indefinite overlapping schwarz methods for time dependent stokes problems abstract a class of indefinite overlapping schwarz methods is introduced and studied for time dependent stokes problems discretized with either mixed finite elements or mixed spectral elements the methods proposed are based on the solution of local time dependent stokes problems on overlapping subdomains and on the solution of a coarse time dependent stokes problem defined on the coarse subdomain mesh the resulting preconditioner is accelerated by an appropriate krylov space method producing a very efficient scalable and parallel solver

the optimal symmetrical points for polynomial interpolation of real functions in the tetrahedron abstract the main result of this paper is the computation of the mean optimal symmetrical interpolation points in the tetrahedron up to degree 9 this interpoation set has the smallest lebesgue constant known today

an iterative solver for p version finite elements in three dimensions abstract a new iterative method for systems arising from p version finite elements is presented and computational experience reported for conforming solid serendipity elements of order up to p 8 and real world structures with 5000 elements and over 1 million degrees of freedom the method uses a domain decomposition technique with each element taken to be a subdomain and a coarse space consisting of p 1 elements to overcome the i o bottleneck only access to a fast matrix vector multiplication instead of stiffness data is needed during iterations in terms of wall clock time the method outperforms on ibm rs 6000 an earlier method running on cray xmp

from electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex the electrostatic interpretation of the jacobi gauss quadrature points is exploited to obtain interpolation points suitable for approximation of smooth functions defined on a simplex moreover several new estimates based on extensive numerical studies for approximation along the line using jacobi gauss lobatto quadrature points as the nodal sets are presented the electrostatic analogy is extended to the two dimensional case with the emphasis being on nodal sets inside a triangle for which two very good matrices of nodal sets are presented the matrices are evaluated by computing the lebesgue constants and they share the property that the nodes along the edges of the simplex are the gauss lobatto quadrature points of the chebyshev and legendre polynomials respectively this makes the resulting nodal sets particularly well suited for integration with conventional spectral methods and supplies a new nodal basis for h p finite element methods

low energy basis preconditioning for elliptic substructured solvers based on unstructured spectral hp element discretization abstract the development and application of three dimensional unstructured hierarchical spectral hp element algorithms has highlighted the need for efficient preconditioning for elliptic solvers building on the work of bica ph d thesis courant institute new york university 1997 we have developed an efficient preconditioning strategy for substructured solvers based on a transformation of the expansion basis to a low energy basis in this numerically derived basis the strong coupling between expansion modes in the original basis is reduced thus making it amenable to block diagonal preconditioning the efficiency of the algorithm is maintained by developing the new basis on a symmetric reference element and ignoring in the preconditioning step the role of the jacobian of the mapping from the reference to the global element by applying an additive schwarz block preconditioner to the low energy basis combined with a coarse space linear vertex solver we have observed reductions in execution time of up to three times for tetrahedral elements and 10 times for prismatic elements when compared to a standard diagonal preconditioner full details of the implementation and validation of the tetrahedral and prismatic element preconditioning strategy are set out below

improvements of the efficiency of some three step iterative like newton methods an improvement of the local order of convergence is presented to increase the efficiency of the iterative method with an appropriate number of evaluations of the function and its derivative the third and fourth order of known two step like newton methods have been improved and the efficiency has also been increased

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

existence and uniqueness of time periodic physically reasonable navier stokes flow past a body let be a three dimensional exterior domain of class c2 0 1 assume that a navier stokes liquid is moving in under the action of a body force f that is time periodic of period t and that the velocity of the liquid is zero at spatial infinity in this paper we show that if f satisfies suitable conditions and its norm in appropriate function spaces is sufficiently small there is at least one time periodic strong solution furthermore the velocity field v of such a solution decays to zero for large x as x 1 and its spatial gradient decays as x 2 both uniformly in time in addition the pressure p decays like x 2 and its gradient like x 3 for almost all t 0 t in the special case where f is time independent these solutions are also time independent and coincide with finn s physically reasonable solutions 4 moreover we show that our time periodic solutions are unique in a very large class namely the class of time periodic weak solutions satisfying the energy inequality and with corresponding pressure fields verifying mild summability conditions in 0 t

