remarks on the semilinear schrodinger equation abstract we investigate the initial value problem for some semi linear schrodinger equations in two space dimensions with exponential growth

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

instability for standing waves of nonlinear klein gordon equations via mountain pass arguments we introduce mountain pass type arguments in the context of orbital instability for klein gordon equations our aim is to illustrate on two examples how these arguments can be useful to simplify proofs and derive new results of orbital stability instability for a power type nonlin earity we prove that the ground states of the associated stationary equation are minimizers of the functional action on a wide variety of constraints for a general nonlinearity we extend to the dimension n 2 the classical in stability result for stationary solutions of nonlinear klein gordon equations proved in 1985 by shatah in dimension n 3

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

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

simple measure for complexity a measure of complexity is proposed based on appropriately defined notions of order and disorder which has a considerable degree of flexibility in its dependence on these concepts the possible functional dependencies which result encompass those of many earlier definitions of complexity the proposed measure is in principle easy to calculate and has the property of an intensive thermodynamic quantity with appropriate choices of parameters it behaves similarly to effective measure complexity for the logistic map it is also a generalization of the normalized complexity of lopez ruiz et al but does not suffer from over universality

on the use of inexact subdomain solvers for bddc algorithms the standard bddc balancing domain decomposition by constraints preconditioner is shown to be equivalent to a preconditioner built from a partially subassembled finite element model this results in a system of linear algebraic equations which is much easier to solve in parallel than the fully assembled model the cost is then often dominated by that of the problems on the subdomains an important role is also played both in theory and practice by an averaging operator and in addition exact dirichlet solvers are used on the subdomains in order to eliminate the residual in the interior of the subdomains the use of inexact solvers for these problems and even the replacement of the dirichlet solvers by a trivial extension are considered it is established that one of the resulting algorithms has the same eigenvalues as the standard bddc algorithm and the connection of another with the feti dp algorithm with a lumped preconditioner is also considered multigrid methods are used in the experimental work and under certain assumptions it is established that the iteration count essentially remains the same as when exact solvers are used while considerable gains in the speed of the algorithm can be realized since the cost of the exact solvers grows superlinearly with the size of the subdomain problems while the multigrid methods are linear

coupling of boundary element and wave based methods for the efficient solution of complex multiple scattering problems a novel hybrid method for the efficient solution of complex acoustic multiple scattering problems is proposed in this paper the wave based method and the boundary element method are coupled to benefit from the strengths of both the former is an indirect trefftz approach which has a faster convergence rate and lower computational load compared to element based methods when applied on geometries of moderate complexity the latter is the state of the art technique for unbounded acoustic problems and can handle very complex geometries the idea behind the hybrid method is to take advantage of the fast wbm solution for scatterers of moderate complexity and take advantage of the bem s capability for handling scatterers of high complexity for the bem part the indirect variational formulation is used which allows modeling of open boundary problems zero thickness walls in addition the wbm makes it possible to easily add heterogeneities domain inclusions to the problem therefore the hybrid method does not only aim for better efficiency but also for extending the variety of configurations that can be tackled by both methods with ease the accuracy and the performance of the method are demonstrated with three examples both in 2d and 3d it is shown that when complex and simple scatterers coexist the hybrid method is more efficient than the wbm the bem and the fast multipole bem while it is able to provide accuracy of similar level

incorporation of multipoint constraints into the balancing domain decomposition method and its parallel implementation the domain decomposition method ddm is a major solution algorithm that is used to parallelize the finite element method in the case of implicit structural analysis using the ddm the substructuring based iterative linear solver is a powerful tool when an effective preconditioner such as the balancing domain decomposition bdd method is used in the present study a method by which to incorporate a set of linear multipoint constraints mpc into the bdd method is proposed in this method when an mpc is enforced on the internal degrees of freedom dofs in some subdomains the dofs are converted into interface dofs that is all of the dofs constrained by mpcs become interface dofs then the interface problem with the set of mpcs for the interface dofs is solved by the conjugate projected gradient method in order to combine the above procedure with the preconditioner used in the bdd method the effect of the mpcs for the interface dofs is also imposed on the coarse grid problem of the bdd method using the penalty method a parallel implementation of the present method is also described some illustrative examples are solved and good convergence and parallel performance of the present method are demonstrated copyright 2006 john wiley sons ltd

