efficient and robust schur complement approximations in the augmented lagrangian preconditioner for the incompressible laminar flows this paper introduces three new schur complement approximations for the augmented lagrangian preconditioner the incompressible navier stokes equations discretized by a stabilized finite element method are utilized to evaluate these new approximations of the schur complement a wide range of numerical experiments in the laminar context determines the most efficient schur complement approximation and investigates the effect of the reynolds number mesh anisotropy and refinement on the optimal choice furthermore the advantage over the traditional schur complement approximation is exhibited

block diagonal and constraint preconditioners for nonsymmetric indefinite linear systems part i theory we study block diagonal preconditioners and an efficient variant of constraint preconditioners for general two by two block linear systems with zero 2 2 block we derive block diagonal preconditioners from a splitting of the 1 1 block of the matrix from the resulting preconditioned system we derive a smaller so called related system that yields the solution of the original problem solving the related system corresponds to an efficient implementation of constraint preconditioning we analyze the properties of both classes of preconditioned matrices in particular their spectra using analytical results we show that the related system matrix has the more favorable spectrum which in many applications translates into faster convergence for krylov subspace methods we show that fast convergence depends mainly on the quality of the splitting a topic for which a substantial body of theory exists our analysis also provides a number of new relations between block diagonal preconditioners and constraint preconditioners for constrained problems solving the related system produces iterates that satisfy the constraints exactly just as for systems with a constraint preconditioner finally for the lagrange multiplier formulation of a constrained optimization problem we show how scaling nonlinear constraints can dramatically improve the convergence for linear systems in a newton iteration our theoretical results are confirmed by numerical experiments on a constrained optimization problem we consider the general nonsymmetric nonsingular case our only additional requirement is the nonsingularity of the schur complement type matrix derived from the splitting that defines the preconditioners in particular the 1 2 block need not equal the transposed 2 1 block and the 1 1 block might be indefinite or even singular this is the first paper in a two part sequence in the second paper we will study the use of our preconditioners in a variety of applications

an accurate and efficient method for the incompressible navier stokes equations using the projection method as a preconditioner the projection method is a widely used fractional step algorithm for solving the incompressible navier stokes equations despite numerous improvements to the methodology however imposing physical boundary conditions with projection based fluid solvers remains difficult and obtaining high order accuracy may not be possible for some choices of boundary conditions in this work we present an unsplit linearly implicit discretization of the incompressible navier stokes equations on a staggered grid along with an efficient solution method for the resulting system of linear equations since our scheme is not a fractional step algorithm it is straightforward to specify general physical boundary conditions accurately however this capability comes at the price of having to solve the time dependent incompressible stokes equations at each timestep to solve this linear system efficiently we employ a krylov subspace method preconditioned by the projection method in our implementation the subdomain solvers required by the projection preconditioner employ the conjugate gradient method with geometric multigrid preconditioning the accuracy of the scheme is demonstrated for several problems including forced and unforced analytic test cases and lid driven cavity flows these tests consider a variety of physical boundary conditions with reynolds numbers ranging from 1 to 30000 the effectiveness of the projection preconditioner is compared to an alternative preconditioning strategy based on an approximation to the schur complement for the time dependent incompressible stokes operator the projection method is found to be a more efficient preconditioner in most cases considered in the present work

preconditioning harmonic unsteady potential flow calculations this paper considers finite element discretizations of the helmholtz equation and its generalization arising from harmonic acoustic perturbations to a nonuniform steady potential flow a novel elliptic positive definite preconditioner with a multigrid implementation is used to accelerate the iterative convergence of krylov subspace solvers both theory and numerical results show that for a model 1 d helmholtz test problem the preconditioner clusters the discrete system s eigenvalues and lowers its condition number to a level independent of grid resolution for the 2 d helmholtz equation grid independent convergence is achieved using a quasi minimal residual krylov solver significantly outperforming the popular symmetric successive over relaxation preconditioner impressive results are also presented on more complex domains including an axisymmetric aircraft engine inlet with nonstagnant mean flow and modal boundary conditions

optimal control of quasilinear boldsymbol h mathbf curl elliptic partial differential equations in magnetostatic field problems this paper examines the mathematical and numerical analysis for optimal control problems governed by quasilinear boldsymbol h mathbf curl elliptic partial differential equations we consider a mathematical model involving isotropic materials with magnetic permeability depending strongly on the magnetic field due to the physical and mathematical nature of the problem it is necessary to include divergence free constraints on the state and the control the divergence free control constraint is treated as an explicit variational equality constraint whereas a lagrange multiplier is included in the state equation to deal with the divergence free state constraint we investigate the sensitivity analysis of the control to state operator and establish the associated optimality conditions here the key tool for proving the kkt theory is the helmholtz decomposition an important consequence of the optimality system is a higher regularity result for the optimal control which we prove under the assumption

high order methods for the approximation of the incompressible navier stokes equations in a moving domain in this paper we address the numerical approximation of the incompressible navier stokes equations in a moving domain by the spectral element method and high order time integrators we present the arbitrary lagrangian eulerian ale formulation of the incompressible navier stokes equations and propose a numerical method based on the following kernels a lagrange basis associated with fekete points in the spectral element method context bdf time integrators an ale map of high degree and an algebraic linear solver in particular the high degree ale map is appropriate to deal with a computational domain whose boundary is described with curved elements finally we apply the proposed strategy to a test case c 2011 elsevier b v all rights reserved

