domain decomposition algorithms for mixed methods for second order elliptic problems domain decomposition algorithms for mixed finite element methods for linear second order elliptic problems in r 2 and r 3 are developed a convergence theory for two level and multilevel schwarz methods applied to the algorithms under consideration is given it is shown that the condition number of these iterative methods is bounded uniformly from above in the same manner as in the theory of domain decomposition methods for conforming and nonconforming finite element methods for the same differential problems numerical experiments are presented to illustrate the present techniques

multigrid method and multilevel additive preconditioner for mixed element method for non self adjoint and indefinite problems in this paper a v cycle multigrid method is proposed for mixed element method for non self adjoint and indefinite second order elliptic problems and the uniform convergence of the v cycle multigrid method is proven under minimal regularity assumption meanwhile a multilevel additive preconditioner is given for these problems and an optimal convergence rate for preconditioned gmres method is obtained under minimal regularity assumption

asymptotic expansions for large closed queueing networks with multiple job classes a closed bcmp queuing network consisting of r job classes chains k 1 single server fixed rate nodes and m sub j class j jobs j 1 2 r is considered asymptotic expansions are constructed for the partition function under assumptions 1 k 1 2 m sub j 1 for each j and 3 k m sub j o 1 analytic expressions for performance measures such as the mean queue length are also given the approach employs the ray method and the method of matched asymptotic expansions numerical comparisons illustrate the accuracy of the approximations

multivariate rational approximants for multiclass closed queuing networks closed markovian networks of queues with multiclass customers and having a product form equilibrium state probability distribution are useful in the performance evaluation and design of computer and telecommunication systems therefore the efficient computation of the normalizing function the key element of the solution in product form has attracted considerable effort we consider a network that consists of one infinite server is station and one processor sharing ps or fcfs single server station we use multivariate newton pade approximants computed from data for small numbers of customers in each class to estimate the normalizing function for a larger population in the network the effectiveness and tremendous gain in computing time of this procedure are illustrated through various numerical experiments

overlapping schwarz methods for mixed linear elasticity and stokes problems overlapping schwarz preconditioners are introduced and studied for saddle point problems with a penalty term such as stokes equations and mixed formulations of linear elasticity these preconditioners are based on the solution of local saddle point problems on overlapping subdomains and the solution of a coarse saddle point problem numerical experiments show that these are parallel and scalable preconditioners since the rate of convergence of the preconditioned operator is independent of the mesh size h the number of subdomains n and the penalty parameter

a parallel domain decomposition algorithm of mixed element equation for second order elliptic dirichlet boundary value problem a parallel domain decomposition algorithm of schwarz type is introduced to solve the mixed finite element equation for the dirichlet boundary value problem of a second order partial differential equation of elliptic type and convergence of this algorithm is analyzed

numerical simulation of singular solutions to the two dimensional cubic schrodinger equation numerical simulations of the cubic nonlinear schrodinger equation are presented in dimension n 2 the emphasis is on a detailed mechanism of blow up the numerical results indicate that the blow up is not restricted to the case where the problem is posed in the entire r2 space but also occurs with periodic boundary conditions the structure of the solution as the singular time t is approached has been investigated using examples with and without radial symmetry and or periodicity in most cases we observe that the amplitude develops something like the peaked self similar profile predicted by zakharov and synakh 38 which has a t t 2 3 maximum amplitude and a t t 2 3 width

asymptotic expansions for closed markovian networks with state dependent service rates a method is presented for calculating the partition function and from it performance measures for closed markovian stochastic networks with queuing centers in which the service or processing rate depends on the center s state or load the analysis on which this method is based is new and a major extension of our earlier work on load independent queuing networks the method gives asymptotic expansions for the partition function in powers of 1 n where n is a parameter that reflects the size of the network the expansions are particularly useful for large networks with many classes each class having many customers the end result is a decomposition by which expansion coefficients are obtained exactly by linear combinations of partition function values of small network constructs called pseudonetworks effectively computable bounds are given for errors arising from the use of a finite number of expansion terms this method is important because load dependence is at once an essential element of sophisticated network models of computers computer communications and switching teletraffic and manufacturing systems and the cause of very intensive computations in conventional techniques with this method very large load dependent networks can be analyzed whereas previously only small networks were computationally tractable

