the computations of the critical groups with an application to elliptic resonant problems at a higher eigenvalue in this paper we study the existence of a nontrivial solution of the elliptic problem which is a resonant type both at the origin and at infinity with the same higher eigenvalue and the nonlinear term is permitted to be unbounded our main ingredient is the computation of the critical groups

multiplicity results for asymptotically linear elliptic problems at resonance in this paper several new multiplicity results for asymptotically linear elliptic problem at resonance are obtained via morse theory and minimax methods some new observations on the critical groups of a local linking type critical point are used to deal with the resonance case at 0

multiple periodic solutions of ordinary differential equations with double resonance abstract in this paper we study by morse theory the existence of multiple periodic solutions of a class of ordinary differential equation with double resonance at infinity between two consecutive eigenvalues and with resonance at origin

the analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems we prove some new estimates for the convergence of multigrid algorithms applied to nonsymmetric and indefinite elliptic boundary value problems we provide results for the so called symmetric multigrid schemes we show that for the variable v script cycle and the w script cycle schemes multigrid algorithms with any amount of smoothing on the finest grid converge at a rate that is independent of the number of levels or unknowns provided that the initial grid is sufficiently fine we show that the v script cycle algorithm also converges under appropriate assumptions on the coarsest grid but at a rate which may deteriorate as the number of levels increases this deterioration for the v script cycle may occur even in the case of full elliptic regularity finally the results of numerical experiments are given which illustrate the convergence behavior suggested by the theory

multigrid methods for two dimensional maxwell s equations on graded meshes in this paper we investigate the numerical solution for two dimensional maxwell s equations on graded meshes the approach is based on the hodge decomposition the solution u of maxwell s equations is approximated by solving standard second order elliptic problems quasi optimal error estimates for both u and i u in the l 2 norm are obtained on graded meshes we prove the uniform convergence of the w cycle and full multigrid algorithms for the resulting discrete problem the performance of these methods is illustrated by numerical results

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

efficient iterative solution of the three dimensional helmholtz equation we examine preconditioners for the discrete indefinite helmholtz equation on a three dimensional box shaped domain with sommerfeld like boundary conditions the preconditioners are of two types the first is derived by discretization of a related continuous operator that differs from the original only in its boundary conditions the second is derived by a block toeplitz approximation to the descretized problem the resulting preconditioning matrices allow the use of fast transform methods and differ from the discrete helmholtz operator by an operator of low rank we present experimental results demonstrating that when these methods are combined with krylov subspace iteration convergence rates depend only mildly on both the wave number and discretization mesh size in addition the methods display high efficiencies in an implementation on an ibm sp 2 parallel computer

algebraic study of multigrid methods for symmetric definite problems abstract we present a convergence theory based on a simple algebraic assumption which permits us to analyze multilevel iterative methods for symmetric positive definite linear systems coarse grid problems are derived variationally we prove fast convergence of v and w cycles with any positive number of smoothing steps under a discrete analogue of the h 2 and h 1 ga regularity assumption respectively and for a wide class of smoothings including arbitrarily ordered gauss seidel and steepest descent

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

new convergence estimates for multigrid algorithms in this paper new convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems our theory relates the convergence of multigrid algorithms to a regularity and approximation parameter cap alpha epsilon 0 1 and the number of relaxations m we show that for the symmetric and nonsymmetric nu cycles the multigrid iteration converges for any positive m at a rate which deteriorates no worse than 1 cj sup 1 sup cap alpha sup sup cap alpha where j is the number of grid levels we then define a generalized nu cycle algorithm which involves exponentially increasing for example doubling the number of smoothings on successively coarser grids we show that the resulting symmetric and nonsymmetric multigrid iterations converge for any cap alpha with rates that are independent of the mesh size the theory is presented in an abstract setting which can be applied to finite element multigrid and finite difference multigrid methods

