variants of the deteriorated pss preconditioner for saddle point problems two new preconditioners which can be viewed as variants of the deteriorated positive definite and skew hermitian splitting preconditioner are proposed for solving saddle point problems the corresponding iteration methods are proved to be convergent unconditionally for cases with positive definite leading blocks the choice strategies of optimal parameters for the two iteration methods are discussed based on two recent optimization results for extrapolated cayley transform which result in faster convergence rate and more clustered spectrum compared with some preconditioners of similar structures the new preconditioners have better convergence properties and spectrum distributions in addition more practical preconditioning variants of the new preconditioners are considered numerical experiments are presented to illustrate the advantages of the new preconditioners over some similar preconditioners to accelerate gmres c 2018 elsevier ltd all rights reserved

computational formulation for periodic vibration of geometrically nonlinear structures part 2 numerical strategy and examples abstract in this paper the numerical strategy for solving the matrix amplitude equation with parameter is discussed in detail the matrix amplitude equation is the result of application of the galerkin method for analysis of periodic solutions of the geometrically nonlinear structures a theoretical background of the proposed method of analysis is given in a companion paper by lewandowski lewandowski r 1996 computational formulation for periodic vibration of geometrically nonlinear structures part 1 theoretical background international journal of solids and structures34 15 1925 1947 an example application of general theory is also presented numerical results are given and discussed to show some interesting behaviors of nonlinear structures and to illustrate the numerical efficiency and validity of the suggested method

a block product preconditioner for saddle point problems abstract in this paper a block product bp preconditioner is established for saddle point problems spectral properties of the bp preconditioned matrix are investigated a strategy for practical choice of quasi optimal parameter is given numerical results on saddle point linear systems arising from stokes problems and weighted least square problems show that the proposed bp preconditioner is more economic to implement within krylov subspace acceleration than some extensively studied preconditioners

substructure preconditioners for elliptic saddle point problems domain decomposition preconditioners for the linear systems arising from mixed finite element discretizations of second order elliptic boundary value problems are proposed the preconditioners are based on subproblems with either neumann or dirichlet boundary conditions on the interior boundary the preconditioned systems have the same structure as the nonpreconditioned systems in particular we shall derive a preconditioned system with conditioning independent of the mesh parameter h the application of the minimum residual method to the preconditioned systems is also discussed

free and forced vibration analysis of a nonlinear system with cyclic symmetry application to a simplified model abstract this work program is devoted to studying the nonlinear dynamics of a structure with cyclic symmetry under conditions of geometric nonlinearity through the use of the harmonic balance method hbm in order to study the influence of nonlinearity due to the large deflection of blades a simplified model has been developed this approach leads to a system of linearly coupled second order nonlinear differential equations in which nonlinearity appears via cubic terms periodic solutions in both the free and forced cases are sought by applying hbm coupled with an arc length continuation method solution stability has been investigated using floquet s theorem in addition to featuring similar and nonsimilar nonlinear modes the unforced system is known to contain localized nonlinear modes that arise from branching point bifurcation at certain vibration amplitudes in the forced case these nonlinear modes give rise to a complex dynamic behavior many bifurcations can take place thus leading to strong or weak localization that may or may not be stable in this study special attention has been paid to the influence of excitation on dynamic responses several cases of excitation have been analyzed herein localized excitation and low engine order excitation in the case of low engine order excitation sensitivity of the response to a perturbation of this excitation type has been investigated and it has been shown that for a localized or sufficiently detuned excitation several solutions can coexist some of which are represented by closed curves in the frequency amplitude domain these various solutions overlap when increasing the force amplitude leading to forced nonlinear localization because closed curves are not tied up with the basic nonlinear solution they can easily be overlooked in this study they have been calculated using a sequential continuation with the force amplitude as a parameter

a sufficient condition for the convergence of the inexact uzawa algorithm for saddle point problems in this paper the convergence property of the inexact uzawa algorithm for solving symmetric indefinite linear systems is studied a simple sufficient condition for the convergence of the inexact uzawa algorithm is obtained two examples and numerical experiments illustrating the conclusion are provided

painleve analysis of the non linear schrodinger family of equations the authors apply the painleve test to the generalised derivative non linear schrodinger equation iut uxx iauu ux ibu2ux cu3u 2 where u denotes the complex conjugate of u and a b and c are real constants to determine under what conditions the equation might be completely integrable it is shown that apart from a trivial multiplicative factor this equation possesses the painleve property for partial differential equations as formulated by weiss tabor and carnevale 1983 only if c 1 4b 2b a when then this relation holds this is equivalent under a gauge transformation to the derivative non linear schrodinger equation dnls of kaup and newell which is known to be completely integrable or else to a linear equation