domain decomposition methods for large linearly elliptic three dimensional problems abstract the idea of solving large problems using domain decomposition technique appears particularly attractive on present day large scale parallel computers but the performance of such techniques used on a parallel computer depends on both the numerical efficiency of the proposed algorithm and the efficiency of its parallel implementation the approach proposed herein splits the computational domain in unstructured subdomains of arbitrary shape and solves for unknown on the interface using the associated trace operator the steklov poincare operator on the continuous level or the schur complement matrix after a finite element discretization and a preconditioned conjugate gradient method this algorithm involves the solution of dirichlet and of neumann problems defined on each subdomain and which can be solved in parallel this method has been implemented on a cray 2 computer using multitasking and on an intel hypercube it was tested on a large scale industrial ill conditioned three dimensional linear elasticity problem which gives a fair indication of its performance in a real life environment in such situations the proposed method appears operational and competitive on both machines compared to standard techniques it yields faster results with far less memory requirements

stability of solitary waves for nonlinear schrodinger equations with inhomogeneous nonlinearities we show that the ground state solitary waves of the critical nonlinear schrodinger equation i t t r v 4 d 0 in dimension d 2 are orbitally stable as 0i f v 0 v 4 0 g d v 0 2 where gd is a constant that depends only on d 2002 elsevier science b v all rights reserved

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

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

offdiagonal complexity a computationally quick complexity measure for graphs and networks abstract a vast variety of biological social and economical networks shows topologies drastically differing from random graphs yet the quantitative characterization remains unsatisfactory from a conceptual point of view motivated from the discussion of small scale free networks a biased link distribution entropy is defined which takes an extremum for a power law distribution this approach is extended to the node node link cross distribution whose nondiagonal elements characterize the graph structure beyond link distribution cluster coefficient and average path length from here a simple and computationally cheap complexity measure can be defined this offdiagonal complexity odc is proposed as a novel measure to characterize the complexity of an undirected graph or network while both for regular lattices and fully connected networks odc is zero it takes a moderately low value for a random graph and shows high values for apparently complex structures as scale free networks and hierarchical trees the odc approach is applied to the helicobacter pylori protein interaction network and randomly rewired surrogates

power law distributions resulting from finite resources abstract an elementary stochastic model termed the normalization model is put forward which does demonstrate that power laws generically occur in systems with finite resources the model is capable to exhibit power law distributions with arbitrary power law exponents nevertheless for a large fraction of the parameter space power law exponents near unity are obtained as an application of the normalization mechanism we consider a network growth saturation model this model extends the scale free network model sf to include the fact of finite resources in the network growth saturation model the scale free property holds only for the growth period within the stationary regime we obtain power law distributions of the weight of the edges among the vertices we conjecture that this pattern will be found in the internet if it reaches the steady state

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

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 scalable substructuring method by lagrange multipliers for plate bending problems we present a new lagrange multiplier based domain decomposition method for solving iteratively systems of equations arising from the finite element discretization of plate bending problems the proposed method is essentially an extension of the finite element tearing and interconnecting substructuring algorithm to the biharmonic equation the main idea is to enforce continuity of the transversal displacement field at the subdomain crosspoints throughout the preconditioned conjugate gradient iterations the resulting method is proved to have a condition number that does not grow with the number of subdomains but rather grows at most polylogarithmically with the number of elements per subdomain these optimal properties hold for numerous plate bending elements that are used in practice including the hsieh clough tocher element the discrete kirchhoff triangle and a class of nonlocking elements for the reissner mindlin plate models computational experiments are reported and shown to confirm the theoretical optimal convergence properties of the new domain decomposition method computational efficiency is also demonstrated with the numerical solution in 45 iterations and 105 seconds on a 64 processor ibm sp2 of a plate bending problem with almost one million degrees of freedom