norm estimates of product operators with application to domain decomposition in this paper we consider norm estimates for operators of the product form e i t j i t 1 i t 0 in an abstract hilbert space by imposing appropriate assumptions on t j explicit norm estimates for e are established we then apply the abstract theory to convergence analysis of domain decomposition for nonsymmetric and indefinite second order elliptic problems

convergence estimates for multigrid algorithms without regularity assumptions a new technique for proving rate of convergence estimates of multi grid algorithms for symmetric positive definite problems will be given in this paper the standard multigrid theory requires a regularity and approxima tion assumption in contrast the new theory requires only an easily verified approximation assumption this leads to convergence results for multigrid re finement applications problems with irregular coefficients and problems whose coefficients have large jumps in addition the new theory shows why it suffices to smooth only in the regions where new nodes are being added in multigrid refinement applications

multiplicity of solutions for a class of the quasilinear elliptic equations at resonance abstract in this paper by morse theory we obtain the existence and multiplicity for a class of the quasilinear elliptic equations at resonance

an algebraic theory for multigrid methods for variational problems a convergence theory is developed for multigrid methods for symmetric positive definite problems in a variational setting the theory is based on a single algebraic approximation assumption which is satisfied for finite element discretizations of elliptic boundary value problems although the theory can be applied to problems without any continuous background as well in contrast to previous results we prove fast convergence with any positive number of smoothing steps for v and w cycles under discrete analogues of the h 2 and h 1 alpha regularity assumptions respectively we analyze a wide class of smoothers including arbitrary symmetric and nonsymmetric preconditioned iterations arbitrarily ordered gauss seidel steepest descent chebyshev iteration and conjugate gradients our estimates exhibit the usual asymptotic behavior for a large number of smoothing steps

fourier analysis of a robust multigrid method for convection diffusion equations we consider a two grid method for solving 2d convection diffusion problems the coarse grid correction is based on approximation of the schur complement as a preconditioner of the schur complement we use the exact schur complement of modified fine grid equations we assume constant coefficients and periodic boundary conditions and apply fourier analysis we prove an upper bound for the spectral radius of the two grid iteration matrix that is smaller than one and independent of the mesh size the convection diffusion ratio and the flow direction i e we have a strong robustness result numerical results illustrating the robustness of the corresponding multigrid w cycle are given

a learning approach to the two person decentralized team problem with incomplete information a learning approach to the two person decentralized team problem with incomplete information and with a 2x2 payoff matrix is considered it is shown that if the payoff matrix is unimodal there exists a proper choice of parameters of the learning algorithm that will ensure asymptotically an expected payoff as close to maximum payoff as desired it is shown that a team problem with multimodal payoff matrix gives rise to interesting class of open problems

applications of local linking to critical point theory we prove the existence of nontrivial critical points of functionals with a local linking at 0 applications to elliptic boundary value problems and to hamiltonian systems are given c 1995 academic press inc

multigrid convergence for nonsymmetric indefinite variational problems and one smoothing step we prove that the multigrid method for linear systems arising from nonsymmetric and indefinite elliptic variational problems converges for all sufficiently small h with one smoothing step only previous proofs required that the number of smoothing steps be within certain bounds which were quite difficult to compute our theory is based on sharp energy norm convergence estimates for the symmetric definite case and lower order terms are treated as perturbations the resulting estimates are so sharp that we obtain even the convergence of the v cycle

