Showing papers in "Mathematics of Computation in 1999"
TL;DR: This paper provides a detailed convergence analysis of the multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale.
Abstract: We propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients. The main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution. This is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions. As a consequence, the base functions are adapted to the local properties of the differential operator. In this paper, we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain a useful asymptotic solution structure. The issue of boundary conditions for the base functions will be discussed. Our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution, independently of the small scale in the homogenization limit. Application of our method to problems with continuous scales is also considered.
573 citations
TL;DR: This paper defines the parameterized integer relation construction algorithm PSLQ(τ), where the parameter τ can be freely chosen in a certain interval, and proves that PSLZ(τ) constructs a relation in less than O(n 3 + n 2 logM x ) iterations.
Abstract: In this paper we define the parameterized integer relation construction algorithm PSLQ(tau), where the parameter tau can be freely chosen in a certain interval.
299 citations
TL;DR: This work combines the theory of radial basis functions with the field of Galerkin methods to solve partial differential equations and shows convergence and error estimates for smooth problems in arbitrary dimensions.
Abstract: We combine the theory of radial basis functions with the field of Galerkin methods to solve partial differential equations. After a general description of the method we show convergence and derive error estimates for smooth problems in arbitrary dimensions.
258 citations
TL;DR: A simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection-diffusion problems in any spatial dimension by properly averaging the PDE coefficients on element edges.
Abstract: A simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection-diffusion problems in any spatial dimension by properly averaging the PDE coefficients on element edges. The resulting finite element stiffness matrix is an M-matrix under some mild assumption for the underlying (generally unstructured) finite element grids. As a consequence the proposed edge-averaged finite element scheme is particularly interesting for the discretization of convection dominated problems. This scheme admits a simple variational formulation, it is easy to analyze, and it is also suitable for problems with a relatively smooth flux variable. Some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems.
237 citations
TL;DR: This work provides sets of parameters for multiplicative linear congruential generators (MLCGs) of different sizes and good performance with respect to the spectral test and lists of multipliers a such that the MLCG with modulus m and multiplier a has a good lattice structure in dimensions 2 to 32.
Abstract: We provide sets of parameters for multiplicative linear congruential generators (MLCGs) of different sizes and good performance with respect to the spectral test. For ` = 8, 9, . . . , 64, 127, 128, we take as a modulus m the largest prime smaller than 2`, and provide a list of multipliers a such that the MLCG with modulus m and multiplier a has a good lattice structure in dimensions 2 to 32. We provide similar lists for power-of-two moduli m = 2`, for multiplicative and non-multiplicative LCGs.
226 citations
TL;DR: This work takes a fresh look at the construction of the finite spaces, viewing them from the angle of differential forms, and arrives at a fairly canonical procedure to construct conforming finite element subspaces of function spaces related to differential forms.
Abstract: The mixed variational formulation of many elliptic boundary value problems involves vector valued function spaces, like, in three dimensions, H(curl; Ω) and H(Div; Ω) Thus finite element subspaces of these function spaces are indispensable for effective finite element discretization schemes Given a simplicial triangulation of the computational domain Ω, among others, Raviart, Thomas and Nedelec have found suitable conforming finite elements for H(Div; Ω) and H(curl; Ω) At first glance, it is hard to detect a common guiding principle behind these approaches We take a fresh look at the construction of the finite spaces, viewing them from the angle of differential forms This is motivated by the well-known relationships between differential forms and differential operators: div, curl and grad can all be regarded as special incarnations of the exterior derivative of a differential form Moreover, in the realm of differential forms most concepts are basically dimension-independent Thus, we arrive at a fairly canonical procedure to construct conforming finite element subspaces of function spaces related to differential forms In any dimension we can give a simple characterization of the local polynomial spaces and degrees of freedom underlying the definition of the finite element spaces With unprecedented ease we can recover the familiar H(Div; Q)- and H(curl; Ω)-conforming finite elements, and establish the unisolvence of degrees of freedom In addition, the use of differential forms makes it possible to establish crucial algebraic properties of the canonical interpolation operators and representation theorems in a single sweep for all kinds of spaces
209 citations
TL;DR: This paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit n-cube to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains.
Abstract: This paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit n-cube. These bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains. This covers also elliptic boundary value problems, although this study is primarily motivated by our previous analysis of wavelet methods for pseudo-differential equations with special emphasis on boundary integral equations. In this case it is natural to model the boundary surface as a union of parametric images of the unit cube. It will be shown how to construct wavelet bases on the surface which are composed of wavelet bases defined on each surface patch. Here the relevant properties are the validity of norm equivalences in certain ranges of Sobolev scales, as well as appropriate moment conditions.
175 citations
TL;DR: The results of a computer search for maximally equidistributed combined linear feedback shift register (or Tausworthe) random number generators, whose components are trinomials of degrees slightly less than 32 or 64 are given.
Abstract: We give the results of a computer search for maximally equidistributed combined linear feedback shift register (or Tausworthe) random number generators, whose components are trinomials of degrees slightly less than 32 or 64. These generators are fast and have good statistical properties.
