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

strictly local growth of penrose patterns non locality in the sense of penrose does not imply that no local growth algorithms can exist which generate global tilings in an aperiodic species the unsuccessful search until now for the species of penrose tilings sp provided the motivation to develop growth algorithms which accept violation of matching rules these are described and statistical data are given

lower bounds for three dimensional nonoverlapping domain decomposition algorithms lower bounds for the condition numbers of the preconditioned systems are obtained for the wire basket preconditioner and the neumann neumann preconditioner in three dimensions they show that the known upper bounds are sharp

algebraic multigrid amg experiences and comparisons algebraic multigrid methods are designed for the solution of sparse linear systems of equations using multigrid principles in contrast to standard multigrid methods amg does not take advantage of the origin of a particular system of equations at hand nor does it exploit any underlying geometrical situation fully automatically and based solely on algebraic information contained in the given matrix amg constructs a sequence of grids and corresponding operators a special amg algorithm will be presented for a wide range of problems including certain problems which do not have a continuous background this algorithm yields an iterative method which exhibits a convergence behavior typical for multigrid 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

finite element simulation of laminar swirling decaying flow induced by means of a tangential inlet in an annulus abstract this work is dedicated to the numerical modeling of hydrodynamics of an annular swirling decaying flow induced by means of a single tangential inlet in laminar flow regime complete navier stokes equations have been solved by a finite element method using the fidap package we have shown that a correct numerical modeling is strongly linked to the mesh generation the design of which is crucial indeed the tangential inlet induces recirculating flows whose shapes and sizes are intimately depending on both the axial location from the swirl generator and the azimuthal position with respect to the tangential inlet axis the swirl decay along the flow path has been characterized using various profiles of both tangential and axial velocity components streamlines angles and local swirl intensity near both cylinders of the annulus the mean swirl intensity in the whole annular gap thickness has been found to decrease exponentially from the entrance section

on block diagonal and schur complement preconditioning we study symmetric positive definite linear systems with a 2 by 2 block matrix preconditioned by inverting directly one of the diagonal blocks and suitably preconditioning the other using an approximate version of young s property a we show that the condition number of the schur complement is smaller than the condition number obtained by the block diagonal preconditioning we also get bounds on both condition numbers from a strengthened cauchy inequality for systems arising from the finite element method the bounds do not depend on the number of elements and can be obtained from element by element computations the results are applied to thep version finite element method where the first block of variables consists of degrees of freedom of a low order

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

improving jacobi and gauss seidel iterations abstract when convergent jacobi or gauss seidel iterations can be applied to solve systems of linear equations a natural question is how convergence rates are affected if the original system is modified by performing some gaussian elimination we prove that if the initial iteration matrix is nonnegative then such elimination improves convergence our results extend those contained in 4

a review of algebraic multigrid abstract since the early 1990s there has been a strongly increasing demand for more efficient methods to solve large sparse unstructured linear systems of equations for practically relevant problem sizes classical one level methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an efficient solution of even larger problems this paper gives a review of the first hierarchical and purely matrix based approach algebraic multigrid amg amg can directly be applied for instance to efficiently solve various types of elliptic partial differential equations discretized on unstructured meshes both in 2d and 3d since amg does not make use of any geometric information it is a plug in solver which can even be applied to problems without any geometric background provided that the underlying matrix has certain properties

effective renormalization group algorithm for transport in oil reservoirs abstract a real space renormalization group algorithm is used to evaluate macroscopic permeabilities the algorithm was tested on many different distributions with large spatial correlations and anisotropies and the results are very close to the expected exact values

