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

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

robust feti dp methods for heterogeneous three dimensional elasticity problems iterative substructuring methods with lagrange multipliers are considered for heterogeneous linear elasticity problems with large discontinuities in the material stiffnesses in particular results for algorithms belonging to the family of dual primal feti methods are presented the core issue of these algorithms is the construction of an appropriate global problem in order to obtain a robust method which converges independently of the material discontinuities in this article several necessary and sufficient conditions arising from the theory are numerically tested and confirmed furthermore results of numerical experiments are presented for situations which are not covered by the theory such as curved edges and material discontinuities not aligned with the interface and an attempt is made to develop rules for these cases

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

an algebraically partitioned feti method for parallel structural analysis algorithm description an algebraically partitioned feti method for the solution of structural engineering problems on parallel computers is presented the present algorithm consists of three attributes an explicit generation of the orthogonal null space matrix associated with the interface nodal forces the floating subdomain rigid body modes computed from the subdomain static equilibrium equation of the classical force method and the identification of redundant interface force constraint operator that emanates when the interface force computations are localized comparisons of the present method with the previously developed differentially partitioned feti method are offered in terms of the saddle point formulations at the end of the paper a companion paper reports implementation details and numerical performance of the proposed algorithm 1997 john wiley sons ltd

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

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

a strict error bound with separated contributions of the discretization and of the iterative solver in non overlapping domain decomposition methods this paper deals with the estimation of the distance between the solution of a static linear mechanic problem and its approximation by the finite element method solved with a non overlapping domain decomposition method feti or bdd we propose a new strict upper bound of the error which separates the contribution of the iterative solver and the contribution of the discretization numerical assessments show that the bound is sharp and enables us to define an objective stopping criterion for the iterative solver

fast estimation of discretization error for fe problems solved by domain decomposition this paper presents a strategy for a posteriori error estimation for substructured problems solved by non overlapping domain decomposition methods we focus on global estimates of the discretization error obtained through the error in constitutive relation for linear mechanical problems our method allows to compute error estimate in a fully parallel way for both primal bdd and dual feti approaches of non overlapping domain decomposition whatever the state converged or not of the associated iterative solver results obtained on an academic problem show that the strategy we propose is efficient in the sense that correct estimation is obtained with fully parallel computations they also indicate that the estimation of the discretization error reaches sufficient precision in very few iterations of the domain decomposition solver which enables to consider highly effective adaptive computational strategies

arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders we prove a form of arnold diffusion in the a priori stable case let h 0 p epsilon h 1 theta p t quad theta in mathbb t n p in b n t in mathbb t mathbb r mathbb t be a nearly integrable system of arbitrary degrees of freedom n geqslant 2 with a strictly convex h0 we show that for a generic epsilon h 1 there exists an orbit theta p satisfying p t p 0 l h 1 0 where l h 1 is independent of epsilon the diffusion orbit travels along a codimension 1 resonance and the only obstruction to our construction is a finite set of additional resonances

comparison of different feti preconditioners for elastoplasticity this paper illustrates parallel solution of elastoplastic problems with hardening based on the tfeti domain decomposition method with several preconditioning strategies we consider von mises plasticity with isotropic hardening using the return mapping concept for one time step to treat nonlinearity and nonsmoothness we use the semismooth newton method in each newton iteration we solve a linear system of equations using the tfeti domain decomposition method with lumped dirichlet or no preconditioner our permon software is used for the numerical experiments the observed times and numbers of iterations are backed up by the regular condition number estimates both preconditioners accelerate the solution process in terms of the convergence rate for the worst conditioned benchmark the most expensive dirichlet preconditioner gives the lowest computational times this lets us suggest the dirichlet preconditioner as the default choice for engineering problems

a two step two field hybrid method for the static and dynamic analysis of substructure problems with conforming and non conforming interfaces abstract the need for assembling independent finite element substructure solutions arises in several engineering and scientific problems including the design and analysis of complex structural systems component mode synthesis global local analysis adaptive refinement and parallel processing in this paper we discuss the solution of such problems by a two field hybrid method where the substructures are jointed with low order polynomial or piece wise polynomial lagrange multipliers and present a rayleigh ritz based smoothing procedure for improving the accuracy of the computed coupled solution in the presence of various substructure heterogeneities we consider both conforming and nonconforming substructure meshes and demonstrate the benefits of the proposed two step solution method with several examples from structural mechanics

