a continuous framework for feti dp with a mesh independent condition number for the dual problem abstract we present an abstract setting for a variation of the feti dp method applied to second order elliptic problems in this setting the matrix iteration for the dual problem has a mesh independent condition number this work improves the ideas recently introduced by bernardi et al c bernardi t chacon rebollo e chacon vera a feti method with a mesh independent condition number for the iteration matrix comput methods appl mech engrg 197 13 16 2008 1410 1429 the continuity restrictions for the displacement field across interfaces are weakly imposed using the natural trace norm instead of pointwise continuity although a slight coupling between subdomains is introduced still a parallel computation is possible some numerical tests are also performed to illustrate the method

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

discrete solitons in nonlinear schrodinger lattices with a power law nonlinearity abstract we study the discrete nonlinear schrodinger lattice model with the onsite nonlinearity of the general form u 2 u we systematically verify the conditions for the existence and stability of discrete solitons in the one dimensional version of the model predicted by means of the variational approximation va and demonstrate the following monostability of fundamental solitons fss in the case of the weak nonlinearity 2 1 3 68 bistability in a finite range of values of the soliton s power for 3 68 2 1 5 and the presence of a threshold minimum norm of the fs for 2 1 5 we also perform systematic numerical simulations to study higher order solitons in the same general model i e bound states of the fss while all in phase bound states are unstable stability regions are identified for antisymmetric double solitons and their triple counterparts these numerical findings are supplemented by an analytical treatment of the stability problem which allows quantitively accurate predictions for the stability features of such multipulses when these waveforms are found to be unstable we show by means of direct simulations that they self trap into a persistent lattice breather or relax into a stable fs or sometimes decay completely

embedded solitons in lagrangian and semi lagrangian systems we develop the technique of the variational approximation va for solitons in two directions first one may have a physical model which does not admit the usual lagrangian representation as some terms were discarded for various reasons for instance the second harmonic generation shg model considered here which includes the kerr nonlinearity lacks the usual lagrangian representation if one ignores the kerr nonlinearity of the second harmonic as compared to that of the fundamental however we show that with a natural modification one may still apply the va to those seemingly flawed systems as efficiently as it applies to their fully lagrangian counterparts we call such models that do not admit the usual lagrangian representation semi lagrangiansystems second we show that upon adding an infinitesimal tail that does not vanish at infinity to a usual soliton ansatz one can obtain an analytical criterion which within the framework of va gives a condition for finding embedded solitons ess i e isolated truly localized solutions existing inside the continuous spectrum of the radiation modes the criterion takes a form of orthogonality of the radiation mode in the infinite tail to the soliton core to test the criterion we have applied it to both the semi lagrangian truncated version of the shg model and to the same model in its full form in the former case the criterion combined with va for the soliton proper yields an exact solution for the es in the latter case the criterion selects the es with a relative error 1 2003 elsevier b v all rights reserved

generalized browder s and weyl s theorems for left and right multiplication operators abstract the main objective of this work is to study generalized browder s and weyl s theorems for the multiplication operators l a and r b and for the elementary operator a b l a r b

generalized anderson s inequality abstract let h be a separable infinite dimensional complex hilbert space and let b h denote the algebra of operators on h into itself we study the elementary operator b h b h defined by x axb cxd where a and c respectively b and d are nonzero normal commuting operators we prove that i for all n the kernel of and for all b or ii for all n c the von neumann schatten class 1 b such that c if and only if n n n n 0

the nsbeti method an extension of the feti method to non symmetrical bem fem coupled problems summary the finite element tearing and interconnecting feti method is recognized as an effective domain decomposition tool to achieve scalability in the solution of partitioned second order elasticity problems in the boundary element tearing and interconnecting beti method a direct extension of the feti algorithm to the bem the symmetric galerkin bem formulation is used to obtain symmetric system matrices making possible to apply the same feti conjugate gradient solver in this work we propose a new beti variant labeled nsbeti that allows to couple substructures modeled with the fem and or non symmetrical bem formulations the method connects non matching bem and fem subdomains using localized lagrange multipliers and solves the associated non symmetrical flexibility equations with a bi cgstab iterative algorithm scalability issues of nsbeti in bem bem and combined bem fem coupled problems are also investigated copyright 2012 john wiley sons ltd

a feti preconditioner for two dimensional edge element approximations of maxwell s equations on nonmatching grids a class of feti methods for the mortar approximation of a vector field problem in two dimensions is proposed edge element discretizations of lowest degree are considered the method proposed can be employed with geometrically conforming and nonconforming partitions our numerical results show that its condition number increases only with the number of unknowns in each subdomain and is independent of the number of subdomains and the size of the problem

