribution and alignment principles several challenging issues in the numerical solution are addressed particularly the radiation diffusion coefficient depends on the energy density highly nonlinearly this nonlinearity is treated using a predictor corrector and lagged diffusion strategy moreover the nonnegativity of the energy density is maintained using a cutoff method which has been known in literature to retain the accuracy and convergence order of finite difference approximation for parabolic equations numerical examples with multi material multiple spot concentration situations are presented numerical results show that the method works well for radiation diffusion equations and can produce numerical solutions of good accuracy it is also shown that a two level mesh movement strategy can significantly improve the efficiency of the computation

local multilevel methods for adaptive finite element methods for nonsymmetric and indefinite elliptic boundary value problems in this paper we propose some local multilevel algorithms for solving linear systems arising from adaptive finite element approximations of nonsymmetric and indefinite elliptic boundary value problems two types of local smoothers are constructed one is based on the original nonsymmetric problems and the other is defined in terms of the associated symmetric problems it is shown that the local multilevel methods for the nonsymmetric and indefinite elliptic boundary value problems are optimal which means that the convergence rates of the local multilevel methods are independent of mesh sizes and mesh levels provided that the coarsest grid is sufficiently fine numerical experiments are reported to confirm our theory

convergence analysis of the local defect correction method for diffusion equations this paper is concerned with the convergence analysis of the local defect correction ldc method for diffusion equations we derive a general expression for the iteration matrix of the method we consider the model problem of poisson s equation on the unit square and use standard five point finite difference discretizations on uniform grids it is shown via both an upper bound for the norm of the iteration matrix and numerical experiments that the rate of convergence of the ldc method is proportional to h 2 with h the grid size of the global coarse grid

renormalized measure of regular languages this paper modifies the signed real measure of regular languages which has been reported in recent literature for analysis and synthesis of discrete event supervisory control laws a new concept of renormalized measure is introduced to eliminate a user selectable parameter in the present version of the language measure the concept of measure renormalization is illustrated by an example

a projection method for locally refined grids a numerical method for the solution of the two dimensional euler equatons for incompressible flow on locally refined grids is presented the method is a second order godunov projection method adapted from bell colella and glaz second order accuracy of the numerical method in time and space is established through numerical experiments the main contributions of this work concern the formulation and implementation of a projection for refined grids a discussion of the adjointness relation between gradient and divergence operators for a refined grid mac projection is presented and a refined grid approximate projection is developed an efficient multigrid method which exactly solves the projection is developed and a method for casting certain approximate projections as mac projections on refined grids is presented

the role of the strengthened cauchy buniakowskii schwarz inequality in multilevel methods the basic theory of the strengthened cauchy buniakowskii schwarz inequality and its applications in multilevel methods for the solution of linear systems arising from finite element or finite difference discretisation of elliptic partial differential equations is surveyed proofs are given both in a finite element context and in purely algebraic form