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

inexact feti dp methods inexact feti dp domain decomposition methods are considered preconditioners based on formulations of feti dp as a saddle point problem are used which allow for an inexact solution of the coarse problem a positive definite reformulation of the preconditioned saddle point problem which also allows for approximate solvers is considered as well in the formulation that iterates on the original feti dp saddle point system it is also possible to solve the local neumann subdomain problems inexactly given good approximate solvers for the local and coarse problems convergence bounds of the same quality as for the standard feti dp methods are obtained numerical experiments which compare the convergence of the inexact methods with that of standard feti dp are shown for 2d and 3d elasticity using gmres and cg as krylov space methods based on parallel computations a comparison of one variant of the inexact feti dp algorithms and the standard feti dp method is carried out and similar parallel performance is achieved parallel scalability of the inexact variant is also demonstrated it is shown that for a very large number of subdomains and a very large coarse problem the inexact method can be superior copyright 2006 john wiley sons ltd

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

efficient preconditioning for the p version finite element method in two dimensions parallel preconditioners are formulated and analyzed for systems of equations arising from the p version finite element method applied to second order self adjoint elliptic boundary value problems in two dimensions by using new theoretical results for polynomial spaces it is proved that the condition number of the preconditioned system grows as log 2 p where p is the degree of the polynomial space numerical results are presented showing that the condition number indeed grows very slowly with p

the contraction number of a multigrid method for solving the poisson equation the treatment of a multigrid method in the framework of numerical analysis elucidates that regularity of the solution is not necessary for the convergence of the multigrid algorithm but only for fast convergence for the linear equations which arise from the discretization of the poisson equation a convergence factor 0 5 is established independent of the shape of the domain and of the regularity of the solution

self avoiding walks in quenched random environments the self avoiding walk in a quenched random environment is studied using realspace and field theoretic renormalization and flory arguments these methods indicate that the system is described for d de by a strong disorder fixed point corresponding to a glass state in which the polymer is confined to the lowest energy path this fixed point is characterized by scaling laws for the size of the walk l n with n the number of steps and the fluctuations in the free energy af l the bound 1 co pore and suggests that this inequality holds for d 2 and 3 although pure cannot be excluded particularly for d 2 for d d c there is a transition between strong and weak disorder phases at which pure the strong disorder fixed point for saws on percolation clusters is discussed the analogy with directed walks is emphasized

efficient algebraic multigrid algorithms and their convergence in this paper efficient algorithms for the algebraic multigrid amg method are advanced and amg methods are proposed for improving the convergence rate a new convergence theorem of amg is given and a complete theoretical study of new algorithms is subsequently presented various numerical results demonstrate the efficiency of the new approaches

finite element simulation of mass transfer in laminar swirling decaying flow induced by means of a tangential inlet in an annulus abstract this work is devoted to the numerical modeling of mass transfer on both cylinders of an annular cell subjected to swirling laminar decaying flow induced by means of a single tangential inlet greater in diameter e 12 5 mm than the annular gap thickness e 7 mm the complete diffusion convection equation has been solved in a three dimensional frame by means of the fidap finite element package using the velocity fields previously predicted by the same numerical technique as for the flow field we have shown that a correct prediction of mass transfer coefficients in such a flow is strongly linked to an adequate mesh generation especially when mass transfer occurs on the inner wall of the annulus near which recirculating eddies exist numerical results have been found to be in good agreement with experimental data previously obtained on both cylinders of the annulus a simulation study of the influence of the schmidt number on the overall mass transfer on the outer cylinder has shown that the mean sherwood number varies as sc 0 44 even for forced convection the numerical simulation of mass transfer is not directly obtained from the simulation of the dynamics of the flow due to the difference in mass and momentum boundary layer thicknesses the mesh refinement has to be adapted to schmidt number

polymers on disordered hierarchical lattices a nonlinear combination of random variables the problem of directed polymers on disordered hierarchical and hypercubic lattices is considered for the hierarchical lattices the problem can be reduced to the study of the stable laws for combining random variables in a nonlinear way we present the results of numerical simulations of two hierarchical lattices finding evidence of a phase transition in one case for a limiting case we extend the perturbation theory developed by derrida and griffiths to nonzero temperature and to higher order and use this approach to calculate thermal and geometrical properties overlaps of the model in this limit we obtain an interpolation formula allowing one to obtain the noninteger moments of the partition function from the integer moments we obtain bounds for the transition temperature for hierarchical and hypercubic lattices and some similarities between the problem on the two different types of lattice are discussed

