deflation of conjugate gradients with applications to boundary value problems a method for improving the convergence of the standard conjugate gradient method is given this method involves the use of auxiliary subspaces it is shown how such subspaces may be constructed for boundary value problems and an analysis of convergence for second order problems is presented

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

the condition number of the schur complement in domain decomposition it is shown that for elliptic boundary value problems of order 2m the condition number of the schur complement matrix that appears in nonoverlapping domain decomposition methods is of order d 1 h 2m 1 where d measures the diameters of the subdomains and h is the mesh size of the triangulation the result holds for both conforming and nonconforming finite elements

preconditioning methods for local discontinuous galerkin discretizations a multilevel interior penalty method is used as an efficient preconditioner for the schur complement of the local discontinuous galerkin ldg discretization of a poisson problem the method is then used in a block triangular preconditioner of the ldg saddle point system the block preconditioner is of the same efficiency as the schur complement version finally the block preconditioner is extended to the discretization of the stokes problem by the ldg method again the preconditioned saddle point problem can be solved in about as many steps as the schur complement the influence of several parameters on the performance of these methods is investigated

an efficient method for solving highly anisotropic elliptic equations solving elliptic pdes in more than one dimension can be a computationally expensive task for some applications characterized by a high degree of anisotropy in the coefficients of the elliptic operator such that the term with the highest derivative in one direction is much larger than the terms in the remaining directions the discretized elliptic operator often has a very large condition number taking the solution even further out of reach using traditional methods this paper will demonstrate a solution method for such ill behaved problems the high condition number of the d dimensional discretized elliptic operator will be exploited to split the problem into a series of well behaved one and d 1 dimensional elliptic problems this solution technique can be used alone on sufficiently coarse grids or in conjunction with standard iterative methods such as conjugate gradient to substantially reduce the number of iterations needed to solve the problem to a specified accuracy the solution is formulated analytically for a generic anisotropic problem using arbitrary coordinates hopefully bringing this method into the scope of a wide variety of applications

gait based human recognition by classification of cyclostationary processes on nonlinear shape manifolds we study the problem of analyzing and classifying human gait by modeling it as a stochastic process on a shape space we consider gait as a evolution of human silhouettes as seen in video sequences and focus on their shapes more specifically we define a shape space of planar closed curves and model a human gait as a stochastic process on this space due to the periodic nature of human walk this process is naturally constrained to be cyclostationary that is its mean path is assumed to be cyclic we compare two subjects using a metric that quantifies differences between average gait cycles of each subject this computation uses several tools from differential geometry of the shape space including computation of geodesics estimation of means of observed shapes interpolation between observed shapes and temporal registration of two gait cycles finally we apply a nearest neighbor classifier using the gait metric to perform human recognition and present results from an experiment involving 26 sub

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

a finite element method for laplace and helmholtz type boundary value problems with singularities laplace and helmholtz type boundary value problems with singularities are considered a sequence of approximations to the exact boundary conditions at an artificial boundary is given then the original problem is reduced to a boundary value problem in a domain away from the singularities furthermore finite element approximations are applied to this problem and error estimates are obtained finally some numerical examples show the effectiveness of this method

which are the important modes of a subsystem a linearly behaving vibrational substructure or more generally a linear dynamic subsystem attached to a main structure or a main dynamic system is considered after discretization the substructure is represented by afinite typically large number of degrees of freedom n and hence also by n s eigenmodes in order to reduce the computational effort it is common to apply modal reduction to the subsystem such that only n r modes out of the total number of n modes are retained where n r n s the following question then arises which n r modes should be retained in structural dynamics it is traditional to retain those modes associated with the lowest frequencies in this paper the question is answered by solving an appropriate optimization problem the most important modes of the subsystem are shown to be those whose coupling matrices which are defined in a particular way have the highest norm this leads to a simple and effective algorithm for optimal modal reduction the new criterion for modal importance is explained both mathematically and physically and is demonstrated by numerical examples