a taxonomy and comparison of parallel block multi level preconditioners for the incompressible navier stokes equations in recent years considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible navier stokes equations based on preconditioned krylov methods these include physics based methods such as simple and purely algebraic preconditioners based on the approximation of the schur complement all these techniques can be represented as approximate block factorization abf type preconditioners the goal is to decompose the application of the preconditioner into simplified sub systems in which scalable multi level type solvers can be applied in this paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the navier stokes system first presented in a quarteroni f saleri a veneziani factorization methods for the numerical approximation of navier stokes equations computational methods in applied mechanical engineering 188 2000 505 526 this taxonomy illuminates the similarities and differences among these preconditioners and the central role played by efficient approximation of certain schur complement operators we then present a parallel computational study that examines the performance of these methods and compares them to an additive schwarz domain decomposition dd algorithm results are presented for two and three dimensional steady state problems for enclosed domains and inflow outflow systems on both structured and unstructured meshes the numerical experiments are performed using mpsalsa a stabilized finite element code

choice of the perfectly matched layer boundary condition for frequency domain maxwell s equations solvers we show that the performance of frequency domain solvers of maxwell s equations is greatly affected by the kind of the perfectly matched layer pml used in particular we demonstrate that using the stretched coordinate pml sc pml results in significantly faster convergence speed than using the uniaxial pml upml such a difference in convergence behavior is explained by an analysis of the condition number of the coefficient matrices additionally we develop a diagonal preconditioning scheme that significantly improves solver performance when upml is used

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

a sweeping preconditioner for time harmonic maxwell s equations with finite elements this paper is concerned with preconditioning the stiffness matrix resulting from finite element discretizations of maxwell s equations in the high frequency regime the moving pml sweeping preconditioner first introduced for the helmholtz equation on a cartesian finite difference grid is generalized to an unstructured mesh with finite elements the method dramatically reduces the number of gmres iterations necessary for convergence resulting in an almost linear complexity solver numerical examples including electromagnetic cloaking simulations are presented to demonstrate the efficiency of the proposed method

iterative techniques for time dependent stokes problems abstract in this paper we consider solving the coupled systems of discrete equations which arise from implicit time stepping procedures for the time dependent stokes equations using a mixed finite element spatial discretization at each time step a two by two block system corresponding to a perturbed stokes problem must be solved although there are a number of techniques for iteratively solving this type of block system to be effective they require a good preconditioner for the resulting pressure operator schur complement in contrast to the time independent stokes equations where the pressure operator is well conditioned the pressure operator for the perturbed system becomes more ill conditioned as the time step is reduced and or the reynolds number is increased in this paper we shall describe and analyze preconditioners for the resulting pressure systems these preconditioners give rise to iterative rates of convergence which are independent of both the mesh size h as well as the time step and reynolds number parameter k

eigenvalue bounds for the schur complement with a pressure convection diffusion preconditioner in incompressible flow computations if the stationary navier stokes system or an implicit time discretization of the evolutionary navier stokes system is linearized by a picard iteration and discretized in space by a mixed finite element method there arises a saddle point system which may be solved by a krylov subspace method or an uzawa type approach for each of these resolution methods it is necessary to precondition the schur complement associated to the saddle point problem in question in the work at hand we give upper and lower bounds of the eigenvalues of this schur complement under the assumption that it is preconditioned by a pressure convection diffusion matrix

modeling studies and efficient numerical methods for proton exchange membrane fuel cell abstract in this paper a three dimensional 3d nonisothermal multiphysics two phase steady state transport model and its efficient numerical methods are systematically studied for a full proton exchange membrane fuel cell pemfc in the sense of efficiency and accuracy the conservation equations of mass momentum species charge and energy are fully addressed in view of nonisothermality and multiphase characteristics in addition from an accurate numerical discretization s point of view we present some new formulations for species equations by investigating the interactions among the species in a framework of the combined finite element upwind finite volume method some efficient numerical methods are developed in terms of kirchhoff transformation for the sake of a fast and convergent numerical simulation the 3d simulations demonstrate that the convergent solutions can be attained within 80 nonlinear iterations in contrast to the oscillating and nonconvergent iterations conducted by commercial flow solvers or in house code with standard finite element volume methods numerical convergence tests are carried out to verify the efficiency and accuracy of our numerical algorithms and techniques

on parameter choice and iterative convergence for stabilised discretisations of advection diffusion problems abstract in this work we consider the design of robust and efficient finite element approximation methods for solving advection diffusion equations specifically we consider the stabilisation of discrete approximations using uniform grids which do not resolve boundary layers as might arise using a multi level or multigrid iteration strategy to solve the discrete problem our analysis shows that when using supg streamline upwind finite element methodology there is a symbiotic relationship between best solution approximation and fast convergence of smoothers based on the standard gmres iteration we also show that stabilisation based on simple artificial diffusion perturbation terms an approach often advocated by multigrid practitioners is less appealing