acceleration and stabilization properties of minimal residual smoothing technique in multigrid we analyze the standard multigrid method accelerated by a minimal residual smoothing mrs technique we show that mrs can accelerate the convergence of the slow residual components thus accelerates the overall multigrid convergence we prove that under certain hypotheses mrs stabilizes the divergence of certain slow residual components and thus stabilizes the divergent multigrid iteration the analysis is customarily conducted on the two level method

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

on friedrichs poincare type inequalities friedrichs and poincare type inequalities are important and widely used in the area of partial differential equations and numerical analysis most of their proofs appearing in references are the argument of reduction to absurdity in this paper we give direct proofs of friedrichs type inequalities in h 1 and poincare type inequalities in some subspaces of w 1 p the dependencies of the inequality coefficients on the domain and some sub domains are illustrated explicitly

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

self avoiding walks on random fractal environments self avoiding random walks saws are studied on several hierarchical lattices in a randomly disordered environment an analytical method to determine whether their fractal dimensiondsaw is affected by disorder is introduced using this method it is found that for some lattices dsaw is unaffected by weak disorder while for othersdsaw changes even for infinitestimal disorder a weak disorder exponent is defined and calculated analytically measures the dependence of the variance in the partition function or in the effective fugacity per step v l on the end to end distance of the saw l for lattices which are stable against weak disorder 0 a phase transition exists at a critical valuev v which separates weak and strong disorder phases the geometrical properties which contribute to the value of are discussed

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

ensemble level multiscale finite element and preconditioner for channelized systems and applications we consider multiscale flow in porous media we assume that we can characterize the ensemble of all possible flow scenarios that is we can describe all possible permeability configurations needed for the simulations we construct coarse basis functions that can provide inexpensive coarse approximations that are 1 adequate for all possible flow scenarios in the given ensemble 2 robust with respect to the small scales and high variations in each flow scenario the coarse approximations developed here can be used as a multiscale finite element method or as the coarse solver in a two level domain decomposition iterative method the methods presented here extend to the ensemble case some of the results in j galvis y efendiev domain decomposition preconditioners for multiscale flows in high contrast media reduced dimension coarse spaces siam multiscale model simul 8 5 2010 1621 1644 and y efendiev j galvis x h wu multiscale finite element methods for high contrast problems using local spectral basis functions j comput phys 230 4 2011 937 955 specifically ensembles of permeability fields with high contrast channels and inclusions are considered our main objective here is to construct special multiscale basis functions for the whole ensemble of flow scenarios the coarse basis functions are pre computed for selected or constructed permeability fields with certain topological properties this procedure is a preprocessing step and it avoids constructing basis functions or computing upscaling parameters for each permeability realization then for any permeability the solution of elliptic equation can be project to the space spanned by these pre computed basis functions we apply this coarse multiscale solver to the design of two level domain decomposition preconditioner numerical experiments show that the ensemble level multiscale finite element method converges to the reference solution numerical experiments also show that the ensemble level domain decomposition preconditioner condition number is independent of the high contrast in the coefficient

a note on multigrid for the three dimensional poisson equation in cylindrical coordinates abstract following young and dauwalder 1965 iyengar and manohar 1988 derived high order difference methods for the solution of the three dimensional poisson equation and heat equation in polar cylindrical coordinates in this note we implement and compare s and v cycles in the multigrid context for the fourth order method derived by iyengar and manohar 1988 for the solution of the three dimensional poisson equation defect correction is also studied suitable restrictions and prolongations for the three dimensional problems are given it is seen that the s cycle is more efficient than the v cycle and defect correction is more economical than the direct implementation of the fourth order method