an overlapping schwarz method for spectral element solution of the incompressible navier stokes equations efficient solution of the navier stokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems for unsteady incompressible flows the pressure operator is the leading contributor to stiffness as the characteristic propagation speed is infinite in the context of operator splitting formulations it is the pressure solve which is the most computationally challenging despite its elliptic origins we examine several preconditioners for the consistent l2poisson operator arising in the pn pn 2spectral element formulation of the incompressible navier stokes equations we develop a finite element based additive schwarz preconditioner using overlapping subdomains plus a coarse grid projection operator which is applied directly to the pressure on the interior gauss points for large two dimensional problems this approach can yield as much as a fivefold reduction in simulation time over previously employed methods based upon deflation

deflation accelerated preconditioning of the poisson neumann schur problem on long domains with a high order discontinuous element based collocation method a combination of block jacobi and deflation preconditioning is used to solve a high order discontinuous element based collocation discretization of the schur complement of the poisson neumann system as arises in the operator splitting of the incompressible navier stokes equations the preconditioners and deflation vectors are chosen to mitigate the effects of ill conditioning due to highly elongated domains typical of simulations of strongly non hydrostatic environmental flows and to achieve generalized minimum residual method gmres convergence independent of the size of the number of elements in the long direction the ill posedness of the poisson neumann system manifests as an inconsistency of the schur complement problem but it is shown that this can be accounted for with appropriate projections out of the null space of the schur complement matrix without affecting the accuracy of the solution the block jacobi preconditioner is shown to yield gmres convergence independent of the polynomial order and only weakly dependent on the number of elements within a subdomain in the decomposition the combined deflation and block jacobi preconditioning is compared with two level non overlapping block jacobi preconditioning of the schur problem and while both methods achieve convergence independent of the grid size deflation is shown to require half as many gmres iterations and 25 less wall clock time for a variety of grid sizes and domain aspect ratios the deflation methods shown to be effective for the two dimensional poisson neumann problem are extensible to the three dimensional problem assuming a fourier discretization in the third dimension a fourier discretization results in a two dimensional helmholtz problem for each fourier component that is solved using deflated block jacobi preconditioning on its schur complement here again deflation is shown to be superior to two level non overlapping block jacobi preconditioning requiring about half as many gmres iterations and 15 less time

a domain decomposition method for discontinuous galerkin discretizations of helmholtz problems with plane waves and lagrange multipliers a nonoverlapping domain decomposition dd method is proposed for the iterative solution of systems of equations arising from the discretization of helmholtz problems by the discontinuous enrichment method this discretization method is a discontinuous galerkin finite element method with plane wave basis functions for approximating locally the solution and dual lagrange multipliers for weakly enforcing its continuity over the element interfaces the primal subdomain degrees of freedom are eliminated by local static condensations to obtain an algebraic system of equations formulated in terms of the interface lagrange multipliers only as in the feti h and feti dph dd methods for continuous galerkin discretizations this system of lagrange multipliers is iteratively solved by a krylov method equipped with both a local preconditioner based on subdomain data and a global one using a coarse space numerical experiments performed for two and three dimensional acoustic scattering problems suggest that the proposed dd based iterative solver is scalable with respect to both the size of the global problem and the number of subdomains copyright 2009 john wiley sons ltd

an efficient wave based method for 2d acoustic problems containing corner singularities abstract the wave based method is an efficient alternative prediction technique based on a trefftz approach which as compared to element based methods can be applied in a wider frequency range at a reasonable cost this paper discusses the use of this method for the particular case of 2d acoustic problems where singularities are present in the corners of the acoustic domain the conventional set of expansion functions used to approximate the dynamic pressure field is extended with special purpose functions that incorporate the singular behaviour in the region around the singular corners this enrichment comes with a beneficial convergence rate while only a small increase of computational effort is incurred this is illustrated for various problem settings including different combinations of both homogeneous and non homogeneous boundary conditions and bounded as well as unbounded problems are considered

a schwarz alternating algorithm for elliptic boundary value problems in an infinite domain with a concave angle in this paper we investigate a schwarz alternating algorithm for elliptic boundary value problems in an infinite domain with a concave angle which is an overlapping domain decomposition method based on natural boundary reduction nbr the algorithm is constructed and its convergence is discussed the finite element and natural boundary element method are alternatively applied to solve a bounded subdomain and a typical unbounded subdomain it has overcome the difficulty met in standard domain decomposition method for problems with unbounded domains the convergence rate is analyzed in details for a typical domain finally some numerical examples are presented to show the effectiveness of this method