non linear steady state vibration of frames by finite element method a finite element method based on the virtual work principle to determine the steady state response of frams in free or forced periodic vibration is introduced the axial and flexural deformations are coupled by mean of the induced axial force along the element the spatial discretization of the deformations is achieved by the usual finite element method and the time discretization by fourier coefficients of the nodal displacements no unconventional element matrices are needed after applying the harmonic balance method a set of non linear algebraic equations of the fourier coefficients is obtained these equations are solved by the newtonian iteration method in terms of the fourier coefficient increments nodal damping can easily be included by a diagonal damping matrix the direct numerical determination of the fourier coefficient increments is difficult owing to the presence of peaks loops and discontinuities of slope along the amplitude frequency response curves parametric construction of the response curves using the phase difference between the response and excitation is recommended to provide more points during the rapid change of the phase i e at resonance for undamped natural vibration the method of selective coefficients adopted numerical examples on the duffing equation a hinged hinged beam a clamped hinged beam a ring and a frame are given for reasonably accurate results it is shown that the number of finite elements must be sufficient to predict at least the linear mode at the frequency of interest and the number of harmones considered must satisfy the conditions of completeness and balanceability which are discussed in detail

some uzawa methods for steady incompressible navier stokes equations discretized by mixed element methods some uzawa type iterative methods are devised to solve steady incompressible navier stokes equations discretized by mixed element methods compared with the most existing iterative methods these methods require no numerical solution of any saddle point system at each iteration step after some novel and technical derivation it is proved that the methods converge geometrically with a contraction number independent of the finite element mesh size h even for regular triangulations a series of numerical experiments are reported to show the computational performance and accuracy of our methods proposed

bifurcations and internal resonances in space curved rods abstract the paper discusses a geometrically non linear model of space curved beams the timoshenko type model is used which includes the shear effects and rotatory inertia the motion of the system is described by a non linear matrix equation which accounts for non linearities up to the second order to solve the non linear problem discretization methods based on the galerkin method are used and the resulting non linear eigenvalue problem is then solved by the continuation methods the analysis of the discretized problem allows the study of the bifurcations of solutions the post bifurcation analysis is used to explain the phenomenon of internal resonance which is of much importance in the prediction of the dynamic response of the analyzed systems the approach is illustrated by numerical examples which consider the free vibration of beams and circular arches with different boundary conditions

convergence behavior of generalized parameterized uzawa method for singular saddle point problems in this paper we will seek the least squares solution for singular saddle point problems the parameterized uzawa pu method is further studied and a generalized pu gpu proper splitting is proposed the convergence behavior of the corresponding gpu iteration is studied it is proved that the gpu iteration method can converge to the best least squares solutions of the singular saddle point problems in addition we prove that the gpu preconditioned gmres for singular saddle point problems will also determine the least squares solution at breakdown the eigenvalue distributions of the gpu preconditioned matrix are derived numerical experiments are presented which show that the convergence behavior of the singular preconditioning is significantly better than that of the corresponding nonsingular case and demonstrate that the gpu iteration has better convergence behavior than the pu iteration both as a solver and a preconditioner of gmres

geometrical non linear steady state forced periodic vibration of plates part ii stability study and analysis of multi modal response the steady state geometrically non linear periodic vibration of rectangular thin plates under harmonic external excitations is analyzed using the hierarchical finite element and the harmonic balance methods modal coupling due to internal resonance is detected and the consequent multi modal and multi frequency response is demonstrated the stability of the obtained solutions is investigated by studying the evolution of perturbations to the solutions and using floquet s theory

uzawa type algorithms for nonsymmetric saddle point problems in this paper we consider iterative algorithms of uzawa type for solving linear nonsymmetric saddle point problems specifically we consider systems written as usual in block form where the upper left block is an invertible linear operator with positive definite symmetric part such saddle point problems arise for example in certain finite element and finite difference discretizations of navier stokes equations oseen equations and mixed finite element discretization of second order convection diffusion problems we consider two algorithms each of which utilizes a preconditioner for the operator in the upper left block convergence results for the algorithms are established in appropriate norms the convergence of one of the algorithms is shown assuming only that the preconditioner is spectrally equivalent to the inverse of the symmetric part of the operator the other algorithm is shown to converge provided that the preconditioner is a sufficiently accurate approximation of the inverse of the upper left block applications to the solution of steady state navier stokes equations are discussed and finally the results of numerical experiments involving the algorithms are presented

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

a geometrically non linear model of rotating shafts with internal resonance and self excited vibration a geometrically non linear model of the rotating shaft is introduced which includes karman non linearity non linear curvature effects large displacements and rotations as well as gyroscopic effects through applying timoshenko type assumptions the shear effects are also included in the model convenient matrix descriptions are used in order to facilitate the analysis based on galerkin and continuation methods the model is used to analyze the phenomenon of internal resonance the influence of some of the system s parameters on the amplitude and frequency of self excited vibration is investigated

geometrical non linear steady state forced periodic vibration of plates part i model and convergence studies a model for the steady state geometrically non linear periodic vibration of thin rectangular plates under harmonic external excitation is presented the equations of motion in the time domain are derived by applying the principle of virtual work and the hierarchical finite element method hfem these equations are transformed into the frequency domain by the harmonic balance method hbm and are solved by a continuation method the convergence properties of the model are discussed by applying it to isotropic and to composite laminated plates