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

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

parallel computation of flow in heterogeneous media modelled by mixed finite elements abstract in this paper we describe a fast parallel method for solving highly ill conditioned saddle point systems arising from mixed finite element simulations of stochastic partial differential equations pdes modelling flow in heterogeneous media each realisation of these stochastic pdes requires the solution of the linear first order velocity pressure system comprising darcy s law coupled with an incompressibility constraint the chief difficulty is that the permeability may be highly variable especially when the statistical model has a large variance and a small correlation length for reasonable accuracy the discretisation has to be extremely fine we solve these problems by first reducing the saddle point formulation to a symmetric positive definite spd problem using a suitable basis for the space of divergence free velocities the reduced problem is solved using parallel conjugate gradients preconditioned with an algebraically determined additive schwarz domain decomposition preconditioner the result is a solver which exhibits a good degree of robustness with respect to the mesh size as well as to the variance and to physically relevant values of the correlation length of the underlying permeability field numerical experiments exhibit almost optimal levels of parallel efficiency the domain decomposition solver doug http www maths bath ac uk parsoft used here not only is applicable to this problem but can be used to solve general unstructured finite element systems on a wide range of parallel architectures

an overlapping schwarz algorithm for raviart thomas vector fields with discontinuous coefficients overlapping schwarz methods form one of two major families of domain decomposition methods in this paper a two level overlapping schwarz method for raviart thomas vector fields is developed and analyzed the global component of the preconditioner is based on local energy minimizing discrete harmonic extensions of coarse interface values and the local components use local direct solvers on overlapping subdomains the algorithm can be implemented algebraically using submatrices of a fully assembled stiffness matrix of the entire finite element problem moreover it gives a condition number bound which depends only on the size of the local subproblems the condition number of our method is shown to be bounded by c 1 log h h 1 h delta where h is the diameter of the subdomain and h and delta are the sizes of the elements and the overlap between subdomains respectively the bound of the condition number is independent of the values and jumps of the coefficients across the interface

unstructured additive schwarz conjugate gradient method for elliptic problems with highly discontinuous coefficients this paper concerns the iterative solution of symmetric elliptic problems with piecewise constant coefficients in two or three space dimensions discretized by linear finite element methods on unstructured triangular or tetrahedral meshes the effect of the discontinuous coefficients is studied by first postulating that there are d fixed regions of the domain where the coefficient takes constant positive values a a1 ad and then considering certain positive coefficient sequences a m in which some of these values approach 0 or infty as m rightarrow infty we consider the performance of additive schwarz domain decomposition preconditioners constructed from local solves on automatically generated subdomains together with a global solve on some coarser grid assuming no relationship between the regions on which the coefficient function is constant and either the subdomains or the coarse grid we show that the preconditioned conjugate gradient method converges in both the energy and the euclidean norms with a number of iterations which grows only logarithmically in the size of the maximum jump cal j m max a k m a l m k l 1 ldots d as m rightarrow infty the result is obtained by a careful analysis of the preconditioned matrix it cannot be obtained by the usual procedure of estimating condition numbers a simple example is given in which the condition number of both the original and the preconditioned stiffness matrices degrade linearly in cal j m recent results of chan smith and zou have shown that by using this preconditioner together with the conjugate gradient method the number of iterations can be bounded independently of the mesh diameter provided the subdomains have overlap commensurate with the size of the coarse mesh our results now show that this method is also highly resilient to discontinuous coefficients even if no attention is paid to the coefficient discontinuity in the construction of the solver

different self avoiding walks on percolation clusters a small cell real space renormalization group study we calculate the average number of stepsn for edge to edge normal and indefinitely growing self avoiding walks saws on two dimensional critical percolation clusters using the real space renormalization group approach with small h cells our results are of the formn al d saw b wherel is the end to end distance similarly to several deterministic fractals the fractal dimensionsd saw for these three different kinds of saws are found to be equal and the differences between them appear in the amplitudesa and in the correction termsb this behavior is atributed to the hierarchical nature of the critical percolation cluster

