a comparison of preconditioned iterative techniques using rapid operator application against direct solution methods the p version of the finite element method which uses high order hierarchic basis functions within an element allows for ease of adaptive computation and high accuracy with relatively few degrees of freedom however the solution time and storage requirements for large 3 d problems can be very high when direct methods are used to factor global matrices iterative techniques with rapid operator application have minimal storage requirements because neither element nor global matrices are formed but may not converge rapidly enough we study an approach that attempts to combine the two methods by using direct factorization of the matrix corresponding to low level polynomials as a preconditioner to an iterative method at a higher polynomial level this resulting solution scheme converges more rapidly than an unpreconditioned scheme but has much lower storage requirements than direct methods we illustrate the effectiveness of this scheme on some large three dimensional problems in structural analysis

proof verification and the hardness of approximation problems we show that every language in np has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof if a string is in the language then there exists a proof such that the verifier accepts with probability 1 i e for every choice of its random string for strings not in the language the verifier rejects every provided proof with probability at least 1 2 our result builds upon and improves a recent result of arora and safra 1998 whose verifiers examine a nonconstant number of bits in the proof though this number is a very slowly growing function of the input length as a consequence we prove that no max snp hard problem has a polynomial time approximation scheme unless np p the class max snp was defined by papadimitriou and yannakakis 1991 and hard problems for this class include vertex cover maximum satisfiability maximum cut metric tsp steiner trees and shortest superstring we also improve upon the clique hardness results of feige et al 1996 and arora and safra 1998 and show that there exists a positive e such that approximating the maximum clique size in an n vertex graph to within a factor of n e is np hard

assessment of rod behaviour theories used in spectral finite element modelling abstract in this work different theories of rods have been discussed and compared the investigated theories are widely used in spectral finite element modelling of rod behaviour associated with propagation of symmetric longitudinal waves these are various single two mode and three mode theories including the elementary classical love and mindlin herrmann approaches as well as new two three and four mode theories proposed by the authors dispersion curves associated with each theory obtained by the use of hamilton s principle have been presented and discussed in the paper the investigation programme carried out by the authors aimed to show major differences and similarities between the rod theories and to discuss certain numerical aspects of their application great attention has been paid on properties limitations as well as difficulties associated with the use of the theories the results obtained from a wide program on numerical tests allowed the authors to draw certain general conclusions that are valid not only in the field of the spectral finite element method but also in the field of dynamics of engineering rod structures

dynamical behavior of reaction diffusion neural networks and their synchronization arising in modeling epileptic seizure a numerical simulation study abstract excessive synchronizations of neurons in the brain networks can be a reason for some episodic disorders such as epilepsy in this paper we simulate neural dynamic models and their synchronizations numerically by a new numerical algorithm in order to overcome the complexity of the problem a suitable combination of the legendre spectral element method and operator splitting technique is proposed to convert a complex system of partial differential equations into sparse linear algebraic systems the simulations are explored in different aspects such as accuracy computing the wavefront and angular speeds computing time the condition number of obtained algebraic systems to demonstrate the proposed scheme effectiveness the numerical simulations and comparisons confirm the method is suitable in terms of accuracy and speed

optimal decomposition of the domain in spectral methods for wave like phenomena a strategy for determining the optimal number of grid points and subdomains in a spectral method with domain decomposition on a serial computer is presented the rapidly growing computational cost for large numbers of grid points in each subdomain is balanced against the exponential convergence for spectral approximation of smooth functions and the optimum is found as the number of grid points and subdomains that gives the minimal computational cost for a given accuracy the typical length scale of the problem is found to influence the number of subdomains but not the number of grid points within each subdomain

a chebyshev collocation algorithm for the solution of advection diffusion equations an original method to precondition chebyshev collocated advection diffusion operators is proposed the preconditioner based on an upwind finite difference approximation is tested on one and two dimensional problems the preconditioner behaves efficiently for advection dominated cases and produces rates of convergence independent of the number of nodes finally an ad hoc designed multi domain strategy is introduced

