quantifying spatiotemporal chaos in rayleigh benard convection using large scale parallel numerical simulations we explore spatiotemporal chaos in rayleigh benard convection in a cylindrical domain with experimentally relevant boundary conditions we use the variation of the spectrum of lyapunov exponents and the leading order lyapunov vector with system parameters to quantify states of high dimensional chaos in fluid convection we explore the relationship between the time dynamics of the spectrum of lyapunov exponents and the pattern dynamics for chaotic dynamics we find that all of the lyapunov exponents are positively correlated with the leading order lyapunov exponent and we quantify the details of their response to the dynamics of defects the leading order lyapunov vector is used to identify topological features of the fluid patterns that contribute significantly to the chaotic dynamics our results show a transition from boundary dominated dynamics to bulk dominated dynamics as the system size is increased the spectrum of lyapunov exponents is used to compute the variation of the fractal dimension with system parameters to quantify how the underlying high dimensional strange attractor accommodates a range of different chaotic dynamics

deflation of conjugate gradients with applications to boundary value problems a method for improving the convergence of the standard conjugate gradient method is given this method involves the use of auxiliary subspaces it is shown how such subspaces may be constructed for boundary value problems and an analysis of convergence for second order problems is presented

overlapping schwarz preconditioners for spectral element methods in nonstandard domains and heterogeneous media overlapping schwarz preconditioners are constructed and numerically studied for gauss lobatto legendre gll spectral element discretizations of heterogeneous elliptic problems on nonstandard domains defined by gordon hall transfinite mappings the results of several test problems in the plane show that the proposed preconditioners retain the good convergence properties of overlapping schwarz preconditioners for standard affine gll spectral elements i e their convergence rate is independent of the number of subdomains of the spectral degree in the case of generous overlap and of the discontinuity jumps in the coefficients of the elliptic operator while in the case of small overlap the convergence rate depends on the inverse of the overlap size

on poisson lie structure on the external algebra of the classical lie groups the general expression for the bicovariant bracket for odd generators of the external algebra on a poisson lie group is given the graded poisson lie structures derived before for gl n and sl n are shown to be the special cases of this bracket the formula is universal and can be applied to any matrix lie group

the use of pointwise interpolation in domain decomposition methods with nonnested meshes in this paper the author develops a new technique and a corresponding theory for schwarz type overlapping domain decomposition methods for solving large sparse linear systems which arise from finite element discretization of elliptic partial differential equations the theory provides an optimal convergence of an additive schwarz algorithm that is constructed with a nonnested coarse space and a not necessarily shape regular subdomain partitioning the theory is also applicable to the graph partitioning algorithms recently developed by cai and saad overlapping domain decomposition algorithms for general sparse matrices preprint 93 027 univ of minnesota and farhat and lesoine internat j numer methods engrg 36 1993 pp 745 764 for problems defined on unstructured meshes

nonlinear laplacian spectral analysis of rayleigh benard convection the analysis of physical datasets using modern methods developed in machine learning presents unique challenges and opportunities these datasets typically feature many degrees of freedom which tends to increase the computational cost of statistical methods and complicate interpretation in addition physical systems frequently exhibit a high degree of symmetry that should be exploited by any data analysis technique the classic problem of rayleigh benard convection in a periodic domain is an example of such a physical system with trivial symmetries this article presents a technique for analyzing the time variability of numerical simulations of two dimensional rayleigh benard convection at large aspect ratio and intermediate rayleigh number the simulated dynamics are highly unsteady and consist of several convective rolls that are distributed across the domain and oscillate with a preferred frequency intermittent extreme events in the net heat transfer as quantified by the time weighted probability distribution function of the nusselt number are a hallmark of these simulations nonlinear laplacian spectral analysis nlsa is a data driven method which is ideally suited for the study of such highly nonlinear and intermittent dynamics but the trivial symmetries of the rayleigh benard problem such as horizontal shift invariance can mask the interesting dynamics to overcome this issue the vertical velocity is averaged over parcels of similar temperature and height which substantially compresses the size of the dataset and removes trivial horizontal symmetries this isothermally averaged dataset which is shown to preserve the net convective heat flux across horizontal surfaces is then used as an input to nlsa the analysis generates a small number of orthogonal modes which describe the spatiotemporal variability of the heat transfer a regression analysis shows that the extreme events of the net heat transfer are primarily associated with a family of modes with fat tailed probability distributions and low frequency temporal power spectra on the other hand the regular oscillation of the heat transfer is associated with a pair of modes with nearly uniform probability distributions physical mechanisms for the regular oscillation and the extreme heat transfer events are hypothesized based on an analysis of the spatio temporal structure of these modes finally proposals are made for this approach to be applied to the study of other problems in turbulent convection including three dimensional rayleigh benard convection and moist atmospheric convection