a numerically scalable domain decomposition method for the solution of frictionless contact problems we present a domain decomposition method with lagrange multipliers for solving iteratively frictionless contact problems this method which is based on the feti method and therefore is named here the feti c method incorporates a coarse contact system that guides the iterative prediction of the active zone of contact we demonstrate numerically that this method is numerically scalable with respect to both the problem size and the number of subdomains copyright 2001 john wiley sons ltd

the closure of the range of an elementary operator abstract let b h denote the algebra of operators on a hilbert space h and let b b h be the elementary operator defined by x axb cxd a necessary condition for 1 0 b h b h is that 0 is an isolated point of the spectrum of we prove a sufficient condition for 1 0 b h b h applied to the case in which the hyponormal a b and normal c d satisfy certain conditions it is seen that the condition 0 is isolated in the set s a b c and d is sufficient for 1 0 b h b h

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

dual primal feti methods for three dimensional elliptic problems with heterogeneous coefficients in this paper certain iterative substructuring methods with lagrange multipliers are considered for elliptic problems in three dimensions the algorithms belong to the family of dual primal finite element tearing and interconnecting feti methods which recently have been introduced and analyzed successfully for elliptic problems in the plane the family of algorithms for three dimensions is extended and a full analysis is provided for the new algorithms particular attention is paid to finding algorithms with a small primal subspace since that subspace represents the only global part of the dual primal preconditioner it is shown that the condition numbers of several of the dual primal feti methods can be bounded polylogarithmically as a function of the dimension of the individual subregion problems and that the bounds are otherwise independent of the number of subdomains the mesh size and jumps in the coefficients these results closely parallel those of other successful iterative substructuring methods of primal as well as dual type

a parallel implementation of dual primal feti methods for three dimensional linear elasticity using a transformation of basis dual primal feti methods for linear elasticity problems in three dimensions are considered these are nonoverlapping domain decomposition methods where some primal continuity constraints across subdomain boundaries are required to hold throughout the iterations whereas most of the constraints are enforced by lagrange multipliers an algorithmic framework for dual primal feti methods is described together with a transformation of basis to implement the primal constraints numerical results obtained from a parallel implementation of these algorithms applied to a model benchmark problem with structured meshes and to problems with more complicated geometries from industrial and biological applications using unstructured meshes are provided these results show that the presented dual primal feti algorithms are numerical and parallel scalable

variational approximation and the use of collective coordinates we consider propagating spatially localized waves in a class of equations that contain variational and nonvariational terms the dynamics of the waves is analyzed through a collective coordinate approach motivated by the variational approximation we show that there is a natural choice of projection onto collective variables for reducing the governing nonlinear partial differential equation pde to coupled ordinary differential equations odes this projection produces odes whose solutions are exactly the stationary states of the effective lagrangian that would be considered in applying the variational approximation method we illustrate our approach by applying it to a modified fisher equation for a traveling front containing a non constant coefficient nonlinear term we present numerical results that show that our proposed projection captures both the equilibria and the dynamics of the pde much more closely than previously proposed projections

optimal convergence properties of the feti domain decomposition method abstract the finite element tearing and interconnecting feti method is a practical and efficient domain decomposition dd algorithm for the solution of self adjoint elliptic partial differential equations for large scale structural problems discretized with shell and beam elements this method was found to outperform popular iterative algorithms and direct solvers on both serial and parallel computers and to compare favorably with leading dd methods in this paper we discuss some numerical properties of the feti method that were not addressed before in particular we show that the mathematical treatment of the floating subdomains and the specific conjugate projected gradient algorithm that characterize the feti method are equivalent to the construction and solution of a coarse problem that propagates the error globally accelerates convergence and ensures a performance that is independent of the number of subdomains we also show that when the interface problem is optimally preconditioned and the mesh is partitioned into well structured subdomains with good aspect ratios the performance of the feti method is also independent of the mesh size however we also argue that the feti and other leading dd methods for unstructured problems lose in practice these scalability properties when the mesh contains junctures with rotational degrees of freedom or the decomposition is irregular and characterized by arbitrary subdomain aspect ratios finally we report that for realistic problems optimal preconditioners are not necessarily computationally efficient and can be outperformed by non optimal ones

an optimization based domain decomposition method for partial differential equations abstract an optimization based domain decomposition method for the solution of partial differential equations is presented the crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains the constraints are the partial differential equations the existence of optimal solutions for the optimization problem is shown as is the convergence to the exact solution of the given problem we then derive an optimality system of partial differential equations from which solutions of the domain decomposition problem may be determined finite element approximations to solutions of the optimality system are defined and analyzed as well as an eminently parallelizable gradient method for solving the optimality system then the results of some numerical experiments and some concluding remarks are given the latter includes the extension of the method to nonlinear problems such as the navier stokes equations

schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations

bddc and feti dp without matrices or vectors abstract the connection between the bddc balancing domain decomposition by constraints algorithm and the feti dp finite element tearing and interconnecting dual primal algorithm is discussed in the language of function spaces operators and linear functionals

a numerical study of feti algorithms for mortar finite element methods the finite element tearing and interconnecting feti method is an iterative substructuring method using lagrange multipliers to enforce the continuity of the finite element solution across the subdomain interface mortar finite elements are nonconforming finite elements that allow for a geometrically nonconforming decomposition of the computational domain into subregions and at the same time for the optimal coupling of different variational approximations in different subregions we present a numerical study of feti algorithms for elliptic self adjoint equations discretized by mortar finite elements several preconditioners which have been successful for the case of conforming finite elements are considered we compare the performance of our algorithms when applied to classical mortar elements and to a new family of biorthogonal mortar elements and we discuss the differences between enforcing mortar conditions instead of continuity conditions for the case of matching nodes across the interface our experiments are carried out for both two and three dimensional problems and include a study of the relative costs of applying different preconditioners for mortar elements

on the convergence of a dual primal substructuring method in the dual primal feti method introduced by farhat et al 1999 the domain is decomposed into non overlapping subdomains but the degrees of freedom on crosspoints remain common to all subdomains adjacent to the crosspoint the continuity of the remaining degrees of freedom on subdomain interfaces is enforced by lagrange multipliers and all degrees of freedom are eliminated the resulting dual problem is solved by preconditioned conjugate gradients we give an algebraic bound on the condition number assuming only a single inequality in discrete norms and use the algebraic bound to show that the condition number is bounded by c 1 log 2 h h for both second and fourth order elliptic selfadjoint problems discretized by conforming finite elements as well as for a wide class of finite elements for the reissner mindlin plate model

an additive schwarz preconditioner for the feti method a new additive schwarz preconditioner for the finite element tearing and interconnecting feti method is analyzed in this paper this preconditioner has the unique feature that the coefficient matrix of its coarse grid problem is mesh independent for a model second order heterogeneous elliptic boundary value problem in two dimensions the condition number of the preconditioned system is shown to be bounded by c 1 ln h h 2 where h is the mesh size h is the typical diameter of the subdomains and the constant c is independent of h h the number of subdomains and the coefficients of the boundary value problem

a domain decomposition method for isogeometric multi patch problems with inexact local solvers abstract in isogeometric analysis the computational domain is often described as multi patch where each patch is given by a tensor product spline nurbs parametrization in this work we propose a feti like solver where local inexact solvers exploit the tensor product structure at the patch level to this purpose we extend to the isogeometric framework the so called all floating variant of feti that allows us to use the fast diagonalization method at the patch level we construct then a preconditioner for the whole system and prove its quasi robustness with respect to the local mesh size h and patch size h precisely the condition number of the preconditioned system is bounded by the square of the logarithm of h h our numerical tests confirm the theory and also show a favourable dependence of the computational cost of the method from the spline degree p

on the validity of the variational approximation in discrete nonlinear schrodinger equations abstract the variational approximation is a well known tool to approximate localized states in nonlinear systems in the context of a discrete nonlinear schrodinger equation with a small coupling constant we prove error estimates for the variational approximations of site symmetric bond symmetric and twisted discrete solitons this is shown for various trial configurations which become increasingly more accurate as more parameters are taken it is also shown that the variational approximation yields the correct spectral stability result and controls the oscillatory dynamics of stable discrete solitons for long but finite time intervals

elementary operators finite ascent range closure and compactness abstract given banach space operators a i b i b x 1 i 2 let ab b b x denote the elementary operator ab x a 1 x b 1 a 2 x b 2 then ab has finite ascent 1 for a number of fairly general choices of the operators a i and b i this information is applied to prove some necessary and sufficient conditions for the range of ab to be closed and in deciding conditions on the tuples a 1 a 2 and b 1 b 2 so that ab n x compact for some integer n 1 and operator x implies ab x compact this generalizes some well known results of anderson and foias 4 and yosun 25 also considered is the question what is a necessary and sufficient condition on the tuples a 1 a 2 b 1 b 2 and ab for ab n to be compact for some integer n 1

a robin type domain decomposition method with red black partition in this paper we present a new robin type nonoverlapping domain decomposition dd preconditioner the unknown variables to be solved in this preconditioned algebraic system are the robin transmission data on the interface which are different from the well known dd methods like substructuring nonoverlapping dd method and feti method through choosing suitable parameter on each subdomain boundary and using the tool of energy estimate for the second order elliptic problem we prove that the condition number of the preconditioned system is c 1 log frac h h 2 where h is the coarse mesh size and h is the fine mesh size moreover in this paper we always assume that there is a red black partition for the whole domain omega numerical results are given to illustrate the efficiency of our dd preconditioner