elements finis mixtes incompressibles pour l equation de stokes dans ℝ3 we introduce some new families of finite element approximation for the stationary stokes and navier stokes equations in a bounded domain in 3 these elements can used tetahedrons or cubes the approximation satisfie exactly the incompressibility condition

on convergence and semi convergence of ssor like methods for augmented linear systems abstract in this paper we analyze the convergence and semi convergence of a class of ssor like methods with four real functions for augmented systems the class takes most existed ssor like methods as its special cases for nonsingular systems we obtain the minimum of convergence factors of all the ssor like methods in the class and study when it can be reached by the convergence factors of methods in the class by considering the equivalence of methods we show that most of the existed ssor like methods have the same minimum of convergence factors

on local hermitian and skew hermitian splitting iteration methods for generalized saddle point problems in this paper we first present a local hermitian and skew hermitian splitting lhss iteration method for solving a class of generalized saddle point problems the new method converges to the solution under suitable restrictions on the preconditioning matrix then we give a modified lhss mlhss iteration method and further extend it to the generalized saddle point problems obtaining the so called generalized mlhss gmlhss iteration method numerical experiments for a model navier stokes problem are given and the results show that the new methods outperform the classical uzawa method and the inexact parameterized uzawa method

on pss based constraint preconditioners for singular nonsymmetric saddle point problems for singular nonsymmetric saddle point problems a class of singular constraint preconditioners and the corresponding preconditioning iteration methods are proposed based on the positive definite and skew symmetric splitting pss of 1 1 block of the coefficient matrix which can be categorized into a recently proposed generalized constraint preconditioner gcp for singular nonsymmetric saddle point problems the convergence analysis of the preconditioned iteration method is presented and the convergence conditions are derived some numerical experiments are implemented to demonstrate the feasibility and effectiveness of the pss based constraint preconditioning both as a preconditioned iteration method and a preconditioner of the krylov subspace method

a generalization of the local hermitian and skew hermitian splitting iteration methods for the non hermitian saddle point problems abstract for large sparse saddle point problems jiang and cao studied a class of local hermitian and skew hermitian splitting lhss iteration methods see m q jiang y cao on local hermitian and skew hermitian splitting iteration methods for generalized saddle point problems j comput appl math 231 2009 973 982 in this paper we generalized these methods and propose a class of generalized local hermitian and skew hermitian splitting glhss iteration schemes for solving the non hermitian saddle point problems we derive conditions for guaranteeing the convergence of these iterative methods with different choices of the parameter matrices the generalized iterative methods lead to a series of existing and new iterative methods numerical experiments for a model stokes problem are provided further show that the new iteration methods are feasible and effective

corrected uzawa methods for solving large nonsymmetric saddle point problems abstract in this paper we consider the solution of linear systems of large nonsymmetric saddle point problems by two nonlinear iterative methods which are similar in some respects to the uzawa type methods and are called corrected uzawa methods their convergence rates are analyzed the results of numerical experiments are presented when we apply them to navier stokes equations discretized by mixed finite elements

aor uzawa iterative method for a class of complex symmetric linear system of equations abstract in this paper based on accelerated overrelaxation aor method and uzawa method we present aor uzawa iterative method for solving a broad class of complex symmetric linear systems we investigate its convergence properties in addition the resulting aor uzawa preconditioner is used to precondition krylov subspace methods such as the restarted generalized minimal residual gmres method for solving the real equivalent formulation of the complex symmetric system finally two numerical examples are given to show the effectiveness of the proposed method

balancing domain decomposition for mixed finite elements the rate of convergence of the balancing domain decomposition method applied to the mixed finite element discretization of second order elliptic equations is analyzed the balancing domain decomposition method introduced by mandel in 24 is a substructuring method that involves at each iteration the solution of a local problem with dirichlet data a local problem with neumann data and a coarse grid problem to propagate information globally and to insure the consistency of the neumann problems it is shown that the condition number grows at worst like the logarithm squared of the ratio of the subdomain size to the element size in both two and three dimensions and for elements of arbitrary order the bounds are uniform with respect to coefficient jumps of arbitrary size between subdomains the key component of our analysis is the demonstration of an equivalence between the norm induced by the bilinear form on the interface and the h 1 2 norm of an interplant of the boundary data computational results from a message passing parallel implementation on an intel delta machine demonstrate the scalability properties of the method and show almost optimal linear observed speed up for up to 64 processors

on semi convergence of the generalized shift splitting iteration method for singular nonsymmetric saddle point problems recently a class of generalized shift splitting iteration methods were proposed for solving nonsingular nonsymmetric saddle point problems cao et al 2015 in this paper the generalized shift splitting iteration method is extended to solve singular nonsymmetric saddle point problems semi convergence of the generalized shift splitting iteration method is carefully analyzed which shows that the iterative sequence generated by the generalized shift splitting iteration method converges to a solution of the singular saddle point problem unconditionally numerical results verify the robustness and efficiency of the generalized shift splitting iteration method