parallel solvers for virtual element discretizations of elliptic equations in mixed form abstract the aim of this paper is twofold on the one hand we numerically test the performance of mixed virtual elements in three dimensions to solve the mixed formulation of three dimensional elliptic equations on polyhedral meshes on the other hand we focus on the parallel solution of the linear system arising from such discretization considering both direct and iterative parallel solvers in the latter case we develop two block preconditioners one based on the approximate schur complement and one on a regularization technique both these topics are numerically validated by several parallel tests performed on a linux cluster more specifically we show that the proposed virtual element discretization recovers the expected theoretical convergence rates and we analyze the performance of the direct and iterative parallel solvers taken into account

a stabilized explicit lagrange multiplier based domain decomposition method for parabolic problems a fully explicit stabilized domain decomposition method for solving moderately stiff parabolic partial differential equations pdes is presented writing the semi discretized equations as a differential algebraic equation dae system where the interface continuity constraints between subdomains are enforced by lagrange multipliers the method uses the runge kutta chebyshev projection scheme to integrate the dae explicitly and to enforce the constraints by a projection with mass lumping techniques and node to node matching grids the method is fully explicit without solving any linear system a stability analysis is presented to show the extended stability property of the method the method is straightforward to implement and to parallelize numerical results demonstrate that it has excellent performance

schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations

an iterative substructuring method for raviart thomas vector fields in three dimensions the iterative substructuring methods also known as schur complement methods form one of two important families of domain decomposition algorithms they are based on a partitioning of a given region on which the partial differential equation is defined into nonoverlapping substructures the preconditioners of these conjugate gradient methods are then given in terms of local problems defined on individual substructures and pairs of substructures and in addition a global problem of low dimension an iterative method of this kind is introduced for the lowest order raviart thomas finite elements in three dimensions and it is shown that the condition number of the relevant operator is independent of the number of substructures and grows only as the square of the logarithm of the number of unknowns associated with an individual substructure the theoretical bounds are confirmed by a series of numerical experiments

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

a remark on the blowing up of solutions to the cauchy problem for nonlinear schrodinger equations on considere des solutions de iu t u u p 1 u u o u 0 ou x appartient a un domaine lisse cr n et on demontre que sous de bonnes conditions sur p n et u 0 h 2 h 0 1 u t l 2 eclate en un temps fini

nested bddc for a saddle point problem we propose a nested bddc for a class of saddle point problems the method solves for both flux and pressure variables the fluxes are resolved in three steps the coarse solve is followed by subdomain solves and last we look for a divergence free flux correction and pressure variables using conjugate gradients with a multilevel bddc preconditioner because the coarse solve in the first step has the same structure as the original problem we can use this procedure recursively and solve a hierarchy of coarse problems only approximately utilizing the coarse problems known from the bddc the resulting algorithm thus first performs several upscaling steps and then solves a hierarchy of problems that have the same structure but increase in size while sweeping down the levels using the same components in the first and in the third step on each level and also reusing the components from the higher levels because the coarsening can be quite aggressive the number of levels can be kept small and the additional computational cost is significantly reduced due to the reuse of the components we also provide the condition number bound and numerical experiments confirming the theory

closed queueing networks under congestion nonbottleneck independence and bottleneck convergence we analyze the behavior of closed multiclass product form queueing networks when the number of customers grows to infinity and remains proportionate on each route or class first we focus on the stationary behavior and prove the conjecture that the stationary distribution at nonbottleneck queues converges weakly to the stationary distribution of an ergodic open product form queueing network which is geometric this open network is obtained by replacing bottleneck queues with per route poissonian sources whose rates are uniquely determined by the solution of a strictly concave optimization problem we strengthen such results by also proving convergence of the first moment of the queue lengths of nonbottleneck stations then we focus on the transient behavior of the network and use fluid limits to prove that the amount of fluid or customers on each route eventually concentrates on the bottleneck queues only and that the long term proportions of fluid in each route and in each queue solve the dual of the concave optimization problem that determines the throughputs of the previous open network