the convergence of multi level methods for solving finite element equations in the presence of singularities the known convergence proofs for multi level methods assume the quasi uniformity of the family of domain triangulations used such triangulations are not suitable for problems with singularities caused by re entrant corners and abrupt changes in the boundary condi tions in this paper it is shown that families of properly refined grids yield the same convergence behavior of multi level methods for such singular problems as quasi uniform subdivisions do for h2 regular problems 1 the continuous problem its discretization and the multi level method using multi level techniques 1 3 4 6 7 8 10 it is possible to solve the large systems of linear equations arising in connection with finite element methods with an amount of work roughly proportional to the number of unknowns this property makes multi level methods at least theoretically superior to all other solution methods including fast solvers based on fft like algorithms which may be directly applied to special problems only the convergence of multi level methods was proved for example by nicolaides 10 bank and dupont 4 and hackbusch 8 all these proofs assume a certain amount of elliptic regularity of the continuous problem to be solved approximately and quasi uniform subdivisions of the domain in finite elements assuming h1 a regularity such quasi uniform triangulations and a jacobi like smoothing procedure bank and dupont 4 and hackbusch 8 showed the following result the rate of convergence of a full iteration step of the multi level method behaves like o m 0 2 uniformly in the number of levels for a growing number m of smoothing steps per level in the optimal case a 1 the problem has to be h2 regular this means for example that the region is not allowed to have re entrant corners if this condition is violated the convergence rate of the multi level procedure decreases and in addition the approximation properties of the finite element discretization itself change for the worse because of the presence of singularities in the solution not captured by the quasi uniform grids the strongly nonuniform systematically re fined triangulations suitable for these problems are not included in the theory so far the aim of the present paper is to fill this gap

nonconforming multigrid method for nonsymmetric and indefinite problems abstract the convergence of w cycle multigrid algorithms applied to nonsymmetric and indefinite elliptic boundary value problems using p 1 nonconforming finite elements is proved

a new approach to nonlinear partial differential equations abstract the author s decomposition method for the solution of operator equations which may be nonlinear and or stochastic is generalized to multidimensional cases

convergence analysis of a multigrid method for convection diffusion equations this paper is concerned with the convergence analysis of robust multigrid methods for convection diffusion problems we consider a finite difference discretization of a 2d model convection diffusion problem with constant coefficients and dirichlet boundary conditions for the approximate solution of this discrete problem a multigrid method based on semicoarsening matrix dependent prolongation and restriction and line smoothers is applied for a multigrid w cycle we prove an upper bound for the contraction number in the euclidean norm which is smaller than one and independent of the mesh size and the diffusion convection ratio for the contraction number of a multigrid v cycle a bound is proved which is uniform for a class of convection dominated problems the analysis is based on linear algebra arguments only

the analysis of smoothers for multigrid algorithms the purpose of this paper is to provide a general technique for defining and analyzing smoothing operators for use in multigrid algorithms the smoothing operators considered are based on subspace decomposition and include point line and block versions of jacobi and gauss seidel iteration as well as generalizations we shall show that these smoothers will be effective in multigrid algorithms provided that the subspace decomposition satisfies two simple conditions in many applications these conditions are trivial to verify

sharp estimates for multigrid rates of convergence with general smoothing and acceleration in this paper we prove the convergence of the multilevel iterative method for solving linear equations that arise from elliptic partial differential equations our theory is presented entirely in terms of the generalized condition number kappa of the matrix a and the smoothing matrix b this leads to a completely algebraic analysis of the method as an iterative technique for solving linear equations the properties of the elliptic equation and discretization procedure enter only when we seek to estimate kappa just as in the case of most standard iterative methods here we consider the fundamental two level iteration and the v and w cycles of the j level iteration j 2 we prove that the v and w cycles converge even when only one smoothing iteration is used we present several examples of the computation of kappa using both fourier analysis and standard finite element techniques we compare the predictions of our theorems with the actual rate of convergence our analysis also shows that

analysis and comparison of geometric and algebraic multigrid for convection diffusion equations the discrete convection diffusion equations obtained from streamline diffusion finite element discretization are solved on both uniform meshes and adaptive meshes estimates of error reduction rates for both geometric multigrid gmg and algebraic multigrid amg are established on uniform rectangular meshes for a model problem our analysis shows that gmg with line gauss seidel smoothing and bilinear interpolation converges if h gg epsilon 2 3 and amg with the same smoother converges more rapidly than gmg if the interpolation constant beta in the approximation assumption of amg satisfies beta ll frac h sqrt epsilon alpha hs 1mm mbox where alpha big begin smallmatrix 1 h sqrt varepsilon 2 h ge sqrt varepsilon end smallmatrix on unstructured triangular meshes the performance of gmg and amg both as solvers and as preconditioners for gmres are evaluated numerical results show that gmres with amg preconditioning is a robust and reliable solver on both type of meshes