updating the inverse of a matrix the sherman morrison woodbury formulas relate the inverse of a matrix after a small rank perturbation to the inverse of the original matrix the history of these fomulas is presented and various applications to statistics networks structural analysis asymptotic analysis optimization and partial differential equations are discussed the sherman morrison woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix this paper surveys the history of these formulas and we examine some applications where these formulas are helpful

weak boundary conditions for wave propagation problems in confined domains formulation and implementation using a variational multiscale method abstract we propose a new approach to the enforcement of dirichlet neumann or robin boundary conditions in finite element computations of wave propagation problems the key idea is to enforce the boundary conditions weakly as part of the variational formulation due to the hyperbolic structure of the problem considered the variational formulation does not require any penalty parameters in contrast with what typically happens in elliptic or advection diffusion parabolic problems this article presents the implementation of the proposed boundary condition framework using a variational multiscale method for the wave equation in mixed form we conclude with an extensive set of tests to validate the robustness and accuracy of the proposed approach

corner singularities for elliptic problems integral equations graded meshes quadrature and compressed inverse preconditioning we take a fairly comprehensive approach to the problem of solving elliptic partial differential equations numerically using integral equation methods on domains where the boundary has a large number of corners and branching points use of non standard integral equations graded meshes interpolatory quadrature and compressed inverse preconditioning are techniques that are explored developed mixed and tested on some familiar problems in materials science the recursive compressed inverse preconditioning the major novelty of the paper turns out to be particularly powerful and when it applies eliminates the need for mesh grading completely in an electrostatic example for a multiphase granular material with about two thousand corners and triple junctions and a conductivity ratio between phases up to a million we compute a common functional of the solution with an estimated relative error of 10 1 2 in another example five times as large but with a conductivity ratio of only a hundred we achieve an estimated relative error of 10 1 4

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

two level additive schwarz methods for a discontinuous galerkin approximation of second order elliptic problems we present some two level nonoverlapping and overlapping additive schwarz methods for a discontinuous galerkin method for solving second order elliptic problems it is shown that the condition numbers of the preconditioned systems are of the order o frac h h for the nonoverlapping schwarz methods and of the order o frac h delta for the overlapping schwarz methods where h and h stand for the fine mesh size and the coarse mesh size respectively and delta denotes the size of the overlaps between subdomains numerical experiments are provided to gauge the efficiency of the methods and to validate the theory

a comparison of deflation and coarse grid correction applied to porous media flow in this paper we compare various preconditioners for the numerical solution of partial differential equations we compare a coarse grid correction preconditioner used in domain decomposition methods with a so called deflation preconditioner we prove that the effective condition number of the deflated preconditioned system is always for all deflation vectors and all restrictions and prolongations below the condition number of the system preconditioned by the coarse grid correction this implies that the conjugate gradient method applied to the deflated preconditioned system is expected always to converge faster than the conjugate gradient method applied to the system preconditioned by the coarse grid correction numerical results for porous media flows emphasize the theoretical results

preconditioned spectral multi domain discretization of the incompressible navier stokes equations in this paper we present a preconditioned multi domain algorithm applied to the elliptic kernels arising from the spectral collocation of the incompressible navier stokes equations in three space dimensions with one homogeneous direction the technique based on the iterative solution of the schur complement matrix allows for efficient numerical solution of the operators in complex geometries consisting of a collection of non overlapping rectangular subdomains the method is shown to be nearly optimal in terms of condition number behavior in a double path of refinement strategy i e whenever both the number of legendre modes and the number of subdomains are significantly increased it is thus well suited for engineering applications in the fields of direct numerical simulation and large eddy simulation of turbulence