a class of uzawa pss iteration methods for nonsingular and singular non hermitian saddle point problems in this paper based on the positive definite and skew hermitian splitting iteration scheme and the uzawa iteration method we propose a class of uzawa pss iteration methods for solving nonsingular and singular non hermitian saddle point problems with 1 1 block of the saddle point matrix being non hermitian positive definite we derive conditions of the new iteration method for guaranteeing the convergence for nonsingular saddle point problems and its semi convergence for singular saddle point problems respectively numerical experiments of a model navier stokes problem are presented to illustrate the effectiveness of the uzawa pss iteration method numerical results show that the new method is much efficient than the uzawa hss iteration method for solving both the nonsingular saddle point problems yang and wu 2014 28 and the singular saddle point problems yang li and wu 2015 30

computational formulation for periodic vibration of geometrically nonlinear structures part 1 theoretical background abstract a general computational formulation for geometrically nonlinear structures excited by harmonic forces and executing periodic motion in a steady state is presented the equations of both continuous and discretized models are reformulated to obtain the motion equation in a more suitable form to a further analysis the multi harmonic solution of motion equation is written in a form of truncated fourier series next the galerkin ritz and the harmonic balance method are discussed in a context of their equivalency in derivation of the matrix amplitude equation the matrix amplitude equation as well as the associated tangent matrix are given in an explicit form the stability of steady state solution is discussed by using the floquet theory the numerical algorithm and an example application are described in a companion paper by lewandowski lewandowski r computational formulation for periodic vibration of geometrically nonlinear structures part 2 numerical strategy and numerical examples international journal of solids and structures in preparation

a particle partition of unity method for the solution of elliptic parabolic and hyperbolic pdes in this paper we present a meshless discretization technique for instationary convection diffusion problems it is based on operator splitting the method of characteristics and a generalized partition of unity method we focus on the discretization process and its quality the method may be used as an h version or a p version even for general particle distributions the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid we discuss the implementational aspects of the proposed method furthermore we present the results of numerical examples where we considered instationary convection diffusion instationary diffusion linear advection and elliptic problems

application of the ritz method to the analysis of non linear free vibrations of beams this paper presents an analytical solution for geometrically non linear free vibrations of beams with elastically supported ends in the horizontal direction the equation of motion is obtained by employing hamilton s principle and assuming that horizontal inertia forces can be neglected the ritz method with a continuum solution and an iterative procedure are used for determining the frequencies and non linear modes of vibrations the orthogonality conditions for these modes are also discussed numerical results for various beam boundary conditions are presented and compared with available results

on structure oriented hybrid two stage iteration methods for the large and sparse blocked system of linear equations in this paper we first present a class of structure oriented hybrid two stage iteration methods for solving the large and sparse blocked system of linear equations as well as the saddle point problem as a special case and the new methods converge to the solution under suitable restrictions for instance when the coefficient matrix is positive stable matrix generally numerical experiments for a model generalized saddle point problem are given and the results show that our new methods are feasible and efficient and converge faster than the classical uzawa method

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

on generalized local hermitian and skew hermitian splitting iterative method for block two by two linear systems abstract for large sparse saddle point problems whose 1 2 and 2 1 blocks are the transpose of each other zhu studied a generalized local hermitian and skew hermitian splitting glhss iterative method see zhu 2012 in this paper the glhss method is extended to the block 2 2 linear system which allows that the 1 2 block is not equal to the transpose of the 2 1 block or the 2 2 block is non zero with different choices of the parameter matrices the existing methods are included and the new algorithms for solving the block 2 2 linear system are obtained the conditions for guaranteeing the convergence of the new iterative method are studied and the correctness of the existing corollaries are questioned numerical experiments are provided to show that the proposed method is feasible and effective and it is not necessary to introduce some parameter matrices in some cases

the hamiltonian structures of the nonlinear schrodinger equation in the classical limit using madelung s hydrodynamical variables it is shown that the bi hamiltonian structure of the nonlinear schrodinger equation goes over to a bi hamiltonian structure of the shallow water wave equations in the classical limit

inexact uzawa algorithms for variational inequalities of the second kind in this paper we discuss inexact uzawa algorithms and inexact non linear uzawa algorithms to solve discretized variational inequalities of the second kind we prove convergence results for the algorithms numerical examples are included to show the effectiveness of the algorithms

landau lifshitz and higher order nonlinear systems gauge generated from nonlinear schrodinger type equations new landau lifshitz ll and higher order nonlinear systems gauge generated from nonlinear schrodinger ns type equations are presented the consequences of gauge equivalence between different dynamical systems are discussed the gauge connections among various ll and ns equations are found and depicted through a schematic representation

on a galerkin lagrange multiplier method for the stationary navier stokes equations a galerkin lagrange multiplier formulation is used for the numerical solution of the stationary navier stokes equations in order to avoid the construction of zero divergence elements the formulation is based on different approximating spaces for the velocity field and the pressure optimal rate of convergence estimates are derived moreover a galerkin newton scheme for the solution of the nonlinear equations is shown to be quadratically locally convergent another scheme is shown to be linearly globally convergent

an example of a miura map the shallow water wave equations the linear hamiltonian structure of modified shallow water wave equations goes to the non linear and second hamiltonian structure of the bi hamiltonian shallow water wave equations a similar situation must exist for gas dynamics according to a result of dubrovin and novikov 1983