a parallel geometric multigrid method for finite elements on octree meshes in this article we present a parallel geometric multigrid algorithm for solving variable coefficient elliptic partial differential equations on the unit box with dirichlet or neumann boundary conditions using highly nonuniform octree based conforming finite element discretizations our octrees are 2 1 balanced that is we allow no more than one octree level difference between octants that share a face edge or vertex we describe a parallel algorithm whose input is an arbitrary 2 1 balanced fine grid octree and whose output is a set of coarser 2 1 balanced octrees that are used in the multigrid scheme also we derive matrix free schemes for the discretized finite element operators and the intergrid transfer operations the overall scheme is second order accurate for sufficiently smooth right hand sides and material properties its complexity for nearly uniform trees is mathcal o frac n n p log frac n n p mathcal o n p log n p where n is the number of octree nodes and n p is the number of processors our implementation uses the message passing interface standard we present numerical experiments for the laplace and navier linear elasticity operators that demonstrate the scalability of our method our largest run was a highly nonuniform 8 billion unknown elasticity calculation using 32 000 processors on the teragrid system ranger at the texas advanced computing center our implementation is publically available in the dendro library which is built on top of the petsc library from argonne national laboratory

nontrivial critical points for asymptotically quadratic function abstract in this paper we study the existence of nontrivial critical points as applications we study the existence of nontrivial solutions of the dirichelet problem for nonlinear elliptic pde and periodic boundary value problem for nonlinear ope

a unified convergence theory for abstract multigrid or multilevel algorithms serial and parallel multigrid methods are analyzed in the style of standard iterative methods a basic error bound is derived in terms of residuals on neighboring levels the terms in this bound derive from the iterative methods used as smoothers on each level and the operators used to go from a level to the next coarser level this bound is correct whether the underlying operator is symmetric or nonsymmetric definite or indefinite and singular or nonsingular this paper allows any iterative method as a smoother or rougher in the multigrid cycle while standard multigrid error analysis typically assumes a specific multigrid cycle e g a v w or f cycle analysis for arbitrary multigrid cycles including adaptively chosen ones is provided this theory applies directly to aggregation disaggregation methods used to solve systems of linear equations

adomian s method of decomposition critical review and examples of failure abstract adomian s method of decomposition is considered in application to initial boundary value problems for the one space dimensional spatially homogeneous heat conduction equation it is shown that the fundamental equation of the method is well defined only for certain restricted types of boundary conditions within the class of such boundary conditions examples are given such that the fundamental equation fails to have a unique solution and such that the sequence produced by iteration of this equation is divergent the latter is a counterexample to a published assertion of convergence

convergence and optimality of adaptive mixed finite element methods the convergence and optimality of adaptive mixed finite element methods for the poisson equation are established in this paper the main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality a quasi orthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result this discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation

multiple periodic solutions of superlinear ordinary differential equations with a parameter abstract we study the existence of multiple 2 periodic solutions of ordinary differential equation x x f t x with superlinear terms via homological linking and morse theory

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

a w cycle algorithm for a weakly over penalized interior penalty method we study a w cycle multigrid algorithm for a weakly over penalized interior penalty method for the two dimensional poisson problem on convex polygonal domains

preconditioning projection nonconforming element method for the lowest order raviart thomas mixed triangular element method this paper is concerned with the h 1 condition number and the b h singular value distribution of preconditioned operators b h 1a h 0 h 1 where a h is the projection nonconforming element discretization of the nonself adjoint and possibly indefinite second order elliptic operator a b h is the nonconforming element discretization of the self adjoint and positive definite second order elliptic operator b the projection nonconforming element method is equivalent to the lowest order raviart thomas mixed triangular element method for second order elliptic problems it is proved that the h 1 condition number of operators b h 1a h 0 h 1 are uniformly bounded and the b h singular values cluster in a positive finite interval finally a simple v cycle multigrid implementation of b h 1 is 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