a spectral multidomain penalty method model for the simulation of high reynolds number localized incompressible stratified turbulence a spectral multidomain penalty method model has been developed for the simulation of high reynolds number localized stratified turbulence this is the first time that a penalty method with a particular focus on subdomain interface treatment has been used with a multidomain scheme to simulate incompressible flows the temporal discretization ensures maximum temporal accuracy by combining third order stiffly stable and backward differentiation schemes with a high order boundary condition for the pressure in the non periodic vertical direction a spectral multidomain discretization is used and its stability for under resolved simulations at high reynolds numbers is ensured through use of penalty techniques spectral filtering and strong adaptive interfacial averaging the penalty method is implemented in different formulations for both the explicit non linear term advancement and the implicit treatment of the viscous terms the multidomain model is validated by comparing results of simulations of the mid to late time stratified turbulent wake with non zero net momentum to the corresponding laboratory data for a towed sphere the model replicates correctly the characteristic vorticity and internal wave structure of the stratified wake and exhibits robust agreement with experiments in terms of the temporal power laws in the evolution of mean profile characteristic velocity and lengthscales

a finite element method for solving helmholtz type equations in waveguides and other unbounded domains a finite element method is described for solving helmholtz type boundary value problems in unbounded regions including those with infinite boundaries typical examples include the propagation of acoustic or electromagnetic waves in waveguides the radiation condition at infinity is based on separation of variables and differs from the classical sommerfeld radiation condition it is shown that the problem may be replaced by a boundary value problem on a fixed bounded domain the behavior of the solution near infinity is incorporated in a nonlocal boundary condition this problem is given a weak or variational formulation and the finite element method is then applied it is proved that optimal error estimates hold

two level additive schwarz preconditioners for nonconforming finite element methods two level additive schwarz preconditioners are developed for the nonconforming p1 finite element approximation of scalar second order symmetric positive definite elliptic boundary value problems the morley finite element approximation of the biharmonic equation and the divergence free nonconforming p1 finite element approximation of the stationary stokes equations the condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of generous overlap

multigrid solver for the inner problem in domain decomposition methods for p fem from the literature it is known that the conjugate gradient method with domain decomposition preconditioners is one of the most efficient methods for solving systems of linear algebraic equations resulting from p version finite element discretizations of elliptic boundary value problems the ingredients of such a preconditioner are a preconditioner for the schur complement a preconditioner related to the dirichlet problems in the subdomains and an extension operator from the boundaries of the subdomains into their interior in the case of poisson s equation we propose a preconditioner for the problems in the subdomains which can be interpreted as the stiffness matrix resulting from an h version finite element discretization of a degenerate operator for solving the corresponding systems of finite element equations a multigrid algorithm with a special line smoother is used we prove that the convergence rate of the multigrid method is independent of the discretization parameter the proof is based on the strengthened cauchy inequality the theoretical result is confirmed by numerical examples

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

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

a fast schur complement method for the spectral element discretization of the incompressible navier stokes equations the weak formulation of the incompressible navier stokes equations in three space dimensions is discretized with spectral element approximations and gauss lobatto legendre quadratures the uzawa algorithm is applied to decouple the velocities from the pressure the equation that results for the pressure is solved by an iterative method within each pressure iteration a helmholtz operator has to be inverted this can efficiently be done by separating the equations for the interior nodes from the equations at the interfaces according to the schur method fast diagonalization techniques are applied to the interior variables of the spectral elements several ways to deal with the resulting interface problem are discussed finally a comparison is made with a more classical method c 1995 academic press inc

the post processing approach in the finite element method part 2 the calculation of stress intensity factors abstract in the context of a model problem the authors describe post processing techniques for the calculation of generalized stress intensity factors they discuss two broad classes of methods one involving an influence function and the other related to the well known energy release principle of fracture mechanics an error analysis is sketched and two numerical examples are given to illustrate the effectivity of the techniques this is the second in a series of three papers author

overlapping schwarz methods for unstructured spectral elements a parallel and scalable domain decomposition method for unstructured and hybrid spectral element discretizations of elliptic problems is introduced and studied the spectral elements are affine images of the reference triangle or square in two dimensions and of the reference tetrahedron pyramid prism or cube in three dimensions the method is based on overlapping schwarz techniques applied to the schur complement of the discrete system and is implemented as a preconditioner for a krylov space method numerical results in two and three dimensions show that the iteration counts of our method are bounded by a constant independent of the spectral degree and the number of subdomains the resulting elliptic solver can be used in navier stokes simulations using the spectral element code nektar