non linear free periodic vibrations of open cylindrical shallow shells this paper is concerned with the non linear free periodic vibrations of thin open cylindrical and shallow shells vibrating in the geometrically non linear regime a multi degree of freedom model with hierarchical basis functions is adopted and the principle of the virtual work is used to define the time domain equations of motion these equations are transformed into the frequency domain by the harmonic balance method and are finally solved by an arc length continuation method shells of different thicknesses and of different curvature radius are analysed and the variation of the non linear natural frequencies of these shells with the vibration amplitude are investigated in some detail the variation of the mode shapes with the vibration amplitude is demonstrated it is found that both softening and hardening spring effects occur and that the number of couplings between vibration modes is rather large in undamped shells

on the steady state response and stability of non linear oscillators using harmonic balance abstract the harmonic balance hb method is used to study the steady state periodic response as well as its stability of a softening duffing oscillator the objectives of the work are 1 to show how the qualitative nature of the hb solution changes if one uses two harmonics rather than one in the approximate solution and 2 to show that the analysis of the mathieu equation which governs local stability of the steady state response must be based on an approximate rather than an exact analysis of this mathieu equation thus floquet theory incorrectly describes the stability of the approximate solution

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 semi analytical approach to the non linear dynamic response problem of beams at large vibration amplitudes part ii multimode approach to the steady state forced periodic response abstract the semi analytical approach to the non linear dynamic response of beams based on multimode analysis has been presented in part i of this series of papers azrar et al 1999 journal of sound and vibration 224 183 207 1 the mathematical formulation of the problem and single mode analysis have been studied the objective of this paper is to take advantage of applying this semi analytical approach to the large amplitude forced vibrations of beams various types of excitation forces such as harmonic distributed and concentrated loads are considered the governing equation of motion is obtained and can be considered as a multi dimensional form of the duffing equation using the harmonic balance method the equation of motion is converted into non linear algebraic form techniques of solution based on iterative incremental procedures are presented the non linear frequency and the non linear modes are determined at large amplitudes of vibration the basic function contribution coefficients to the displacement response for various beam boundary conditions are calculated the percentage of participation for each mode in the response is presented in order to appraise the relation to higher modes contributing to the solution also the percentage contributions of the higher modes to the bending moment near to the clamps are given in order to determine accurately the error introduced in the non linear bending stress estimated by different approximations solutions obtained in the jump phenomena region have been determined by a careful selection of the initial iteration at each frequency the non linear deflection shapes in various regions of the solution the corresponding axial force ratios and the bending moments are presented in order to follow the behaviour of the beam at large vibration amplitudes the numerical results obtained here for the non linear forced response are compared with those from the linear theory with available non linear results based on various approaches and with the single mode analysis

on quasi shift splitting iteration method for a class of saddle point problems abstract motivated by a technique of scale splitting for matrix we put forward a quasi shift splitting named after qss iteration method to address a class of large scale sparse saddle point problems this novel method also naturally leads to the corresponding qss preconditioner in addition an extrapolated variant of qss named after eqss iteration method is constructed further some useful convergence properties of the qss and eqss iteration methods and spectral properties of qss preconditioner are studied finally a saddle point equations stem from model navier stokes problem is supplied to illustrate the effectiveness of our new two methods and qss preconditioner

on hss based sequential two stage method for non hermitian saddle point problems abstract for large sparse saddle point problems with symmetric positive definite 1 1 block li et al studied an efficient iterative method see li et al 2011 25 by making use of the same preconditioning technique and a new matrix splitting based on the hermitian and skew hermitian splitting hss of the 1 1 block of the preconditioned non hermitian saddle point systems an efficient sequential two stage method is proposed for solving the non hermitian saddle point problems theoretical analysis shows the proposed iterative method is convergent and that the spectral radius of iterative matrix monotonically decreases and tends to 0 as the iterative parameter approaches infinity numerical experiments arising from naiver stokes problem are provided to show that the new iterative method is feasible effective and robust

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

the convergence factor of preconditioned algorithms of the arrow hurwicz type the convergence properties of the arrow hurwicz algorithm for solving mixed finite element problems are studied using a purely algebraic technique the convergence factor of the algorithm will be estimated the iteration parameters depending on the smallest and greatest eigenvalues of some generalized eigenvalue problems can be chosen optimally for special cases the convergence factor of the preconditioned arrow hurwicz algorithm is independent of the discretization parameter finally a quite general method that allows us to construct preconditioning operators of the arrow hurwicz algorithm for a wide class of problems is presented

a class of preconditioned generalized local pss iteration methods for non hermitian saddle point problems in this paper a class of new methods based on the positive definite and skew hermitian splitting scheme called preconditioned generalized local positive definite and skew hermitian splitting pglpss methods are considered to solve non hermitian saddle point problems the convergence properties of the proposed methods are analyzed which prove that the pglpss methods are convergent if the iteration parameters and parameter matrices satisfy appropriate conditions some numerical experiments are provided to verify the efficiency of the proposed method showing the competitiveness and efficiency of this novel method over other testing methods whether it served as a preconditioned iteration method or as a preconditioner to the krylov subspace method

residual reduction algorithms for nonsymmetric saddle point problems in this paper we introduce and analyze uzawa algorithms for non symmetric saddle point systems convergence for the algorithms is established based on new spectral results about schur complements a new uzawa type algorithm with optimal relaxation parameters at each new iteration is introduced and analyzed in a general framework numerical results supporting the efficiency of the algorithms are presented for finite element discretization of steady state navier stokes equations