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

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

stable boundary element domain decomposition methods for the helmholtz equation in this paper we present a stable boundary element domain decomposition method to solve boundary value problems of the helmholtz equation via a tearing and interconnecting approach a possible non uniqueness of the solution of local boundary value problems due to the appearance of local eigensolutions is resolved by using modified interface conditions of robin type which results in a galerkin boundary element discretization which is robust for all local wave numbers numerical examples confirm the stability of the proposed approach

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

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

balancing neumann neumann methods for incompressible stokes equations this paper describes the unerlying mathematical model and the balancing neumann neumann methods are introduced and studied for incompressible stokes equations discretized with mixed finite or spectral elements with discontinuous pressures after decomposing the original domain of the problem into nonoverlapping subdomains the interior unknowns which are the interior velocity component and all except the constant pressure component of each subdomain problem are implicitly eliminated the resulting saddle point schur complement is solved with a krylov space method with a balancing neumann neumann preconditioner based on the solution of a coarse stokes problem with a few degrees of freedom per subdomain and on the solution of local stokes problems with natural neumann velocity and essential boundary conditions on the subdomains this preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively the condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and the inverse of the inf sup constants of the discrete problem and of the coarse subproblem numerical results show that the method is quite fast they are also fully consistent with the theory

an extension of the feti domain decomposition method for incompressible and nearly incompressible problems incompressible and nearly incompressible problems are treated herein with a mixed finite element formulation in order to avoid ill conditioning that prevents accuracy in pressure estimation and can lead to a lack of convergence for iterative solution algorithms a multilevel dual domain decomposition method is then chosen as an iterative algorithm the original feti and feti dp methods are extended to deal with such problems when the discretization of the pressure field is discontinuous throughout the elements a dedicated augmentation of the algorithms is proposed and the different methods are compared with several preconditioners for bidimensional test cases the resulting approaches are both optimal and numerically scalable and their costs are estimated with a complexity analysis

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

blowup of h1 solutions for a class of the focusing inhomogeneous nonlinear schrodinger equation abstract we consider a class of focusing inhomogeneous nonlinear schrodinger equation i t u u x b u u 0 u 0 u 0 h 1 r d with 0 b min 2 d and where 4 2 b d and 4 2 b d 2 if d 3 and if d 1 2 in the mass critical case we prove that if u 0 has negative energy and satisfies either x u 0 l 2 or u 0 is radial with d 2 then the corresponding solution blows up in finite time moreover when d 1 we prove that if the initial data not necessarily radial has negative energy then the corresponding solution blows up in finite time in the mass and energy intercritical case we prove the finite time blowup for radial negative energy initial data as well as the finite time blowup below ground state for radial initial data in dimensions d 2 this result extends the one of farah in j evol equ 16 193 208 2016 where the blowup below ground state was proved for data in the virial space h 1 l 2 x 2 d x with d 1

neumann neumann domain decomposition preconditioners for linear quadratic elliptic optimal control problems we present a class of domain decomposition dd preconditioners for the solution of elliptic linear quadratic optimal control problems our dd preconditioners are extensions of neumann neumann dd preconditioners which have been successfully applied to the solution of single pdes the dd preconditioners are based on a decomposition of the optimality conditions for the elliptic linear quadratic optimal control problem into smaller subdomain optimality conditions with dirichlet boundary conditions for the states and the adjoints on the subdomain interfaces these subdomain optimality conditions are coupled through neumann interface conditions for the states and the adjoints this decomposition leads to a schur complement system in which the unknowns are the state and adjoint variables on the subdomain interfaces the schur complement operator is the sum of subdomain schur complement operators the application of which is shown to correspond to the solution of subdomain elliptic linear quadratic optimal control problems which are essentially smaller copies of the original optimal control problem we show that under suitable conditions the application of the inverse of the subdomain schur complement operators requires the solution of a subdomain elliptic linear quadratic optimal control problem with neumann interface conditions for the state the subdomain schur complement operators are analyzed in the variational setting of the problem as well as the algebraic setting obtained after a finite element discretization of the problem definiteness properties of the algebraic form of the subdomain schur complement operator s are studied numerical tests show that the dependence of these preconditioners on mesh size and subdomain size is comparable to its counterpart applied to elliptic equations only these tests also show that the preconditioners are insensitive to the size of the control regularization parameter