development of a stokes flow solver robust to large viscosity jumps using a schur complement approach with mixed precision arithmetic we develop an iterative solution technique for solving stokes flow problems with smooth and discontinuous viscosity structures using a three dimensional staggered grid finite difference discretization two preconditioned iterative methodologies are applied to the saddle point arising from the discrete stokes problem they consist of a velocity pressure coupled approach fc and a decoupled schur complement approach sc within both of these methods we utilize either the scaled bfbt or an identity matrix scaled by the local cell viscosity lv to define a preconditioner for the schur complement additionally we propose to use a mixed precision krylov kernel to improve the convergence by reducing round off error in this approach standard double precision is used during the application of the preconditioner whilst higher precision arithmetic is used to define the matrix vector product dot products and norms required by the krylov method in our krylov kernel we utilize quad precision arithmetic which is emulated via the double double precision method we consider several simplified geodynamic problems with a viscosity contrast to demonstrate the robustness and scalability of our solution methods through a careful choice of stopping conditions we are able to quantitatively compare the residuals between the sc and fc approaches we examine the trade off relationship between the number of outer iterations required for convergence and the computational cost per iteration for the each solution methods we find that it is advantageous to use the fc approach utilizing relaxed tolerances for solution of the sub problems combined with the lv preconditioner we also observed that in general the sc approach is more robust than fc and that bfbt is more robust than lv when used in our numerical experimental in addition our mixed precision method produces improved convergence rates of arnoldi type krylov subspace methods without a drastic increasing the computational time the usage of a high precision krylov kernel is found to be useful for the solver associated with the velocity sub problem

a relaxed dimensional factorization preconditioner for the incompressible navier stokes equations in this paper we introduce a relaxed dimensional factorization rdf preconditioner for saddle point problems properties of the preconditioned matrix are analyzed and compared with those of the closely related dimensional splitting ds preconditioner recently introduced by benzi and guo 7 numerical results for a variety of finite element discretizations of both steady and unsteady incompressible flow problems indicate very good behavior of the rdf preconditioner with respect to both mesh size and viscosity

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

a systematic study of the sensitivity of partial volume correction methods for the quantification of perfusion from pseudo continuous arterial spin labeling mri abstract arterial spin labeling asl mri is a non invasive technique for the quantification of cerebral perfusion and pseudo continuous arterial spin labeling pcasl has been recommended as the standard implementation by a recent consensus of the community due to the low spatial resolution of asl images perfusion quantification is biased by partial volume effects consequently several partial volume correction pvec methods have been developed to reduce the bias in gray matter gm perfusion quantification the efficacy of these methods relies on both the quality of the asl data and the accuracy of partial volume estimates here we systematically investigate the sensitivity of different pvec methods to variability in both the asl data and partial volume estimates using simulated pcasl data and in vivo pcasl data from a reproducibility study we examined the pvec methods in two ways the ability to preserve spatial details and the accuracy of gm perfusion estimation judging by the root mean square error rmse between simulated and estimated gm cbf the spatially regularized method was superior in preserving spatial details compared to the linear regression method rmse of 1 2 vs 5 1 in simulation of gm cbf with short scale spatial variations the linear regression method was generally less sensitive than the spatially regularized method to noise in data and errors in the partial volume estimates rmse 6 3 vs 23 4 for snr 5 simulated data but this could be attributed to the greater smoothing introduced by the method analysis of a healthy cohort dataset indicates that pvec using either method improves the repeatability of perfusion quantification within subject coefficient of variation reduced by 5 after pvec

multi space reduced basis preconditioners for parametrized stokes equations abstract we introduce a two level preconditioner for the efficient solution of large scale saddle point linear systems arising from the finite element fe discretization of parametrized stokes equations this preconditioner extends the multi space reduced basis msrb preconditioning method proposed in dal santo et al 2018 it combines an approximated block fine grid preconditioner with a reduced basis rb solver which plays the role of coarse component a sequence of rb spaces constructed either with an enriched velocity formulation or a petrov galerkin projection is built each rb coarse component is defined to perform a single iteration of the iterative method at hand the flexible gmres fgmres algorithm is employed to solve the resulting preconditioned system and targets small tolerances with a very small iteration count and in a very short time numerical test cases for stokes flows in three dimensional parameter dependent geometries are considered to assess the numerical properties of the proposed technique in different large scale computational settings

orthogonal error methods in this paper we discuss the conjugate gradient method generalized to nonsymmetric matrices constructing a class of algorithms that includes both the conjugate gradient algorithms as described in 12 and the orthogonal residual algorithms cf elman 10 we call this class orthogonal error algorithms and characterize the class of matrices for which finite term orthogonal error algorithms exist

an optimal preconditioner for a class of saddle point problems with a penalty term iterative methods are considered for a class of saddle point problems with a penalty term arising from finite element discretizations of certain elliptic problems an optimal preconditioner which is independent of the discretization and the penalty parameter is constructed this approach is then used to design an iterative method with a convergence rate bounded independently of the lame parameters occurring in the equations of linear elasticity

preconditioning for the steady state navier stokes equations with low viscosity we introduce a preconditioner for the linearized navier stokes equations that is effective when either the discretization mesh size or the viscosity approaches zero for constant coefficient problems with periodic boundary conditions we show that the preconditioning yields a system with a single eigenvalue equal to 1 so that performance is independent of both viscosity and mesh size for other boundary conditions we demonstrate empirically that convergence depends only mildly on these parameters and we give a partial analysis of this phenomenon we also show that some expensive subsidiary computations required by the new method can be replaced by inexpensive approximate versions of these tasks based on iteration with virtually no degradation of performance

efficient preconditioning of the linearized navier stokes equations for incompressible flow we outline a new class of robust and efficient methods for solving subproblems that arise in the linearization and operator splitting of navier stokes equations we describe a very general strategy for preconditioning that has two basic building blocks a multigrid v cycle for the scalar convection diffusion operator and a multigrid v cycle for a pressure poisson operator we present numerical experiments illustrating that a simple implementation of our approach leads to an effective and robust solver strategy in that the convergence rate is independent of the grid robust with respect to the time step and only deteriorates very slowly as the reynolds number is increased

fast iterative solution of stabilised stokes systems part ii using general block preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems part i of this work described a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems a j wathen and d j silvester siam j numer anal 30 1993 pp 630 649 using simple arguments estimates for the eigenvalue distribution of the discrete stokes operator on which the convergence rate of the iteration depends are easily derived part i discussed the important case of diagonal preconditioning scaling this paper considers the more general class of block preconditioners where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables it is shown that provid