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

block lu preconditioners for symmetric and nonsymmetric saddle point problems in this paper a general purpose block lu preconditioner for saddle point problems is presented the main difference between the approach presented here and that of other studies is that an explicit accurate approximation of the schur complement matrix is efficiently computed this is used to compute a preconditioner to the schur complement matrix which in turn defines a preconditioner for the global system a number of different variants are developed and results are reported for a few linear systems arising from cfd applications

analysis of the inexact uzawa algorithm for saddle point problems in this paper we consider the so called inexact uzawa algorithm for iteratively solving linear block saddle point problems such saddle point problems arise for example in finite element and finite difference discretizations of stokes equations the equations of elasticity and mixed finite element discretization of second order problems we consider both the linear and nonlinear variants of the inexact uzawa iteration we show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block in the case of nonlinear iteration the inexact uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear uzawa algorithm applications to the stokes equations and mixed finite element discretization of second order elliptic problems are discussed and finally the results of numerical experiments involving the algorithms are presented

modified unsymmetric sor method for saddle point problems in this paper we propose a modified unsymmetric sor musor method for solving singular or nonsingular saddle point problem which is an extension of the ssor iteration method for the musor method we prove its convergence for solving nonsingular saddle point problem and semi convergence for solving singular saddle point problem under suitable conditions moreover we study the choices of the optimal parameters numerical experiments are presented which support our theoretical results and reveal that the new method is effective and feasible for solving both singular and nonsingular saddle point problems

asymmetric solutions in large amplitude free periodic vibrations of plates when the in plane inertia is neglected the equations of motion that represent free large amplitude periodic vibrations of perfect plates only contain cubic nonlinear terms it is here shown that unlike often assumed these equations admit asymmetric solutions for that purpose the time domain equations of motion are mapped into the frequency domain by the harmonic balance method keeping the constant term and the second harmonic in the truncated fourier expansion of the transverse displacements a test plate is analysed and a bifurcation due to a 1 2 internal resonance is found this bifurcation results in a branch of solutions where the equilibrium position is not the flat one and where odd and even harmonics coexist

cascadic multilevel algorithms for symmetric saddle point systems abstract in this paper we introduce a multilevel algorithm for approximating variational formulations of symmetric saddle point systems the algorithm is based on availability of families of stable finite element pairs and on the availability of fast and accurate solvers for symmetric positive definite systems on each fixed level an efficient solver such as the gradient or the conjugate gradient algorithm for inverting a schur complement is implemented the level change criterion follows the cascade principle and requires that the iteration error be close to the expected discretization error we prove new estimates that relate the iteration error and the residual for the constraint equation the new estimates are the key ingredients in imposing an efficient level change criterion the first iteration on each new level uses information about the best approximation of the discrete solution from the previous level the theoretical results and experiments show that the algorithms achieve optimal or close to optimal approximation rates by performing a non increasing number of iterations on each level even though numerical results supporting the efficiency of the algorithms are presented for the stokes system the algorithms can be applied to a large class of boundary value problems including first order systems that can be reformulated at the continuous level as symmetric saddle point problems such as the maxwell equations

on a new class of completely integrable nonlinear wave equations ii multi hamiltonian structure the multi hamiltonian structure of a class of nonlinear wave equations governing the propagation of finite amplitude waves is discussed infinitely many conservation laws had earlier been obtained for these equations starting from a primary hamiltonian formulation of these equations the necessary and sufficient conditions for the existence of bi hamiltonian structure are obtained and it is shown that the second hamiltonian operator can be constructed solely through a knowledge of the first hamiltonian function the recursion operator which first appears at the level of bi hamiltonian structure gives rise to an infinite sequence of conserved hamiltonians it is found that in general there exist two different infinite sequences of conserved quantities for these equations the recursion relation defining higher hamiltonian structures enables one to obtain the necessary and sufficient conditions for the existence of the k 1 st hamiltonian operator which depends on the kth hamiltonian function the infinite sequence of conserved hamiltonians are common to all the higher hamiltonian structures the equations of gas dynamics are discussed as an illustration of this formalism and it is shown that in general they admit tri hamiltonian structure with two distinct infinite sets of conserved quantities the isothermal case of 1 is an exceptional one that requires separate treatment this corresponds to a specialization of the equations governing the expansion of plasma into vacuum which will be shown to be equivalent to poisson s equation in nonlinear acoustics

