the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

a fast iterative method for solving stationary heat conduction problems on regions partitioned into rectangles the paper deals with some finite element approximation of stationary heat conduction problems on regions which can be partitioned into rectangular subregions by a special superelement technique employing fast elimination of the inner nodal parameters the original finite element problem is reduced to a smaller problem which is only connected with the nodes on the boundary of the superelements to solve the reduced system of finite element equations an efficient iterative algorithm is proposed this algorithm is based either on the conjugate gradient method or the tshebysheff method using a special matrix by vector multiplication procedure the explicit form of the matrix is not used the presented numerical method is asymptotically optimal with respect to the memory requirement as well as to the operation count

explicit preconditioned domain decomposition schemes for solving nonlinear boundary value problems abstract a new class of inner outer iterative procedures in conjunction with picard newton methods based on explicit preconditioning iterative methods for solving nonlinear systems is presented explicit preconditioned iterative schemes based on the explicit computation of a class of domain decomposition generalized approximate inverse matrix techniques are presented for the efficient solution of nonlinear boundary value problems on multiprocessor systems applications of the new composite scheme on characteristic nonlinear boundary value problems are discussed and numerical results are given

nonlinear simulation of electromagnetic fields with domain decomposition methods on mind parallel computers abstract an efficient domain decomposition dd algorithm for solving nonlinear boundary value problems on multiple instruction multiple data parallel computers is presented the linearization is done by newton s method and the linear system is solved by cg with dd preconditioning further the dd approach allows finite element and galerkin boundary element methods to be combined in a unifled variational problem test computations on a transputer based system were performed for both model problems and a real life electric motor

a finite element capacitance method for elliptic problems on regions partitioned into subregions a method is given for the solution of linear equations arising in the finite element method applied to a general elliptic problem this method reduces the original problem to several subproblems of the same form considered on subregions and an auxiliary problem very efficient iterative methods with the preconditioning operator and using fft are developed for the auxiliary problem

weighted max norms splittings and overlapping additive schwarz iterations summary weighted max norm bounds are obtained for algebraic additive schwarz iterations with overlapping blocks for the solution of ax b when the coefficient matrix a is an m matrix the case of inexact local solvers is also covered these bounds are analogous to those that exist using a norms when the matrix a is symmetric positive definite a new theorem concerningp regular splittings is presented which provides a useful tool for the a norm bounds furthermore a theory of splittings is developed to represent algebraic additive schwarz iterations this representation makes a connection with multisplitting methods with this representation and using a comparison theorem it is shown that a coarse grid correction improves the convergence of additive schwarz iterations when measured in weighted max norm

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

on energy inequality smoothness and large time behavior in l 2 for weak solutions of the navier stokes equations in exterior domains soit un domaine exterieur de r n n 3 4 a frontiere lisse sur le cylindre d espace temps 0 on considere le probleme de navier stokes non stationnaire u t u u u p f dans 0 u 0 dans 0 u 0 sur 0 u x 0 a x avec des donnees prescrites f et a

an age dependent population model and its operator abstract an age dependent single population model in the form of a partial differential equation is discussed properties of the associated age dependent operator and the corresponding semigroup are obtained we then use these properties to discuss the stability of the null solution of this model the method proposed here may be useful for some other models

discrete interpolation norms with applications we describe norm representations for interpolation spaces generated by finite dimensional subspaces of hilbert spaces these norms are products of integer and noninteger powers of the grammian matrices associated with the generating pair of spaces for the interpolation space we include a brief description of some of the algorithms which allow the efficient computation of matrix powers we consider in some detail the case of fractional sobolev spaces both for positive and negative indices together with applications arising in preconditioning techniques numerical experiments are included

domain decomposition using spectral expansions of steklov poincare operators a new domain decomposition method is described for solving elliptic boundary value problems on a region partitioned into nonoverlapping subdomains which are discretized independently using a galerkin finite element method the interdomain continuity requirements on the solution are imposed using an independent low dimensional set of interfacial basis functions which are separately derived for each subdomain by solving an eigenvalue problem for the steklov poincare operator on the complementary region the efficient numerical implementation of this method is described here for the two subdomain case and its convergence properties are studied for some test examples in specific examples involving constant coefficient problems on rectangular regions partitioned into rectangular subdomains the pointwise error of the discrete solution in the subdomain interior converges exponentially in the number of interfacial modes so that the present method is effectively a direct solution method in other examples

