some results on sparse block factorization iterative methods abstract we consider the general sparse block factorization iterative methods as beauwens and ben bouzid did we develop some existence and convergence theorems for general nonsingular h matrices

on sparse block factorization iterative methods a synthetic formalism is proposed for the description of sparse block factorization iterative methods it is used to develop existence convergence and comparison theorems and to compare block factorization schemes against point factorization schemes

preconditioning indefinite discretization matrices the finite element discretization of many elliptic boundary value problems leads to linear systems with positive definite and symmetric coefficient matrices many efficient preconditioners are known for these systems we show that these preconditioning matrices can also be used for the linear systems arising from boundary value problems which are potentially indefinite due to lower order terms in the partial differential equation our main tool is a careful algebraic analysis of the condition numbers and the spectra of perturbed matrices which are preconditioned by the same matrices as in the unperturbed case

a general incomplete block matrix factorization method abstract a general method for incomplete factorization of m matrices in block matrix form is presented the method generalizes previous methods for pointwise incomplete factorization methods of varga 1960 of wo nicki 1973 and of meijerink and van der vorst 1977 as well as a method for tridiagonal block matrices also we present a new and more general result on the existence of so called modified incomplete factorizations which satisfy a generalized rowsum criterion

continuum modeling of supply chain networks using discontinuous galerkin methods using a connectivity matrix we establish a continuum modeling approach with partial differential equations of conservation laws for simulating materials flow in supply chain networks a number of existing and new constitutive relationships for modeling velocity are summarized or proposed to effectively treat strong advection components within the modeling system we apply discontinuous galerkin dg methods for solving production flow in a supply chain network in addition a number of dg properties are analyzed for treating network flow in particular a nearly optimal error estimate is obtained using a new estimating technique that utilizes two physical meaningful assumptions on the connectivity matrix numerical examples are provided to simulate a single node a serial supply chain and an entire network as well as to investigate the influence of influx variation and node shut down to the profiles of work in progress wip and outflux it is shown that the proposed modeling approach is applicable to a large number of scenarios including re entrant lines and the proposed dg algorithm is robust and accurate for predicting wip and outflux behaviors

polynomial identities of algebras in positive characteristic the verbally prime algebras are well understood in characteristic 0 while over a field of positive characteristic p 2 little is known about them in previous papers we discussed some sharp differences between these two cases for the characteristic and we showed that the so called tensor product theorem is in part no longer valid in the second case in this paper we study the gelfand kirillov dimension of the relatively free algebras of verbally prime and related algebras we compute the gk dimensions of several algebras and thus obtain a new proof of the fact that the algebras m1 1 e and e e are not pi equivalent in characteristic p 2 furthermore we show that the following algebras are not pi equivalent in positive characteristic ma b e e and ma b e ma b e e and mc d e e when a b c d a b

incomplete block factorization preconditioning for indefinite elliptic problems the application of the finite difference method to approximate the solution of an indefinite elliptic problem produces a linear system whose coefficient matrix is block tridiagonal and symmetric indefinite such a linear system can be solved efficiently by a conjugate residual method particularly when combined with a good preconditioner we show that specific incomplete block factorization exists for the indefinite matrix if the mesh size is reasonably small and that this factorization can serve as an efficient preconditioner some efforts are made to estimate the eigenvalues of the preconditioned matrix numerical results are also given

pointwise error estimates of discontinuous galerkin methods with penalty for second order elliptic problems in this paper discontinuous galerkin methods with penalty for solving second order elliptic problems are considered error estimates are studied for these methods in particular optimal localized pointwise error estimates are established on quasi uniform grids in rn n ge 2

a discontinuous hp finite element method for convection diffusion problems this paper presents a new method which exhibits the best features of both finite volume and finite element techniques special attention is given to the issues of conservation flexible accuracy and stability the method is elementwise conservative the order of polynomial approximation can be adjusted element by element and the stability is not based on the introduction of artificial diffusion but on the use of a very particular finite element formulation with discontinuous basis functions the method supports h p and hp approximations and can be applied to any type of meshes including non matching grids a priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy

a priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems we analyze three discontinuous galerkin approximations for solving elliptic problems in two or three dimensions in each one the basic bilinear form is nonsymmetric the first one has a penalty term on edges the second has one constraint per edge and the third is totally unconstrained for each of them we prove hp error estimates in the h1 norm optimal with respect to h the mesh size and nearly optimal with respect to p the degree of polynomial approximation we establish these results for general elements in two and three dimensions for the unconstrained method we establish a new approximation result valid on simplicial elements l2 estimates are also derived for the three methods

an algebraic multilevel multigraph algorithm we describe an algebraic multilevel multigraph algorithm many of the multilevel components are generalizations of algorithms originally applied to general sparse gaussian elimination indeed general sparse gaussian elimination with minimum degree ordering is a limiting case of our algorithm our goal is to develop a procedure which has the robustness and simplicity of use of sparse direct methods yet offers the opportunity to obtain the optimal or near optimal complexity typical of classical multigrid methods