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

block preconditioners based on approximate commutators this paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible navier stokes equations we consider the pressure convection diffusion preconditioners proposed by kay loghin and wathen siam j sci comput 24 2002 pp 237 256 and silvester elman kay and wathen j comput appl math 128 2001 pp 261 279 numerous theoretical and numerical studies have demonstrated mesh independent convergence on several problems and the overall efficacy of this methodology a drawback however is that it requires the construction of a convection diffusion operator denoted f p projected onto the discrete pressure space this means that integration of this idea into a code that models incompressible flow requires a sophisticated understanding of the discretization and other implementation issues something often held only by the developers of the model as an alternative we consider automatic ways of computing f p based on purely algebraic considerations the new methods are closely related to the bfbt preconditioner of elman siam j sci comput 20 1999 pp 1299 1316 we use the fact that the preconditioner is derived from considerations of commutativity between the gradient and convection diffusion operators together with methods for computing sparse approximate inverses to generate the required matrix f p automatically we demonstrate that with this strategy the favorable convergence properties of the preconditioning methodology are retained

asymptotic behavior of solutions toward the rarefaction waves to the cauchy problem for the scalar diffusive dispersive conservation laws abstract in this paper we investigate the asymptotic behavior of solutions to the cauchy problem for the scalar diffusive dispersive conservation laws where the far field states are prescribed especially we deal with the generalized models for korteweg de vries burgers kuramoto equation when the corresponding riemann problem for the hyperbolic part admits a riemann solution which consists of single rarefaction wave it is proved that the solution of the cauchy problem tends toward the rarefaction wave as time goes to infinity

performance and robustness of block constraint preconditioners in finite element coupled consolidation problems block constraint preconditioners are a most recent development for the iterative solution to large scale often ill conditioned coupled consolidation problems a major limitation to their practical use however is the somewhat difficult selection of a number of user defined parameters at least 4 in a more or less optimal way the present paper investigates the robustness of three variants of the block constraint preconditioning in relation to the above parameters a theoretical analysis of the eigenspectrum of the preconditioned matrix provides relatively simple bounds of the eigenvalues as a function of these parameters a number of test problems used to validate the theoretical results show that both the mixed constraint preconditioner mcp combined with the symmetric quasi minimal residual sqmr solver and the mcp triangular variant t mcp combined with the bi conjugate gradient stabilized bi cgstab are efficient and robust tools for the solution to difficult finite element discretized coupled consolidation problems moreover the practical selection of the user defined parameters is relatively easy as a stable behavior is observed for a wide range of fill in degree values the theoretical bounds on the eigenspectrum of the preconditioned matrix may help to suggest the most appropriate parameter combination copyright 2009 john wiley sons ltd

an adaptive perfectly matched layer technique for 3 d time harmonic electromagnetic scattering problems an adaptive perfectly matched layer pml technique for solving the time harmonic electromagnetic scattering problems is developed the pml parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates combined with the adaptive finite element method the adaptive pml technique provides a complete numerical strategy to solve the scattering problem in the framework of fem which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the thickness of the pml absorbing layer numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method

mixed constraint preconditioning in computational contact mechanics contact mechanics can be addressed numerically by finite elements using either a penalty formulation or the lagrange multipliers the penalty approach leads to a linearized symmetric positive definite system which can prove severely ill conditioned with the iterative solution to large 3d problems requiring expensive preconditioners to accelerate or even to allow for convergence in the present paper a mixed constraint preconditioner mcp is developed and implemented for the iterative solution of contact problems based on the penalty approach a theoretical bound for the eigenspectrum of the preconditioned matrix is provided depending on the user specified parameters that control the preconditioner fill in the performance of mcp combined with the preconditioned conjugate gradient is studied in two real 3d test cases describing the geomechanics of faulted porous media and compared to a well established preconditioner such as the incomplete cholesky decomposition illt with optimal fill in numerical results show that typically mcp outperforms illt especially in the most challenging problems where very large penalty parameters must be used finally a modification aimed at simplifying the mcp implementation and increasing its robustness is developed and experimented with showing that the modified mcp can be viewed as an efficient and viable alternative to both mcp and illt especially in problems where the size of the contact surfaces is large with respect to the global domain

nonlinear stability of strong rarefaction waves for the generalized kdv burgers kuramoto equation with large initial perturbation abstract in was shown in ruan et al 2008 3 that rarefaction waves for the generalized kdv burgers kuramoto equation are nonlinearly stable provided that both the strength of the rarefaction waves and the initial perturbation are sufficiently small the main purpose of this work is concerned with nonlinear stability of strong rarefaction waves for the generalized kdv burgers kuramoto equation with large initial perturbation in our results we do not require the strength of the rarefaction waves to be small and when the smooth nonlinear flux function satisfies certain growth condition at infinity the initial perturbation can be chosen arbitrarily in h 1 r while for a general smooth nonlinear flux function we need to ask for the l 2 norm of the initial perturbation to be small but the l 2 norm of the first derivative of the initial perturbation can be large and consequently the h 1 r norm of the initial perturbation can also be large