tracing complex singularities with spectral methods abstract a numerical method for investigating the possibility of blow up after a finite time is introduced for a large class of nonlinear evolution problems with initial data analytic in the space variable s the solutions have for any t 0 complex space singularities at the edge of an analyticity strip of width t loss of regularity corresponds to the vanishing of t numerical integration by high resolution spectral methods reveals the large wavenumber behavior of the fourier transform of the solutions from which t is readily obtained its time evolution can be traced down to about one mesh length by extrapolation of t such numerical experiments provide evidence suggesting finite time blow up or all time regularity the method is tested on the inviscid and viscous burgers equations and is applied to the one dimensional nonlinear schrodinger equation with quartic potential and to the two dimensional incompressible euler equation all with periodic boundary conditions in the latter case evidence is found suggesting that existing all time regularity results can be substantially sharpened

parallel computation of flow in heterogeneous media modelled by mixed finite elements abstract in this paper we describe a fast parallel method for solving highly ill conditioned saddle point systems arising from mixed finite element simulations of stochastic partial differential equations pdes modelling flow in heterogeneous media each realisation of these stochastic pdes requires the solution of the linear first order velocity pressure system comprising darcy s law coupled with an incompressibility constraint the chief difficulty is that the permeability may be highly variable especially when the statistical model has a large variance and a small correlation length for reasonable accuracy the discretisation has to be extremely fine we solve these problems by first reducing the saddle point formulation to a symmetric positive definite spd problem using a suitable basis for the space of divergence free velocities the reduced problem is solved using parallel conjugate gradients preconditioned with an algebraically determined additive schwarz domain decomposition preconditioner the result is a solver which exhibits a good degree of robustness with respect to the mesh size as well as to the variance and to physically relevant values of the correlation length of the underlying permeability field numerical experiments exhibit almost optimal levels of parallel efficiency the domain decomposition solver doug http www maths bath ac uk parsoft used here not only is applicable to this problem but can be used to solve general unstructured finite element systems on a wide range of parallel architectures

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

adaptive multilevel techniques for mixed finite element discretizations of elliptic boundary value problems we consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations by a well known postprocessing technique the discrete problem is equivalent to a modified nonconforming discretization which is solved by preconditioned cg iterations using a multilevel preconditioner in the spirit of bramble pasciak and xu designed for standard nonconforming approximations local refinement of the triangulations is based on an a posteriori error estimator which can be easily derived from superconvergence results the performance of the preconditioner and the error estimator is illustrated by several numerical examples

domain decomposition preconditioners for p and hp finite element approximation of stokes equations abstract domain decomposition preconditioning techniques are developed in the context of hp finite element approximation of the stokes problem two basic types of preconditioner are considered a block diagonal scheme based on decoupling the velocity and pressure components and a scheme based on an indefinite system similar to the original stokes system for each type of scheme theoretical estimates are obtained for the location of the eigenvalues of the preconditioned operators in terms of the polynomial degree the mesh sizes on the coarse and fine grids and the inf sup constant for the method theoretical estimates show that the growth of the bounds is modest as the mesh is refined and the polynomial order is increased the preconditioners are shown to be applicable to various iterative schemes for the stokes systems the theoretical bounds are compared with actual quantities obtained in practical computations for several representative problems

schwarz preconditioners for the spectral element discretization of the steady stokes and navier stokes equations the q n q n 2 spectral element discretization of the stokes equation gives rise to an ill conditioned indefinite symmetric linear system for the velocity and pressure degrees of freedom we propose a domain decomposition method which involves the solution of a low order global and several local problems related to the vertices edges and interiors of the subdomains the original system is reduced to a symmetric equation for the velocity which can be solved with the conjugate gradient method we prove that the condition number of the iteration operator is bounded from above by c 1 log n 3 beta n where c is a positive constant independent of the degree n and the number of subdomains and beta n is the inf sup condition of the pair q n q n 2 we also consider the stationary navier stokes equations in each newton step a non symmetric indefinite problem is solved using a schwarz preconditioner by using an especially designed low order global space and the solution of local problems analogous to those decribed above for the stokes equation we are able to present a complete theory for the method we prove that the number of iterations of the gmres method at each newton step is bounded from above by c 1 log n 3 beta n the constant c does not depend on the number of subdomains or n and it does not deteriorate as the newton iteration proceeds