incorporation of linear multipoint constraints in domain decomposition based iterative solvers part ii blending feti dp and mortar methods and assembling floating substructures two approaches are presented for extending the feti dp domain decomposition based iterative method to the solution of problems with linear multipoint constraints lmpcs both cases of arbitrary lmpcs and lmpcs generated by the mortar method for enforcing a weak continuity of the solution across non matching finite element interfaces are considered in the first approach the lmpcs are addressed during the preconditioning step but enforced only at convergence in the second approach the lmpcs are enforced at each iteration through the solution of an auxiliary coarse problem both extended feti dp solvers are benchmarked on a parallel processor for a wide range of constrained structural mechanics problems the obtained results reveal that both solvers are numerically scalable and complement each other from the memory and cpu performance viewpoints

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

an adaptive bddc algorithm in variational form for mortar discretizations abstract a balancing domain decomposition by constraints bddc algorithm with adaptive primal constraints in variational form is introduced and analyzed for high order mortar discretization of two dimensional elliptic problems with high varying and random coefficients some vector valued auxiliary spaces and operators with essential properties are defined to describe the variational algorithm and the coarse space is formed by using a transformation operator on each interface compared with the adaptive bddc algorithms for conforming galerkin approximations our algorithm is simpler because there have not any continuity constraints at subdomain vertices in the mortar method involved in this paper the condition number of the preconditioned system is proved to be bounded above by a user defined tolerance and a constant which is dependent on the maximum number of interfaces per subdomain and independent of the mesh size and the contrast of the given coefficients numerical results show the robustness and efficiency of the algorithm for various model problems

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

a neumann neumann domain decomposition algorithm for solving plate and shell problems we present a new neumann neumann type preconditioner of large scale linear systems arising from plate and shell problems the advantage of the new method is a smaller coarse space than those of earlier methods of the authors this improves parallel scalability a new abstract framework for neumann neumann preconditioners is used to prove almost optimal convergence properties of the method the convergence estimates are independent of the number of subdomains coefficient jumps between subdomains and depend only polylogarithmically on the number of elements per subdomain we formulate and prove an approximate parametric variational principle for reissner mindlin elements as the plate thickness approaches zero which makes the results applicable to a large class of nonlocking elements in everyday engineering use the theoretical results are confirmed by computational experiments on model problems as well as examples from real world engineering practice

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

simultaneous feti and block feti robust domain decomposition with multiple search directions domain decomposition methods often exhibit very poor performance when applied to engineering problems with large heterogeneities in particular for heterogeneities along domain interfaces the iterative techniques to solve the interface problem are lacking an efficient preconditioner recently a robust approach named feti geneo was proposed where troublesome modes are precomputed and deflated from the interface problem the cost of the feti geneo is however high we propose in this paper techniques that share similar ideas with feti geneo but where no pre processing is needed and that can be easily and efficiently implemented as an alternative to standard domain decomposition methods in the block iterative approaches presented in this paper the search space at every iteration on the interface problem contains as many directions as there are domains in the decomposition those search directions originate either from the domain wise preconditioner in the simultaneous feti method or from the block structure of the right hand side of the interface problem block feti we show on 2d structural examples that both methods are robust and provide good convergence in the presence of high heterogeneities even when the interface is jagged or when the domains have a bad aspect ratio the simultaneous feti was also efficiently implemented in an optimized parallel code and exhibited excellent performance compared to the regular feti method