171 citations
TL;DR: An iteration-by-subdomain procedure is proven to converge, showing that the preconditioner implicitly defined by the iterative procedure is optimal, and proving a regularity theorem for Dirichlet and Neumann harmonic fields.
Abstract: The time-harmonic Maxwell equations are considered in the low-frequency case. A finite element domain decomposition approach is proposed for the numerical approximation of the exact solution. This leads to an iteration-by-subdomain procedure, which is proven to converge. The rate of convergence turns out to be independent of the mesh size, showing that the preconditioner implicitly defined by the iterative procedure is optimal. For obtaining this convergence result it has been necessary to prove a regularity theorem for Dirichlet and Neumann harmonic fields.
161 citations
TL;DR: The census contains descriptions of all hyperbolic 3-manifolds obtained by gluing the faces of at most seven ideal tetrahedra and various geometric and topological invariants are calculated for these manifolds.
Abstract: The census provides a basic collection of noncompact hyperbolic 3-manifolds of finite volume. It contains descriptions of all hyperbolic 3-manifolds obtained by gluing the faces of at most seven ideal tetrahedra. Additionally, various geometric and topological invariants are calculated for these manifolds. The findings are summarized and a listing of all manifolds appears in the microfiche supplement.
160 citations
TL;DR: Newton's method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation andumerical results are reported to illustrate the convergence behaviour of various algorithms.
Abstract: We study iterative methods for finding the maximal Hermitian positive definite solutions of the matrix equations X + A * X -1 A = Q and X - A * X -1 A = Q, where Q is Hermitian positive definite. General convergence results are given for the basic fixed point iteration for both equations. Newton's method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation. Numerical results are reported to illustrate the convergence behaviour of various algorithms.
TL;DR: An improvement of the above result by BRENT et al., is able to improve these estimates and to show that the k th prime is greater than k(ln k + ln ln k - 1) for k ≥ 2.
Abstract: ROSSER and SCHOENFELD have used the fact that the first 3,500,000 zeros of the RIEMANN zeta function lie on the critical line to give estimates on ψ(x) and θ(x) With an improvement of the above result by BRENT et al, we are able to improve these estimates and to show that the k th prime is greater than k(ln k + ln ln k - 1) for k ≥ 2 We give further results without proof
TL;DR: The Markov inequality for spherical harmonics is essential to this analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonic.
Abstract: We study Sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the n-sphere. The interpolation knots are scattered. Our approach partly follows the general theory of Golomb and Weinberger and related estimates. These error estimates are then based on series expansions of smooth functions in terms of spherical harmonics. The Markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics.
TL;DR: A complete discussion of the optimal (global and local) order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear Volterra integral equations with algebraic or logarithmic singularities in their kernels is presented.
Abstract: Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear Volterra integral equations with algebraic or logarithmic singularities in their kernels.
TL;DR: It is proved that there exists a unique solution for sufficiently smooth initial data in the Cahn-Hilliard equation and an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions is proved.
Abstract: We consider the Cahn-Hilliard equation with a logarithmic free energy and non-degenerate concentration dependent mobility. In particular we prove that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions. Finally some numerical experiments are presented.
TL;DR: In this article, a Hermite spectral scheme is constructed for Burgers equation and the stability and convergence of the proposed scheme are proved strictly, and the techniques used in this paper are also applicable to other nonlinear problems in unbounded domains.
Abstract: Hermite approximation is investigated. Some inverse inequalities, imbedding inequalities and approximation results are obtained. A Hermite spectral scheme is constructed for Burgers equation. The stability and convergence of the proposed scheme are proved strictly. The techniques used in this paper are also applicable to other nonlinear problems in unbounded domains.
TL;DR: In this paper, a proper condition which makes Newton's method converge and an exact estimate for the radius of the ball of the inverse function theorem are given in a Banach space under the hypothesis that the derivative satisfies some kind of weak Lipschitz condition.
Abstract: Under the hypothesis that the derivative satisfies some kind of weak Lipschitz condition, a proper condition which makes Newton's method converge, and an exact estimate for the radius of the ball of the inverse function theorem are given in a Banach space. Also, the relevant results on premises of Kantorovich and Smale types are improved in this paper.
TL;DR: The goal of this paper is to define a procedure for bounding the error in a conforming finite element method that is fully explicit and can be computed locally.
Abstract: The goal of this paper is to define a procedure for bounding the error in a conforming finite element method. The new point is that this upper bound is fully explicit and can be computed locally. Numerical tests prove the efficiency of the method. It is presented here for the case of the Poisson equation and a first order finite element approximation.
TL;DR: If additional smoothness requirements and boundary conditions are met, the well-known approximation orders of scattered data interpolants by radial functions can roughly be doubled.
Abstract: If additional smoothness requirements and boundary conditions are met, the well-known approximation orders of scattered data interpolants by radial functions can roughly be doubled
TL;DR: A systematic numerical search for genus two curves defined over the rationals such that their Jacobians are simple and have endomorphism ring equal to the ring of integers of a quartic CM field finds 19 non-isomorphic such curves.