internal modes of discrete solitons near the anti continuum limit of the dnls equation discrete solitons of the discrete nonlinear schrodinger dnls equation are compactly supported in the anti continuum limit of the zero coupling between lattice sites eigenvalues of the linearization of the dnls equation at the discrete soliton determine its spectral stability small eigenvalues bifurcating from the zero eigenvalue near the anti continuum limit were characterized earlier for this model here we analyze the resolvent operator and prove that it is bounded in the neighborhood of the continuous spectrum if the discrete soliton is simply connected in the anti continuum limit this result rules out the existence of internal modes neutrally stable eigenvalues of the discrete spectrum near the anti continuum limit

a feti method with a mesh independent condition number for the iteration matrix abstract we introduce a framework for feti methods using ideas from the decomposition via lagrange multipliers of h 0 1 derived by raviart thomas p a raviart j m thomas primal hybrid finite element methods for second order elliptic equations math comput 31 138 1977 391 413 and complemented with the detailed work on polygonal domains developed by grisvard p grisvard singularities in boundary value problems rech math appl 22 1992 we compute the action of the lagrange multipliers using the natural h 00 1 2 scalar product therefore no consistency error appears our analysis allows to deal with cross points and floating subdomains in a natural manner as a byproduct we obtain that the condition number for the iteration matrix is independent of the mesh size and there is no need for preconditioning this result improves the standard asymptotic bound for this condition number given by 1 log h h 2 where h and h are the characteristic subdomain size and element size respectively shown by mandel tezaur in j mandel r tezaur convergence of a substructuring method with lagrange multipliers numer math 73 1996 473 487 numerical results that confirm our theoretical analysis are presented

feti and neumann neumann iterative substructuring methods connections and new results the feti and neumann neumann families of algorithms are among the best know and most severely tested domain decomposition methods for elliptic partial differential equations they are iterative substructuring methods and have many algorithmic components in common but there are also differences the purpose of this paper is to further unify the theory for these two families of methods and to introduce a new family of feti algorithms bounds on the rate of convergence which are uniform with respect to the coefficients of a family of elliptic problems with heterogeneous coefficients are established for these new algorithms the theory for a variant of the neumann neumann algorithm is also redeveloped stressing similarities to that for the feti methods

feti domain decomposition methods for scalar advection diffusion problems abstract in this paper we show that iterative substructuring methods of finite element tearing and interconnecting type can be successfully employed for the solution of linear systems arising from the finite element approximation of scalar advection diffusion problems using similar ideas as those of a recently developed neumann neumann method we propose a one level algorithm and a class of two level algorithms obtained by suitably modifying the local problems on the subdomains we present some numerical results for some significant test cases our methods appear to be optimal for flows without closed streamlines and possibly very small values of the viscosity they also show very good performances for rotating flows and moderate reynolds numbers therefore the algorithms proposed appear to be well suited for many convection dominated problems of practical interest

extended preconditioners for the feti method applied to constrained problems an efficient and simple approach for handling linear multipoint constraints in a class of substructure based solvers namely the finite element tearing and interconnecting feti method is proposed previously it was argued that multipoint constraints should be handled in feti by adding a second level iteration on a coarse grid such that the feti iterates satisfy the multipoint constraints exactly the procedure presented here does not require an additional coarse grid but instead takes account of the multipoint constraints in the preconditioning step the preconditioning strategy is shown to be mechanically consistent and to incur only a small additional computational cost this strategy is simpler and computationally less expensive than the two level feti procedure its numerical scalability even for highly heterogeneous problems is demonstrated in several test examples

an iterative substructuring method for coupled fluid solid acoustic problems a fast parallel iterative method is proposed for the solution of linear equations arising from finite element discretization of the time harmonic coupled fluid solid systems in fluid pressure and solid displacement formulation the fluid and the solid domains are decomposed into nonoverlapping subdomains continuity of the solution is enforced by lagrange multipliers the system is augmented by duplicating the degrees of freedom on the wet interface the original degrees of freedom are then eliminated and the resulting system is solved by iterations preconditioned by a coarse space correction in each iteration the method requires the solution of one independent local acoustic problem per subdomain and the solution of a global problem with several degrees of freedom per subdomain computational results show that the method is scalable with the problem size frequency and the number of subdomains the method generalizes the feti h method for the helmholtz equation to coupled fluid elastic scattering the number of iterations is about same as for the feti h method for the related helmholtz problem with neumann boundary condition instead of an elastic scatterer if enough coarse space functions are used convergence behavior is explained from the spectrum of the iteration operator and from numerical near decoupling of the equations in the fluid and in the solid regions