asymptotic behavior for the navier stokes equations in 2d exterior domains we show that the lp spatial temporal decay rates of solutions of incompressible flow in an 2d exterior domain when a domain has a boundary pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary to overcome the difficulty we adopt the ideas in he xin c he z xin weighted estimates for nonstationary navier stokes equations in exterior domain methods appl anal 7 3 2000 443 458 and our previous results h o bae b j jin asymptotic behavior of stokes solutions in 2d exterior domains j math fluid mech in press h o bae b j jin temporal and spatial decay rates of navier stokes solutions in exterior domains submitted for publication for the spatial decay rate estimate we first extend temporal decay rate result of the navier stokes solutions for general lp space when the initial velocity is in l 2 lr 1 r q 1 r q

bifurcation of time periodic solutions of the mckendrick equations with applications to population dynamics abstract a local two parameter bifurcation theorem concerning the bifurcation from steady states of time periodic solutions of a nonlinear system of partial integro differential equations is proved a hopf bifurcation theorem is derived as a corollary by means of independent and dependent variable changes this theorem is applicable to the general mckendrick equations governing the growth of an age structured population with the added feature here of a possible gestation period the theorem is based on a fredholm theory developed in the paper for the associated linear equations an application is given to an age structured population whose fecundity is density and age dependent and it is shown that for a sufficiently narrow age specific reproductive and resource consumption window steady state instabilities accompanied by sustained time periodic oscillations occur when the birth modulus surpasses a critical value

weighted estimates for the incompressible fluid in exterior domains for a strong solution u x t of the navier stokes equations in exterior domain in rn where n 2 3 we study the time decay of x u t lp for n when a domain has a boundary pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary to overcome the difficulty we adopt the ideas in h o bae b j jin temporal and spatial decay rates of navier stokes solutions in exterior domains bull korean math soc 44 3 2007 547 567 h o bae b j jin asymptotic behavior for the navier stokes solutions in 2d exterior domains j funct anal 240 2006 508 529 and we will extend bae and jin s results by modifying their methods

block preconditioning and domain decomposition methods ii abstract domain decomposition methods for the solution of partial differential equations are attractive on parallel processors because each processor can work independently on a large subtask the corresponding stiffness matrix takes a sparse block structure for which preconditioned iterative methods can be used when solving linear systems with the stiffness matrix for domains decomposed in strips we get a blocktridiagonal structure for which a new block lu preconditioner was presented in an earlier report 5 by the authors an alternative method and also the one more commonly used for substructuring methods is based on approximation of the schur complement matrix this approximation is frequently done by various difference methods see 6 11 and 16 in the present paper we examine methods based on algebraic approximation methods this is similar to methods used by chan 9

estimation in gaussian graphical models using tractable subgraphs a walk sum analysis graphical models provide a powerful formalism for statistical signal processing due to their sophisticated modeling capabilities they have found applications in a variety of fields such as computer vision image processing and distributed sensor networks in this paper we present a general class of algorithms for estimation in gaussian graphical models with arbitrary structure these algorithms involve a sequence of inference problems on tractable subgraphs over subsets of variables this framework includes parallel iterations such as embedded trees serial iterations such as block gauss seidel and hybrid versions of these iterations we also discuss a method that uses local memory at each node to overcome temporary communication failures that may arise in distributed sensor network applications we analyze these algorithms based on the recently developed walk sum interpretation of gaussian inference we describe the walks ldquocomputedrdquo by the algorithms using walk sum diagrams and show that for iterations based on a very large and flexible set of sequences of subgraphs convergence is guaranteed in walk summable models consequently we are free to choose spanning trees and subsets of variables adaptively at each iteration this leads to efficient methods for optimizing the next iteration step to achieve maximum reduction in error simulation results demonstrate that these nonstationary algorithms provide a significant speedup in convergence over traditional one tree and two tree iterations