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

dynamics around simple resonant tori in nearly integrable hamiltonian systems abstract by an application of the k a m theory we derive an accurate normal form valid in the vicinity of partially hyperbolic tori which arise close to simple resonances in nearly integrable hamiltonian systems this normal form allows to detect orbits homoclinic to a persistent torus moreover it also gives precise estimates on the times of transition around the stable and unstable manifolds of these tori hence we provide an efficient tool to compute the speed of drift of orbits shadowing a chain of hyperbolic tori associated to a simple resonant curve in the action space

hyperbolicity versus partial hyperbolicity and the transversality torsion phenomenon abstract in this paper we describe a process to create hyperbolicity in the neighbourhood of a homoclinic orbit to a partially hyperbolic torus for three degrees of freedom hamiltonian systems the transversality torsion phenomenon

positive versus compact support solutions to a singular elliptic problem abstract the semilinear elliptic problem u k x u q u p u 0 in u 0 on is considered in this paper where is a bounded domain of r n n 2 with c 2 boundary 0 q p 1 and k x as x we mainly study the effect of the blow up rate of k x near to the existence of positive and compact support solutions furthermore an optimal compact support principle is given for a class of elliptic differential inequalities

the two level feti method part ii extension to shell problems parallel implementation and performance results the two level feti method presented in part i of this work is a mathematically optimal and scalable substructuring algorithm for solving iteratively large scale systems of equations arising from the finite element discretization of static and dynamic plate bending problems in this paper we extend the two level feti method to shell problems describe a variant approach revisit the preconditioning problem address the efficient implementation of the corresponding iterative solvers on massively parallel processors highlight the computational price of mathematical optimality and discuss its consequences and report some impressive performance results on two paragon xp s and ibm sp2 massively parallel processors for several realistic plate and shell problems

adaptive coarse spaces for feti dp in three dimensions an adaptive coarse space approach including a condition number bound for dual primal finite element tearing and interconnecting feti dp methods applied to three dimensional problems with coefficient jumps inside subdomains and across subdomain boundaries is presented the approach is based on a known adaptive coarse space approach enriched by a small number of additional local edge eigenvalue problems these edge eigenvalue problems serve to make the method robust and permit a condition number bound which depends only on the tolerance of the local eigenvalue problems and some properties of the domain decomposition the introduction of the edge eigenvalue problems thus turns a well known condition number indicator for feti dp and balancing domain decomposition by constraints bddc methods into a condition number estimate numerical results are presented for linear elasticity and heterogeneous materials supporting our theoretical findings the problems considered include those with random coefficients an

an analysis of a feti dp algorithm on irregular subdomains in the plane in the theory for domain decomposition algorithms of the iterative substructuring family each subdomain is typically assumed to be the union of a few coarse triangles or tetrahedra this is an unrealistic assumption in particular if the subdomains result from the use of a mesh partitioner in which case they might not even have uniformly lipschitz continuous boundaries the purpose of this study is to derive bounds for the condition number of these preconditioned conjugate gradient methods which depend only on a parameter in an isoperimetric inequality two geometric parameters characterizing john and uniform domains and the maximum number of edges of any subdomain a related purpose is to explore to what extent well known technical tools previously developed for quite regular subdomains can be extended to much more irregular subdomains some of these results are valid for any john domain while an extension theorem which is needed in this study requires that the subdomains have complements which are uniform the results so far are complete only for problems in two dimensions details are worked out for a feti dp algorithm and numerical results support the findings some of the numerical experiments illustrate that care must be taken when selecting the scaling of the preconditioners in the case of irregular subdomains

a feti based domain decomposition technique for time dependent first order systems based on a dae approach we present a novel partitioned coupling algorithm to solve first order time dependent non linear problems e g transient heat conduction the spatial domain is partitioned into a set of totally disconnected subdomains the continuity conditions at the interface are modeled using a dual schur formulation where the lagrange multipliers represent the interface fluxes or the reaction forces that are required to maintain the continuity conditions the interface equations along with the subdomain equations lead to a system of differential algebraic equations daes for the resulting equations a numerical algorithm is developed which includes choosing appropriate constraint stabilization techniques the algorithm first solves for the interface lagrange multipliers which are subsequently used to advance the solution in the subdomains the proposed coupling algorithm enables arbitrary numeric schemes to be coupled with different time steps i e it allows subcycling in each subdomain this implies that existing software and numerical techniques can be used to solve each subdomain separately the coupling algorithm can also be applied to multiple subdomains and is suitable for parallel computers we present examples showing the feasibility of the proposed coupling algorithm copyright 2008 john wiley sons ltd

