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

domain decomposition for least squares finite element methods for the stokes equations in this paper we consider the solutions of the discrete linear system resulting from least squares finite element discretization applied to the stokes equations additive and multiplicative domain decomposition methods are used to solve the linear system and the uniform convergence results are established for both methods

a parallel domain decomposition algorithm of mixed element equation for second order elliptic dirichlet boundary value problem a parallel domain decomposition algorithm of schwarz type is introduced to solve the mixed finite element equation for the dirichlet boundary value problem of a second order partial differential equation of elliptic type and convergence of this algorithm is analyzed

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

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

an application of the abstract multilevel theory to nonconforming finite element methods a multilevel product algorithm and its additive version or a multilevel preconditioner are proposed for the crouzeix raviart nonconforming finite element methods for second order elliptic equations the authors show that the preconditioned matrix has a condition number bounded uniformly from above if the domain can be partitioned into overlapping convex subdomains a uniform convergence for the multilevel product algorithm is also established the results are regularity free for the continuous problem and can be applied to problems with rough solutions and local grid refinement numerical results that illustrate the theory are presented

on schwarz alternating methods for the incompressible navier stokes equations the schwarz alternating method can be used to solve linear elliptic boundary value problems on domains which consist of two or more overlapping subdomains the solution is approximated by an infinite sequence of functions which result from solving a sequence of elliptic boundary value problems in each of the subdomains this paper considers four schwarz alternating methods for the n dimensional steady viscous incompressible navier stokes equations n leq 4 it is shown that the schwarz sequences converge to the true solution provided that the reynolds number is sufficiently small

a wavelet based method for solving discrete first kind fredholm equations with piecewise constant solutions the inverse problem of finding piecewise constant solutions to discrete fredholm integral equations of the first kind arises in many applied fields e g in geophysics this equation is usually an ill posed problem when it is considered in a hilbert space framework requiring regularization techniques to control arbitrary error amplifications and to get adequate solutions in this work we describe an iterative regularizing method for computing piecewise constant solutions to first kind discrete fredholm integral equations the algorithm involves two main steps at each iteration 1 approximating the solution using a new signal reconstruction algorithm from its wavelet maxima which involves a previous step of detecting discontinuities by estimation of its local holder exponents and 2 obtaining a regularized solution of the original equation using the a priori knowledge and the above approximation in order to check the behaviour of the proposed technique we have carried out a statistical study from a high number of simulations obtaining excellent results their comparisons with the results coming from using classical tikhonov regularization by the multiresolution support total variation tv regularization and piecewise polynomial truncated singular value decomposition pp tsvd algorithm serve to illustrate the advantages of the new method copyright 2003 john wiley sons ltd

rate of convergence for some constraint decomposition methods for nonlinear variational inequalities some general subspace correction algorithms are proposed for a convex optimization problem over a convex constraint subset one of the nontrivial applications of the algorithms is the solving of some obstacle problems by multilevel domain decomposition and multigrid methods for domain decomposition and multigrid methods the rate of convergence for the algorithms for obstacle problems is of the same order as the rate of convergence for jump coefficient linear elliptic problems in order to analyse the convergence rate we need to decompose a finite element function into a sum of functions from the subspaces and also satisfying some constraints a special nonlinear interpolation operator is introduced for decomposing the functions

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

multilevel schwarz methods we consider the solution of the algebraic system of equations which result from the discretization of second order elliptic equations a class of multilevel algorithms are studied using the additive schwarz framework we establish that the condition number of the iteration operators are bounded independent of mesh sizes and the number of levels this is an improvement on dryja and widlund s result on a multilevel additive schwarz algorithm as well as bramble pasciak and xu s result on the bpx algorithm some multiplicative variants of the multilevel methods are also considered we establish that the energy norms of the corresponding iteration operators are bounded by a constant less than one which is independent of the number of levels for a proper ordering the iteration operators correspond to the error propagation operators of certain v cycle multigrid methods using gauss seidel and damped jacobi methods as smoothers respectively