asymptotic behavior for the stokes flow and navier stokes equations in half spaces abstract using the solution formula in ukai 1987 27 for the stokes equations we find asymptotic profiles of solutions in l 1 r n n 2 for the stokes flow and non stationary navier stokes equations since the projection operator p l 1 r n l 1 r n is unbounded we use a decomposition for p u u to overcome the difficulty and prove that the decay rate for the first derivatives of the strong solution u of the navier stokes system in l 1 r n is controlled by t 1 2 1 t n 2 2 for any t 0

coupling boundary integral and finite element methods for the oseen coupled problem abstract in this paper we represent an oseen coupled problem and related numerical method for solving the nonstationary navier stokes problem in an unbounded domain the oseen coupled problem consists of a coupling between the navier stokes problem in an inner region and oseen problem in an outer region the related numerical method consists of coupling the boundary integral and the finite element method to solve the coupled problem the variational formulation of the coupled problem and its well posedness are obtained the optimal error estimates between the numerical solution of the oseen coupled problem and the exact solution of the navier stokes problem are provided

spatial stability of 3d exterior stationary navier stokes flows abstract in this paper we study the stability of stationary solutions w for the navier stokes flows in an exterior domain with zero velocity at infinity with suitable assumptions of w by the works of chen 1993 kozono ogawa 1994 and borchers miyakawa 1995 if u 0 w l r l 3 then one can obtain u t w p o t 3 2 1 r 1 p for 1 r p u t w p o t 3 2 1 r 1 p 1 2 for 1 r p 3 where u x t is a solution of the navier stokes equations with the initial condition u 0 in this paper we will prove that for any 0 3 if x u 0 w belongs to l r then one has x u t w l p o t 3 2 1 r 1 p 2 for p 3 r 3 r

convergence rate of a schwarz multilevel method for the constrained minimization of nonquadratic functionals in l badea convergence rate of a multiplicative schwarz method for strongly nonlinear inequalities v barbu i lasiecka d tiba and c varsan eds kluwer academic publishers boston 2003 the convergence of a subspace correction method applied to the constrained minimization of a functional in a general reflexive banach space has been proved provided that the convex set verifies a certain assumption this assumption is weaker than that in which the convex set is decomposed according to the space decomposition as a sum of subsets in the sobolev spaces the proposed method becomes a multiplicative schwarz method for the solution of the variational inequalities coming from the minimization of nonquadratic functionals we prove in this paper that this assumption holds for the one two and multilevel multiplicative schwarz methods in the finite element space and we explicitly write the constants in the error estimations depending on the overlapping and mesh parameters our error estimates are similar with those obtained for the minimization of quadratic functionals in l badea x c tai and j wang siam j numer anal 2003 pp 1052 1073 or with those obtained for the one obstacle problem in x c tai numer math 93 2003 pp 755 786

analysis of non overlapping domain decomposition algorithms with inexact solves in this paper we construct and analyze new non overlapping domain decomposition preconditioners for the solution of second order elliptic and parabolic boundary value problems the preconditioners are developed using uniform preconditioners on the subdomains instead of exact solves they exhibit the same asymptotic condition number growth as the corresponding preconditioners with exact subdomain solves and are much more efficient computationally moreover this asymptotic condition number growth is bounded independently of jumps in the operator coefficients across subdomain boundaries we also show that our preconditioners fit into the additive schwarz framework with appropriately chosen subspace decompositions condition numbers associated with the new algorithms are computed numerically in several cases and compared with those of the corresponding algorithms in which exact subdomain solves are used

algebraic theory of multiplicative schwarz methods the convergence of multiplicative schwarz type methods for solving linear systems when the coefficient matrix is either a nonsingular m matrix or a symmetric positive definite matrix is studied using classical and new results from the theory of splittings the effect on convergence of algorithmic parameters such as the number of subdomains the amount of overlap the result of inexact local solves and of coarse grid corrections global coarse solves is analyzed in an algebraic setting results on algebraic additive schwarz are also included