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

extending substructure based iterative solvers to multiple load and repeated analyses direct solvers currently dominate commercial finite element structural software but do not scale well in the fine granularity regime targeted by emerging parallel processors substructure based iterative solvers often called also domain decomposition algorithms lend themselves better to parallel processing but must overcome several obstacles before earning their place in general purpose structural analysis programs one such obstacle is the solution of systems with many or repeated right hand sides such systems arise for example in multiple load static analyses and in implicit linear dynamics computations direct solvers are well suited for these problems because after the system matrix has been factored the multiple or repeated solutions can be obtained through relatively inexpensive forward and backward substitutions on the other hand iterative solvers in general are ill suited for these problems because they often must restart from scratch for every different right hand side in this paper we present a methodology for extending the range of applications of domain decomposition methods to problems with multiple or repeated right hand sides basically we formulate the overall problem as a series of minimization problems over k orthogonal and supplementary subspaces and tailor the preconditioned conjugate gradient algorithm to solve them efficiently the resulting solution method is scalable whereas direct factorization schemes and forward and backward substitution algorithms are not we illustrate the proposed methodology with the solution of static and dynamic structural problems and highlight its potential to outperform forward and backward substitutions on parallel computers as an example we show that for a linear structural dynamics problem with 11640 degrees of freedom every time step beyond time step 15 is solved in a single iteration and consumes 1 0 second on a 32 processor ipsc 860 system for the same problem and the same parallel processor a pair of forward backward substitutions at each step consumes 15 0 seconds

incorporation of linear multipoint constraints in substructure based iterative solvers part 1 a numerically scalable algorithm we consider the iterative solution by a class of substructuring methods of the large scale systems of equations arising from the nite element discretization of structural models with an arbitrary set of linear multipoint constraints we present a methodology for generalizing to such problems numerically scalable substructure based iterative solvers without interfering with their formulations and their well established local and global preconditioners we apply this methodology to the feti method and show that the resulting algorithm is numerically scalable with respect to both the substructure and problem sizes 1998 john wiley sons ltd

on a singular nonlinear semilinear elliptic problem where k x c2 b v9 a p 0 1 and l is a real parameter such singular elliptic problems arise in the contexts of chemical heterogeneous catalysts nonnewtonian fluids and also the theory of heat conduction in electrically conducting materials see 3 5 8 9 for a detailed discussion obviously 1 1 cannot have a solution u c2 v9 if k x is not vanishing near v however under various appropriate assumptions on k x we will obtain classical solutions of 1 1 for l belonging to a certain range and we will also obtain some uniqueness criteria here a classical solution is a solution u of 1 1 which belongs to c2 v mc v9 with u 0 in v we also study the boundary behaviour of solutions of 1 1 and we will show that the solution u of 1 1 lies in a certain holder class the special case when k x is negative and l 0 has been studied by several authors the existence and uniqueness of the solution were established by crandall rabinowitz and tartar 6 del pino 7 gomes 10 and lazer and mckenna

dual primal feti methods for linear elasticity dual primal feti methods are nonoverlapping domain decomposition methods where some of the continuity constraints across subdomain boundaries are required to hold throughout the iterations as in primal iterative substructuring methods while most of the constraints are enforced by lagrange multipliers as in one level feti methods these methods are used to solve the large algebraic systems of equations that arise in elliptic finite element problems the purpose of this article is to develop strategies for selecting these constraints which are enforced throughout the iterations such that good convergence bounds are obtained that are independent of even large changes in the stiffness of the subdomains across the interface between them the algorithms are described in terms of a change of basis that has proven to be quite robust in practice a theoretical analysis is provided for the case of linear elasticity and condition number bounds are established that are uniform with respect to arbitrarily large jumps in the young s modulus of the material and otherwise depend only polylogarithmically on the number of unknowns of a single subdomain the strategies have already proven quite successful in large scale implementations of these iterative methods 2006 wiley periodicals inc