a bddc algorithm with adaptive primal constraints for staggered discontinuous galerkin approximation of elliptic problems with highly oscillating coefficients a bddc balancing domain decomposition by constraints algorithm for a staggered discontinuous galerkin approximation is considered after applying domain decomposition method a global linear system on the subdomain interface unknowns is obtained and solved by the conjugate gradient method combined with a preconditioner to construct a preconditioner that is robust to the coefficient variations a generalized eigenvalue problem on each subdomain interface is solved and primal unknowns are selected from the eigenvectors using a predetermined tolerance by the construction of the staggered discontinuous galerkin methods the degrees of freedom on subdomain interfaces are shared by only two subdomains and hence the construction of primal unknowns are simplified the resulting bddc algorithm is shown to have the condition number bounded by the predetermined tolerance a modified algorithm for parameter dependent problems is also introduced where the primal unknowns are only computed in an offline stage numerical results are included to show the performance of the proposed method and to verify the theoretical estimate

random tilings concepts and examples we introduce a concept for random tilings which comprising the conventional one is also applicable to tiling ensembles without height representation in particular we focus on the random tiling entropy as a function of the tile densities in this context and under rather mild assumptions we prove a generalization of the first random tiling hypothesis which connects the maximum of the entropy with the symmetry of the ensemble explicit examples are obtained through the re interpretation of several exactly solvable models this also leads to a counterexample to the analogue of the second random tiling hypothesis about the form of the entropy function near its maximum

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

infinite compressibility states in the hierarchical reference theory of fluids i analytical considerations in its customary formulation for one component fluids the hierarchical reference theory yields a quasilinear partial differential equation pde for an auxiliary quantity f that can be solved even arbitrarily close to the critical point reproduces non trivial scaling laws at the critical singularity and directly locates the binodal without the need for a maxwell construction in the present contribution we present a systematic exploration of the possible types of behavior of the pde for thermodynamic states of diverging isothermal compressibility t as the renormalization group theoretical momentum cutoff approaches zero by purely analytical means we identify three classes of asymptotic solutions compatible with infinite t characterized by uniform or slowly varying bounds on the curvature of f by monotonicity of the build up of diverging t and by stiffness of the pde in part of its domain respectively these scenarios are analzyed and discussed with respect to their numerical properties a seeming contradiction between two of these alternatives and an asymptotic solution derived earlier parola et al phys rev e 48 3321 1993 is easily resolved

quasi monte carlo methods for elliptic pdes with random coefficients and applications we devise and implement quasi monte carlo methods for computing the expectations of nonlinear functionals of solutions of a class of elliptic partial differential equations with random coefficients our motivation comes from fluid flow in random porous media where relevant functionals include the fluid pressure velocity at any point in space or the breakthrough time of a pollution plume being transported by the velocity field our emphasis is on situations where a very large number of random variables is needed to model the coefficient field as an alternative to classical monte carlo we here employ quasi monte carlo methods which use deterministically chosen sample points in an appropriate usually high dimensional parameter space each realization of the pde solution requires a finite element fe approximation in space and this is done using a realization of the coefficient field restricted to a suitable regular spatial grid not necessarily the same as the fe grid in the statistically homogeneous case the corresponding covariance matrix can be diagonalized and the required coefficient realizations can be computed efficiently using fft in this way we avoid the use of a truncated karhunen loeve expansion but introduce high nominal dimension in parameter space numerical experiments with 2 dimensional rough random fields high variance and small length scale are reported showing that the quasi monte carlo method consistently outperforms the monte carlo method with a smaller error and a noticeably better than o n 1 2 convergence rate where n is the number of samples moreover the rate of convergence of the quasi monte carlo method does not appear to degrade as the nominal dimension increases examples with dimension as high as 10 6 are reported

nonstandard coarse spaces and schwarz methods for elliptic problems with discontinuous coefficients using non conforming elements two level domain decomposition methods are developed for a simple nonconforming approximation of second order elliptic problems a bound is established for the condition number of these iterative methods that grows only logarithmically with the number of degrees of freedom in each subregion this bound holds for two and three dimensions and is independent of jumps in the value of the coefficients and number of subregions we introduce face coarse spaces and isomorphisms to map between conforming and nonconforming spaces