additive schwarz methods for elliptic mortar finite element problems two variants of the additive schwarz method for solving linear systems arising from the mortar finite element discretization on nonmatching meshes of second order elliptic problems with discontinuous coefficients are designed and analyzed the methods are defined on subdomains without overlap and they use special coarse spaces resulting in algorithms that are well suited for parallel computation the condition number estimate for the preconditioned system in each method is proportional to the ratio h h where h and h are the mesh sizes and it is independent of discontinuous jumps of the coefficients for one of the methods presented the choice of the mortar nonmortar side is independent of the coefficients

iterative methods for overflow queuing models ii preconditioned conjugate gradient methods are employed to find the steady state probability distribution of markovian queuing networks that have overflow capacity different singular preconditioners that can be handled by separation of variables are discussed the resulting preconditioned systems are nonsingular numerical results show that the number of iterations required for convergence grows very slowly with the queue sizes

l2 decay for the navier stokes flow in halfspaces on etudie le comportement asymptotique dans l 2 des solutions faibles des equations de navier stokes dans des demi espaces

an iterative method for elliptic problems on regions partitioned into substructures some new preconditioners for discretizations of elliptic boundary problems are studied with these preconditioners the domain under consideration is broken into subdomains and preconditioners are defined which only require the solution of matrix problems on the subdomains analytic estimates are given which guarantee that under appropriate hypotheses the preconditioned iterative procedure converges to the solution of the discrete equations with a rate per iteration that is independent of the number of unknowns numerical examples are presented which illustrate the theoretically predicted iterative convergence rates

between homogeneous and inhomogeneous navier stokes systems the issue of stability abstract we construct large velocity vector solutions to the three dimensional inhomogeneous navier stokes system the result is proved via the stability of two dimensional solutions with constant density under the assumption that initial density is point wisely close to a constant key elements of our approach are estimates in the maximal regularity regime and the lagrangian coordinates considerations are done in the whole r 3

analysis of preconditioners for domain decomposition domain decomposition is a class of techniques that are designed to solve elliptic problems on irregular domains and on multiprocessor systems typically a domain is decomposed into many smaller regular subdomains and the capacitance system governing the interface unknowns is solved by some version of the preconditioned conjugate gradient method in this paper we show that for a simple model problem poisson s equation on a rectangle decomposed into two smaller rectangles the capacitance system can be inverted exactly by fast fourier transform an exact eigen decomposition of the capacitance matrix also makes it possible to relate and compare the various preconditioners that have been proposed in the literature for example we show that in the limit as the aspect ratio of the two rectangles tend to infinity the preconditioner proposed by golub and mayers becomes exact but the one proposed by dryja does not both preconditioners however are poor when the aspect ratio is small

balancing domain decomposition for problems with large jumps in coefficients the balancing domain decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent the abstract theory introduced recently by the first named author is used to develop condition number bounds for conforming linear elements in two and three dimensions the bounds are independent of arbitrary coefficient jumps between subdomains and of the number of subdomains and grow only as the squared logarithm of the mesh size h computational experiments for two and three dimensional problems confirm the theory

uniform convergence of the schwarz alternating method for solving singularly perturbed advection diffusion equations our purpose in this paper is to study the convergence rate of some variants of the schwarz alternating method for solving two types of singularly perturbed advection diffusion equations one in which the advection term first order term dominates and the other in which the lowest order term zeroth order term dominates for such problems boundary or other layer regions may be present for singular perturbation problems in which the advection dominates we choose two overlapping subregions one corresponding to an outer or elliptic region and the other corresponding to an inner or hyperbolic region our main result for this problem shows that if the subdomains can be chosen to follow the flow i e if the boundary interface of one of the subdomains corresponds to an outflow boundary for the streamlines of the flow then the schwarz iterates converge in the maximum norm with an error reduction factor per iteration that exponentially decays with increasing overlap or decreasing diffusion for the singular perturbation problem in which the lowest order term dominates we consider many overlapping subregions with no special requirement on outflow or inflow boundaries our main result here which is valid for two variants of the classical schwarz algorithm shows that the error reduction factor per iteration decays exponentially in the maximum norm with increasing overlap and increasing dominance of the zeroth order term for both problems our proofs rely on the maximum principle and the construction of barrier functions