a compact support principle for singular elliptic differential inequalities with a gradient term abstract in this paper a compact support principle is established for the elliptic differential inequality u u p k x f u u 0 for x large where p 1 k x 0 and f satisfies conditions f1 f3 below the main feature of this note is the presence of the gradient term u p and the singular coefficient function k x the result is optimal in some sense related to the power p and the decaying rate of k x at and the proof is based on finding appropriate comparison functions we also give a similar result in a bounded domain of r n n 2

arnold diffusion in a priori chaotic symplectic maps we assume that a symplectic real analytic map has an invariant normally hyperbolic cylinder and an associated transverse homoclinic cylinder we prove that generically in the real analytic category the boundaries of the invariant cylinder are connected by trajectories of the map

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

symbolic dynamics and arnold diffusion we consider hyperbolic tori of three degrees of freedom initially hyperbolic hamiltonian systems we prove that if the stable and unstable manifold of a hyperbolictorus intersec t transversaly then there exists a hyperbolic invariant set near a homoclinic orbit on which the dynamics is conjugated to a bernoulli shift the proof is based on a new geometricodynamical feature of partially hyperbolic systems the transversality torsion phenomenon which produces complete hyperbolicity from partial hyperbolicity we deduce the existence of infinitely many hyperbolicperiodicorbits near the given torus the relevanc e of these results for the instability of near integrable hamiltonian systems is then discussed for a given transition chain we construct chain of hyperbolic periodic orbits then we easily prove the existence of periodic orbits of arbitrarily high period close to such chain using standard results on hyperbolicsets r 2002 elsevier science usa all rights reserved

branches of positive and free boundary solutions for some singular quasilinear elliptic problems we study the existence and multiplicity of solutions strictly positive or nonnegative having a free boundary the boundary of the set where the solution vanishes of some one dimensional quasilinear problems of eigenvalue type with possibly singular nonlinear terms

a feti dp method for the parallel iterative solution of indefinite and complex valued solid and shell vibration problems the dual primal finite element tearing and interconnecting feti dp domain decomposition method ddm is extended to address the iterative solution of a class of indefinite problems of the form k 2 m u f and a class of complex problems of the form k 2 m i d u f where k m and d are three real symmetric matrices arising from the finite element discretization of solid and shell dynamic problems i is the imaginary complex number and a is a real positive number a key component of this extension is a new coarse problem based on the free space solutions of navier s equations of motion these solutions are waves and therefore the resulting ddm is reminiscent of the feti h method for this reason it is named here the feti dph method for a practically large a range feti dph is shown numerically to be scalable with respect to all of the problem size substructure size and number of substructures the cpu performance of this iterative solver is illustrated on a 40 processor computing system with the parallel solution for various a ranges of several large scale indefinite or complex valued systems of equations associated with shifted eigenvalue and forced frequency response structural dynamics problems

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

hopf boundary maximum principle violation for semilinear elliptic equations abstract we consider elliptic equations with non lipschitz nonlinearity u u 1 u u 1 u in a smooth bounded domain r n with dirichlet boundary conditions here 0 1 we prove the existence of a weak nonnegative solution which does not satisfy the hopf boundary maximum principle provided that is large enough and n max 2 2 1 1 1 1

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

strict bounding of quantities of interest in computations based on domain decomposition abstract this paper deals with bounding the error on the estimation of quantities of interest obtained by finite element and domain decomposition methods the proposed bounds are written in order to separate the two errors involved in the resolution of reference and adjoint problems on the one hand the discretization error due to the finite element method and on the other hand the algebraic error due to the use of the iterative solver beside practical considerations on the parallel computation of the bounds it is shown that the interface conformity can be slightly relaxed so that local enrichment or refinement are possible in the subdomains bearing singularities or quantities of interest which simplifies the improvement of the estimation academic assessments are given on 2d static linear mechanic problems