model reduction for dynamical systems with local nonlinearities many kinds of nonlinear engineering systems have a large number of degrees of freedom while nonlinear components are spatially localized even though the local nonlinearities constitute only a small part of the system the dynamic behavior of the system is wholly nonlinear and the analysis of the whole model is very expensive to reduce the computational effort signie cantly two model reduction schemes are proposed to reduce the size of the nonlinear model before the analysis is performed one is dee ned on the system level and the other is on the substructure level they are both based on the dynamic condensation technique in the former scheme the dynamic condensation technique is implemented into the linear part of the whole system directly whereas it is applied to the linear e exible substructures in the later scheme the accuracy and stability of the former is usually better than the latter however it is more computationally expensive than the latter two numerical examples are also included the results show that the present two schemes are feasible and efe cient to reduce the degrees of freedom of the model with local nonlinearities

optimization of nonlinear structural resonance using the incremental harmonic balance method abstract we present an optimization procedure for tailoring the nonlinear structural resonant response with time harmonic loads a nonlinear finite element method is used for modeling beam structures with a geometric nonlinearity and the incremental harmonic balance method is applied for accurate nonlinear vibration analysis an optimization procedure based on a gradient based algorithm is developed and we use the adjoint method for efficient computation of design sensitivities we consider several examples in which we find optimized beam width distributions that minimize or maximize fundamental or super harmonic resonant responses

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

multigrid methods for variational problems general theory for the v cycle in two earlier papers siam j numer anal 19 1982 pp 924 929 21 1984 pp 255 262 we developed an algebraic convergence theory for a class of multigrid methods applied to positive definite self adjoint linear operator equations the purpose of the present paper is to extend these results by eliminating an earlier approximation order restriction developing additional rate estimates and allowing for very general relaxation schemes including those that are nonstationary and nonsymmetric these results apply to most well known iterative methods and preconditioners

non linear vibration of beams with internal resonance by the hierarchical finite element method the hierarchical nite element hfem and the harmonic balance methods hbm are used to investigate the geometrically non linear free and steady state forced vibrations of uniform slender beams the beam analogue of von kah rmah n s non linear strain displacement relationships are employed and the middle plane in plane displacements are included in the model the equations of motion are developed by applying the principle of virtual work and are solved by a continuation method 1 3 and 1 5 internal resonances are discovered and their consequences are discussed the convergence properties of the hfem are analyzed and it is demonstrated that the hfem model requires far fewer degrees of freedom than the h version of the fem models presented in the literature 1999 academic press

a preconditioned glhss iteration method for non hermitian singular saddle point problems in this paper we propose a new preconditioned generalized local hermitian and skew hermitian splitting glhss iteration method for solving the non hermitian saddle point problems the semi convergence of this method is discussed theoretical analysis shows that the semi convergence of this new method can be guaranteed by suitable choices of the parameters and parameter matrices numerical examples are used to illustrate the theoretical results and examine the numerical effectiveness of the glhss iteration method served either as a preconditioner for gmres or as a solver

free vibration of structures with cubic non linearity remarks on amplitude equation and rayleigh quotient in the paper several aspects concerning the formulation of and solution to amplitude equations for free vibration of systems with cubic non linearity are discussed in detail the use of several possible constraint equations to achieve the non trivial solution to the amplitude equation is also presented the well known concept of the rayleigh quotient is extended to the non linear case of interest some general remarks on the formulation of the non linear amplitude equation and the non linear rayleigh quotient are also presented moreover the introduced rayleigh quotient is used to develop a new and suitably efficient method to solve amplitude equations the accuracy of solutions obtained by means of some widely used methods of analysis of free vibration problems is also presented it is found that the galerkin method gives the most accurate results

analysis of the relaxed deteriorated pss preconditioner for singular saddle point linear systems abstract the relaxed deteriorated pss rdpss preconditioner proposed by cao et al 13 is further generalized to solve singular saddle point linear systems properties of the rdpss splitting and convergence of the corresponding iteration method are studied by using the moore penrose inverse the rdpss iteration method and the rdpss preconditioned gmres are both proved to converge to the least squares solution of the singular saddle point linear system the rdpss preconditioned matrix is also analyzed results about eigenvalue distributions are derived moreover the rdpss preconditioner is generalized to a class of more general preconditioners the corresponding convergence and eigenvalue distribution results are analyzed numerical experiments are presented to illustrate the effectiveness of the rdpss preconditioner and its variants to accelerate gmres for solving singular saddle point linear systems

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

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

dynamic analysis of frames with viscoelastic dampers modelled by rheological models with fractionalderivatives frame structures with viscoelastic dampers mounted on them are considered in this paper viscoelastic ve dampers are modelled using two three parameter fractional rheological models the structures are treated as elastic linear systems the equation of motion of the whole system structure with dampers is written in terms of state space variables the resulting matrix equation of motion is the fractional differential equation the proposed state space formulation is new and does not require matrices with huge dimensions the paper is devoted to determine the dynamic properties of the considered structures the nonlinear eigenvalue problem is formulated from which the dynamic parameters of the system can be determined the continuation method is used to solve the nonlinear eigenvalue problem moreover results of typical calculations are presented