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

approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle the main results of this paper are the analysis of the quality of approximation of polynomial interpolation and the computation of the approximate optimal interpolation points in the triangle we introduce various norms for the interpolation operator computational results indicate that for a given polynomial degree the set that minimizes the mean l2 norm of the interpolation operator is close to the smallest lebesgue constant interpolation set in particular for the triangle this set gives the smallest lebesgue constant currently known

adams like techniques for zero finder methods two families of zero finding iterative methods for nonlinear equations are presented we derive them solving an initial value problem using adams like multistep techniques namely adams methods have been used to solve the problem that consists in a differential equation in what appears the inverse function of the one which zero will be computed and the condition given by the value attained by it at the initial approximation performing this procedure several methods of different local orders of convergence have been obtained

parallel multilevel series k way partitioning scheme for irregular graphs in this paper we present a parallel formulation of a multilevel k way graph partitioning algorithm a key feature of this parallel formulation is that it is able to achieve a high degree of concurrency while maintaining the high quality of the partitions produced by the serial multilevel k way partitioning algorithm in particular the time taken by our parallel graph partitioning algorithm is only slightly longer than the time taken for re arrangement of the graph among processors according to the new partition experiments with a variety of finite element graphs show that our parallel formulation produces high quality partitionings in a short amount of time for example a 128 way partitioning of graphs with one million vertices can be computed in a little over two seconds on a 128 processor cray t3d furthermore the quality of the partitions produced is comparable edge cuts within 5 to those produced by the serial multilevel k way algorithm thus our parallel algorithm makes it feasible to perform frequent repartitioning of graphs in dynamic computations without compromising the partitioning quality

a weighted variant family of newton s method with accelerated third order convergence abstract an improvement to the iteration of newton s method that generalizes previous known improvements of it are presented a symbolic computation that allows us to find the best coefficients with regard to the local order of convergence is given furthermore a set of functions have been tested and all the methods considered present third theoretical and computational order of convergence

spectral collocation on triangular elements we consider the poisson problem on a segment of the unit disc and on triangles on the segment we transform the poisson problem by means of polar coordinates in these new coordinates we have a problem in a rectangle which can easily be mapped onto the square here standard chebyshev collocation techniques can be applied then the segment is mapped onto a triangle where the same spectral scheme may be used by numerical tests we observed the expected high spectral accuracy due to the corner singularity a singular behaviour of the solution can be expected here we improved the accuracy by auxiliary mapping techniques further it is shown that finite difference preconditioning can be successfully applied in order to construct an efficient iterative solver finally a domain decomposition technique is applied to the patching of a rectangular and a triangular element

on the uniqueness of time periodic solutions to the navier stokes equations in unbounded domains we present a uniqueness theorem for time periodic solutions to the navier stokes equations in unbounded domains thus far results on the uniqueness of time periodic solutions to the navier stokes equations in unbounded domain roughly speaking have only found that a small time periodic ln solution is unique within the class of solutions which have sufficiently small l ln norm in this paper we show that a small time periodic ln solution is unique within the class of all time periodic ln solutions which contains large solutions we also consider the uniqueness of solutions in weak ln space the proof of the present uniqueness theorem is based on the method of dual equations

quasi optimal schwarz methods for the conforming spectral element discretization the spectral element method is used to discretize self adjoint elliptic equations in three dimensional domains the domain is decomposed into hexahedral elements and in each of the elements the discretization space is the set of polynomials of degree n in each variable a conforming galerkin formulation is used the corresponding integrals are computed approximately with gauss lobatto legendre gll quadrature rules of order n and a lagrange interpolation basis associated with the gll nodes is used fast methods are developed for solving the resulting linear system by the preconditioned conjugate gradient method the conforming finite element space on the gll mesh consisting of piecewise q1 or p1 functions produces a stiffness matrix kh that is known to be spectrally equivalent to the spectral element stiffness matrix kn kh is replaced by a preconditioner tilde k h which is well adapted to parallel computer architectures the preconditioned operator is then tilde k h 1 k n techniques for nonregular meshes are developed which make it possible to estimate the condition number of tilde k h 1 k n where tilde k h is a standard finite element preconditioner of kh based on the gll mesh two finite element based preconditioners the wirebasket method of smith and the overlapping schwarz algorithm for the spectral element method are given as examples of the use of these tools numerical experiments performed by pahl are briefly discussed to illustrate the efficiency of these methods in two dimensions

zero finder methods derived using runge kutta techniques abstract in this paper some families of zero finding iterative methods for nonlinear equations are presented the key idea to derive them is to solve an initial value problem applying runge kutta techniques more explicitly these methods are used to solve the problem that consists in a differential equation in what appears the inverse function of the one which zero will be computed and the condition given by the value attained by it at the initial approximation carrying out this procedure several families of different orders of local convergence are obtained furthermore the efficiency of these families are computed and two new families using like newton s methods that improve the most efficient one are also given

a hummel seebeck family of iterative methods with improved convergence and efficiency several improvements to a family of iterative methods deduced from a hummel seebeck theorem are presented a symbolic computation that allows to find the best coefficients respect to the local order of convergence is also given the theoretical and computational order of convergence for all methods is increased furthermore the efficiency of these methods applied to the functions tested is improved adapting the strategy presented here a new iteration function with a new evaluation of the function is obtained and using adaptive multi precision arithmetic a smaller cost is got the numerical results computed carrying out this procedure with a floating point system representing 1000 decimal digits support this theory