p refinement and p threads abstract p type refinement leads to exponential decay of numerical errors for sufficiently smooth solutions and has been used effectively in turbulence and structural mechanics simulations in the context of spectral and hp finite element discretizations however it induces a computational cost of o p d 1 in d dimensions which is higher than lower order methods in this paper we demonstrate that by employing multi threading within mpi processes we manage to counter balance the cost increase associated with p refinement this approach reduces effectively the wall clock time and keeps it essentially constant as the polynomial order is increased while achieving exponential convergence rate since the number of threads within mpi processes can be dynamically adjusted through thread library functions the algorithm can be readily adapted for dynamic p refinement the resulting hybrid mpi threads dual level parallelism is particularly suitable for modern supercomputers consisting of symmetric multiprocessor nodes we demonstrate this approach in simulations of two three dimensional fluid dynamics problems

an overlapping schwarz method for spectral element solution of the incompressible navier stokes equations efficient solution of the navier stokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems for unsteady incompressible flows the pressure operator is the leading contributor to stiffness as the characteristic propagation speed is infinite in the context of operator splitting formulations it is the pressure solve which is the most computationally challenging despite its elliptic origins we examine several preconditioners for the consistent l2poisson operator arising in the pn pn 2spectral element formulation of the incompressible navier stokes equations we develop a finite element based additive schwarz preconditioner using overlapping subdomains plus a coarse grid projection operator which is applied directly to the pressure on the interior gauss points for large two dimensional problems this approach can yield as much as a fivefold reduction in simulation time over previously employed methods based upon deflation

a polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensions iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems a p version finite element method based on continuous piecewise q p functions is considered for second order elliptic problems in three dimensions this special method can also be viewed as a conforming spectral element method an iterative method is designed for which the condition number of the relevant operator grows only in proportion to 1 log p 2 this bound is independent of jumps in the coefficient of the elliptic problem across the interfaces between the subregions numerical results are also reported which support the theory

balancing neumann neumann methods for incompressible stokes equations this paper describes the unerlying mathematical model and the balancing neumann neumann methods are introduced and studied for incompressible stokes equations discretized with mixed finite or spectral elements with discontinuous pressures after decomposing the original domain of the problem into nonoverlapping subdomains the interior unknowns which are the interior velocity component and all except the constant pressure component of each subdomain problem are implicitly eliminated the resulting saddle point schur complement is solved with a krylov space method with a balancing neumann neumann preconditioner based on the solution of a coarse stokes problem with a few degrees of freedom per subdomain and on the solution of local stokes problems with natural neumann velocity and essential boundary conditions on the subdomains this preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively the condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and the inverse of the inf sup constants of the discrete problem and of the coarse subproblem numerical results show that the method is quite fast they are also fully consistent with the theory

preconditioning legendre spectral collocation approximations to elliptic problems this work deals with the h 1 condition numbers and the distribution of the tilde beta n m singular values of the preconditioned operators tilde beta n m 1 w n m hat a n m hat a n m is the matrix representation of the legendre spectral collocation discretization of the elliptic operator a defined by au delta u a 1 u x a 2 u y a 0 u in omega the unit square with boundary conditions u 0 on gamma 0 frac partial u partial nu a alpha u on gamma 1 tilde beta n m is the stiffness matrix associated with the finite element discretization of the positive definite elliptic operator b defined by bv delta v b 0 v in omega with boundary conditions v 0 on gamma 0 frac partial v partial nu b beta v on gamma 1 the finite element space is either the space of continuous functions which are bilinear on the rectangles determined by the legendre gauss lobatto lgl points or the s

characterization of the domain chaos convection state by the largest lyapunov exponent using numerical integrations of the boussinesq equations in rotating cylindrical domains with realistic boundary conditions we have computed the value of the largest lyapunov exponent lambda1 for a variety of aspect ratios and driving strengths we study in particular the domain chaos state which bifurcates supercritically from the conducting fluid state and involves extended propagating fronts as well as point defects we compare our results with those from egolf et al nature 404 733 2000 who suggested that the value of lambda1 for the spiral defect chaos state of a convecting fluid was determined primarily by bursts of instability arising from short lived spatially localized dislocation nucleation events we also show that the quantity lambda1 is not intensive for aspect ratios gamma over the range 20 gamma 40 and that the scaling exponent of lambda1 near onset is consistent with the value predicted by the amplitude equation formalism

isogeometric block feti dp preconditioners for the stokes and mixed linear elasticity systems abstract the aim of this work is to construct and analyze a feti dp type domain decomposition preconditioner for isogeometric discretizations of the stokes and mixed linear elasticity systems this method extends to the isogeometric analysis context the preconditioner previously proposed by tu and li 2015 for finite element discretizations of the stokes system the resulting isogeometric feti dp algorithm is proven to be scalable in the number of subdomains and has a quasi optimal convergence rate bound which is polylogarithmic in the ratio of subdomain and element sizes extensive two dimensional numerical experiments validate the theory investigate the behavior of the preconditioner with respect to both the spline polynomial degree and regularity and show its robustness with respect to domain deformation material incompressibility and presence of elastic coefficient discontinuities across subdomain interfaces