a further study on nonlinear vibration of initially stressed plates abstract numerical solutions of an arbitrary initially stressed plate based on various displacement fields under nonlinear vibration are presented nonlinear partial differential equations of plate vibration are formulated from lo s high order transverse shear and transverse normal deformation plate theory the higher order terms for lo s displacement field are neglected to obtain various simpler forms of equations such as the first order theory and other higher order theories for isotropic plate these nonlinear partial equations of different forms are then transformed by the galerkin method to ordinary nonlinear differential equations the runge kutta method is used to obtain the ratio of nonlinear frequency to linear frequencies of vibration by using these equations the nonlinear vibration of simply supported initially stress plates with various plate theories are investigated the frequency responses of nonlinear vibration are sensitive of the vibration amplitude poisson ratio initial stress and plate theory the study concludes that the surprising discrepancies exist among the various plate theories which indicate the transverse normal strain nonuniform shear stress and initial stress have great effect on the vibration behavior of plate under nonlinear vibration

new choices of preconditioning matrices for generalized inexact parameterized iterative methods for large sparse saddle point problems chen and jiang recently studied a class of generalized inexact parameterized iterative methods see f chen y l jiang a generalization of the inexact parameterized uzawa methods for saddle point problems appl math comput 206 2008 765 771 in this paper the methods are modified and some choices of preconditioning matrices are given these preconditioning matrices have advantages in solving large sparse linear system numerical experiments of a model stokes problem are presented

pressure schur complement preconditioners for the discrete oseen problem we consider several preconditioners for the pressure schur complement of the discrete steady oseen problem two of the preconditioners are well known from the literature and the other is new supplemented with an appropriate approximate solve for an auxiliary velocity subproblem these approaches give rise to a family of the block preconditioners for the matrix of the discrete oseen system in the paper we critically review possible advantages and difficulties of using various schur complement preconditioners we recall existing eigenvalue bounds for the preconditioned schur complement and prove such for the newly proposed preconditioner these bounds hold both for lbb stable and stabilized finite elements results of numerical experiments for several model two dimensional and three dimensional problems are presented in the experiments we use lbb stable finite element methods on uniform triangular and tetrahedral meshes one particular conclusion is that in spite of essential improvement in comparison with simple scaled mass matrix preconditioners in certain cases none of the considered approaches provides satisfactory convergence rates in the case of small viscosity coefficients and a sufficiently complex e g circulating advection vector field

splittings of symmetric matrices and a question of ortega a complete answer is given to a problem posed in 1988 by ortega concerning convergent splittings of symmetric matrices

the uzawa pps iteration methods for nonsingular and singular non hermitian saddle point problems abstract in this paper based on the positive definite and positive semidefinite splitting pps iteration scheme we establish a class of uzawa pps iteration methods for solving nonsingular and singular non hermitian saddle point problems with the 1 1 part of the coefficient matrix being non hermitian positive definite theoretical analyses show that the convergence and semi convergence properties of the proposed methods can be guaranteed under suitable conditions furthermore we consider acceleration of the uzawa pps methods by krylov subspace like gmres methods and discuss the spectral properties of the corresponding preconditioned matrix numerical experiments are given to confirm the theoretical results which show that the feasibility and effectiveness of the proposed methods and preconditioners

analysis of some mixed finite element methods related to reduced integration we prove error estimates for the following two mixed finite element methods related to reduced integration a method for stokes problem using rectangular elements with piecewise bilinear approximations for the velocities and piecewise constants for the pressure and one method for a plate problem using bilinear approximations for transversal displacement and rotations and piecewise constants for the shear stress the main idea of the proof in the case of stokes problem is to combine a weak babuska brezzi type stability estimate for the pressure with a superapproximability property for the velocities a similar technique is used in the case of the plate problem

on parameter acceleration methods for saddle point problems for solving saddle point problems parameter acceleration methods which include uzawa type methods are investigated by many researchers in the literature in this paper we introduce the inexact uzawa method with another parameter acceleration that is the so called momentum acceleration method for solving saddle point problems we discuss the convergence conditions of the inexact uzawa iteration with momentum acceleration and give the optimal momentum factors which minimize the spectral radii of the associated iteration matrices numerical results demonstrate the effectiveness of the inexact uzawa method with momentum acceleration and the mixed parameter acceleration methods for solving saddle point problems

simple like preconditioners for saddle point problems from the steady navier stokes equations the semi implicit method for pressure linked equations simple preconditioner is further generalized to a class of simple like sl preconditioners for solving saddle point problems from the steady navier stokes equations the sl preconditioners can be also viewed as a generalization of the relaxed deteriorated pss rdpss preconditioner proposed by cao et al 2015 convergence analysis of the corresponding sl iteration is presented and the optimal iteration parameter is obtained by minimizing the spectral radius of the sl iteration matrix moreover krylov subspace acceleration of the sl preconditioning is studied the sl preconditioned saddle point matrix is analyzed results about eigenvalue and eigenvector distributions and the minimal polynomial are derived numerical experiments from leaky two dimensional lid driven cavity problems are given which show the advantages of the sl preconditioned gmres over the dpss and rdpss preconditioned ones for solving saddle point problems