subdomain methods in structural mechanics the principle of subdomain methods has been introduced many years ago1 and such methods have been used with success in recent years in this paper we discuss some methods used to solve the condensed system resulting from the schur complement method we will present the hybrid method and a mechanical application the contact problem between two deformable solids in the case of such a problem we propose two methods one for the contact with friction and one for the contact without friction finally the results of validation tests for both proposed approaches are given

the efficient parallel solution of pdes abstract the report presents some results in solving finite element equations via a parallel version of the preconditioned cg method parpcg we use a nonoverlapping domain decomposition and construct preconditioners based on additive and multiplicative schwarz methods asm msm as components in the preconditioner multigrid methods hierarchical bases new extension techniques and modified bps and bpx preconditioners for handling the unknowns at the coupling nodes on the boundaries between subdomains are used the scale up efficiency e g an increasing number of processors causes an increasing problem size of the algorithm by doubling the number of processors is larger than 95 even the practical not relevant speed up efficiency e g an increasing number of processors and a constant problem size reaches 80 by doubling the number of processors

domain decomposition algorithms for fourth order nonlinear elliptic eigenvalue problems we study domain decomposition methods for fourth order plate problems the well known von karman equations are used as our model problem by exploiting the symmetry of the domain the solution of the original problem can be obtained by solving those associated reduced problems which are defined on subdomains with appropriate boundary conditions we show how nonoverlapping and overlapping domain decomposition methods can be used to solve the reduced problems for the linearized von karman equation we present preconditioners using both fourier analysis and probing techniques for the interface systems which are similar to those derived by chan et al finally we compare the efficiency of various domain decomposition preconditioners for solving the von karman equations

multilevel preconditioners for elliptic problems by substructuring we consider the construction of preconditioners for reduced schur problems for solving finite element elliptic equations which are obtained by block gaussian elimination of the unknowns corresponding to a number of subregions the preconditioners are based on approximations of the considered elliptic problem on a sequence of meshes and are derived in a purely algebraic manner it is proved that the proposed algebraic multilevel preconditioner is of nearly optimal order that is its relative condition number with respect to the corresponding reduced schur complement is proportional to l 2 where l is the number of discretization levels used this condition number is bounded uniformly with respect to the number of subregions the way the region is partitioned for example into strips or into boxes and the discontinuity of the coefficients of the elliptic operator if they are constants on the elements from the initial triangulation the data transport except on the coarsest level is of local type and is proportional to the number of nodes on the interface boundary

on the conjugate gradient solution of the schur complement system obtained from domain decomposition for a linear system arising from a finite element discretization on a quasi uniform grid of a second order elliptic partial differential equation it is shown that the condition number of the schur

geometry related convergence results for domain decomposition algorithms for general second order elliptic partial differential equations the schwarz alternating procedure is proved to converge at a rate independent of the aspect ratio for l shaped t shaped and c shaped domains these results cover both continuous and discrete versions of the schwarz algorithm moreover they apply to the nonoverlapping schur complement algorithms with the preconditioner proposed in chan siam j numer anal 24 1987 pp 382 390 in particular it is shown that the condition number of the preconditioned interface operator is bounded by 2 for all l shaped and t shaped domains this improves similar geometry independent convergence results for the schur complement algorithms obtained previously by chan and resasco advances in computer methods for partial differential equations vi r vichnevetsky and r s stepleman eds pp 317 322

a defect equation approach for the coupling of subdomains in domain decomposition methods a defect equation for the coupling of nonlinear subproblems defined in nonoverlapped subdomains arise in domain decomposition methods is presented numerical solutions of defect equations by means of quasi newton methods are considered

convergence of the sinc method for a fourth order ordinary differential equation with an application we analytically show the exponential convergence of the sinc method applied to fourth order ordinary differential equations odes linear numerical examples are included to confirm our analytical results the solution obtained with the sinc method is compared with the one obtained with a second and a sixth order finite difference scheme and with the ode boundary value solver colnew in the case of the linear equations we then compute the steady state solution of the one dimensional cahn hilliard equation to illustrate our results