two level additive schwarz methods for a discontinuous galerkin approximation of second order elliptic problems we present some two level nonoverlapping and overlapping additive schwarz methods for a discontinuous galerkin method for solving second order elliptic problems it is shown that the condition numbers of the preconditioned systems are of the order o frac h h for the nonoverlapping schwarz methods and of the order o frac h delta for the overlapping schwarz methods where h and h stand for the fine mesh size and the coarse mesh size respectively and delta denotes the size of the overlaps between subdomains numerical experiments are provided to gauge the efficiency of the methods and to validate the theory

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

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

an anisotropic h adaptive finite element method for compressible navier stokes equations abstract application of the finite element method with significantly stretched elements to solve compressible navier stokes equations is presented stretched elements are introduced in an adaptive process as a result of selective directional subsecting of elements an adaptive strategy of such anisotropic h type refinements was developed it is based on minimization of the interpolation errors reduction of the residual error indicators and an adaptive recovery of the viscous fluxes with the possibility to control their accuracy an iterative solver with the convergence characteristics independent of the aspect ratio of elements is presented the solver is the gmres algorithm with the domain decomposition schwarz type preconditioner the proposed techniques are used to solve a benchmark problem a hypersonic high reynolds number flow with strong shock boundary layer interaction

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

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

parallel galerkin domain decomposition procedures based on the streamline diffusion method for convection diffusion problems based upon the streamline diffusion method parallel galerkin domain decomposition procedures for convection diffusion problems are given these procedures use implicit method in the sub domains and simple explicit flux calculations on the inter boundaries of sub domains by integral mean method or extrapolation method to predict the inner boundary conditions thus the parallelism can be achieved by these procedures the explicit nature of the flux calculations induces a time step limitation that is necessary to preserve stability artificial diffusion parameters d are given by analysis optimal order error estimate is derived in a norm which is stronger than l 2 norm for these procedures this error estimate not only includes the optimal h 1 norm error estimate but also includes the error estimate along the streamline direction b u u which cannot be achieved by standard finite element method experimental results are presented to confirm theoretical results

an additive analysis of multiplicative schwarz methods we present an analysis of multiplicative schwarz methods for symmetric positive definite problems that is based on the theory of additive schwarz preconditioners and discuss applications to multigrid methods and domain decomposition methods

global and uniform convergence of subspace correction methods for some convex optimization problems this paper gives some global and uniform convergence estimates for a class of subspace correction based on space decomposition iterative methods applied to some unconstrained convex optimization problems some multigrid and domain decomposition methods are also discussed as special examples for solving some nonlinear elliptic boundary value problems

convergence estimates for product iterative methods with applications to domain decomposition in this paper we consider iterative methods for the solution of symmetric positive definite problems on a space which are defined in terms of products of operators defined with respect to a number of subspaces the simplest algorithm of this sort has an error reducing operator which is the product of orthogonal projections onto the complement of the subspaces new normreduction estimates for these iterative techniques will be presented in an abstract setting applications are given for overlapping schwarz algorithms with many subregions for finite element approximation of second order elliptic problems

parallel galerkin domain decomposition procedures for wave equation parallel galerkin domain decomposition procedures for wave equation are given these procedures use implicit method in the sub domains and simple explicit flux calculation on the inter boundaries of sub domains by integral mean method or extrapolation method thus the parallelism can be achieved by these procedures the explicit nature of the flux prediction induces a time step constraint that is necessary to preserve the stability l 2 norm error estimates are derived for these procedures experimental results are presented to confirm the theoretical results

schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems part i algorithms and numerical results we describe sequential and parallel algorithms based on the schwarz alternating method for the solution of mixed finite element discretizations of elliptic problems using the raviart thomas finite element spaces these lead to symmetric indefinite linear systems and the algorithms have some similarities with the traditional block gauss seidel or block jacobi methods with overlapping blocks the indefiniteness requires special treatment the sub blocks used in the algorithm correspond to problems on a coarse grid and some overlapping subdomains and is based on a similar partition used in an algorithm of dryja and widlund for standard elliptic problems if there is sufficient overlap between the subdomains the algorithm converges with a rate independent of the mesh size the number of subdomains and discontinuities of the coefficients extensions of the above algorithms to the case of local grid refinement is also described convergence theory for these algorithms will be presented in a subsequent paper

multigrid convergence for second order elliptic problems with smooth complex coefficients the finite element method when applied to a second order partial differential equation in divergence form can generate operators that are neither hermitian nor definite when the coefficient function is complex valued for such problems under a uniqueness assumption we prove the continuous dependence of the exact solution and its finite element approximations on data provided that the coefficients are smooth and uniformly bounded away from zero then we show that a multigrid algorithm converges once the coarse mesh size is smaller than some fixed number providing an efficient solver for computing discrete approximations numerical experiments while confirming the theory also reveal pronounced sensitivity of gauss seidel iterations on the ordering of the unknowns for certain problems

maximum norm analysis of overlapping nonmatching grid discretizations of elliptic equations in this paper we provide a maximum norm analysis of a finite difference scheme defined on overlapping nonmatching grids for second order elliptic equations we consider a domain which is the union of p overlapping subdomains where each subdomain has its own independently generated grid the grid points on the subdomain boundaries need not match the grid points from adjacent subdomains to obtain a global finite difference discretization of the elliptic problem we employ standard stable finite difference discretizations within each of the overlapping subdomains and the different subproblems are coupled by enforcing continuity of the solutions across the boundary of each subdomain by interpolating the discrete solution on adjacent subdomains if the subdomain finite difference schemes satisfy a strong discrete maximum principle and if the overlap is sufficiently large we show that the global discretization converges in optimal order corresponding to the largest truncation errors of the local interpolation maps and discretizations our discretization scheme and the corresponding theory allows any combination of lower order and higher order finite difference schemes in different subdomains we describe also how the resulting linear system can be solved iteratively by a parallel schwarz alternating method or a schwarz preconditioned krylov subspace iterative method several numerical results are included to support the theory

convergence of algebraic multigrid based on smoothed aggregation we prove aconvergence estimate for the algebraic multigrid method with prolongations defined by aggregation using zero energy modes followed by a smoothing the method input is the problem matrix and a matrix of the zero energy modes the estimate depends only polylogarithmically on the mesh size and requires only a weak approximation property for the aggregates which can be a priori verified computationally construction of the prolongator in the case of a general second order system is described and the assumptions of the theorem are verified for a scalar problem discretized by linear conforming finite elements

schwarz methods of neumann neumann type for three dimensional elliptic finite element problems several domain decomposition methods of neumann neumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite elements we will only consider problems in three dimensions several new variants of the basic algorithm are introduced in a schwarz method framework that provides tools which have already proven very useful in the design and analysis of other domain decomposition and multi level methods the neumann neumann algorithms have several advantages over other domain decomposition methods the subregions which define the subproblems only share the boundary degrees of freedom with their neighbors the subregions can also be of quite arbitrary shape and many of the major components of the preconditioner can be constructed from subprograms available in standard finite element program libraries in its original form however the algorithm lacks a mechanism for global transportation of information and its performance therefore suffers when the number of subregions increases in the new variants of the algorithms considered in this paper the preconditioners include global components of low rank to overcome this difficulty bounds are established for the condition number of the iteration operator which are independent of the number of subregions and depend only polylogarithmically on the number of degrees of freedom of individual local subproblems results are also given for problems with arbitrarily large jumps in the coefficients across the interfaces separating the subregions 1995 john wiley sons inc

