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

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

overlapping schwarz methods for mixed linear elasticity and stokes problems overlapping schwarz preconditioners are introduced and studied for saddle point problems with a penalty term such as stokes equations and mixed formulations of linear elasticity these preconditioners are based on the solution of local saddle point problems on overlapping subdomains and the solution of a coarse saddle point problem numerical experiments show that these are parallel and scalable preconditioners since the rate of convergence of the preconditioned operator is independent of the mesh size h the number of subdomains n and the penalty parameter

a preconditioner for substructuring based on constrained energy minimization a preconditioner for substructuring based on constrained energy minimization concepts is presented the preconditioner is applicable to both structured and unstructured meshes and offers a straightforward approach for the iterative solution of second and fourth order structural mechanics problems the approach involves constraints associated with disjoint sets of nodes on substructure boundaries these constraints provide the means for preconditioning at both the substructure and global levels numerical examples are presented that demonstrate the good performance of the method in terms of iterations compute time and condition numbers of the preconditioned equations

some properties of fractal interpolation functions on sierpinski gasket abstract in this paper we discuss some basic properties of uniform fractal interpolation functions fifs which is a special class of fifs on sierpinski gasket we firstly study the min max property of uniform fifs then we present a necessary and sufficient condition such that uniform fifs have finite energy normal derivative and laplacian of uniform fifs are also discussed

bddc algorithms for incompressible stokes equations the purpose of this paper is to extend the balancing domain decomposition by constraints bddc algorithm to saddle point problems that arise when mixed finite element methods are used to approximate the system of incompressible stokes equations the bddc algorithms are iterative substructuring methods which form a class of domain decomposition methods based on the decomposition of the domain of the differential equations into nonoverlapping subdomains they are defined in terms of a set of primal continuity constraints which are enforced across the interface between the subdomains and which provide a coarse space component of the preconditioner sets of such constraints are identified for which bounds on the rate of convergence can be established that are just as strong as previously known bounds for the elliptic case in fact the preconditioned operator is effectively positive definite which makes the use of a conjugate gradient method possible a close connection is also established between the bddc

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

domain decomposition for less regular subdomains overlapping schwarz in two dimensions in the theory of domain decomposition methods it is often assumed that each subdomain is the union of a small set of coarse triangles or tetrahedra in this study extensions to the existing theory which accommodate subdomains with much less regular shapes are presented the subdomains are required only to be john domains attention is focused on overlapping schwarz preconditioners for problems in two dimensions with a coarse space component of the preconditioner which allows for good results even for coefficients which vary considerably it is shown that the condition number of the domain decomposition method is bounded by c 1 h delta 1 log h h 2 where the constant c is independent of the number of subdomains and possible jumps in coefficients between subdomains numerical examples are provided which confirm the theory and demonstrate very good performance of the method for a variety of subregions including those obtained when a mesh partitioner is used for the domain decomposition

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

an interface strip domain decomposition preconditioner in this paper we consider domain decomposition preconditioners based on a vertex oriented vo decomposition of the computational domain in element oriented eo decompositions each element of the grid belongs to a different domain while in vo decompositions each vertex belongs to a different subdomain based on vo decompositions we present some preconditioners for the solution of the original system as well as for that of the schur complement system theoretical properties are investigated for a finite element approximation of an elliptic problem numerical results and comparison with state of the art preconditioners are also reported the numerical results presented here show the effectiveness of the proposed preconditioners and their good scalability properties

isogeometric block feti dp preconditioners for the stokes and mixed linear elasticity systems abstract the aim of this work is to construct and analyze a feti dp type domain decomposition preconditioner for isogeometric discretizations of the stokes and mixed linear elasticity systems this method extends to the isogeometric analysis context the preconditioner previously proposed by tu and li 2015 for finite element discretizations of the stokes system the resulting isogeometric feti dp algorithm is proven to be scalable in the number of subdomains and has a quasi optimal convergence rate bound which is polylogarithmic in the ratio of subdomain and element sizes extensive two dimensional numerical experiments validate the theory investigate the behavior of the preconditioner with respect to both the spline polynomial degree and regularity and show its robustness with respect to domain deformation material incompressibility and presence of elastic coefficient discontinuities across subdomain interfaces

abstract robust coarse spaces for systems of pdes via generalized eigenproblems in the overlaps coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations pdes however it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the pde coefficients especially for systems of pdes here we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities we achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence we prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous pdes and systems of pdes that confirm this property