a preconditioner for the feti dp formulation with mortar methods in two dimensions in this paper we consider a dual primal feti feti dp method for elliptic problems on nonmatching grids the feti dp method is a domain decomposition method that uses lagrange multipliers to match solutions continuously across subdomain boundaries in the sense of dual primal variables we use the mortar matching condition as the continuity constraints for the feti dp formulation we construct a preconditioner for the feti dp operator and show that the condition number of the preconditioned feti dp operator is bounded by c max i 1 ldots n left 1 log left h i h i right right 2 where h i and h i are sizes of domain and mesh for each subdomain respectively and c is a constant independent of h i s and h i s we allow jumps of coefficients of elliptic problems across subdomain boundaries numerical results are included

nonlinear feti dp and bddc methods new nonlinear feti dp dual primal finite element tearing and interconnecting and bddc balancing domain decomposition by constraints domain decomposition methods are introduced in all these methods in each iteration local nonlinear problems are solved on the subdomains the new approaches can significantly reduce communication and show a significantly improved performance especially for problems with localized nonlinearities compared to a standard newton krylov feti dp or bddc approach moreover the coarse space of the nonlinear feti dp methods can be used to accelerate the newton convergence it is also found that the new nonlinear feti dp and nonlinear bddc methods are not as closely related as in the linear context numerical results for the p laplace operator are presented

instability of nonlinear bound states we establish a sharp instability theorem for the bound states of lowest energy of the nonlinear klein gordon equation utt u f u 0 and the nonlinear schrodinger equation iut u f u 0

information theoretic description of networks we present a new information theoretic approach for network characterizations it is developed to describe the general type of networks with n nodes and l directed and weighted links i e it also works for the simpler undirected and unweighted networks the new information theoretic measures for network characterizations are based on a transmitter receiver analogy of effluxes and influxes based on these measures we classify networks as either complex or non complex and as either democracy or dictatorship networks directed networks in particular are furthermore classified as either information spreading and information collecting networks the complexity classification is based on the information theoretic network complexity measure medium articulation ma it is proven that special networks with a medium number of links l n1 5 show the theoretical maximum complexity ma logn 2 2 a network is complex if its ma is larger than the average ma of appropriately randomized networks ma mar a network is of the democracy type if its redundancy r h b a it is an information collecting network otherwise an information spreading network finally different real networks directed and undirected weighted and unweighted are classified according to our general scheme

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

blow up solutions for the inhomogeneous schrodinger equation with l2 supercritical nonlinearity abstract this paper studies blow up solutions for the inhomogeneous schrodinger equation with l 2 supercritical nonlinearity in terms of strauss arguments in strauss 1977 22 we find a new compactness lemma for radial symmetric functions thus we use it to derive the best constants of two generalized gagliardo nirenberg type inequalities moreover we obtain a more precisely sharp criteria of blow up and global existence and derive the weak concentration phenomenon of blow up solutions by the variational 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

balancing domain decomposition for problems with large jumps in coefficients the balancing domain decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent the abstract theory introduced recently by the first named author is used to develop condition number bounds for conforming linear elements in two and three dimensions the bounds are independent of arbitrary coefficient jumps between subdomains and of the number of subdomains and grow only as the squared logarithm of the mesh size h computational experiments for two and three dimensional problems confirm the theory