convergence estimates for product iterative methods with applications to domain decomposition in this paper we consider iterative methods for the solution of symmetric positive definite problems on a space which are defined in terms of products of operators defined with respect to a number of subspaces the simplest algorithm of this sort has an error reducing operator which is the product of orthogonal projections onto the complement of the subspaces new normreduction estimates for these iterative techniques will be presented in an abstract setting applications are given for overlapping schwarz algorithms with many subregions for finite element approximation of second order elliptic problems

a numerical scheme based on mean value solutions for the helmholtz equation on triangular grids a numerical treatment for the dirichlet boundary value problem on regular triangular grids for homogeneous helmholtz equations is presented which also applies to the convection diffusion problems the main characteristic of the method is that an accuracy estimate is provided in analytical form with a better evaluation than that obtained with the usual finite difference method besides this classical method can be seen as a truncated series approximation to the proposed method the method is developed from the analytical solutions for the dirichlet problem on a ball together with an error evaluation of an integral on the corresponding circle yielding o h 4 accuracy some numerical examples are discussed and the results are compared with other methods with a consistent advantage to the solution obtained here

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

asymptotic expansion for large closed queuing networks in this paper a new asymptotic method is developed for analyzing closed bcmp queuing networks with a single class chain consisting of a large number of customers a single infinite server queue and a large number of single server queues with fixed state independent service rates asymptotic approximations are computed for the normalization constant partition function starting directly from a recursion relation of buzen the approach of the authors employs the ray method of geometrical optics and the method of matched asymptotic expansions the method is applicable when the servers have nearly equal relative utilizations or can be divided into classes with nearly equal relative utilizations numerical comparisons are given that illustrate the accuracy of the asymptotic approximations

schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems part ii convergence theory in this paper we discuss bounds for the convergence rates of several domain decomposition algorithms to solve symmetric indefinite linear systems arising from mixed finite element discretizations of elliptic problems the algorithms include schwarz methods and iterative refinement methods on locally refined grids the implementation of schwarz and iterative refinement algorithms have been discussed in part i a discussion on the stability of mixed discretizations on locally refined grids is included and quantiative estimates for the convergence rates of some iterative refinement algorithms are also derived

calculating normalization constants of closed queueing networks by numerically inverting their generating functions a new algorithm is developed for calculating normalization constants partition functions and moments of product form steady state distributions of closed queuing networks and related models the essential idea is to numerically invert the generating function of the normalization constant and related generating functions appearing in expressions for the moments it is known that the generating function of the normalization constant often has a remarkably simple form but numerical inversion evidently has not been considered before for p dimensional transforms as occur with queuing networks having p closed chains the algorithm recursively performs p one dimensional inversions the required computation grows exponentially in the dimension but the dimension can often be reduced by exploiting conditional decomposition based on special structure for large populations the inversion algorithm is made more efficient by computing large sums using euler summation the inversion algorithm also has a very low storage requirement a key ingredient in the inversion algorithm is scaling an effective static scaling is developed for multichain closed queuing networks with only single server and optionally infinite server queues an important feature of the inversion algorithm is a self contained accuracy check which allows the results to be verified in the absence of alternative algorithms

the blow up problem for a quasilinear schrodinger equation consider the blow up results for w2 2 rn solutions of a quasilinear schrodinger equation iut u u p 2u u 2 u 0 u t 0 u0 x x rn when x u0 l2 rn we show that the w2 2 rn solutions must blow up for any 0 r and some restriction on p we also show that the radial symmetric solutions in w2 2 rn must blow up at finite time without assuming x u0 l2 rn

growth and oscillations of solutions of nonlinear schrodinger equation we study the nonlinear schrodinger equation in ann cube n 1 2 3 under dirichlet boundary conditions treating it as a dynamical system in a function space formed by sufficiently smooth functions ofx we show that this space contains a distinguished small subset mathfrak a which is a recursion subset for the dynamical system and describe the dynamics of the equation in terms of the trajectory s recurrence to mathfrak a we use this description to estimate from below the space and time space oscillations of solutions in terms of a quantity similar to the reynolds number of classical hydrodynamics