a simple explicit implicit finite element tearing and interconnecting transient analysis algorithm summary a simple explicit implicit finite element tearing and interconnecting feti algorithm afeti ei algorithm is presented for partitioned transient analysis of linear structural systems the present algorithm employs two decompositions first the total system is partitioned via spatial or domain decomposition to obtain the governing equations of motions for each partitioned domain second for each partitioned subsystem the governing equations are modally decomposed into the rigid body and deformational equations the resulting rigid body equations are integrated by an explicit integrator for its stability is not affected by step size restriction on account of zero frequency contents 0 the modally decomposed partitioned deformation equations of motion are integrated by an unconditionally stable implicit integration algorithm it is shown that the present afeti ei algorithm exhibits unconditional stability and that the resulting interface problem possesses the same solution matrix profile as the basic feti static problems the present simple dynamic algorithm as expected falls short of the performance of the feti dp but offers a similar performance of implicit two level feti d algorithm with a much cheaper coarse solver hence its simplicity may offer relatively easy means for conducting parallel analysis of both static and dynamic problems by employing the same basic scalable feti solver especially for research mode numerical experiments copyright 2011 john wiley sons ltd

the two level feti method for static and dynamic plate problems part i an optimal iterative solver for biharmonic systems abstract we present a lagrange multiplier based substructuring method for solving iteratively large scale systems of equations arising from the finite element discretization of static and dynamic plate bending problems the proposed method is essentially an extension of the feti domain decomposition algorithm to fourth order problems the main idea is to enforce exactly the continuity of the transverse displacement field at the substructure corners throughout the preconditioned conjugate projected gradient iterations this results in a two level feti substructuring method where the condition number of the preconditioned interface problem does not grow with the number of substructures and grows at most polylogarithmically with the number of elements per substructure these theoretically proven optimal convergence properties of the new feti method are numerically demonstrated for several finite element plate bending static and transient problems the two level iterative solver presented in this paper is applicable to a large family of biharmonic time independent as well as time dependent systems it is also extendible to shell problems

regularity of an elliptic problem with a singular nonlinearity we study a semilinear boundary value problem with the feature that the vanishing boundary value makes the equation singular we prove that the positive solution is in general holder continuous up to the boundary and has even better regularity in some special cases

a simple and efficient extension of a class of substructure based preconditioners to heterogeneous structural mechanics problems several domain decomposition methods with lagrange multipliers have been recently designed for solving iteratively large scale systems of finite element equations while these methods differ typically by implementational details they share in most cases the same substructure based preconditioners that were originally developed for the feti method the success of these preconditioners is due to the fact that for homogeneous structural mechanics problems they ensure a computational performance that scales with the problem size in this paper we address the suboptimal behaviour of these preconditioners in the presence of material and or discretization heterogeneities we propose a simple and virtually no cost extension of these preconditioners that exhibits scalability even for highly heterogeneous systems of equations we consider several intricate structural analysis problems and demonstrate numerically the optimal performance delivered by the new preconditioners for problems with discontinuities copyright 1999 john wiley sons ltd

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

the mosaic of high performance domain decomposition methods for structural mechanics formulation interrelation and numerical efficiency of primal and dual methods a multitude of domain decomposition methods ddm for structural mechanics is available in the literature today a unified framework for formulating primal and dual ddm is thus presented in this paper aiming at providing a mathematical platform for a uniform treatment of high performance ddm in structural mechanics a novel approach for developing new ddm from existing methods is also proposed and is applied to dual and primal methods in the field of the feti methods this approach leads to a new category of methods derived from existing feti variants furthermore two alternative formulations of the balancing domain decomposition method are described while interrelations between the introduced and existing methods are established finally comparative numerical tests demonstrate the differences in the computational performance of the methods in question

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

a vector dual primal finite element tearing and interconnecting method for solving 3 d large scale electromagnetic problems a lagrange multiplier based non overlapping domain decomposition method referred to as the dual primal finite element tearing and interconnecting feti dp is formulated for the finite element simulation of large three dimensional 3 d electromagnetic problems this formulation extends the feti dp for solving the scalar helmholtz equation to the solution of the vector curl curl wave equation using edge based finite elements it enforces the field continuity explicitly along the edges shared by more than two subdomains and implicitly at the interfaces between two subdomains through the use of lagrange multipliers with the aid of a direct sparse solver for each subdomain system the large global problem is reduced to a much smaller interface problem from which a neumann boundary condition is obtained at the interfaces between all the subdomains this neumann boundary condition is then used to calculate the field within each subdomain it is shown that the resulting feti dpem method is scalable with respect to the size of finite elements and the number of subdomains it is also scalable with respect to the size of the subdomains when the subdomains with its surfaces enclosed by perfect magnetic conductors cannot support any resonant modes the feti dpem method is applied to the electromagnetic simulation of array type structures where the geometrical redundancy is fully exploited to speedup the simulation and reduce the memory requirement numerical results for the simulation of finite antenna arrays and photonic bandgap devices are presented to demonstrate the application accuracy efficiency and capability of the feti dpem method