a spectrally accurate method for overlapping grid solution of incompressible navier stokes equations an overlapping mesh methodology that is spectrally accurate in space and up to third order accurate in time is developed for solution of unsteady incompressible flow equations in three dimensional domains the ability to decompose a global domain into separate but overlapping subdomains eases mesh generation procedures and increases flexibility of modeling flows with complex geometries the methodology employs implicit spectral element discretization of equations in each subdomain and explicit treatment of subdomain interfaces with spectrally accurate spatial interpolation and high order accurate temporal extrapolation and requires few if any iterations yet maintains the global accuracy and stability of the underlying flow solver the overlapping mesh methodology is thoroughly validated using two dimensional and three dimensional benchmark problems in laminar and turbulent flows the spatial and temporal convergence is documented and is in agreement with the nominal order of accuracy of the solver the influence of long integration times as well as inflow outflow global boundary conditions on the performance of the overlapping grid solver is assessed in a turbulent benchmark of fully developed turbulent pipe flow the turbulent statistics with the overlapping grids is validated against published available experimental and other computation data scaling tests are presented that show near linear strong scaling even for moderately large processor counts a novel overlapping grid method for incompressible flow equations is developed the overlapping grid method is based on spectral element discretization the method maintains global spectral accuracy with polynomial refinement an explicit interface extrapolation scheme for temporal coupling is proposed the scheme allows for high order temporal accuracy with only a few iterations

chebyshev pseudospectral solution of the stokes equations using finite element preconditioning the stokes equations are solved by a chebyshev pseudospectral method on a rectangular domain as the resulting system of algebraic equations is very difficult to factorize a preconditioning is designed using a finite element technique the fem solver constitutes the masterpiece of a richardson iteration process several finite elements are investigated the 9 nodes lagrangian element b q 2 b q 1 the b q 1 b q 0 element and the b q 1 b q 1 element due to brezzi and pitkaranta an eigenvalue analysis is carried out in order to pinpoint the characteristic features of each precondition it is shown that the b q 2 b q 1 element yields the best convergence results the power of this choice is demonstrated on theoretical solutions and on the regularized square cavity problem

overlapping schwarz methods on unstructured meshes using non matching coarse grids we consider two level overlapping schwarz domain decomposition methods for solving the finite element problems that arise from discretizations of elliptic problems on general unstructured meshes in two and three dimensions standard finite element interpolation from the coarse to the fine grid may be used our theory requires no assumption on the substructures that constitute the whole domain so the substructures can be of arbitrary shape and of different size the global coarse mesh is allowed to be non nested to the fine grid on which the discrete problem is to be solved and neither the coarse mesh nor the fine mesh need be quasi uniform in addition the domains defined by the fine and coarse grid need not be identical the one important constraint is that the closure of the coarse grid must cover any portion of the fine grid boundary for which neumann boundary conditions are given in this general setting our algorithms have the same optimal convergence rate as the usual two level overlapping domain decomposition methods on structured meshes the condition number of the preconditioned system depends only on the possibly small overlap of the substructures and the size of the coarse grid but is independent of the sizes of the subdomains

an analysis of the fractional step method the fractional step method for solving the incompressible navier stokes equations in primitive variables is analyzed as a block lu decomposition in this formulation the issues involving boundary conditions for the intermediate velocity variables and the pressure are clearly resolved in addition it is shown that poor temporal accuracy first order is not due to boundary conditions but due to the method itself a generalized block lu decomposition that overcomes this difficulty is presented allowing arbitrarily high temporal order of accuracy the generalized decomposition is shown to be useful for a wide range of problems including steady problems technical issues such as stability and the appropriate pressure update scheme are also addressed numerical simulations of the steady incompressible navier stokes equations in a square domain confirm the theoretical results 18 refs 4 figs

stable interval observers in bbc for linear systems with time varying input bounds this technical note deals with the design of stable interval observers and estimators for continuous time linear dynamic systems under uncertain initial states and uncertain inputs enclosed within time varying zonotopic bounds no monotony assumption such as cooperativity is required in the vector field the interval observer stability directly derives from the stability of the observer state matrix where any poles real or complex single or multiple are handled in the same way

a spectral collocation method for the navier stokes equations abstract a fourier chebyshev spectral method for the incompressible navier stokes equations is described it is applicable to a variety of problems including some with fluid properties which vary strongly both in the normal direction and in time in this fully spectral algorithm a preconditioned iterative technique is used for solving the implicit equations arising from semi implicit treatment of pressure mean advection and vertical diffusion terms the algorithm is tested by applying it to hydrodynamic stability problems in channel flow and in external boundary layers with both constant and variable viscosity