a parallel fictitious domain multigrid preconditioner for the solution of poisson s equation in complex geometries abstract a parallel multilevel preconditioner based on domain decomposition and fictitious domain methods has been presented for the solution of the poisson equation in complicated geometries rectangular blocks with matching grids on interfaces on a structured rectangular mesh have been used for the decomposition of the problem domain sloping sides or curved boundary surfaces are approximated using stepwise surfaces formed by the grid cells a seven point stencil based on the central difference scheme has been used for the discretization of the laplacian for both interior and boundary grid points and this results in a symmetric linear algebraic system for any type of boundary condition the preconditioned conjugate gradient method has been used for the solution of this symmetric system the multilevel preconditioner for the cg is based on a v cycle multigrid applied to the poisson equation on a fictitious domain formed by the union of the rectangular blocks used for the domain decomposition numerical results are presented for two typical poisson problems in complicated geometries one related to heat conduction and the other one arising from the les dns of incompressible turbulent flow over a packed array of spheres these results clearly show the efficiency and robustness of the proposed approach

generalized capacitance matrix theorems and algorithm for solving linear systems the capacitance matrix method has been widely used as an efficient numerical tool for solving the boundary value problems on irregular regions initially this method was based on the sherman morrison woodbury formula an expression for the inverse of the matrix a uvt with bf a in re n times n and bf u v in re n times p extensions of this method reported in literature have made restrictive assumptions on the matrices a and a uvt in this paper we present several theorems which are generalizations of the capacitance matrix theorem in cite buzbee and are suited for very general matrices a and a uvt a generalized capacitance matrix algorithm is developed from these theorems and holds the promise of being applicable to more general problems in addition it gives ample freedom to choose the matrix a for developing very efficient numerical algorithms we demonstrate the usefulness of our generalized capacitance matrix algorithm by applying it to efficiently solve large sparse linear systems in several computer vision problems namely the surface reconstruction and interpolation and the shape from orientation problem

an operator splitting scheme with a distributed lagrange multiplier based fictitious domain method for wave propagation problems we propose a novel fictitious domain method based on a distributed lagrange multiplier technique for the solution of the time dependent problem of scattering by an obstacle we study discretizations that include a fully conforming approach as well as mixed finite element formulations utilizing the lowest order nedelec edge elements in 2d on rectangular grids we also present a symmetrized operator splitting scheme for the scattering problem which decouples the operator that propagates the wave from the operator that enforces the dirichlet condition on the boundary of an obstacle a new perfectly matched layer pml model is developed to model the unbounded problem of interest this model is based on a formulation of the wave equation as a system of first order equations and uses a change of variables approach that has been developed to construct pml s for maxwell s equations we present an analysis of our fictitious domain approach for a one dimensional wave problem based on calculations of reflection coefficients we demonstrate the advantages of our fictitious domain approach over the staircase approximation of the finite difference method we also demonstrate some important properties of the distributed multiplier approach that are not shared by a boundary multiplier fictitious domain approach for the same problem numerical results for two dimensional wave problems that validate the effectiveness of the different methods are presented

on stability of steady circular flows in a two dimensional exterior disk we study the stability of some exact stationary solutions to the two dimensional navier stokes equations in an exterior domain to the unit disk these stationary solutions are known as a simple model of the flow around a rotating obstacle while their stability has been open due to the difficulty arising from their spatial decay in a scale critical order in this paper we affirmatively settle this problem for small solutions that is we will show that if these exact solutions are small enough then they are asymptotically stable with respect to small l2 perturbations

l q l r estimates for solutions of the nonstationary stokes equations in an exterior domain and the navier stokes initial value problems inl q spaces

energy decay for a weak solution of the navier stokes equation with slowly varying external forces we consider the navier stokes system with slowly decaying external forces formula we show that the energy norm of a weak solution has non uniform decay u t 2 0 t under suitable conditions on the data f and a which make the energy of solution bounded in time also we show the exact rate of the decay uniform decay of the energy u t 2 c 1 t e for external forces with a given explicit decay rate

a capacitance matrix method for dirichlet problem on polygon region an efficient algorithm for the solution of linear equations arising in a finite element method for the dirichlet problem is given the cost of the algorithm is proportional ton 2log2 n n 1 h where the cost of solving the capacitance matrix equations isnlog2 n on regular grids andn 3 2log2 n on irregular ones