on the existence of cook semantics in siam j comput 7 1978 pp 70 90 cook defines the operational semantics of a programming language in the following way a function is introduced which takes a program r and a state sigma and yields a possibly infinite row of intermediate states as a result this row is meant to be the trace resulting from executing program r starting in state sigma this function is characterized by a number of equations however it is not immediately clear whether these equations have a solution in this paper we show for a simple language the most sophisticated feature of which is that it has parameterless procedures that the corresponding equations have a unique solution the techniques used here can also be applied to other languages described in the same way for instance to the language in cook s paper

convergence rate analysis of an asynchronous space decomposition method for convex minimization we analyze the convergence rate of an asynchronous space decomposition method for constrained convex minimization in a reflexive banach space this method includes as special cases parallel domain decomposition methods and multigrid methods for solving elliptic partial differential equations in particular the method generalizes the additive schwarz domain decomposition methods to allow for asynchronous updates it also generalizes the bpx multigrid method to allow for use as solvers instead of as preconditioners possibly with asynchronous updates and is applicable to nonlinear problems applications to an overlapping domain decomposition for obstacle problems are also studied the method of this work is also closely related to relaxation methods for nonlinear network flow accordingly we specialize our convergence rate results to the above methods the asynchronous method is implementable in a multiprocessor system allowing for communication and computation delays among the processors

additive schwarz methods for parabolic problems based on domain decomposition we give an additive schwarz domain decomposition method for semi linear parabolic problems we consider the dependence of convergence rate of this algorithm on parameters of time step and space mesh we give the error estimate which tell us that the convergence of the approximate solution is independent of the iteration number at each time level

analysis of a multigrid algorithm for time harmonic maxwell equations this paper considers a multigrid algorithm suitable for efficient solution of indefinite linear systems arising from finite element discretization of time harmonic maxwell equations in particular a backslash multigrid cycle is proven to converge at rates independent of refinement level if certain indefinite block smoothers are used the method of analysis involves comparing the multigrid error reduction operator with that of a related positive definite multigrid operator this idea has previously been used in multigrid analysis of indefinite second order elliptic problems however the maxwell application involves a nonelliptic indefinite operator with the help of a few new estimates the earlier ideas can still be applied some numerical experiments with lowest order nedelec elements are also reported

local multilevel methods for adaptive finite element methods for nonsymmetric and indefinite elliptic boundary value problems in this paper we propose some local multilevel algorithms for solving linear systems arising from adaptive finite element approximations of nonsymmetric and indefinite elliptic boundary value problems two types of local smoothers are constructed one is based on the original nonsymmetric problems and the other is defined in terms of the associated symmetric problems it is shown that the local multilevel methods for the nonsymmetric and indefinite elliptic boundary value problems are optimal which means that the convergence rates of the local multilevel methods are independent of mesh sizes and mesh levels provided that the coarsest grid is sufficiently fine numerical experiments are reported to confirm our theory

numerical analysis of vanka type solvers for steady stokes and navier stokes flows we consider vanka type smoothers for solving stokes and navier stokes problems in each iteration step this smoother requires the solution of several small local subproblems over finite element blocks it is shown that for particular choices for the blocks the algorithm always converges to the solution of the stokes problem and under suitable conditions to the solution of the navier stokes problem the convergence properties are analyzed and numerical examples are presented

domain decomposition for a least square finite element method for second order elliptic problem in this paper we consider the solutions of the discrete linear system resulting from least square mixed finite element discretization applied to a second order elliptic problems additive and multiplicative domain decomposition methods are used to solve the linear system and the uniform convergence results are established for both methods

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

analysis of a parallel schwarz algorithm for elliptic problems a parallel algorithm which is derived directly from the schwarz alternating algorithm with an overlapping domain decomposition for solving boundary value problems of elliptic partial differential equations is analyzed a new general analysis framework different from lions ones is established to describe the recursive relations of error functions between approximate solutions and the exact solution at the different iterative steps some abstract theories on convergence and convergent rate of the synchronously parallel algorithm are established under different conditions

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