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

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

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

non compact symmetric spaces and the toda molecule equations it has been shown by olshanetsky and perelomov that the toda molecule equations associated with any lie groupg describe special geodesic motions on the riemannian non compact symmetric space which is the quotient of the normal real form ofg gn by its maximal compact subgroup this is explained in more detail and it is shown that the fundamental poisson bracket relation involving the lax operatora and leading to the yang baxter equation and integrability properties is a direct consequence of the fact that the iwasawa decomposition forgn endows the symmetric space with a hidden group theoretic structure

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

an alternative coarse space for irregular subdomains and an overlapping schwarz algorithm for scalar elliptic problems in the plane in earlier work on domain decomposition methods for elliptic problems in the plane an assumption that each subdomain is triangular or a union of a few coarse triangles has often been made this is similar to what is required in geometric multigrid theory and is unrealistic if the subdomains are produced by a mesh partitioner in an earlier paper coauthored with axel klawonn the authors introduced a coarse subspace for an overlapping schwarz method with one degree of freedom for each subdomain vertex and one for each subdomain edge a condition number bound proportional to 1 log h h 2 1 h delta was established assuming only that the subdomains are john domains here h delta measures the relative overlap between neighboring subdomains and h h the maximum number of elements across individual subdomains we were also able to relate the rate of convergence to a parameter in an isoperimetric inequality for the subdomains into which the domain of the problem has been partitioned in this paper t

a domain decomposition algorithm for elliptic problems in three dimensions most domain decomposition algorithms have been developed for problems in two dimensions one reason for this is the difficulty in devising a satisfactory easy to implement robust method of providing global communication of information for problems in three dimensions several methods that work well in two dimension do not perform satisfactorily in three dimensions a new iterative substructuring algorithm for three dimensions is proposed it is shown that the condition number of the resulting preconditioned problem is bounded independently of the number of subdomains and that the growth is quadratic in the logarithm of the number of degrees of freedom associated with a subdomain the condition number is also bounded independently of the jumps in the coefficients of the differential equation between subdomains the new algorithm also has more potential parallelism than the iterative substructuring methods previously proposed for problems in three dimensions

bddc preconditioners for spectral element discretizations of almost incompressible elasticity in three dimensions balancing domain decomposition by constraints bddc algorithms are constructed and analyzed for the system of almost incompressible elasticity discretized with gauss lobatto legendre spectral elements in three dimensions initially mixed spectral elements are employed to discretize the almost incompressible elasticity system but a positive definite reformulation is obtained by eliminating all pressure degrees of freedom interior to each subdomain into which the spectral elements have been grouped appropriate sets of primal constraints can be associated with the subdomain vertices edges and faces so that the resulting bddc methods have a fast convergence rate independent of the almost incompressibility of the material in particular the condition number of the bddc preconditioned operator is shown to depend only weakly on the polynomial degree n the ratio h h of subdomain and element diameters and the inverse of the inf sup constants of the subdomains and the underlying mixed formulation while being scalable i e independent of the number of subdomains and robust i e independent of the poisson ratio and young s modulus of the material considered these results also apply to the related dual primal finite element tearing and interconnect feti dp algorithms defined by the same set of primal constraints numerical experiments carried out on parallel computing systems confirm these results

balancing neumann neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity balancing neumann neumann methods are extented to mixed formulations of the linear elasticity system with discontinuous coefficients discretized with mixed finite or spectral elements with discontinuous pressures these domain decomposition methods implicitly eliminate the degrees of freedom associated with the interior of each subdomain and solve iteratively the resulting saddle point schur complement using a hybrid preconditioner based on a coarse mixed elasticity problem and local mixed elasticity problems with natural and essential boundary conditions a polylogarithmic bound in the local number of degrees of freedom is proven for the condition number of the preconditioned operator in the constant coefficient case parallel and serial numerical experiments confirm the theoretical results indicate that they still hold for systems with discontinuous coefficients and show that our algorithm is scalable parallel and robust with respect to material heterogeneities the results on heterogeneous general problems are also supported in part by our theory