domain decomposition and parallel processing of a finite element model of the shallow water equations abstract we present nonoverlapping domain decomposition techniques applied to a two stage numerov galerkin finite element model of the shallow water equations over a limited area domain the schur complement matrix formulation is employed and a modified interface matrix approach is proposed to handle the coupling between subdomains the resulting nonsymmetric schur complement matrices modified interface matrices as well as the subdomain coefficient matrices are solved using preconditioned conjugate gradient squared pcgs non symmetric iterative solvers various stages of the finite element solution are parallelized and the code is implemented ona four processor cray y mp supercomputer applying multitasking techniques in a dedicated environment

some properties of generalized k centrosymmetric h matrices every nxn generalized k centrosymmetric matrix a can be reduced into a 2x2 block diagonal matrix see z liu h cao h chen a note on computing matrix vector products with generalized centrosymmetric centrohermitian matrices appl math comput 169 2 2005 1332 1345 this block diagonal matrix is called the reduced form of the matrix a in this paper we further investigate some properties of the reduced form of these matrices and discuss the square roots of these matrices finally exploiting these properties the development of structure preserving algorithms for certain computations for generalized k centrosymmetric h matrices is discussed

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

periodic mckendrick equations for age structered population growth abstract with the averaged net reproductive rate used as a bifurcation parameter the existence of a local parameterized branch of time periodic solutions of the mckendrick equations is proved under the assumption that the death and fertility rates suffers small amplitude time periodicities the required linear theory is developed and the results are illustrated by means of a simple example in which fertility varies cosinusoidally in time

primal and dual interface concentrated iterative substructuring methods this paper is devoted to the fast solution of interface concentrated finite element equations the interface concentrated finite element schemes are constructed on the basis of a nonoverlapping domain decomposition where a conforming boundary concentrated finite element approximation is used in every subdomain similar to boundary element domain decomposition methods the total number of unknowns per subdomain behaves like o h h d 1 where h h and d denote the usual scaling parameter of the subdomains the average discretization parameter of the subdomain boundaries and the spatial dimension respectively we propose and analyze primal and dual substructuring iterative methods which asymptotically exhibit the same or at least almost the same complexity as the number of unknowns in particular the so called all floating finite element tearing and interconnecting solvers are highly parallel and very robust with respect to large coefficient jumps

a multiplicative schwarz iteration scheme for solving the linear complementarity problem with an h matrix in this paper we consider an algebraic multiplicative schwarz iteration scheme for solving the linear complementarity problem that involves an h matrix we show that the sequence generated by the multiplicative schwarz iteration scheme converges to the unique solution of the problem without any restriction on the initial point for different overlapping sizes the convergence rate of the proposed method is analyzed in an algebraic setting moreover we establish monotone convergence of the proposed method under appropriate conditions numerical results show that efficiency can be achieved by the multiplicative schwarz iteration scheme

overlapping additive and multiplicative schwarz iterations for h matrices abstract in recent years an algebraic framework was introduced for the analysis of convergence of schwarz methods for the solution of linear systems of the form ax b within this framework additive and multiplicative schwarz were shown to converge when the coefficient matrix a is a nonsingular m matrix or a symmetric positive definite matrix in this paper many of these results are extended to the case of a being an h matrix the case of inexact local solves is also considered in addition the two level scheme is studied i e when a coarse grid correction is used in conjunction with the additive or the multiplicative schwarz iterations

global well posedness of the two dimensional exterior navier stokes equations for non decaying data we prove global well posedness of the two dimensional exterior navier stokes equations for bounded initial data with a finite dirichlet integral subject to the non slip boundary condition as an application we construct global solutions for asymptotically constant initial data and arbitrary large reynolds numbers

domain decomposition technique for the continuity equations of semiconductor device models abstract in this paper we use the pcg method to solve the linear systems obtained from the continuity equations of semiconductor models a new kind of preconditioner which is based on the domain decomposition technique is used this kind of preconditioner is suitable for parallel computing and efficient for the linear system with rapidly varied solutions since the class of the linear system we considered is a special kind of an unsymmetric matrix we modify the pcg to this case as well with our modified pcg method we compare the preconditioner which is constructed by using incomplete cholesky decomposition without fill in preconditioner to our preconditioners which are based on the domain decomposition technique we find that the latter is much more efficient than the former for our semiconductor model problem