fast nonsymmetric iterations and preconditioning for navier stokes equations discretization and linearization of the steady state navier stokes equations gives rise to a nonsymmetric indefinite linear system of equations in this paper we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded independently of the mesh size used in the discretization we confirm and supplement these analytic results with a series of numerical experiments indicating that krylov subspace iterative methods for nonsymmetric systems display rates of convergence that are independent of the mesh parameter in addition we show that preconditioning costs can be kept small by using iterative methods for some intermediate steps performed by the preconditioner

a block preconditioner for high order mixed finite element approximations to the navier stokes equations an iterative solver for the linear system arising from a discretization by high order mixed finite elements for the linearized steady state navier stokes problem is presented two mixed finite elements are considered mathbf p k 1 p k and mathbf p k 2 p k the iterative solver is based on the gmres method with right preconditioning where the preconditioner has a block upper triangular form computational results show that the convergence rate seems to be independent of the grid size parameter h and the polynomial degree k but is slightly affected by the increasing of the reynolds number

a relaxed positive definite and skew hermitian splitting preconditioner for saddle point problems for saddle point linear system whose coefficient matrix is non hermitian of positive definite hermitian part pan et al 2006 proposed a positive definite and skew hermitian splitting pss preconditioner in this paper based on the pss preconditioner we present a relaxed positive definite and skew hermitian splitting rpss preconditioner we analyze the spectral properties of the new preconditioned matrix and conclude that the rpss preconditioner has much better properties than pss numerical experiments of a model navier stokes problem are presented to illustrate the effectiveness of the new preconditioner

a posteriori error estimation for stabilized mixed approximations of the stokes equations the practical implementation of the lowest order pl p0 linear velocity constant pressure finite element method for the steady state incompressible navier stokes equations is addressed in this work three different types of a posteriori error indicator are introduced and each is shown to be equivalent to the discretization error our numerical results show that these indicators can be used to drive an adaptive refinement process which is specially tailored to create grids which conform to the requirements of the local stabilization it is also shown that the indicators provide an effective method for detecting local singularities in the flow solution

global structure of solutions toward the rarefaction waves for the cauchy problem of the scalar conservation law with nonlinear viscosity abstract in this paper we investigate the global structure of solutions to the cauchy problem for the scalar viscous conservation law where the far field states are prescribed especially we deal with the case when the viscous diffusive flux v v p is of non newtonian type i e p 0 including a pseudo plastic case i e 0 p 1 when the corresponding riemann problem for the hyperbolic part admits a riemann solution which consists of single rarefaction wave under a condition on nonlinearity of the viscosity it has been recently proved by matsumura yoshida 35 that the solution of the cauchy problem tends toward the rarefaction wave as time goes to infinity for the case p 3 7 without any smallness conditions the new ingredients we obtained are the extension to the stability results in 35 to the case p 1 3 also without any smallness conditions and furthermore their precise time decay estimates

asymptotic stability of the rarefaction wave for the generalized kdv burgers kuramoto equation abstract in this paper we will study the cauchy problem for the generalized kdv burgers kuramoto equation which represents a dissipative stroboscopic and unstable system in physics when the initial data is a small disturbance of a rarefaction wave of the inviscid burgers equation we prove the global existence of the solution to the corresponding cauchy problem and asymptotic stability of the rarefaction wave the analysis is based on a priori estimates and the l 2 energy method

analysis of a ladyzhenskaya model for incompressible viscous flow abstract in this paper we analyze a model for the motion of incompressible viscous flows proposed by ladyzhenskaya we establish a new a priori estimate for the nonstationary model and then obtain more general existence and uniqueness results we also present some uniqueness results for the stationary model the relationship between fluid motion governed by the ladyzhenskaya model and that governed by the navier stokes equations is addressed

analysis on block diagonal and triangular preconditioners for a pml system of an electromagnetic scattering problem abstract we shall propose several block triangular preconditioners for a pml system of an electromagnetic wave scattering problem and analyze the spectral behavior of the preconditioned systems when the pml system is discretized by edge element methods it results in a discrete system with its stiffness matrix being complex symmetric but indefinite which can be formulated into a real symmetric but indefinite saddle point system in order to preserve the symmetry of the coefficient matrix we present block triangular preconditioners with two sided preconditioning for the discrete pml system we will estimate the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrices respectively on the other hand one may also like to apply some iteration methods for nonsymmetric linear systems in applications although the discrete systems are symmetric to this end we propose a block triangular preconditioner to precondition the systems only from one side and analyze the spectrum of the preconditioned systems in addition we have also established a spectral estimate of the preconditioned system by an effective preconditioner that was recently developed in literature numerical experiments are presented to demonstrate the effectiveness and robustness of these new preconditioners and our theoretical predictions on the spectral bounds of the preconditioned systems

a low order galerkin finite element method for the navier stokes equations of steady incompressible flow a stabilization issue and iterative methods a galerkin finite element method is considered to approximate the incompressible navier stokes equations together with iterative methods to solve a resulting system of algebraic equations this system couples velocity and pressure unknowns thus requiring a special technique for handling we consider the navier stokes equations in velocity kinematic pressure variables as well as in velocity bernoulli pressure variables the latter leads to the rotation form of nonlinear terms this form of the equations plays an important role in our studies a consistent stabilization method is considered from a new view point theory and numerical results in the paper address both the accuracy of the discrete solutions and the effectiveness of solvers and a mutual interplay between these issues when particular stabilization techniques are applied 2002 published by elsevier science b v

combining the augmented lagrangian preconditioner with the simple schur complement approximation the augmented lagrangian al preconditioner and its variants have been successfully applied to solve saddle point systems arising from the incompressible navier stokes equations discretized by th