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

hybrid domain decomposition algorithms for compressible and almost incompressible elasticity overlapping schwarz methods are considered for mixed finite element approximations of linear elasticity with discontinuous pressure spaces as well as for compressible elasticity approximated by standard conforming finite elements the coarse components of the preconditioners are based on spaces with a number of degrees of freedom per subdomain which are uniformly bounded which are similar to those previously developed for scalar elliptic problems and domain decomposition methods of iterative substructuring type i e methods based on nonoverlapping decompositions of the domain the local components of the new preconditioners are based on solvers on a set of overlapping subdomains in the current study the dimension of the coarse spaces is smaller than in recently developed algorithms in the compressible case all independent face degrees of freedom have been eliminated while in the almost incompressible case five out of six are not needed in many cases this will result in a reduction of the dimension of the coarse space by about one half compared with that of the algorithm previously considered in addition in spite of using overlapping subdomains to define the local components of the preconditioner values of the residual and the approximate solution need only to be retained on the interface between the subdomains in the iteration of the new hybrid schwarz algorithm the use of discontinuous pressures makes it possible to work exclusively with symmetric positive definite problems and the standard preconditioned conjugate gradient method bounds are established for the condition number of the preconditioned operators the bound for the almost incompressible case grows in proportion to the square of the logarithm of the number of degrees of freedom of individual subdomains and the third power of the relative overlap between the overlapping subdomains and it is independent of the poisson ratio as well as jumps in the lame parameters across the interface between the subdomains numerical results illustrate the findings copyright 2009 john wiley sons ltd

overlapping schwarz methods with a standard coarse space for almost incompressible linear elasticity low order finite element discretizations of the linear elasticity system suffer increasingly from locking effects and ill conditioning when the material approaches the incompressible limit if only the displacement variables are used mixed finite elements using both displacement and pressure variables provide a well known remedy but they yield larger and indefinite discrete systems for which the design of scalable and efficient iterative solvers is challenging two level overlapping schwarz preconditioners for the almost incompressible system of linear elasticity discretized by mixed finite elements with discontinuous pressures are constructed and analyzed the preconditioned systems are accelerated either by a gmres generalized minimum residual method applied to the resulting discrete saddle point problem or by a pcg preconditioned conjugate gradient method applied to a positive definite although extremely ill conditioned reformulation of the problem obtained by eliminating all pressure variab

an overlapping schwarz algorithm for almost incompressible elasticity overlapping schwarz methods are extended to mixed finite element approximations of linear elasticity which use discontinuous pressure spaces the coarse component of the preconditioner is based on a low dimensional space previously developed for scalar elliptic problems and a domain decomposition method of iterative substructuring type i e a method based on nonoverlapping decompositions of the domain while the local components of the preconditioner are based on solvers on a set of overlapping subdomains a bound is established for the condition number of the algorithm which grows in proportion to the logarithm of the number of degrees of freedom in individual subdomains and essentially to the third power of the relative overlap between the overlapping subdomains and which is independent of the poisson ratio as well as jumps in the lame parameters across the interface between the subdomains a positive definite reformulation of the discrete problem makes the use of the standard preconditioned conjugate gradient method straightforward numerical results which include a comparison with problems of compressible elasticity illustrate the findings

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

iterative substructuring methods for spectral element discretizations of elliptic systems i compressible linear elasticity an iterative substructuring method for the system of linear elasticity in three dimensions is introduced and analyzed the pure displacement formulation for compressible materials is discretized with the spectral element method the resulting stiffness matrix is symmetric and positive definite the proposed method provides a domain decomposition preconditioner constructed from local solvers for the interior of each element and for each face of the elements and a coarse global solver related to the wire basket of the elements as in the scalar case the condition number of the preconditioned operator is independent of the number of spectral elements and grows as the square of the logarithm of the spectral degree

a nonoverlapping domain decomposition method for incompressible stokes equations with continuous pressures a nonoverlapping domain decomposition algorithm is proposed to solve the linear system arising from mixed finite element approximation of incompressible stokes equations a continuous finite element space for the pressure is used in the proposed algorithm lagrange multipliers are used to enforce continuity of the velocity component across the subdomain boundary the continuity of the pressure component is enforced in the primal form i e neighboring subdomains share the same pressure degrees of freedom on the subdomain interface and no lagrange multipliers are needed after eliminating all velocity variables and the independent subdomain interior parts of the pressures a symmetric positive semidefinite linear system for the subdomain boundary pressures and the lagrange multipliers is formed and solved by a preconditioned conjugate gradient method a lumped preconditioner is studied and the condition number bound of the preconditioned operator is proved to be independent of the number of subdomains fo