overlapping schwarz methods for unstructured spectral elements a parallel and scalable domain decomposition method for unstructured and hybrid spectral element discretizations of elliptic problems is introduced and studied the spectral elements are affine images of the reference triangle or square in two dimensions and of the reference tetrahedron pyramid prism or cube in three dimensions the method is based on overlapping schwarz techniques applied to the schur complement of the discrete system and is implemented as a preconditioner for a krylov space method numerical results in two and three dimensions show that the iteration counts of our method are bounded by a constant independent of the spectral degree and the number of subdomains the resulting elliptic solver can be used in navier stokes simulations using the spectral element code nektar

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

semi implicit formulations of the navier stokes equations application to nonhydrostatic atmospheric modeling we present semi implicit implicit explicit formulations of the compressible navier stokes equations nse for applications in nonhydrostatic atmospheric modeling the compressible nse in nonhydrostatic atmospheric modeling include buoyancy terms that require special handling if one wishes to extract the schur complement form of the linear implicit problem we present results for five different forms of the compressible nse and describe in detail how to formulate the semi implicit time integration method for these equations finally we compare all five equations and compare the semi implicit formulations of these equations both using the schur and no schur forms against an explicit runge kutta method our simulations show that if efficiency is the main criterion it matters which form of the governing equations you choose furthermore the semi implicit formulations are faster than the explicit runge kutta method for all the tests studied especially if the schur form is used while we have used the spectral element method for discretizing the spatial operators the semi implicit formulations that we derive are directly applicable to all other numerical methods we show results for our five semi implicit models for a variety of problems of interest in nonhydrostatic atmospheric modeling including inertia gravity waves density current i e kelvin helmholtz instabilities and mountain test cases the latter test case requires the implementation of nonreflecting boundary conditions therefore we show results for all five semi implicit models using the appropriate boundary conditions required in nonhydrostatic atmospheric modeling no flux reflecting and nonreflecting boundary conditions nrbcs it is shown that the nrbcs exert a strong impact on the accuracy and efficiency of the models

overlapping schwarz methods with a standard coarse space for almost incompressible linear elasticity low order finite element discretizations of the linear elasticity system suffer increasingly from locking effects and ill conditioning when the material approaches the incompressible limit if only the displacement variables are used mixed finite elements using both displacement and pressure variables provide a well known remedy but they yield larger and indefinite discrete systems for which the design of scalable and efficient iterative solvers is challenging two level overlapping schwarz preconditioners for the almost incompressible system of linear elasticity discretized by mixed finite elements with discontinuous pressures are constructed and analyzed the preconditioned systems are accelerated either by a gmres generalized minimum residual method applied to the resulting discrete saddle point problem or by a pcg preconditioned conjugate gradient method applied to a positive definite although extremely ill conditioned reformulation of the problem obtained by eliminating all pressure variab

projection techniques for iterative solution of ax b with successive right hand sides we present two projection techniques for computing approximate solutions to linear systems of the form ax n b n for a sequence n 1 2 e g such as arises from time discretization of a partial differential equation the inexpensive approximate solutions can be used as initial guesses for iterative solution of the system resulting in significantly reduced computational expense examples of two and three dimensional incompressible navier stokes calculations are presented in which x represents the pressure and a is a discrete poisson operator in flows containing significant dynamic activity these projection techniques lead to as much as a two fold reduction in solution time

spectral element methods for transient viscoelastic flow problems spectral element methods are developed for solving transient flows of viscoelastic fluids the fluids are modeled using the upper convected maxwell and oldroyd b constitutive relationships several temporal schemes for dealing with the time dependent nature of the problems are considered the computation associated with each time step comprises an explicit treatment of the convection and deformation terms and an implicit treatment of the linear terms the generalized three field stokes problem generated in the latter is solved using a nested preconditioned conjugate gradient method the performance of a number of preconditioners is investigated a modified formulation of the stokes problem is presented that ensures a unique pressure approximation with zero mean is obtained at each time step the performance of the schemes and models when applied to the benchmark problem of start up of plane poiseuille flow is investigated

spectral element methods for large scale parallel navier stokes calculations abstract we analyze the computational complexity of a recently developed two level iteration scheme for spectral element solution of the time dependent incompressible navier stokes equations in complex domains we present several algorithmic advances which significantly enhance the scalability of this approach including implementation of an advanced combine operation for degrees of freedom on subdomain edges parallel solution of the fine grained coarse grid problem and implementation of local low order finite element preconditioners for the find grid problem timings on the 512 node intel delta machine show that the combined improvements lead to a fourfold reduction in navier stokes solution time for the particular case of a three dimensional boundary layer calculation consisting of one million degrees of freedom