asymptotic stability of traveling waves for scalar viscous conservation laws with non convex nonlinearity the asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation lawsu t f u x u xx with the initial datau0 which tend to the constant statesu asx stability theorems are obtained in the absence of the convexity off and in the allowance ofs shock speed f u moreover the rate of asymptotics in time is investigated for the casef u s f u if the integral of the initial disturbance over x is small and decays at the algebraic rate as x then the solution approaches the traveling wave at the corresponding rate ast this rate seems to be almost optimal compared with the rate in the casef u2 2 for which an explicit form of the solution exists the rate is also obtained in the casef u s under some additional conditions proofs are given by applying an elementary weighted energy method to the integrated equation of the original one the selection of the weight plays a crucial role in those procedures

the quasar reproducibility study part ii results from a multi center arterial spin labeling test retest study arterial spin labeling asl is a method to measure perfusion using magnetically labeled blood water as an endogenous tracer being fully non invasive this technique is attractive for longitudinal studies of cerebral blood flow in healthy and diseased individuals or as a surrogate marker of metabolism so far asl has been restricted mostly to specialist centers due to a generally low snr of the method and potential issues with user dependent analysis needed to obtain quantitative measurement of cerebral blood flow cbf here we evaluated a particular implementation of asl called quantitative star labeling of arterial regions or quasar a method providing user independent quantification of cbf in a large test retest study across sites from around the world dubbed the quasar reproducibility study altogether 28 sites located in asia europe and north america participated and a total of 284 healthy volunteers were scanned minimal operator dependence was assured by using an automatic planning tool and its accuracy and potential usefulness in multi center trials was evaluated as well accurate repositioning between sessions was achieved with the automatic planning tool showing mean displacements of 1 87 0 95 mm and rotations of 1 56 0 66 degrees mean gray matter cbf was 47 4 7 5 ml 100 g min with a between subject standard variation sd b 5 5 ml 100 g min and a within subject standard deviation sd w 4 7 ml 100 g min the corresponding repeatability was 13 0 ml 100 g min and was found to be within the range of previous studies

sharp error bounds of some krylov subspace methods for non hermitian linear systems1project 19601036 supported by the national natural science foundation of china 1 the convergence of the krylov subspace methods e g full orthogonal method fom and generalized minimal residual method gmres etc for solving large non hermitian linear systems is studied in a unified and detailed way when the coefficient matrix is defective in particular when its spectrum lies in the open right left half plane or is on the real axis related theoretical error bounds are established which reveal some intrinsic relationships between the convergence properties and the eigen characteristics of the coefficient matrix these results not only generalize all the known ones for the diagonalizable matrices in the literature but also sharp the corresponding estimates in jia acta mathematica sinica new series 14 1998 507 518

decay rates for the solutions of model equations for bore propagation we study the large time behaviour of solutions to the korteweg de vries burgers equation with bore like initial data this work relies on the methods of amick bona and schonbeck to obtain sharp rates of temporal decay of certain norms of the solution thus obtaining an improvement over results of naumkin and shishmarev

decay properties of solutions toward a multiwave pattern to the cauchy problem for the scalar conservation law with degenerate flux and viscosity in this paper we study the precise decay rate in time to solutions of the cauchy problem for the one dimensional conservation law with the ostwald de waele type viscosity p laplacian type degenerate viscosity tu x f u x xu p 1 xu where the far field states are prescribed especially we deal with the case when the flux function is convex or concave but linearly degenerate on some interval when the corresponding riemann problem admits a multiwave pattern which consists of the rarefaction waves and the contact discontinuity it has already been proved by yoshida that the solution to the cauchy problem tends toward the linear combination of the rarefaction waves and contact wave for the ostwald de waele type viscosity as time goes to infinity we investigate the decay rate in time of the solution toward the multiwave pattern furthermore we investigate the decay rate in time of the solution for the derivative the proof is given by l1 l2 energy and time weighted lq energy methods under the use of the precise asymptotic properties of the interactions between the nonlinear waves

shift splitting preconditioners for saddle point problems in this paper we first present a shift splitting preconditioner for saddle point problems the preconditioner is based on a shift splitting of the saddle point matrix resulting in an unconditional convergent fixed point iteration based on the idea of the splitting we further propose a local shift splitting preconditioner some properties of the local shift splitting preconditioned matrix are studied these preconditioners extend those studied by bai yin and su for solving non hermitian positive definite linear systems bai et al 2006 finally numerical experiments of a model stokes problem are presented to show the effectiveness of the proposed preconditioners

new preconditioners for saddle point problems we present new preconditioners based on matrix splittings for the saddle point problems the spectral property of one of the preconditioned matrix is studied in detail numerical examples are used to illustrate the efficiency of the new preconditioners

towards discrete velte decompositions and narrow bounds for inf sup constants abstract for the velte decomposition of the function space h 1 0 into three orthogonal subspaces the possibility of discrete equivalents is investigated and some first results are derived for conforming finite element methods for the difference approximation on a staggered grid a full description of the discrete velte decomposition is given in the continuous case we consider the connection of the velte decomposition to an orthogonal decomposition of the pressure space and to the inf sup stability condition of the stokes problem some bounds for inf sup constants in the continuous two dimensional case and numerous discrete inf sup constants have been computed

global stability of strong rarefaction waves for the generalized kdv burgers equation this paper is concerned with the global stability of strong rarefaction waves for the generalized kdv burgers equation in contrast to former results obtained by z a wang and c j zhu stability of the rarefaction wave for the generalized kdv burgers equation acta math sci 22b 3 2002 319 328 ours do not require the strength of the rarefaction waves to be small and the initial disturbance can also be chosen large c 2006 elsevier ltd all rights reserved