artificial damping techniques for scalar waves in the frequency domain abstract an artificial damping algorithm for solving the helmholtz problem is considered when the imaginary part of the wave number is small the problem is known to be difficult to solve in this paper we propose an efficient artificial damping algorithm which can be viewed as a rational iteration each damped problem is solved incompletely by a nonoverlapping domain decomposition method convergence arguments are presented numerical results run on an ncube2 are presented to demonstrate the effectiveness of the method

uniform convergence of multigrid v cycle iterations for indefinite and nonsymmetric problems in this paper an analysis of a multigrid method for nonsymmetric and or indefinite elliptic problems is presented in this multigrid method various types of smoothers may be used one type of smoother considered is defined in terms of an associated symmetric problem and includes point and line jacobi and gauss seidel iterations smoothers based entirely on the original operator are also considered one smoother is based on the normal form that is the product of the operator and its transpose other smoothers studied include point and line jacobi and gauss seidel it is shown that the uniform estimates of j h bramble and j e pasciak math comp 60 1993 pp 447 471 for symmetric positive definite problems carry over to these algorithms more precisely the multigrid iteration for the nonsymmetric and or indefinite problem is shown to converge at a uniform rate provided that the coarsest grid in the multilevel iteration is sufficiently fine but not dependent on the number of multigrid levels

a multigrid method enhanced by krylov subspace iteration for discrete helmholtz equations standard multigrid algorithms have proven ineffective for the solution of discretizations of helmholtz equations in this work we modify the standard algorithm by adding gmres iterations at coarse levels and as an outer iteration we demonstrate the algorithm s effectiveness through theoretical analysis of a model problem and experimental results in particular we show that the combined use of gmres as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as 200 for fixed wave numbers it displays grid independent convergence rates and has costs proportional to the number of unknowns

critical point theorem for asymptotically quadratic functional without compactness in this paper we build an abstract critical point theorem for the asymptotically quadratic functional without compactness application is devoted to the existence of nontrivial periodic solution of second order hamiltonian systems

a comparison of two multilevel iterative methods for nonsymmetric and indefinite elliptic finite element equations we describe and analyze two multilevel iterative procedures for solving linear systems arising from certain finite element discretizations of nonself adjoint and indefinite elliptic partial differential equations we prove convergence and optimal work estimate for both schemes

multiple periodic solutions of the second order hamiltonian systems with superlinear terms abstract we study the existence of 2 periodic solutions of the second order hamiltonian systems x a t x x v x t x with superlinear terms and with saddle structure near the origin some multiplicity results are obtained by using bifurcation method homological linking and morse theory

nontrivial periodic solutions for asymptotically linear resonant difference problem in this paper the existence of nontrivial periodic solution for second order asymptotically linear difference equation at resonance is obtained the methods used here are based on combining the minimax methods and the morse theory especially the observation on the critical groups

a new convergence proof for the multigrid method including the v cycle for a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the v cycle the presented proof applies to procedures with any number of smoothing iterations and to the v cycle

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

inversion of nonlinear stochastic operators abstract the operator theoretic method adomian and malakian j math anal appl 76 1 1980 183 201 recently extended adomian s solutions of nonlinear stochastic differential equations g adomian stochastic systems analysis in applied stochastic processes nonlinear stochastic differential equations j math anal appl 55 1 1976 441 452 on the modeling and analysis of nonlinear stochastic systems in proceeding international conf on mathematical modeling vol 1 pp 29 40 to provide an efficient computational procedure for differential equations containing polynomial exponential and trigonometric nonlinear terms n y the procedure depends on the calculation of certain quantities an and bn this paper generalizes the calculation of the an and bn to much wider classes of nonlinearities of the form n y y essentially the method provides a systematic computational procedure for differential equations containing any nonlinear terms of physical significance this procedure depends on a recurrence rule from which explicit general formulae are obtained for the quantities an and bn for any order n in a convenient form this paper also demonstrates the significance of the iterative series decomposition proposed by adomian for linear stochastic operators in 1964 and developed since 1976 for nonlinear stochastic operators since both the nonlinear and stochastic behavior is quite general the results are extremely significant for applications processes need not for example be limited to either gaussian processes white noise or small fluctuations