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

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

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

analysis of feti dp and bddc for linear elasticity in 3d with almost incompressible components and varying coefficients inside subdomains feti dp dual primal finite element tearing and interconnecting methods are nonoverlapping domain decomposition methods which are used to solve large algebraic systems of equations that arise e g from problems in linear elasticity good convergence bounds for problems of compressible linear elasticity are well known for two and three dimensional problems more recently feti dp and bddc balancing domain decomposition by constraints methods have been developed that are robust also in the regime of homogeneous almost incompressible linear elasticity the coarse space of such methods is large especially in 3d three dimensions and its implementation needs knowledge of geometrical information here the convergence of feti dp methods for problems in 3d with almost incompressible inclusions or compressible inclusions with different material parameters embedded in a compressible matrix material is analyzed for such problems where the material is compressible in the vicinity of the subdomain interface

a coarse space construction based on local dirichlet to neumann maps coarse grid correction is a key ingredient of scalable domain decomposition methods in this work we construct coarse grid space using the low frequency modes of the subdomain dirichlet to neumann maps and apply the obtained two level preconditioners to the extended or the original linear system arising from an overlapping domain decomposition our method is suitable for parallel implementation and its efficiency is demonstrated by numerical examples on problems with large heterogeneities for both manual and automatic partitionings

amg for problems of elasticity algebraic multigrid amg is an efficient iterative method designed as a black box solver that uses the principles of geometric multigrid methods the components necessary for the multigrid cycling process are automatically constructed using information taken from the matrix itself the method does not require uniform grids making it an attractive approach for problems in structures and elasticity here we describe the method and give results for a model plane elasticity problem

convergence and performance of iterative methods for solving variable coefficient convection diffusion equation with a fourth order compact difference scheme abstract we conduct convergence analysis on some classical stationary iterative methods for solving the two dimensional variable coefficient convection diffusion equation discretized by a fourth order compact difference scheme several conditions are formulated under which the coefficient matrix is guaranteed to be an m matrix we further investigate the effect of different orderings of the grid points on the performance of some stationary iterative methods multigrid method and preconditioned gmres three sets of numerical experiments are conducted to study the convergence behaviors of these iterative methods under the influence of the flow directions the orderings of the grid points and the magnitude of the convection coefficients

a new multiscale finite element method for high contrast elliptic interface problems we introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes and without resolving the interfaces a typical application would be the modelling of flow in a porous medium containing a number of inclusions of low or high permeability embedded in a matrix of high respectively low permeability our method is h 1 conforming with degrees of freedom at the nodes of a triangular mesh and requiring the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface but which use standard linear approximation otherwise a key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems under moderate assumptions we prove that our methods have optimal convergence rate of o h in the energy norm and o h 2 in the l 2 norm where h is the coarse mesh diameter and the hidden constants in these estimates are independent of the contrast i e ratio of largest to smallest value of the pde coefficient for standard elements the best estimate in the energy norm would be o h 1 2 e with a hidden constant which in general depends on the contrast the new interior boundary conditions depend not only on the contrast of the coefficients but also on the angles of intersection of the interface with the element edges