comparison of performance of iterative methods for singular and nonsingular saddle point linear systems arising from navier stokes equations abstract we compare the convergence performance of iterative methods for solving singular and nonsingular linear systems arising from discretization and linearization of the steady state navier stokes equations with a combination of analytic and empirical results we study the effects of singularity and nonsingularity on convergence we will show that the convergence performance for solving a singular linear system is better than that for solving a nonsingular linear system

decay properties of solutions toward a multiwave pattern for the scalar viscous conservation law with partially linearly degenerate flux abstract in this paper we study the decay rate in time to solutions toward a multiwave pattern of the cauchy problem for the one dimensional viscous conservation law where the far field states are prescribed especially we deal with the case that the flux function is convex or concave but linearly degenerate on some interval as the corresponding riemann problem admits a riemann solution which consists of rarefaction waves and contact discontinuity it has already been proved by matsumura yoshida that the solution to the cauchy problem tends toward the multiwave pattern which consists of the rarefaction waves and the viscous contact wave as the time goes to infinity we investigate that the decay rate in time is 1 t 1 2 1 2 1 p in the l p norm 2 p and 1 t 1 4 for any 0 in the l norm if the initial perturbation from the corresponding asymptotics satisfies h 1 furthermore if the perturbation satisfies h 1 l 1 the decay rate in time is 1 t 1 2 1 1 p for any 0 in the l p norm 1 p and 1 t 1 2 for any 0 in the l norm the proof is given by l 1 l 2 energy and time weighted l p energy methods and the properties of the interactions between the nonlinear waves we also prove the same results of the decay properties by the similar arguments to the initial boundary value problem on the half space

on parameterized block triangular preconditioners for generalized saddle point problems in this paper we discuss two classes of parameterized block triangular preconditioners for the generalized saddle point problems these preconditioners generalize the common block diagonal and triangular preconditioners we will give distributions of the eigenvalues of the preconditioned matrix and provide estimates for the interval containing the real eigenvalues numerical experiments of a model stokes problem are presented

necessary and sufficient conditions for the existence of a conjugate gradient method we characterize the class cg s of matrices a for which the linear system a bf x bf b can be solved by an s term conjugate gradient method we show that except for a few anomalies the class cg s consists of matrices a for which conjugate gradient methods are already known these matrices are the hermitian matrices a a and the matrices of the form ae i theta di b with b b

an investigation of statistical power for continuous arterial spin labeling imaging at 1 5 t variance estimates can be used in conjunction with scientifically meaningful effect sizes to design experiments with type ii error control here we present estimates of intra and inter subject variances for region of interest roi from resting cerebral blood flow cbf maps obtained using whole brain spin echo echoplanar se epi continuous arterial spin labeling casl imaging on 52 elderly subjects age 70 5 7 9 years 29 males there was substantial intrasubject systematic variability in cbf of gray matter rois corresponding to a range of standard deviations 39 168 ml 100g min this variability was mainly due to two factors 1 an expected inverse relationship between roi volume and intrasubject variance and 2 an increased effective post labeling delay for more superior slices acquired later in the sequence for example intrasubject variance in brodmann area 4 ba 4 was 8 times larger than in hippocampus despite their similar gray matter volumes estimated roi wise power was computed for various numbers of acquired cbf images numbers of subjects and cbf effect sizes for two experimental designs independent samples t test and paired t test the theoretical effects of pulse sequence and field strength on general applicability of these results are discussed

rarefactions and large time behavior for parabolic equations and monotone schemes we consider the large time behavior of monotone semigroups associated with degenerate parabolic equations and monotone difference schemes for an appropriate class of initial data the solution is shown to converge to rarefaction waves at a determined asymptotic rate

rmcp relaxed mixed constraint preconditioners for saddle point linear systems arising in geomechanics abstract a major computational issue in the finite element fe integration of coupled consolidation equations is the repeated solution in time of the resulting discretized indefinite system because of ill conditioning the iterative solution which is recommended in large size 3d settings requires the computation of a suitable preconditioner to guarantee convergence in this paper the coupled system is solved by a krylov subspace method preconditioned by a relaxed mixed constraint preconditioner rmcp which is a generalization based on a parameter of the mixed constraint preconditioner mcp developed in 7 choice of optimal is driven by the spectral distribution of suitable symmetric positive definite spd matrices numerical tests performed on realistic 3d problems reveal that rmcp accelerates krylov subspace solvers by a factor up to three with respect to mcp

the convergence rate of the minimal residual method for the stokes problem discretisation of the classical stokes problem gives rise to symmetric indefinite matrices with eigenvalues which in a precise way are not symmetric about the origin but which do depend on a mesh size parameter convergence estimates for the conjugate residual or minimum residual iterative solution of such systems are given by best minimax polynomial approximations on an inclusion set for the eigenvalues in this paper an analytic convergence estimate for such problems is given in terms of an asymptotically small mesh size parameter

on block preconditioners for nonsymmetric saddle point problems we discuss a class of preconditioning methods for an iterative solution of algebraic nonsymmetric saddle point problems arising from a mixed finite element discretization of partial differential equations in particular the navier stokes equation we prove that block diagonal and block triangular preconditioners based on symmetric positive definite blocks guarantee that the convergence rate of the method is independent of the mesh parameter h