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

convergence analysis of a multigrid method for a convection dominated model problem the paper presents a convergence analysis of a multigrid solver for a system of linear algebraic equations resulting from the discretization of a convection diffusion problem using a finite element method we consider piecewise linear finite elements in combination with a streamline diffusion stabilization we analyze a multigrid method that is based on canonical intergrid transfer operators a direct discretization approach for the coarse grid operators and a smoother of line jacobi type a robust diffusion and h independent bound for the contraction number of the two grid method and the multigrid w cycle are proved for a special class of convection diffusion problems namely with neumann conditions on the outflow boundary dirichlet conditions on the rest of the boundary and a flow direction that is constant and aligned with gridlines our convergence analysis is based on modified smoothing and approximation properties the arithmetic complexity of one multigrid iteration is optimal up to a logarithmic term

multigrid method for solving convection diffusion problems with dominant convection a modification of the multigrid method for the solution of linear algebraic equation systems with a strongly nonsymmetric matrix obtained after difference approximation of the convection diffusion equation with dominant convection is proposed specially created triangular iterative methods have been used as the smoothers of the multigrid method some theoretical and numerical results are presented

multigrid methods for variational problems further results in an earlier paper siam j numer anal 19 1982 pp 924 929 we developed a convergence theory for a class of multigrid methods applied to positive definite self adjoint differential boundary value problems where the multigrid process used w cycling and richardson s iteration in the present paper we extend this theory to include v cycling and more general relaxation schemes such as jacobi and arbitrarily ordered gauss seidel

convergence analysis of the schwarz algorithm and multilevel decomposition iterative methods ii nonselfadjoint and indefinite elliptic problems convergence estimate for the schwarz alternating algorithm applied to nonselfadjoint and indefinite elliptic problems is established for a certain overlapping domain decomposition multilevel iterative algorithms are proposed and analyzed for the finite element method for the same problem with locally refined grids the author shows that the algorithm based on the hierarchical basis decomposition converges with rate bounded by 1 cj 2 where j is the number of levels the author also analyzes the algorithm based on the full multigrid decomposition and shows that the algorithm converges with rate 1 cj 1 for problems of any dimension all the results are established without regularity assumption for the differential equation

iterative schemes for nonsymmetric and indefinite elliptic boundary value problems the purpose of this paper is twofold the first is to describe some simple and robust iterative schemes for nonsymmetric and indefinite elliptic boundary value problems the schemes are based in the sobolev space h omega and require minimal hypotheses the second is to develop algorithms utilizing a coarse grid approximation this leads to iteration matrices whose eigenvalues lie in the right half of the complex plane in fact for symmetric indefinite problems the iteration is reduced to a well conditioned symmetric positive definite system which can be solved by conjugate gradient interation applications of the general theory as well as numerical examples are given 20 refs 8 tabs

the analysis of pmhss multigrid methods for elliptic problems with smooth complex coefficients in this paper we investigate a class of multigrid methods with hss smoothers applied to second order elliptic problems whose dominant coefficient is complex valued these smoothers are based on hermitian skew hermitian splitting hss modified hss mhss and preconditioned mhss pmhss iterations it is shown that the abstract theory for multigrid method can be extended and applied to elliptic problems with complex coefficient under some assumption conditions we prove that multigrid method with pmhss smoother is convergent and optimal numerical results are reported to confirm the theoretical analysis

anharmonic oscillator systems abstract the anharmonic oscillator is solved quickly easily and elegantly by adomian s methods for solution of nonlinear stochastic differential equations emphasizing its applicability to nonlinear deterministic equations as well as stochastic equations no difficulty is encountered in treating the case of the forced anharmonic oscillator or the stochastic case or of any nonlinear oscillating system such as the duffing or van der pol oscillators for example with coefficients as well as forcing functions which are stochastic processes since statistical separability is inherent in the adomian method