robust multigrid methods for nonsmooth coefficient elliptic linear systems we survey the literature on robust multigrid methods which have been developed in recent years for solving second order elliptic pdes with nonsmooth coefficients we highlight the key ideas of designing robust multigrid methods which are able to recover the usual multigrid efficiency for nonsmooth coefficient pdes on structured or unstructured grids in particular we shall describe various approaches for constructing the interpolation and the smoothing operators and the coarse grid points selections

algebraic multigrid theory the symmetric case a rigorous two level theory is developed for general symmetric matrices and nonsymmetric ones using kaczmarz relaxation without assuming any regularity not even any grid structure of the unknowns the theory applies to algebraic multigrid amg processes as well as to the usual geometric multigrid it yields very realistic estimates and precise answers to basic algorithmic questions such as in what algebraic sense does gauss seidel or jacobi kaczmarz line gauss seidel etc relaxation smooth the error when is it appropriate to use block relaxation what algebraic relations must be satisfied by the coarse to fine interpolations what is the algorithmic role of the geometric origin of the problem the theory helps to rigorize local mode analyses and locally analyze cases where the latter is inapplicable

an alternative view on quasicrystalline random tilings we apply a framework for the description of random tilings without height representation which was proposed recently to the special case of quasicrystalline random tilings two important examples are discussed thereby demonstrating the consistency of this alternative description with the conventional one we also clarify the latter by deriving a group theoretic criterion for the validity of the first random tiling hypothesis

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

convergence of algebraic multigrid methods for symmetric positive definite matrices with weak diagonal dominance for symmetric positive definite matrices with weak diagonal dominance the algebraic sense of smooth errors is discussed and convergence of algebraic multigrid amg methods is proved it is found that the results of amg for symmetric positive definite m matrices with weak diagonal dominance also hold without the m matrix assumption the key result is the new formula x email protected ia i email protected j email protected email protected j a i j a j i x 2 email protected email protected j email protected email protected j a i j x email protected email protected ia i j a i j x j 2

domain decomposition for multiscale pdes we consider additive schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic pdes with highly variable coefficients in contrast to standard analyses we do not assume that the coefficients can be resolved by a coarse mesh this situation arises often in practice for example in the computation of flows in heterogeneous porous media in both the deterministic and monte carlo simulated stochastic cases we consider preconditioners which combine local solves on general overlapping subdomains together with a global solve on a general coarse space of functions on a coarse grid we perform a new analysis of the preconditioned matrix which shows rather explicitly how its condition number depends on the variable coefficient in the pde as well as on the coarse mesh and overlap parameters the classical estimates for this preconditioner with linear coarsening guarantee good conditioning only when the coefficient varies mildly inside the coarse grid elements by contrast our new results show that with a good choice of subdomains and coarse space basis functions the preconditioner can still be robust even for large coefficient variation inside domains when the classical method fails to be robust in particular our estimates prove very precisely the previously made empirical observation that the use of low energy coarse spaces can lead to robust preconditioners we go on to consider coarse spaces constructed from multiscale finite elements and prove that preconditioners using this type of coarsening lead to robust preconditioners for a variety of binary i e two scale media model problems moreover numerical experiments show that the new preconditioner has greatly improved performance over standard preconditioners even in the random coefficient case we show also how the analysis extends in a straightforward way to multiplicative versions of the schwarz method

bddc and feti dp preconditioners with adaptive coarse spaces for three dimensional elliptic problems with oscillatory and high contrast coefficients abstract bddc and feti dp algorithms are developed for three dimensional elliptic problems with adaptively enriched coarse components it is known that these enriched components are necessary in the development of robust preconditioners to form the adaptive coarse components carefully designed generalized eigenvalue problems are introduced for each faces and edges and the coarse components are formed by using eigenvectors with their corresponding eigenvalues larger than a given tolerance t o l upper bounds for condition numbers of the preconditioned systems are shown to be c t o l with the constant c depending only on the maximum number of edges and faces per subdomain and the maximum number of subdomains sharing an edge numerical results are presented to test the robustness of the proposed approach

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