variational approximations to homoclinic snaking we investigate the snaking of localized patterns seen in numerous physical applications using a variational approximation this method naturally introduces the exponentially small terms responsible for the snaking structure which are not accessible via standard multiple scales asymptotic techniques we obtain the symmetric snaking solutions and the asymmetric ladder states and also predict the stability of the localized states the resulting approximate formulas for the width of the snaking region show good agreement with numerical results

three level bddc in two dimensions balancing domain decomposition by constraints bddc methods are non overlapping iterative substructuring domain decomposition methods for the solution of large sparse linear algebraic systems arising from discretization of elliptic boundary value problems they are similar to the balancing neumann neumann algorithm however in bddc methods a small number of continuity constraints are enforced across the interface and these constraints form a new coarse global component an important advantage of using such constraints is that the schur complements that arise in the computation will all be strictly positive definite the matrix of the coarse problem is generated and factored by direct solvers at the beginning of the computation however this problem can ultimately become a bottleneck if the number of subdomains is very large in this paper two three level bddc methods are introduced for solving the coarse problem approximately in two dimensional space while still maintaining a good convergence rate estimates of the condition numbers are provided for the two three level bddc methods and numerical experiments are also discussed copyright 2006 john wiley sons ltd

theoretically supported scalable feti for numerical solution of variational inequalities the feti method with a natural coarse grid is combined with recently proposed optimal algorithms for the solution of bound and or equality constrained quadratic programming problems in order to develop a scalable solver for elliptic boundary variational inequalities such as those describing equilibrium of a system of bodies in mutual contact a discretized model problem is first reduced by the duality theory of convex optimization to the quadratic programming problem with bound and equality constraints the latter is then modified by means of orthogonal projectors to the natural coarse grid introduced by farhat mandel and roux comput methods appl mech engrg 115 1994 pp 365 385 finally the classical results on linear scalability for linear problems are extended to boundary variational inequalities the results are validated by numerical experiments the experiments also confirm that the algorithm enjoys the same parallel scalability as its linear counterpart

analysis of feti methods for multiscale pdes in this paper we study a variant of the finite element tearing and interconnecting feti method which is suitable for elliptic pdes with highly heterogeneous multiscale coefficients x in particular coefficients with strong variation within subdomains and or jumps that are not aligned with the subdomain interfaces using energy minimisation and cut off arguments we can show rigorously that for an arbitrary positive coefficient function alpha in l infty omega the condition number of the preconditioned feti system can be bounded by c 1 log h h 2 where h is the subdomain diameter and h is the mesh size and where the function c depends only on the coefficient variation in the vicinity of subdomain interfaces in particular if alpha omega i varies only mildly in a layer i of width near the boundary of each of the subdomains i then c alpha mathcal o h eta 2 independent of the variation of in the remainder i i of each subdomain and independent of any jumps of across subdomain interfaces the quadratic dependence of c on h can be relaxed to a linear dependence under stronger assumptions on the behaviour of in the interior of the subdomains our theoretical findings are confirmed in numerical tests

stability of discrete solitons in nonlinear schrodinger lattices we consider the discrete solitons bifurcating from the anti continuum limit of the discrete nonlinear schrodinger nls lattice the discrete soliton in the anti continuum limit represents an arbitrary finite superposition of in phase or anti phase excited nodes separated by an arbitrary sequence of empty nodes by using stability analysis we prove that the discrete solitons are all unstable near the anti continuum limit except for the solitons which consist of alternating anti phase excited nodes we classify analytically and confirm numerically the number of unstable eigenvalues associated with each family of the discrete solitons

analysis of feti methods for multiscale pdes part ii interface variation in this article we give a new rigorous condition number estimate of the finite element tearing and interconnecting feti method and a variant thereof all floating feti we consider a scalar elliptic equation in a two or three dimensional domain with a highly heterogeneous multiscale diffusion coefficient this coefficient is allowed to have large jumps not only across but also along subdomain interfaces and in the interior of the subdomains in other words the subdomain partitioning does not need to resolve any jumps in the coefficient under suitable assumptions we derive bounds for the condition numbers of one level and all floating feti that are robust with respect to strong variations in the contrast in the coefficient and that are explicit in some geometric parameters associated with the coefficient variation in particular robustness holds for face edge and vertex islands in high contrast media as a central tool we prove and use new weighted poincare and discrete sobolev type inequalities that are explicit in the weight our theoretical findings are confirmed in a series of numerical experiments