domain decomposition methods for nonlinear problems in fluid dynamics abstract the numerical solution of finite element approximations of complicated two and three dimensional nonlinear problems can be a most formidable task in order to overcome this difficulty related to dimensionality domain splitting methods can be very effective particularly in view of obtaining a fast and economical conjugate gradient solver which can be used to precondition the solution of nonlinear problems by optimization methods via nonlinear least squares or weighted residual formulations a new technique of this type will be introduced and analysed and its efficiency will be discussed from numerical experiments concerning the numerical simulation of transonic flows for compressible inviscid fluids and incompressible viscous flows modelled by the navier stokes equations

spectral methods in time for a class of parabolic partial differential equations in this paper we introduce a fully spectral solution for the partial differential equation u t uu x vu x x mu x x x lu x x x x 0 for periodic boundary conditions in space the use of fourier expansion in x admits of a particularly efficient algorithm with respect to expansion of the time dependence in a chebyshev series boundary conditions other than periodic may still be treated with reasonable though lesser efficiency for all cases very high accuracy is attainable at moderate computational cost relative to the expense of variable order finite difference methods in time

modelling of wave propagation in composite plates using the time domain spectral element method abstract this paper presents results of numerical simulation of the propagation of transverse elastic waves corresponding to the a0 mode of lamb waves in a composite plate the problem is solved by the spectral element method spectral plate finite elements with 36 nodes defined at gauss lobatto legendre points are used as a consequence of the selection of lagrange polynomials for element shape functions discrete orthogonality is obtained leading to the diagonal form of the element mass matrix this results in a crucial reduction of numerical operations required for the solution of the equation of motion by time integration numerical calculations have been carried out for various orientations and relative volume fractions of reinforcing fibres within the plate the paper shows how the velocities of transverse elastic waves in composite materials depend on the orientation and the relative volume fraction of the reinforcement

variational multiscale turbulence modelling in a high order spectral element method in the variational multiscale vms approach to large eddy simulation les the governing equations are projected onto an a priori scale partitioning of the solution space this gives an alternative framework for designing and analyzing turbulence models we describe the implementation of the vms les methodology in a high order spectral element method with a nodal basis and discuss the properties of the proposed scale partitioning the spectral element code is first validated by doing a direct numerical simulation of fully developed plane channel flow the performance of the turbulence model is then assessed by several coarse grid simulations of channel flow at different reynolds numbers

grid dispersion and stability criteria of some common finite element methods for acoustic and elastic wave equations purely numerical methods based on finite element approximation of the acoustic or elastic wave equation are becoming increasingly popular for the generation of synthetic seismograms we present formulas for the grid dispersion and stability criteria for some popular finite element methods fem for wave propagation namely classical and spectral fem we develop an approach based on a generalized eigenvalue formulation to analyze the dispersive behavior of these fems for acoustic or elastic wave propagation that overcomes difficulties caused by irregular node spacing within the element and the use of high order polynomials as is the case for spectral fem analysis reveals that for spectral fem of order four or greater dispersion is less than 0 2 at four to five nodes per wavelength and dispersion is not angle dependent new results can be compared with grid dispersion results of some classical finite difference methods fdm used for acoustic or elastic wave propagation analysis reveals that fdm and classical fem require a larger sampling ratio than a spectral fem to obtain results with the same degree of accuracy the staggered grid fdm is an efficient scheme but the dispersion is angle dependent with larger values along the grid axes on the other hand spectral fem of order four or greater is isotropic with small dispersion making it attractive for simulations with long propagation times

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

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

a first error indicator for micro macro simulations of viscoelastic fluids having closed form constitutive equations an a posteriori error indicator for viscoelastic flow calculations is proposed by the author for a recently introduced method this new method is derived from the brownian configuration field approach and can be applied to models for which closed form expressions already exist numerical results are presented in the context of the spectral element method to show the reliability of the error indicator the benchmark problem of the flow of an oldroyd b fluid past a cylinder in a channel is used and we show that this new method is always more accurate than using a constitutive equation

new hardness results for congestion minimization and machine scheduling we study the approximability of two natural np hard problems the first problem is congestion minimization in directed networks in this problem we are given a directed graph and a set of source sink pairs the goal is to route all the pairs with minimum congestion on the network edges the second problem is machine scheduling where we are given a set of jobs and for each job there is a list of intervals on which it can be scheduled the goal is to find the smallest number of machines on which all jobs can be scheduled such that no two jobs overlap in their execution on any machine both problems are known to be o log n log log n approximable via the randomized rounding technique of raghavan and thompson 1987 however until recently only max snp hardness was known for each problem we make progress in closing this gap by showing that both problems are o log log n hard to approximate unless np dtime no log log log n