unconditional stability and long term behavior of transient algorithms for the incompressible navier stokes and euler equations abstract this paper examines the long term behavior dissipativity and unconditional nonlinear stability properties of time integration algorithms for the incompressible navier stokes equations including both direct schemes and fractional step projection methods the algorithms are termed nonlinear unconditionally stable if in the absence of a forcing term but for arbitrary initial conditions the computed kinetic energy decreases for arbitrary step sizes i e l 2 stability such an a priori stability estimate is a characteristic feature of the navier stokes system similarly the algorithms are said to exhibit asymptotic t independent long term dissipative behavior if as the continuum flow generated by the navier stokes equations the algorithmic flow also exhibits an absorbing set and a maximal attractor both direct and fractional step methods are described which inherit these fundamental properties of the navier stokes system for any time step size moreover a subclass of algorithms which retain these strong notions of nonlinear stability and long term dissipative behavior is identified which in addition has the remarkable property of being linear within the time step the key implication is that no iterations are required to compute the solution within a time step a specific member of this class is shown to possess the same absorbing set as the continuum navier stokes system a second order accurate unconditionally stable method is also identified which retains the property of linearity within a time step numerical analysis and computational aspects involved in the implementation of these methods are addressed in detail and illustrated in representative numerical simulations

performance and analysis of saddle point preconditioners for the discrete steady state navier stokes equations we examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady state navier stokes equations with a combination of analytic and empirical results we study the effects of fundamental parameters on convergence we demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers the structure of these distributions is independent of the discretization mesh size but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller these characteristics are directly correlated with the convergence properties of iterative solvers

global stability of the rarefaction wave of a one dimensional model system for compressible viscous gas this paper is concerned with the asymptotic behavior toward the rarefaction wave of the solution of a one dimensional barotropic model system for compressible viscous gas we assume that the initial data tend to constant states atx respectively and the riemann problem for the corresponding hyperbolic system admits a weak continuous rarefaction wave if the adiabatic constant satisfies 1 2 then the solution is proved to tend to the rarefaction wave ast under no smallness conditions of both the difference of asymptotic values atx and the initial data the proof is given by an elementaryl 2 energy method

decay properties of solutions to the cauchy problem for the scalar conservation law with nonlinearly degenerate viscosity abstract in this paper we study the decay rate in time to solutions of the cauchy problem for the one dimensional viscous conservation law where the far field states are prescribed especially we deal with the case that the flux function which is convex and also the viscosity is a nonlinearly degenerate one p laplacian type viscosity as the corresponding riemann problem admits a riemann solution as the constant state or the single rarefaction wave it has already been proved by matsumura nishihara that the solution to the cauchy problem tends toward the constant state or the single rarefaction wave as the time goes to infinity we investigate that the decay rate in time of the corresponding solutions and their derivative these are the first results concerning the asymptotic decay of the solutions and their derivative to the cauchy problem of the scalar conservation law with nonlinear viscosity the proof is given by l 1 l 2 energy and time weighted l q energy methods

navier stokes equations in rotation form a robust multigrid solver for the velocity problem the topic of this paper is motivated by the navier stokes equations in rotation form linearization and application of an implicit time stepping scheme results in a linear stationary problem of oseen type in well known solution techniques for this problem such as the uzawa or schur complement method a subproblem consisting of a coupled nonsymmetric system of linear equations of diffusion reaction type must be solved to update the velocity vector field in this paper we analyze a standard finite element method for the discretization of this coupled system and we introduce and analyze a multigrid solver for the discrete problem both for the discretization method and the multigrid solver the question of robustness with respect to the amount of diffusion and variation in the convection field is addressed we prove stability results and discretization error bounds for the galerkin finite element method we present a convergence analysis of the multigrid method which shows the robustness of the solver results of numerical experiments are presented which illustrate the stability of the discretization method and the robustness of the multigrid solver

test retest reliability and sample size estimates after mri scanner relocation abstract objective many factors can contribute to the reliability and robustness of mri derived metrics in this study we assessed the reliability and reproducibility of three mri modalities after an mri scanner was relocated to a new hospital facility methods twenty healthy volunteers 12 females mean age standard deviation 41 11 years age range 25 66 completed three mri sessions the first session s1 was one week prior to the 3t ge hdxt scanner relocation the second s2 occurred nine weeks after s1 and at the new location a third session s3 was acquired 4 weeks after s2 at each session we acquired structural t1 weighted pseudo continuous arterial spin labelled and diffusion tensor imaging sequences we used longitudinal processing streams to create 12 summary mri metrics including total gray matter gm cortical gm subcortical gm white matter wm and lateral ventricle volume mean cortical thickness total surface area average gray matter perfusion and average diffusion tensor metrics along principal white matter pathways we compared mean mri values and variance at the old scanner location to multiple sessions at the new location using bayesian multi level regression models k fold cross validation allowed identification of important predictors whole brain analyses were used to investigate any regional differences furthermore we calculated within subject coefficient of variation wscv intraclass correlation coefficient icc and dice similarity index si of cortical segmentations across scanner relocation and within site additionally we estimated sample sizes required to robustly detect a 4 difference between two groups across mri metrics results all global mri metrics exhibited little mean difference and small variability bar cortical gray matter perfusion both across scanner relocation and within site repeat t1 and dti derived tissue metrics showed 0 80 and within subject coefficient of variation wscv conclusion cortical gray matter perfusion was the most variable metric investigated necessitating large sample sizes to identify group differences with other metrics showing substantially less variability scanner relocation appeared to have a negligible effect on variability of the global mri metrics tested this manuscript reports within site test retest variability to act as a tool for calculating sample size in future investigations our results suggest that when all other parameters are held constant e g sequence parameters and mri processing the effect of scanner relocation is indistinguishable from within site variability but may need to be considered depending on the question being investigated