on an elementary operator with w hyponormal operator entries abstract let a b b h be w hyponormal operators and let d ab b b h denote either the generalized derivation ab x ax xb or the length two elementary operator ab x axb x we prove that d ab has the single valued extension property and the quasinilpotent part h 0 d ab of d ab at iso d ab equals d ab 1 0 let h d ab denote the space of functions which are analytic on d ab and let h c d ab denote the space of f h d ab which are non constant on every connected component of d ab it is proved that for every h h d ab and f g h c d ab the complement of the weyl spectrum w h d f a g b of h d f a g b in h d f a g b consists of isolated points in h d f a g b which are eigenvalues of finite multiplicity

a dual primal feti method for incompressible stokes equations in this paper a dual primal feti method is developed for incompressible stokes equations approximated by mixed finite elements with discontinuous pressures the domain of the problem is decomposed into nonoverlapping subdomains and the continuity of the velocity across the subdomain interface is enforced by introducing lagrange multipliers by a schur complement procedure the solution of an indefinite stokes problem is reduced to solving a symmetric positive definite problem for the dual variables i e the lagrange multipliers this dual problem is solved by the conjugate gradient method with a dirichlet preconditioner in each iteration step both subdomain problems and a coarse level problem are solved by a direct method it is proved that the condition number of this preconditioned dual problem is independent of the number of subdomains and bounded from above by the square of the product of the inverse of the inf sup constant of the discrete problem and the logarithm of the number of unknowns in the individual subdomains numerical experiments demonstrate the scalability of this new method

multistable solitons in the cubic quintic discrete nonlinear schrodinger equation we analyze the existence and stability of localized solutions in the one dimensional discrete nonlinear schrodinger dnls equation with a combination of competing self focusing cubic and defocusing quintic onsite nonlinearities we produce a stability diagram for different families of soliton solutions that suggests the co existence of infinitely many branches of stable localized solutions bifurcations that occur with an increase in the coupling constant are studied in a numerical form a variational approximation is developed for accurate prediction of the most fundamental and next order solitons together with their bifurcations salient properties of the model which distinguish it from the well known cubic dnls equation are the existence of two different types of symmetric solitons and stable asymmetric soliton solutions that are found in narrow regions of the parameter space the asymmetric solutions appear from and disappear back into the symmetric ones via loops of forward and backward pitchfork bifurcations

a domain decomposition method with lagrange multipliers and inexact solvers for linear elasticity a new domain decomposition method with lagrange multipliers for elliptic problems is introduced it is based on a reformulation of the well known finite element tearing and interconnecting feti method as a saddle point problem with both primal and dual variables as unknowns the resulting linear system is solved with block structured preconditioners combined with a suitable krylov subspace method this approach allows the use of inexact subdomain solvers for the positive definite subproblems it is shown that the condition number of the preconditioned saddle point problem is bounded independently of the number of subregions and depends only polylogarithmically on the number of degrees of freedom of individual local subproblems numerical results are presented for a plane stress cantilever membrane problem

feti dp a dual primal unified feti method part i a faster alternative to the two level feti method the feti method and its two level extension feti 2 are two numerically scalable domain decomposition methods with lagrange multipliers for the iterative solution of second order solid mechanics and fourth order beam plate and shell structural problems respectively the feti 2 method distinguishes itself from the basic or one level feti method by a second set of lagrange multipliers that are introduced at the subdomain cross points to enforce at each iteration the exact continuity of a subset of the displacement field at these specific locations in this paper we present a dual primal formulation of the feti 2 concept that eliminates the need for that second set of lagrange multipliers and unifies all previously developed one level and two level feti algorithms into a single dual primal feti dp method we show that this new feti dp method is numerically scalable for both second order and fourth order problems we also show that it is more robust and more computationally efficient than existing feti solvers particularly when the number of subdomains and or processors is very large copyright 2001 john wiley sons ltd

two level domain decomposition methods with lagrange multipliers for the fast iterative solution of acoustic scattering problems we present two different but related lagrange multiplier based domain decomposition dd methods for solving iteratively large scale systems of equations arising from the finite element discretization of high frequency exterior helmholtz problems the proposed methods are essentially two distinct extensions of the regularized finite element tearing and interconnecting feti method to indefinite or complex problems the first method employs a single lagrange multiplier field to glue the local solutions at the subdomain interface boundaries the second method employs two lagrange multiplier fields for that purpose the key ingredients of both of these feti methods are the regularization of each subdomain matrix by a complex lumped mass matrix defined on the subdomain interface boundary and the preconditioning of the global interface problem by a coarse second level problem constructed with planar waves we show numerically that both methods are scalable with respect to the mesh size the subdomain size and the wavenumber but that the feti method with a single lagrange multiplier field labeled feti h h for helmholtz in this paper delivers superior computational performances we apply the feti h method to the parallel solution on a 24 processor origin 2000 of an acoustic scattering problem with a submarine shaped obstacle and report performance results that highlight the unique efficiency of this dd method for the solution of high frequency acoustic scattering problems