new space time spectral and structured spectral element methods for high order problems abstract we propose new space time spectral and structured spectral element methods for high order problems by matrix decomposition we introduce a family of new basis functions which leads to a diagonal system for the fourth order problems with constant coefficients based on new basis functions in space and a dual petrov galerkin formulation in time we propose a new space time spectral method for a time dependent problem and proved their spectral accuracy both in space and time numerical results demonstrate their high efficiency and coincide well with theoretical analysis further to ultimately promote the numerical performance and efficiency we exploit the idea of locally simultaneous diagonalization to provide new structured spectral element methods for high order problems besides to increase the flexibility a c 1 conforming rectangular spectral method in analogy to argyris or bell triangular elements are proposed for fourth order equations which serves as a preparative work towards conforming quadrilateral spectral elements for high order equations

degenerate solutions of a nonlinear elliptic equation on the sphere abstract we consider the equation u u u q on the round sphere s m m 3 with 0 and 1 q m 2 m 2 by studying the local structure of solutions bifurcating from the constant solution we show that for any isoparametric hypersurface m in the sphere there is a value of and a solution of the equation at which the linearized equation has nontrivial solutions at the same time we obtain new solutions of the equation

towards global sem mantle convection simulations on polyhedral based grids abstract a novel approach to the construction of three dimensional grids in spherical geometries is described the grids are based on a range of underlying regular polyhedra the faces of the polyhedra are arranged into a number of diamonds in the latitudinal and longitudinal directions to facilitate the creation of a structured mesh each polyhedron is completely characterized in terms of the arrangement of diamonds the approach is shown to be very flexible in terms of the meshes that can be generated it also aids comparisons between the grids used in many mantle convection studies a spectral element discretization of poisson s equation is performed to demonstrate the efficacy of the grid generation technique rapidly convergent approximations are obtained that demonstrate that fewer degrees of freedom are required to obtain a desired level of accuracy compared with low order finite element approximations

a weak legendre collocation spectral method for the solution of the incompressible navier stokes equations in unstructured quadrilateral subdomains a weak legendre spectral method is developed for the solution of the primitive variable formulation of the unsteady incompressible navier stokes equations in general two dimensional and axisymmetric geometries a semi implicit projection method is utilized for the temporal approximation and for decoupling the velocity field from the pressure field a series of elliptic boundary value problems arises from the above procedure each of which is spatially discretized by a weak collocation method in multiple nonoverlapping subdomains in particular a modified variational formulation of the partial differential equations is presented which leads after discretization to a weak multidomain approximation of the corresponding problems a weak formalism for the influence matrix technique is also developed which is consistent with the spatial discretization scheme and successfully separate the equations for the internal nodes from the ones governing the interface unknowns a method of dealing with the singularity problem faced by the weak formulation at axisymmetric problems is proposed while a combination of direct methods is studied for tackling effectively the linear algebraic systems resulting from the full discretization exponential convergence is demonstrated for a plethora of stokes and navier stokes simulations

numerical methods for hypersonic boundary layer stability various numerical methods for the solution of linear stability equations for compressible boundary layers are compared both the global and local eigenvalue methods for temporal stability analysis are discussed global methods are used to compute all the eigenvalues of the discretized system when a guess for the desired eigenvalue is available local methods may be used both to purify the eigenvalue and compute the associated eigenfunctions the extension to spatial stability analysis is also considered the discretizations studied include a second order finite difference method a fourth order accurate two point compact difference scheme and a chebyshev spectral collocation method eigenvalue results are presented for mach numbers up to 10 as the mach number increases the performance of the spectral method deteriorates due to the outward movement of the critical layer to alleviate this problem a multi domain spectral collocation method is developed which exhibits better convergence the overall performance of the fourth order compact scheme is excellent our results also indicate that in the limit of vanishing mach number there exist stable discrete modes in addition to the discrete modes of the orr sommerfeld equation