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

a general approach to time periodic incompressible viscous fluid flow problems this article develops a general approach to time periodic incompressible fluid flow problems and semilinear evolution equations it yields on the one hand a unified approach to various classical problems in incompressible fluid flow and on the other hand gives new results for periodic solutions to the navier stokes oseen flow the navier stokes flow past rotating obstacles and in the geophysical setting for ornstein uhlenbeck and various diffusion equations with rough coefficients the method is based on a combination of interpolation and topological arguments as well as on the smoothing properties of the linearized equation

on the existence of vanishing at infinity symmetric solutions to the plane stationary exterior navier stokes problem we study a nonhomogeneous boundary value problem for the steady state navier stokes equations in a two dimensional exterior domain with two orthogonal symmetry axes the existence of a solution which tends to zero uniformly at infinity is proved under suitable parity conditions on the data of the problem the result is obtained for arbitrary values of the flux of the boundary datum

a fast schur complement method for the spectral element discretization of the incompressible navier stokes equations the weak formulation of the incompressible navier stokes equations in three space dimensions is discretized with spectral element approximations and gauss lobatto legendre quadratures the uzawa algorithm is applied to decouple the velocities from the pressure the equation that results for the pressure is solved by an iterative method within each pressure iteration a helmholtz operator has to be inverted this can efficiently be done by separating the equations for the interior nodes from the equations at the interfaces according to the schur method fast diagonalization techniques are applied to the interior variables of the spectral elements several ways to deal with the resulting interface problem are discussed finally a comparison is made with a more classical method c 1995 academic press inc

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

on bifurcating time periodic flow of a navier stokes liquid past a cylinder we provide general sufficient conditions for the existence and uniqueness of branching out of a time periodic family of solutions from steady state solutions to the two dimensional navier stokes equations in the exterior of a cylinder by separating the time independent averaged component of the velocity field from its oscillatory one we show that the problem can be formulated as a coupled elliptic parabolic nonlinear system in appropriate and distinct function spaces with the property that the relevant linearized operators become fredholm of index 0 in this functional setting the notorious difficulty of 0 being in the essential spectrum entirely disappears and in fact it is even meaningless our approach is different and we believe more natural and simpler than those proposed by previous authors discussing similar questions moreover the latter all fail when applied to the problem studied here

the navier stokes equations in the weak l n space with time dependent external force abstract we consider the navier stokes equations with time dependent external force either in the whole time or in positive time with initial data with domain either the whole space the half space or an exterior domain of dimension n ge 3 we give conditions on the external force sufficient for the unique existence of small solutions in the weak l n space bounded for all time in particular this result gives sufficient conditions for the unique existence and the stability of small time periodic solutions or almost periodic solutions this result generalizes the previous result on the unique existence and the stability of small stationary solutions in the weak l n space with time independent external force

multi p methods iterative algorithms for the p version of the finite element analysis motivated by classical multigrid methods which are based on either the finite difference method or the h version of the finite element method we study multi p methods which are based on the p version of the finite element method and hierarchical shape functions we have developed a class of multi p v cycle algorithms standard v cycle modified v cycle and varying v cycle we have also studied nested multi p methods by combining the nested multi p methods with the v cycle methods we obtain the so called accelerated multi p v cycle methods the convergence of multi p v cycle methods have been investigated and an error estimate is given for the nested multi p methods numerical experiments on representative sample problems have been conducted and indicate that our multi p methods are very efficient

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

parallel domain decomposition solver for adaptive hp finite element methods in this paper the development and implementation of highly parallelizable domain decomposition solvers for adaptive hp finite element methods is discussed two level orthogonalization is used to obtain a reduced system which is preconditioned by a coarse grid operator the condition number of the preconditioned system for poisson problems in two space dimensions is proved to be bounded by c 1 hp h 2 1 log p 2 and cp 1 hp h 2 1 log p 2 for different choices of coarse grid operators where h is the subdomain size p is the maximum spectral order h is the size of the smallest element in the subdomain and c is a constant independent of the mesh parameters the work here extends the work of bramble et al math comp 47 1986 pp 103 134 on the h version and babuska et al siam j numer anal 29 1991 pp 624 661 on the p version of the finite element method a preliminary version of this solver was first announced by oden patra and feng in domain decomposition solver for adaptive hp finite elements vii conference on domain decomposition state college pa october 1993 numerical experiments show fast convergence of the solver and good control of the condition number on a variety of discretizations

on time periodic flow of a viscous liquid past a moving cylinder we show existence uniqueness and spatial asymptotic behavior of a two dimensional time periodic flow around a cylinder that moves orthogonal to its axis with a time periodic velocity v the result is proved if the size of the data is sufficiently small and the average of v over a period is not zero