preconditioners for the dual primal feti methods on nonmatching grids numerical study the feti dp method is a substructuring method that uses lagrange multipliers to match the continuity condition on the subdomain boundaries for the feti dp method on non matching grids two different formulations are known with respect to how to employ the mortar matching condition keeping step with the developments of the feti dp methods a variety of preconditioners for the feti dp operator have been developed however there has not been any numerical study for the feti dp method which compares those preconditioners on nonmatching grids while there have been a few papers for numerical study on the comparison of feti preconditioners therefore we present the numerical study of four different preconditioners for two dimensional elliptic problems the numerical results confirm the superiority of the preconditioner by kim and lee 1 for noncomparably nonmatching grids while the superiority of the preconditioner by dryja and widlund 2 is confirmed for matching grids and comparably nonmatching grids

a scalable lagrange multiplier based domain decomposition method for time dependent problems we present a new efficient and scalable domain decomposition method for solving implicitly linear and non linear time dependent problems in computational mechanics the method is derived by adding a coarse problem to the recently proposed transient feti substructuring algorithm in order to propagate the error globally and accelerate convergence it is proved that in the limit for large time steps the new method converges toward the feti algorithm for time independent problems computational results confirm that the optimal convergence properties of the time independent feti method are preserved in the time dependent case we employ an iterative scheme for solving efficiently the coarse problem on massively parallel processors and demonstrate the effective scalability of the new transient feti method with the large scale finite element dynamic analysis on the paragon xp s and ibm sp2 systems of several diffraction grating finite element structural models we also show that this new domain decomposition method outperforms the popular direct skyline solver the coarse problem presented herein is applicable and beneficial to a large class of lagrange multiplier based substructuring algorithms for time dependent problems including the fictitious domain decomposition method

algebraic two level preconditioners for the schur complement method the solution of elliptic problems is challenging on parallel distributed memory computers since their green s functions are global to address this issue we present a set of preconditioners for the schur complement domain decomposition method they implement a global coupling mechanism through coarse space components similar to the one proposed in bramble pasciak and shatz math comp 47 1986 pp 103 134 the definition of the coarse space components is algebraic they are defined using the mesh partitioning information and simple interpolation operators these preconditioners are implemented on distributed memory computers without introducing any new global synchronization in the preconditioned conjugate gradient iteration the numerical and parallel scalability of those preconditioners are illustrated on two dimensional model examples that have anisotropy and or discontinuity phenomena

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

feti based algorithms for contact problems scalability large displacements and 3d coulomb friction theoretical and experimental results concerning feti based algorithms for contact problems of elasticity are reviewed a discretized model problem is first reduced by the duality theory of convex optimization to the quadratic programming problem with bound and equality constraints the latter is then optionally modified by means of orthogonal projectors to the natural coarse space introduced by farhat and roux in the framework of their feti method the resulting problem is then solved either by special algorithms for bound constrained quadratic programming problems combined with penalty that imposes the equality constraints or by an augmented lagrangian type algorithm with the inner loop for the solution of bound constrained quadratic programming problems recent theoretical results are reported that guarantee certain optimality and scalability of both algorithms the results are confirmed by numerical experiments the performance of the algorithm in solution of more realistic engineering problems by basic algorithm is demonstrated on the solution of 3d problems with large displacements or coulomb friction

variational approximations in discrete nonlinear schrodinger equations with next nearest neighbor couplings solitons of a discrete nonlinear schrodinger equation which includes the next nearest neighbor interactions are stud ied by means of a variational approximation and numerical computations a large family of multi humped solutions including those with a nontrivial phase structure which are a feature particular to the next nearest neighbor interac tion model are accurately predicted by the variational approximation bifurcations linking solutions with the trivial and nontrivial phase structures are also captured remarkably well including a prediction of critical parameter values

operators that are orthogonal to the range of a derivation the gateaux differentiability of the schatten p norms and the usual operator norm is used to characterize those operators that are orthogonal to the range of a derivation with respect to these norms

inexact data sparse boundary element tearing and interconnecting methods the boundary element tearing and interconnecting beti methods have recently been introduced as boundary element counterparts of the well established finite element tearing and interconnecting feti methods in this paper we present inexact data sparse versions of the beti methods which avoid the elimination of the primal unknowns and dense matrices however instead of symmetric and positive definite systems we finally have to solve twofold saddle point problems the proposed iterative solvers and preconditioners result in almost optimal solvers whose complexity is proportional to the number of unknowns on the skeleton up to some polylogarithmical factor moreover the solvers are robust with respect to large coefficient jumps