the second generation feti methods and their application to the parallel solution of large scale linear and geometrically non linear structural analysis problems the feti algorithms are a family of numerically scalable domain decomposition methods they have been designed in the early 1990s for solving iteratively and on parallel machines large scale systems of equations arising from the finite element discretization of solid mechanics structural engineering structural dynamics and acoustic scattering problems and for analyzing complex structures obtained from the assembly of substructures with incompatible discrete interfaces in this paper we present the second generation of these methods that operate more efficiently on large numbers of subdomains offer greater robustness better performance and more flexibility for implementation on a wider variety of computational platforms we also report on the application and performance of these methods for the solution of geometrically non linear structural analysis problems we discuss key aspects of their implementation on shared and distributed memory parallel processors benchmark them against optimized direct sparse solvers and highlight their potential with the solution of large scale structural mechanics problems with several million degrees of freedom

the mosaic of high performance domain decomposition methods for structural mechanics part ii formulation enhancements multiple right hand sides and implicit dynamics domain decomposition methods ddm constitute a family of high performance solvers for large scale problems of computational mechanics and of structural mechanics in particular this paper compares popular ddm for structural analysis in both theory and computational performance while it investigates the impact of enhanced mapping operators in their convergence behavior multiple right hand side problems are considered while standard and new ddm solution techniques for implicit dynamics are tested the performed benchmark tests lead to useful conclusions regarding the comparative performance of the methods

a compact support principle for a class of elliptic differential inequalities abstract in this paper we establish a compact support principle for a class of elliptic differential inequalities such as u c x u q for x large where q 0 1 and 1 q n 2q is optimal in the sense that it cannot be replaced by any larger number applying this principle we also consider the existence and nonexistence of compact support solutions for a semilinear elliptic equation

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

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

variational theory and domain decomposition for nonlocal problems abstract in this article we present the first results on domain decomposition methods for nonlocal operators we present a nonlocal variational formulation for these operators and establish the well posedness of associated boundary value problems proving a nonlocal poincare inequality to determine the conditioning of the discretized operator we prove a spectral equivalence which leads to a mesh size independent upper bound for the condition number of the stiffness matrix we then introduce a nonlocal two domain variational formulation utilizing nonlocal transmission conditions and prove equivalence with the single domain formulation a nonlocal schur complement is introduced we establish condition number bounds for the nonlocal stiffness and schur complement matrices supporting numerical experiments demonstrating the conditioning of the nonlocal one and two domain problems 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

on the distribution of free path lengths for the periodic lorentz gas consider the domain and let the free path length be defined as the distribution of values of is studied in the limit as for all it is shown that the value is critical for this problem in other words the limiting behavior of depends only on whether is larger or smaller than

a feti preconditioned conjugate gradient method for large scale stochastic finite element problems in the spectral stochastic finite element method for analyzing an uncertain system the uncertainty is represented by a set of random variables and a quantity of interest such as the system response is considered as a function of these random variables consequently the underlying galerkin projection yields a block system of deterministic equations where the blocks are sparse but coupled the solution of this algebraic system of equations becomes rapidly challenging when the size of the physical system and or the level of uncertainty is increased this paper addresses this challenge by presenting a preconditioned conjugate gradient method for such block systems where the preconditioning step is based on the dual primal finite element tearing and interconnecting method equipped with a krylov subspace reusage technique for accelerating the iterative solution of systems with multiple and repeated right hand sides preliminary performance results on a linux cluster suggest that the proposed solution method is numerically scalable and demonstrate its potential for making the uncertainty quantification of realistic systems tractable