stabilization of spectral methods by finite element bubble functions abstract we show that a standard spectral collocation scheme can be stabilized by adding extra trial test functions with local support bubbles applications are given to a scalar advection diffusion problem and to the stokes problem

spectral element method for linear fan tone noise radiation a numerical method for prediction of acoustic spinning mode radiation from turbofan inlets is presented sound propagation is modeled by the linearized mass conservation equation for irrotational flows and solved in the frequency domain the mean flow through the inlet is obtained as a solution of the full potential equation both the mean flow and the acoustic problem are approximated by galerkin projection in spectral element spaces of continuous piecewise polynomials defined on the same grid the gauss chebyshev lobatto points within the elements are generated via transfinite interpolation and cad projection procedures embedded within the code the linear algebraic systems obtained are then solved using either direct or sparse iterative solvers based on the message passing interface standard for interprocessor communication the singularity appearing in the acoustic integrals on the symmetry axis is treated by the use of a collocation operator based on the gauss chebyshev instead of the gauss chebyshev lobatto points to eliminate reflections from the radiation boundaries a novel frequency domain formulation of the matched layer technique wherein waves entering the layer are exponentially damped is proposed the overall computing procedure is first validated on a tone radiation problem from a semi infinite cylinder and then applied to an experimental jt15d turbofan inlet setup

spectral element fourier method for transitional flows in complex geometries an efficient spectral element fourier method is presented for the direct numerical simulation of transitional internal flows in complex geometries this method is applied for the spatial discretization of the unsteady incompressible three dimensional navier stokes equations in the velocity pressure formulation the resulting discrete equations are solved by a semi implicit method in time where the nonlinear convective term is treated explicitly direct numerical simulations are performed to investigate the spatial structure and temporal evolution of two and three dimensional transitional flows in grooved channels

spectral element method in time for rapidly actuated systems in this paper the spectral element se method is applied in time to find the entire time periodic or transient solution of time dependent differential equations the time periodic solution is computed by enforcing periodicity of the element set of particular interest are periodic forcing functions possessing high frequency content to maintain the spectral accuracy for such forcing functions an h refinement scheme is employed near the semi discontinuity without increasing the number of degrees of freedom time discretization by spectral elements is applied initially to a standard form of a set of linear first order differential equations subject to harmonic excitation and an excitation admitting rapid variation other case studies include the application of the se approach to parabolic and hyperbolic partial differential equations the first order form of these equations is obtained through semi discretization using conventional finite element spectral element and finite difference schemes element clustering h refinement is applied to maintain the high accuracy and efficiency in the region of the forcing function admitting rapid variation the convergence in time of the method is demonstrated in some cases machine precision is obtained with 25 degrees of freedom per cycle finally the method is applied to a weakly nonlinear problem with time periodic solution to demonstrate its future applicability to the analysis of limit cycle oscillations in aeroelastic systems

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

higher order triangular spectral element method with optimized cubature points for seismic wavefield modeling the mass lumped method avoids the cost of inverting the mass matrix and simultaneously maintains spatial accuracy by adopting additional interior integration points known as cubature points to date such points are only known analytically in tensor domains such as quadrilateral or hexahedral elements thus the diagonal mass matrix spectral element method sem in non tensor domains always relies on numerically computed interpolation points or quadrature points however only the cubature points for degrees 1 to 6 are known which is the reason that we have developed a p norm based optimization algorithm to obtain higher order cubature points in this way we obtain and tabulate new cubature points with all positive integration weights for degrees 7 to 9 the dispersion analysis illustrates that the dispersion relation determined from the new optimized cubature points is comparable to that of the mass and stiffness matrices obtained by exact integration simultaneously the lebesgue constant for the new optimized cubature points indicates its surprisingly good interpolation properties as a result such points provide both good interpolation properties and integration accuracy the courantfriedrichslewy cfl numbers are tabulated for the conventional fekete based triangular spectral element tsem the tsem with exact integration and the optimized cubature based tsem otsem a complementary study demonstrates the spectral convergence of the otsem a numerical example conducted on a half space model demonstrates that the otsem improves the accuracy by approximately one order of magnitude compared to the conventional fekete based tsem in particular the accuracy of the 7th order otsem is even higher than that of the 14th order fekete based tsem furthermore the otsem produces a result that can compete in accuracy with the quadrilateral sem qsem the high accuracy of the otsem is also tested with a non flat topography model in terms of computational efficiency the otsem is more efficient than the fekete based tsem although it is slightly costlier than the qsem when a comparable numerical accuracy is required higher order cubature points for degrees 7 to 9 are developed the effects of quadrature rule on the mass and stiffness matrices has been conducted the cubature points have always positive integration weights freeing from the inversion of a wide bandwidth mass matrix the accuracy of the tsem has been improved in about one order of magnitude