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 theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

graph distance measures based on topological indices revisited graph distance measures based on topological indices have been already explored by dehmer et al also inequalities for those graph distance measures have been proved in this paper we continue studying such comparative graph measures based on the well known wiener index graph energy and randic index respectively we prove extremal properties of the graph distance measures for some special classes of graphs to demonstrate useful properties of the measures we also discuss numerical results to conclude the paper we state some open problems

global existence and blow up of solutions for the inhomogeneous nonlinear schrodinger equation in r2 abstract this paper discusses a class of inhomogeneous nonlinear schrodinger equation i t u t x u t x v x u t x p 1 u t x u 0 x u 0 x where t x r r 2 v x satisfies some assumptions by a constrained variational problem we firstly define some cross constrained invariant sets for the inhomogeneous nonlinear schrodinger equation then we obtain some sharp conditions for global existence and blow up of solutions as a consequence it is shown that the solution is globally well posed in h r 1 r 2 with the h 1 norm of the initial data u 0 which is dominated by the minimal value of the constrained variational problem

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

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

a domain decomposition method for the helmholtz equation and related optimal control problems we present an iterative domain decomposition method to solve the helmholtz equation and related optimal control problems the proof of convergence of this method relies on energy techniques this method leads to efficient algorithms for the numerical resolution of harmonic wave propagation problems in homogeneous and heterogeneous media

remarks on the inhomogeneous fractional nonlinear schrodinger equation using a sharp gagliardo nirenberg type inequality well posedness issues of the initial value problem for a fractional inhomogeneous schrodinger equation are investigated

instability of bound states of a nonlinear schrodinger equation with a dirac potential abstract we study analytically and numerically the stability of the standing waves for a nonlinear schrodinger equation with a point defect and a power type nonlinearity a major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves this is overcome by a perturbation method and continuation arguments among others in the case of a repulsive defect we show that the standing wave solution is stable in h rad 1 r and unstable in h 1 r under subcritical nonlinearity further we investigate the nature of instability under critical or supercritical nonlinear interaction we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints in the subcritical radial case unstable bound states cannot collapse but rather narrow down until they reach the stable regime a finite width instability in the nonradial repulsive case all bound states are unstable and the instability is manifested by a lateral drift away from the defect sometimes in combination with a finite width instability or a blowup instability

a scalable feti dp algorithm with non penetration mortar conditions on contact interface by combining feti algorithms of dual primal type with recent results for bound constrained quadratic programming problems we develop an optimal algorithm for the numerical solution of coercive variational inequalities the model problem is discretized using non penetration conditions of mortar type across the potential contact interface and a feti dp algorithm is formulated the resulting quadratic programming problem with bound constraints is solved by a scalable algorithm with a known rate of convergence given in terms of the spectral condition number of the quadratic problem numerical experiments for non matching meshes across the contact interface confirm the theoretical scalability of the algorithm

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

what is a complex graph abstract many papers published in recent years show that real world graphs g n m n nodes m edges are more or less complex in the sense that different topological features deviate from random graphs here we narrow the definition of graph complexity and argue that a complex graph contains many different subgraphs we present different measures that quantify this complexity for instance c 1 e the relative number of non isomorphic one edge deleted subgraphs i e deck size however because these different subgraph measures are computationally demanding we also study simpler complexity measures focussing on slightly different aspects of graph complexity we consider heuristically defined product measures the products of two quantities which are zero in the extreme cases of a path and clique and entropy measures quantifying the diversity of different topological features the previously defined network graph complexity measures medium articulation and offdiagonal complexity odc belong to these two classes we study odc measures in some detail and compare it with our new measures for all measures the most complex graph g c max has a medium number of edges between the edge numbers of the minimum and the maximum connected graph n 1 m g c max n n 1 2 interestingly for some measures c this number scales exactly with the geometric mean of the extremes m g c max n 2 n 1 n 1 5 all graph complexity measures are characterized with the help of different example graphs for all measures the corresponding time complexity is given finally we discuss the complexity of 33 real world graphs of different biological social and economic systems with the six computationally most simple measures including odc the complexities of the real graphs are compared with average complexities of two different random graph versions complete random graphs just fixed n m and rewired graphs with fixed node degrees