a domain decomposition method for problems with structural heterogeneities on the interface application to a passenger ship in this article we extend a domain decomposition method based on the feti dp linear solver to applications such as passenger ship analysis more generally the method is designed for large scale elastic analysis of a structure which exhibits geometrical and structural heterogeneities such as plate and stiffener assemblies in presence of structural details the problem of the structural heterogeneities on the subdomain interfaces arising from the presence of stiffeners or elastic joints on these interfaces is addressed a suited interface connection between subdomains modeled with plate elements in the case of a 3d assembling is proposed and tested the selection of an efficient preconditioner is presented and the performances and results are discussed in terms of convergence rate for several examples

on micro to macro connections in domain decomposition multiscale methods abstract micro to macro connection techniques constitute a key ingredient in the formulation of multiscale strategies in this contribution several connection methods are explored for the concurrent multiscale analysis of brittle heterogeneous materials particularly these techniques are investigated in a domain decomposition strong coupling multiscale framework the structural component under analysis is partitioned into a number of non overlapping domains and a fine scale resolution is assigned therein in an adaptive manner exploiting a zoom in technique mesh refinement is employed in the domains where crack coalescence and growth take place the original contribution presented in this manuscript consists in the study of different strong and weak locality constraints that connect coarse and fine resolution domains standard collocation and average compatibility are considered and serve as a basis for the development of two new interscale links the influence of different locality constraints is studied in terms of the mechanical response and the error distribution is compared to a full fine scale analysis

on dual schur domain decomposition method for linear first order transient problems this paper addresses some numerical and theoretical aspects of dual schur domain decomposition methods for linear first order transient partial differential equations the spatially discrete system of equations resulting from a dual schur domain decomposition method can be expressed as a system of differential algebraic equations daes in this work we consider the trapezoidal family of schemes for integrating the ordinary differential equations odes for each subdomain and present four different coupling methods corresponding to different algebraic constraints for enforcing kinematic continuity on the interface between the subdomains unlike the continuous formulation the discretized formulation of the transient problem is unable to enforce simultaneously the continuity of both the primary variable and its rate along the subdomain interface except for the backward euler method method 1 d continuity is based on the conventional approach using continuity of the primary variable and we show that this method is unstable for a lot of commonly used time integrators including the mid point rule to alleviate this difficulty we propose a new method 2 modified d continuity and prove its stability for coupling all time integrators in the trapezoidal family except the forward euler method 3 v continuity is based on enforcing the continuity of the time derivative of the primary variable however this constraint introduces a drift in the primary variable on the interface we present method 4 baumgarte stabilized which uses baumgarte stabilization to limit this drift and we derive bounds for the stabilization parameter to ensure stability our stability analysis is based on the energy method and one of the main contributions of this paper is the extension of the energy method which was previously introduced in the context of numerical methods for odes to assess the stability of numerical formulations for index 2 differential algebraic equations daes finally we present numerical examples to corroborate our theoretical predictions

melnikov s method and arnold diffusion for perturbations of integrable hamiltonian systems we start with an unperturbed system containing a homoclinic orbit and at least two families of periodic orbits associated with action angle coordinates we use kolmogorov arnold moser kam theory to show that some of the resulting tori persist under small perturbations and use a vector of melnikov integrals to show that under suitable hypotheses their stable and unstable manifolds intersect transversely this transverse intersection is ultimately responsible for arnold diffusion on each energy surface the method is applied to a pendulum oscillator system

feti dp bddc and block cholesky methods the feti dp and bddc algorithms are reformulated using block cholesky factorizations an approach which can provide a useful framework for the design of domain decomposition algorithms for solving symmetric positive definite linear system of equations instead of introducing lagrange multipliers to enforce the coarse level primal continuity constraints in these algorithms a change of variables is used such that each primal constraint corresponds to an explicit degree of freedom with the new formulation of these algorithms a simplified proof is provided that the spectra of a pair of feti dp and bddc algorithms with the same set of primal constraints are essentially the same numerical experiments for a two dimensional laplace s equation also confirm this result copyright 2005 john wiley sons ltd