a legendre spectral element model for sloshing and acoustic analysis in nearly incompressible fluids a new spectral finite element formulation is presented for modeling the sloshing and the acoustic waves in nearly incompressible fluids the formulation makes use of the legendre polynomials in deriving the finite element interpolation shape functions in the lagrangian frame of reference the formulated element uses gauss lobatto legendre quadrature scheme for integrating the volumetric stiffness and the mass matrices while the conventional gauss legendre quadrature scheme is used on the rotational stiffness matrix to completely eliminate the zero energy modes which are normally associated with the lagrangian fe formulation the numerical performance of the spectral element formulated here is examined by doing the inf sup test on a standard rectangular rigid tank partially filled with liquid the eigenvalues obtained from the formulated spectral element are compared with the conventional equally spaced node locations of the h type lagrangian finite element and the predicted results show that these spectral elements are more accurate and give superior convergence the efficiency and robustness of the formulated elements are demonstrated by solving few standard problems involving free vibration and dynamic response analysis with undistorted and distorted spectral elements and the obtained results are compared with available results in the published literature

a parallel repetition theorem we show that a parallel repetition of any two prover one round proof system mip 2 1 decreases the probability of error at an exponential rate no constructive bound was previously known the constant in the exponent in our analysis depends only on the original probability of error and on the total number of possible answers of the two provers the dependency on the total number of possible answers is logarithmic which was recently proved to be almost the best possible u feige and o verbitsky proc 11th annual ieee conference on computational complexity ieee computer society press los alamitos ca 1996 pp 70 76

fully parallel mesh i o using petsc dmplex with an application to waveform modeling large scale pde simulations using high order finite element methods on unstructured meshes are an indispensable tool in science and engineering the widely used open source petsc library offers an

three dimensional mixed explicit implicit generalized galerkin spectral element methods for high speed turbulent compressible flows abstract in high speed flows the interactions of shock waves with turbulent boundary layers are important design considerations because of the complex flowfields resulting in increased adverse pressure gradients skin friction and temperatures unsteadiness and three dimensional flowfield structure are also characteristic of shock wave turbulent boundary layer interactions such physical phenomena require sophisticated numerical schemes in the solution of governing equations the purpose of this paper therefore is to introduce an accurate and efficient approach the mixed explicit implicit generalized galerkin spectral element method mei gg sem with legendre polynomial spectral elements in which flowfield dependent implicitness parameters provide automatically adequate computational requirements for compressible and incompressible flows or high speed and low speed flows this is in contrast to the traditional approach in which all speed regime analysis requires a separate hyperbolic elliptic pressure equation for pressure correction if the flow becomes incompressible in the mei gg sem scheme mesh refinements are carried out adaptively until shock waves are resolved followed then by the adaptive increase of legendre polynomial degrees until turbulence microscales are resolved in which the traditional turbulence modeling is no longer required aimed toward direct numerical simulation in order to demonstrate the validity of the theory and numerical procedure two dimensional flat plate and compression corner high speed flows are investigated followed by a three dimensional sharp leading edged fin for swept shock wave turbulent boundary layer interactions comparisons of the present study with experimental measurements and other numerical studies show favorable agreement

spectral method solution of the stokes equations on nonstaggered grids abstract the stokes equations are solved using spectral methods with staggered and nonstaggered grids numerous ways to avoid the problem of spurious pressure modes are presented including new techniques using the pseudospectral method and a method solving the weak form of the governing equations a variation on the spectral element method developed by patera the pseudospectral methods using nonstaggered grids are simpler to implement and have comparable or better accuracy than the staggered grid formulations three test cases are presented a formulation with an exact solution a formulation with homogeneous boundary conditions and the driven cavity problem the solution accuracy is shown to be greatly improved for the driven cavity problem when the analytical solution of the singular flow behavior in the upper corners is separated from the computational solution