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 computational approach to construct a multivariate complete graph invariant in this paper we present a computational approach for finding complete graph invariants specifically we generate exhaustive sets of connected non isomorphic graphs with 9 and 10 vertices and demonstrate that a 97 dimensional multivariate graph invariant is capable to distinguish each of the non isomorphic graphs furthermore in order to tame the computational complexity of the problem caused by the vast number of graphs e g involving over 10 million networks with 10 vertices we suggest a low dimensional iterative procedure that is based on highly discriminative individual graph invariants we show that also this computational approach leads to a perfect discrimination overall our numerical results prove the existence of such graph invariants for networks with 9 and 10 vertices furthermore we show that our iterative approach has a polynomial time complexity

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

a study on the lumped preconditioner and memory requirements of feti and related primal domain decomposition methods in recent years domain decomposition methods ddms have emerged as advanced solvers in several areas of computational mechanics in particular during the last decade in the area of solid and structural mechanics they reached a considerable level of advancement and were shown to be more efficient than popular solvers like advanced sparse direct solvers the present contribution follows the lines of a series of recent publications on the relationship between primal and dual formulations of ddms in some of these papers the effort to unify primal and dual methods led to a family of ddms that was shown to be more efficient than the previous methods the present paper extends this work presenting a new family of related ddms thus enriching the theory of the relations between primal and dual methods with the primal methods which correspond to the dual ddm that uses the lumped preconditioner the paper also compares the numerical performance of the new methods with that of the previous ones and focuses particularly on memory requirement issues related to the use of the lumped preconditioner suggesting a particularly memory efficient formulation

instability of standing waves of the schrodinger equation with inhomogeneous nonlinearity this paper is concerned with the inhomogeneous nonlinear shrodinger equation inls equation iu t u v x u pu 0 x r n in the critical and supercritical cases p 4 n with n 2 it is shown here that standing wave solutions of inls equation on h 1 r n perturbation are nonlinearly unstable or unstable by blow up under certain conditions on the potential term v with a small 0

multilevel schwarz methods for elliptic problems with discontinuous coefficients in three dimensions multilevel schwarz methods are developed for a conforming finite element approximation of second order elliptic problems we focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions we establish a condition number estimate for the iterative operator which is independent of the coefficients and grows at most as the square of the number of levels we also characterize a class of distributions of the coefficients called quasi monotone for which the weighted l 2 projection is stable and for which we can use the standard piecewise linear functions as a coarse space in this case we obtain optimal methods i e bounds which are independent of the number of levels and subregions we also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas we consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods

a two level domain decomposition method for the iterative solution of high frequency exterior helmholtz problems summary we present a lagrange multiplier based two level domain decomposition method for solving iteratively large scale systems of equations arising from the finite element discretization of high frequency exterior helmholtz problems the proposed method is essentially an extension of the regularized feti finite element tearing and interconnecting method to indefinite problems its two key ingredients are the regularization of each subdomain matrix by a complex interface lumped mass matrix and the preconditioning of the interface problem by an auxiliary coarse problem constructed to enforce at each iteration the orthogonality of the residual to a set of carefully chosen planar waves we show numerically that the proposed method is scalable with respect to the mesh size the subdomain size and the wavenumber we report performance results for a submarine application that highlight the efficiency of the proposed method for the solution of high frequency acoustic scattering problems discretized by finite elements

primal and dual interface concentrated iterative substructuring methods this paper is devoted to the fast solution of interface concentrated finite element equations the interface concentrated finite element schemes are constructed on the basis of a nonoverlapping domain decomposition where a conforming boundary concentrated finite element approximation is used in every subdomain similar to boundary element domain decomposition methods the total number of unknowns per subdomain behaves like o h h d 1 where h h and d denote the usual scaling parameter of the subdomains the average discretization parameter of the subdomain boundaries and the spatial dimension respectively we propose and analyze primal and dual substructuring iterative methods which asymptotically exhibit the same or at least almost the same complexity as the number of unknowns in particular the so called all floating finite element tearing and interconnecting solvers are highly parallel and very robust with respect to large coefficient jumps

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

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

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

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

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