discrete boundary conditions for elasticity problems with singularities discrete boundary conditions on an artificial boundary are obtained for the elasticity problems with singularities an iteration method is designed to solve the boundary value problem by using the discrete boundary conditions some numerical results are given which demonstrate the effectiveness of the numerical method

on boundary integral equations for crack problems a ubiquitous linear boundary value problem in mathematical physics involves solving a partial differential equation exterior to a thin obstacle one typical example is the scattering of scalar waves by a curved crack or rigid strip neumann boundary condition in two dimensions this problem can be reduced to an integrodifferential equation which is often regularized we adopt a more direct approach and prove that the problem can be reduced to a hypersingular boundary integral equation similar reductions will obtain in more complicated situations computational schemes for solving this equation are described with special emphasis on smoothness requirements extensions to three dimensional problems involving an arbitrary smooth bounded crack in an elastic solid are discussed

on an h type mesh refinement strategy based on minimization of interpolation errors abstract an adaptive finite element method is proposed which involves an automatic mesh refinement in areas of the mesh where local errors are determined to exceed a pre assigned limit the estimation of local errors is based on interpolation error bounds and extraction formulas for highly accurate estimates of second derivatives applications to several two dimensional model problems are discussed the results indicate that the method can be very effective for both regular problems and problems with strong singularities

on the accuracy of least squares methods in the presence of corner singularities abstract this paper treats elliptic problems with corner singularities finite element approximations based on variational principles of the least squares type tend to display poor convergence properties in such contexts moreover mesh refinement or the use of special singular elements do not appreciably improve matters here we show that if the least squares formulation is done in appropriately weighted spaces then optimal convergence results in unweighted spaces like l2

a finite element method for domains with corners a new finite element method is devised for the numerical solution of elliptic boundary value problems with geometrical singularities in it the singularity is eliminated form the computational domain in an exact fashion this is in contrast to other common methods such as those which use a refined mesh in the singularity region or those which use special singular finite elements in them the singularity is treated as a part of the numerical scheme the new method is illustrated on an elliptic differential equation in a domain containing a re entrant corner numerical experiments show that the new method yields result which are generally much more accurate than those obtained by using the standard finite element method with mesh refinement in the singularity region both methods require about the same computing time

numerical solution to the time dependent maxwell equations in two dimensional singular domains in this paper we present a method to solve numerically the time dependent maxwell equations in nonsmooth and nonconvex domains indeed the solution is not of regularity h1 in space in general moreover the space of h1 regular fields is not dense in the space of solutions thus an h1 conforming finite element method can fail even with mesh refinement the situation is different than in the case of the laplace problem or of the lame system for which mesh refinement or the addition of conforming singular functions work to cope with this difficulty the singular complement method is introduced this method consists of adding some well chosen test functions these functions are derived from the singular solutions of the laplace problem also the scm preserves the interesting features of the original method easiness of implementation low memory requirements small cost in terms of the cpu time to ascertain its validity some concrete problems are solved numerically

a stable penalty method for the compressible navier stokes equations iii multidimensional domain decomposition schemes this paper concluding the trilogy develops schemes for the stable solution of wave dominated unsteady problems in general three dimensional domains the schemes utilize a spectral approximation in each subdomain and asymptotic stability of the semidiscrete schemes is established the complex computational domains are constructed by using nonoverlapping quadrilaterals in the two dimensional case and hexahedrals in the three dimensional space to illustrate the ideas underlying the multidomain method a stable scheme for the solution of the three dimensional linear advection diffusion equation in general curvilinear coordinates is developed the analysis suggests a novel yet simple stable treatment of geometric singularities like edges and vertices the theoretical results are supported by a two dimensional implementation of the scheme the main part of the paper is devoted to the development of a spectral multidomain scheme for the compressible navier stokes equations on conservation form and a unified approach for dealing with the open boundaries and subdomain boundaries is presented well posedness and asymptotic stability of the semidiscrete scheme is established in a general curvilinear volume with special attention given to a hexahedral domain the treatment includes a stable procedure for dealing with boundary conditions at a solid wall the efficacy of the scheme for the compressible navier stokes equations is illustrated by obtaining solutions to subsonic and supersonic boundary layer flows with various types of boundary conditions the results are found to agree with the solution of the compressible boundary layer equations