the numerical solution of the helmholtz equation for wave propagation problems in underwater acoustics abstract the helmholtz equation k2n2 u 0 with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems a numerical algorithm has been developed and a computer code implemented that can effectively solve this equation in the intermediate frequency range the equation is discretized by using the finite element method thus allowing for the modeling of complicated geometries including interfaces and complicated boundary conditions a global radiation boundary condition is imposed at the far field boundary that is exact for an arbitrary number of propagating modes the resulting large nonselfadjoint system of linear equations with indefinite symmetric part is solved by using the preconditioned conjugate gradient method applied to the normal equations a new preconditioner based on the multigrid method is developed this preconditioner is vectorizable and is extremely effective over a wide range of frequencies provided the number of grid levels is reduced for large frequencies a heuristic argument is given that indicates the superior convergence properties of this preconditioner the relevant limit to analyze convergence is for k increasing and a fixed prescribed accuracy level the efficiency and robustness of the numerical algorithm are confirmed for large acoustic models including interfaces with strong velocity contrasts

on approximate factorization methods for block matrices suitable for vector and parallel processors abstract some existence results for methods based on the approximate factorization of block matrices are proven these methods are based on recursive computations of diagonal block matrices and the approximation of their inverses to preserve sparsity we also discuss a recently proposed 1 inverse free factorization method and present some numerical tests for it

a posteriori error estimates for a discontinuous galerkin approximation of second order elliptic problems several a posteriori error estimators are introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem in addition to residual type estimators we introduce some estimators that are couched in the ideas and techniques of domain decomposition results of numerical experiments are presented

on polynomial preconditioning and asymptotic convergence factors for indefinite hermitian matrices abstract we are concerned with the minimal residual method combined with polynomial preconditioning for solving large linear systems ax b with indefinite hermitian coefficient matrices a the standard approach for choosing the polynomial preconditioner leads to preconditioned systems which are positive definite here we investigate a different strategy which leaves the preconditioned coefficient matrix indefinite more precisely the polynomial preconditioner is designed to cluster the positive negative eigenvalues of a around 1 around some negative constant in particular it is shown that such indefinite polynomial preconditioners can be obtained as the optimal solutions of a certain two parameter family of chebyshev approximation problems the problem of selecting the parameters so that the resulting indefinite polynomial preconditioner speeds up the convergence of the minimal residual method optimally is also addressed for this task we propose an approach based on the concept of asymptotic convergence factors finally some numerical examples of indefinite polynomial preconditioners are given

incomplete factorization based preconditionings for solving the helmholtz equation preconditioning techniques based on incomplete factorization of matrices are investigated to solve highly indefinite complex symmetric linear systems a novel preconditioning is introduced the real part of the matrix is made positive definite or less indefinite by adding properly defined perturbations to the diagonal entries while the imaginary part is unaltered the resulting preconditioning matrix which is obtained by applying standard methods to the perturbed complex matrix turns out to perform significantly better than classical incomplete factorization schemes for realistic values of the gmres restart parameter spectacular reduction of iteration counts is observed a theoretical spectral analysis is provided in which the spectrum of the preconditioner applied to indefinite matrix is related to the spectrum of the same preconditioner applied to a stieltjes matrix extracted from the indefinite matrix results of numerical experiments are reported which display the efficiency of the new preconditioning copyright 2001 john wiley sons ltd

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

on the eigenvalue distribution of a class of preconditioning methods a class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationax b wherea is a symmetric positive definite matrix the methods are based on an incomplete factorization of the matrixa and include both pointwise and blockwise factorization we study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofc 1a wherec is the preconditioning matrix we also show graphic representations of the eigenvalues and present numerical tests of the methods

symmetric and nonsymmetric discontinuous galerkin methods for reactive transport in porous media for solving reactive transport problems in porous media we analyze three primal discontinuous galerkin dg methods with penalty namely symmetric interior penalty galerkin sipg nonsymmetric interior penalty galerkin nipg and incomplete interior penalty galerkin iipg a cut off operator is introduced in dg to treat general kinetic chemistry error estimates in l2 h1 are established which are optimal in h and nearly optimal in p we develop a parabolic lift technique for sipg which leads to h optimal and nearly p optimal error estimates in the l2 l2 and negative norms numerical results validate these estimates we also discuss implementation issues including penalty parameters and the choice of physical versus reference polynomial spaces

analysis of a nonsymmetric discontinuous galerkin method for elliptic problems stability and energy error estimates in this paper we analyze a nonsymmetric discontinuous galerkin method for elliptic problems proposed by oden babuska and baumann our main results are a complete inf sup stability analysis and as a consequence error estimates in a mesh dependent energy norm allowing variable meshsize and order of polynomials the analysis is carried out in two spatial dimensions on an unstructured triangulation