convergence rate estimate for a domain decomposition method we provide a convergence rate analysis for a variant of the domain decomposition method introduced by gropp and keyes for solving the algebraic equations that arise from finite element discretization of nonsymmetric and indefinite elliptic problems with dirichlet boundary conditions in 2 we show that the convergence rate of the preconditioned gmres method is nearly optimal in the sense that the rate of convergence depends only logarithmically on the mesh size and the number of substructures if the global coarse mesh is fine enough

convergence of the sinc overlapping domain decomposition method the sinc collocation overlapping method is developed for two point boundary value problems for second order ordinary differential equations the discrete system is formulated and the bordering algorithm used for the solution of this system is described it is then shown that the convergence rate is exponential even if the solution has boundary singularities the details of the convergence proof are given for a sinc collocation method for two point boundary value problems when the original domain is divided into two subdomains the extension to multiple domains is then straightforward the analytical results are illustrated with numerical examples that exhibit the exponential convergence rate

an additive schwarz method for variational inequalities this paper proposes an additive schwarz method for variational inequalities and their approximations by finite element methods the schwarz domain decomposition method is proved to converge with a geometric rate depending on the decomposition of the domain the result is based on an framework of convergence analysis established for general variational inequalities in hilbert spaces

an interval arithmetic domain decomposition method for a class of elliptic pdes on nonrectangular domains abstract we introduce an interval arithmetic domain decomposition method for linear systems with interval coefficients resulting from the application of difference methods for a class of elliptic boundary value problems on domains with irregular geometry the efficient treatment of such systems is crucial for the efficiency of globally convergent newton like interval methods for the corresponding nonlinear problems

nonlinear dynamics of open marine population with space limited recruitment the case of mortality control in this paper we develop a nonlinear extension for the open marine population model which has been proposed by roughgarden et al ecology 66 1985 54 67 to avoid the negative population density which is a drawback of the original model we introduce a nonlinear mechanism that the mortality rate depends on the size of area occupied by the adult population then we give a rigorous mathematical framework to analyse the model equation and we show sufficient conditions for stability and instability of the steady state our instability result suggests as was proposed by roughgarden et al that there exists a sustained oscillation of the population density

spectral analysis of the anisotropic steklov poincare matrix abstract in this work we analyse the steklov poincare or interface schur complement matrix arising in a domain decomposition method in the presence of anisotropy our problem is formulated such that three types of anisotropy are being considered refinements with high aspect ratios uniform refinements of a domain with high aspect ratio and anisotropic diffusion problems discretized on uniform meshes our analysis indicates a condition number of the interface schur complement with an order ranging from o 1 to o h 2 by relating this behaviour to an underlying scale of fractional sobolev spaces we propose optimal preconditioners which are spectrally equivalent to fractional matrix powers of a discrete interface laplacian numerical experiments to validate the analysis are included extensions to general domains and non uniform meshes are also considered

global ln solution and its decay property for the navier stokes equations in half space r n soit r n x 1 x 2 x n 1 x n x n 0 n 2 dans q r n 0 on considere les equations de navier stokes u t u u u 0 dans q divu 0 dans q u x u n 0 0 u 0 a on demontre l existence et l unicite d une l n solution globale forte pour des donnees initiales petites

preconditioned gradient type iterative methods in a subspace for partial generalized symmetric eigenvalue problems it is shown that a modification of preconditioned gradient type iterative methods for partial generalized eigenvalue problems makes it possible to implement them in a subspace the authors propose such methods and estimate their convergence rate the authors also describe iterative methods for finding a group of eigenvalues propose preconditioners suggest a practical way of computing the initial guess and consider a model example these methods are most effective for finding minimal eigenvalues of simple discretizations of elliptic operators with piecewise constant coefficients in domains composed of rectangles or paral lelepipeds the iterative process is carried out on the interfaces between the subdomains its rate of convergence does not decrease when the mesh gets finer and each iteration has a quite modest cost this process is effective and parallelizable