a fast implementation of the feti dp method feti dp rbs lna and applications on large scale problems with localized non linearities as parallel and distributed computing gradually becomes the computing standard for large scale problems the domain decomposition method dd has received growing attention since it provides a natural basis for splitting a large problem into many small problems which can be submitted to individual computing nodes and processed in a parallel fashion this approach not only provides a method to solve large scale problems that are not solvable on a single computer by using direct sparse solvers but also gives a flexible solution to deal with large scale problems with localized non linearities when some parts of the structure are modified only the corresponding subdomains and the interface equation that connects all the subdomains need to be recomputed in this paper the dual primal finite element tearing and interconnecting method feti dp is carefully investigated and a reduced back substitution rbs algorithm is proposed to accelerate the time consuming preconditioned conjugate gradient pcg iterations involved in the interface problems linear non linear analysis lna is also adopted for large scale problems with localized non linearities based on subdomain linear non linear identification criteria this combined approach is named as the feti dp rbs lna algorithm and demonstrated on the mechanical analyses of a welding problem serial cpu costs of this algorithm are measured at each solution stage and compared with that from the ibm watson direct sparse solver and the feti dp method the results demonstrate the effectiveness of the proposed computational approach for simulating welding problems which is representative of a large class of three dimensional large scale problems with localized non linearities

a dual iterative substructuring method with a penalty term in three dimensions the feti dp method is one of the most advanced dual substructuring methods which introduces lagrange multipliers to enforce the pointwise matching condition on the interface in our previous work for two dimensional problems a dual iterative substructuring method was proposed which is a variant of the feti dp method based on the way to deal with the continuity constraint on the interface the proposed method imposes the continuity not only by the pointwise matching condition on the interface but also by using a penalty term which measures the jump across the interface in this paper a dual substructuring method with a penalty term is extended to three dimensional problems a penalty term with a penalization parameter h is constructed by focusing on the geometric complexity of an interface in three dimensions caused by the coupling among adjacent subdomains for a large h it is shown that the condition number of the resultant dual problem is bounded by a constant independent of both subdomain size h and mesh size h from the implementational viewpoint of the proposed method the difference from the feti dp method is to solve subdomain problems which contain a penalty term with a penalization parameter h to prevent a large penalization parameter from making subdomain problems ill conditioned special attention is paid to establish an optimal preconditioner with respect to a penalization parameter h finally numerical results are presented

automatic spectral coarse spaces for robust finite element tearing and interconnecting and balanced domain decomposition algorithms we introduce spectral coarse spaces for the balanced domain decomposition and the finite element tearing and interconnecting methods these coarse spaces are specifically designed for the two level methods to be scalable and robust with respect to the coefficients in the equation and the choice of the decomposition we achieve this by solving generalized eigenvalue problems on the interfaces between subdomains to identify the modes that slow down convergence theoretical bounds for the condition numbers of the preconditioned operators which depend only on a chosen threshold and the maximal number of neighbors of a subdomain are presented and proved for the finite element tearing and interconnecting method there are two versions of the two level method one based on the full dirichlet preconditioner and the other on the cheaper lumped preconditioner some numerical tests confirm these results copyright 2013 john wiley sons ltd

the feti family of domain decomposition methods for inequality constrained quadratic programming application to contact problems with conforming and nonconforming interfaces two domain decomposition methods with lagrange multipliers for solving iteratively quadratic programming problems with inequality constraints are presented these methods are based on the feti and feti dp substructuring algorithms in the case of linear constraints they do not perform any newton like iteration instead they solve a constrained problem by an active set strategy and a generalized conjugate gradient based descent method equipped with controls to guarantee convergence monotonicity both methods possess the desirable feature of minimizing numerical oscillations during the iterative solution process performance results and comparisons are reported for several numerical simulations that suggest that both methods are numerically scalable with respect to both the problem size and the number of subdomains their parallel scalability is also illustrated on a linux cluster for a complex 1 4 million degree of freedom multibody problem with frictionless contact and nonconforming discrete interfaces

convergence of a substructuring method with lagrange multipliers we analyze the convergence of a substructing iterative method with lagrange multipliers propose recently by farhat and roux the method decomposes finite element discretization of an elliptic boundary value problem into neumann problems on the subdomains plus a coarse problem for the subdomain nullspace components for linear conforming elements and preconditioning by the dirichlet problems on the subdomains we prove the asymptotic bound on the condition number c 1 log h h alpha alpha 2 or 3 where h is the characteristic element size h subdomain size and c is independent of the number of subdomains