spectral element formulations on non conforming grids a comparative study of pointwise matching and integral projection methods pointwise matching pm and integral projection ip methods are two widely used techniques to extend the classical weak formulations to include non conforming grids we present spectral element formulations on polynomial p type and geometric h type non conforming grids using both the pm also known as the constrained approximation and ip also known as the mortar element methods we systematically compare the convergence characteristics of pm and ip methods for diffusion convection and convection diffusion equations consistency errors due to the non conforming formulations of the diffusion equation result in convergence problems for the pm method using the maximum rule both non conforming formulations for the unsteady convection operator result in eigenvalue spectrum with positive real values causing convergence problems due to the consistency errors however small physical diffusion in the convection diffusion equation eliminates these problems resulting in spectral convergence for both methods encouraged by this we present spectral element formulations for incompressible navier stokes equations using pm and ip methods on p type and h type non conforming grids and demonstrate spectral convergence for unsteady and steady test cases results for two dimensional lid driven cavity flow at re 1000 are also presented

chebyshev spectral collocation methods for laminar flow through a channel contraction abstract a chebyshev spectral element method is described for solving the navier stokes equations in a channel contraction the flow region is divided into two semi infinite elements the governing equation for the stream function is solved using a newton linearization the semi infinite elements are treated by means of algebraic type mappings to transform them onto finite domains the stream function is represented by a double chebyshev expansion in each element the coefficients are determined by collocating the linearized equation at each newton step and imposing c3 continuity conditions across the element interface in a collocation sense efficient direct methods based on capacitance matrix ideas are described which take advantage of the structure of the spectral element matrix results are presented for reynolds numbers in the range 0 200 which are in good agreement with previously published work but requiring fewer degrees of freedom only several steps of the newton process are required to achieve a converged solution for re 120 the method converges from a zero initial approximation and thereafter continuation in the reynolds number is used with increments of 10

improving the speed and accuracy of projection type incompressible flow solvers abstract superseding so called first generation incompressible flow solvers of the projection type based on taylor galerkin advection second order pressure damping and element based data structures the current second generation solvers based on high order upwind advection fourth order pressure damping and edge based data structures have now been in use for half a decade and have proven remarkably robust and efficient for many large scale problems in order to achieve higher accuracy and speed these solvers have recently been enhanced in a variety of ways a substepping for advection b implicit treatment of advective terms via sgs and gmres lu sgs iterative solvers c fully implicit time accurate advancement of pressure and velocities and d linelet preconditioning for the pressure poisson equation the combined effect of these third generation improvements leads to speedups of the order of o 1 5 1 10 with similar or even better temporal accuracy as demonstrated on a variety of academic and industrial problems

optimal blended spectral element operators for acoustic wave modeling spectral element methods based on high order polynomials are among the most commonly used techniques for computing accurate simulations of wave propagation phenomena in complex media however to retain computational efficiency very high order polynomials cannot be used and errors such as numerical dispersion and numerical anisotropy cannot be totally avoided in the present work we devise an approach for reducing such errors by considering modified discrete wave operators we analyze consistent and lumped operators together with blended operators weighted averages of consistent and lumped operators furthermore using the operator blending approach and a novel dispersion analysis method we develop optimal spectral element operators that have increased numerical accuracy without resorting to very high order operators the new operators are faster and computationally more efficient than consistent operators our approach is based on the tensor product decomposition of the element matrices into 1d f

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

application of implicit explicit high order runge kutta methods to discontinuous galerkin schemes despite the popularity of high order explicit runge kutta erk methods for integrating semi discrete systems of equations erk methods suffer from severe stability based time step restrictions for very stiff problems we implement a discontinuous galerkin finite element method dgfem along with recently introduced high order implicit explicit runge kutta imex rk schemes to overcome geometry induced stiffness in fluid flow problems the imex algorithms solve the non stiff portions of the domain using explicit methods and isolate and solve the more expensive stiff portions using an l stable stiffly accurate explicit singly diagonally implicit runge kutta method esdirk furthermore we apply adaptive time step controllers based on the embedded temporal error predictors we demonstrate in a number of numerical test problems that imex methods in conjunction with efficient preconditioning become more efficient than explicit methods for systems exhibiting high levels of grid induced stiffness

chebyshev expansion methods for the solution of elliptic partial differential equations abstract in a recent paper haidvogel and zang j comput phys 30 1979 167 described a method based on an expansion in chebyshev polynomials for the solution of poisson s equation in a rectangle they also indicated how the method might be extended to more general equations in this note we give a comparison of the method with the global element method delves and hall j inst math appl 23 1979 223 and give numerical results for those problems treated by haidvogel and zang

approximation algorithms for disjoint paths and related routing and packing problems given a network and a set of connection requests on it we consider the maximum edge disjoint paths and related generalizations and routing problems that arise in assigning paths for these requests we present improved approximation algorithms and or integrality gaps for all problems considered the central theme of this work is the underlying multicommodity flow relaxation applications of these techniques to approximating families of packing integer programs are also presented

mass lumping techniques in the spectral element method on the equivalence of the row sum nodal quadrature and diagonal scaling methods abstract in the context of wave propagation analysis the spectral element method sem in conjunction with a diagonal mass matrix is often the method of choice therefore it is of high importance to investigate the influence of different mass lumping schemes on the accuracy of the numerical results to this end we compare the performance of three established methods including the row sum method the nodal quadrature method and the diagonal scaling method the theoretical analysis of these methods reveals a close connection between them under certain conditions that are discussed in detail in this article we are able to show a direct equivalence between these three approaches in this regard the attainable accuracy of the numerical integration of the mass matrix plays an important role by means of several dynamic benchmark problems we verify the theoretical results and illustrate the convergence properties of the lumped mass matrix sem in comparison to a formulation based on the consistent mass matrix

higher order adaptive finite element methods for orbital free density functional theory in the present work we study various numerical aspects of higher order finite element discretizations of the non linear saddle point formulation of orbital free density functional theory we first investigate the robustness of viable solution schemes by analyzing the solvability conditions of the discrete problem we find that a staggered solution procedure where the potential fields are computed consistently for every trial electron density is a robust solution procedure for higher order finite element discretizations we next study the convergence properties of higher order finite element discretizations of orbital free density functional theory by considering benchmark problems that include calculations involving both pseudopotential as well as coulomb singular potential fields our numerical studies suggest close to optimal rates of convergence on all benchmark problems for various orders of finite element approximations considered in the present study we finally investigate the computational efficiency afforded by various higher order finite element discretizations which constitutes the main aspect of the present work by measuring the cpu time for the solution of discrete equations on benchmark problems that include large aluminum clusters in these studies we use mesh coarse graining rates that are derived from error estimates and an a priori knowledge of the asymptotic solution of the far field electronic fields our studies reveal a significant 100 1000 fold computational savings afforded by the use of higher order finite element discretization alongside providing the desired chemical accuracy we consider this study as a step towards developing a robust and computationally efficient discretization of electronic structure calculations using the finite element basis

a compatible and conservative spectral element method on unstructured grids we derive a formulation of the spectral element method which is compatible on very general unstructured three dimensional grids here compatible means that the method retains discrete analogs of several key properties of the divergence gradient and curl operators the divergence and gradient are anti adjoints the negative transpose of each other the curl is self adjoint and annihilates the gradient operator and the divergence annihilates the curl the adjoint relations hold globally and at the element level with the inclusion of a natural discrete element boundary flux term we then discretize the shallow water equations on the sphere using the cubed sphere grid and show that compatibility allows us to locally conserve mass energy and potential vorticity conservation is obtained without requiring the equations to be in conservation form the conservation is exact assuming exact time integration

on galerkin difference methods energy stable difference methods for hyperbolic initial boundary value problems are constructed using a galerkin framework the underlying basis functions are lagrange functions associated with continuous piecewise polynomial approximation on a computational grid both theoretical and computational evidence shows that the resulting methods possess excellent dispersion properties in the absence of boundaries the spectral radii of the operators for the first and second derivative matrices are bounded independent of discretization order with boundaries the spectral radius of the first order derivative matrix appears to be bounded independent of discretization order and grows only slowly with discretization order for problems in second order form