Abstract: We present the results of a systematic numerical search for genus two curves defined over the rationals such that their Jacobians are simple and have endomorphism ring equal to the ring of integers of a quartic CM field. Including the well-known example y 2 = x 5 - 1 we find 19 non-isomorphic such curves. We believe that these are the only such curves.
TL;DR: It is proved that there is a bound (< 1) for the contraction number of the W-cycle algorithm which is independent of mesh level, provided that the number of smoothing steps is sufficiently large.
Abstract: We consider nonconforming multigrid methods for symmetric positive definite second and fourth order elliptic boundary value problems which do not have full elliptic regularity. We prove that there is a bound (< 1) for the contraction number of the W-cycle algorithm which is independent of mesh level, provided that the number of smoothing steps is sufficiently large. We also show that the symmetric variable V-cycle algorithm is an optimal preconditioner.
TL;DR: This work presents and analyzes four different kinds of error estimators: a residual based estimator, a hierarchical one, error estimator relying on the solution of local subproblems and on a superconvergence result, respectively.
Abstract: We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. In particular, we present and analyze four different kinds of error estimators: a residual based estimator, a hierarchical one, error estimators relying on the solution of local subproblems and on a superconvergence result, respectively. Finally, we examine the relationship between the presented error estimators and compare their local components.
TL;DR: It is initially shown how first (and higher) order single-variate numerical differentiation of higher dimensional observational data can be stabilized with a reduced loss of accuracy than occurs for the corresponding differentiation of one-dimensional data.
Abstract: Finite difference methods, such as the mid-point rule, have been applied successfully to the numerical solution of ordinary and partial differential equations. If such formulas are applied to observational data, in order to determine derivatives, the results can be disastrous. The reason for this is that measurement errors, and even rounding errors in computer approximations, are strongly amplified in the differentiation process, especially if small step-sizes are chosen and higher derivatives are required. A number of authors have examined the use of various forms of averaging which allows the stable computation of low order derivatives from observational data. The size of the averaging set acts like a regularization parameter and has to be chosen as a function of the grid size h. In this paper, it is initially shown how first (and higher) order single-variate numerical differentiation of higher dimensional observational data can be stabilized with a reduced loss of accuracy than occurs for the corresponding differentiation of one-dimensional data. The result is then extended to the multivariate differentiation of higher dimensional data. The nature of the trade-off between convergence and stability is explicitly characterized, and the complexity of various implementations is examined.
TL;DR: Arrays of nodes for which one of these interpolants converges exponentially for analytic functions are given.
Abstract: In 1988 the second author presented experimentally well-conditioned linear rational functions for global interpolation. We give here arrays of nodes for which one of these interpolants converges exponentially for analytic functions.
TL;DR: The Poisson equation −Δu=f with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain Ω with cracks is considered and multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed.
Abstract: We consider the Poisson equation −Δu=f with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain Ω with cracks. Multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed. The convergence rate for the stress intensity factors is\(\mathcal{O}(h^{(3/2) - \in } )\) whenfeL2(Ω) and\(\mathcal{O}(h^{(2 - \in )} )\) whenfeH1(Ω). The convergence rate in the energy norm is\(\mathcal{O}(h^{(1 - \in )} )\) in the first case and\(\mathcal{O}(h)\) in the second case. The costs of these multigrid methods are proportional to the number of elements in the triangulation. The general case wherefeHm(Ω) is also discussed.
TL;DR: The ramifications of the fact that AD is applied not only to the solution of such an algorithm, but to the solutions procedure itself are discussed, which can lead to surprising results when AD tools are applied to variable-stepsize, variable-order ODE integrators.
Abstract: Automatic differentiation (AD) is a technique for automatically augmenting computer programs with statements for the computation of derivatives. This article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations (ODEs), in particular, the ramifications of the fact that AD is applied not only to the solution of such an algorithm, but to the solution procedure itself. This subtle issue can lead to surprising results when AD tools are applied to variable-stepsize, variable-order ODE integrators. The computation of the final time step plays a special role in determining the computed derivatives. We investigate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives.
TL;DR: Local convergence properties of inexact Newton and Newton-like methods for systems of nonlinear equations are investigated, and new sufficient conditions for linear convergence in an arbitrary vector norm are provided.
Abstract: In this paper we investigate local convergence properties of inexact Newton and Newton-like methods for systems of nonlinear equations. Processes with modified relative residual control are considered, and new sufficient conditions for linear convergence in an arbitrary vector norm are provided. For a special case the results are affine invariant.
TL;DR: It is shown that one can evaluate the discrete logarithm in Pic 0 (X) p n by O(n 2 log p) operations in k by generalizing a result of Semaev for elliptic curves to curves of arbitrary genus.
Abstract: Let X be a curve which is defined over a finite field k of characteristic p We show that one can evaluate the discrete logarithm in Pic 0 (X) p n by O(n 2 log p) operations in k This generalizes a result of Semaev for elliptic curves to curves of arbitrary genus
TL;DR: This work presents a general framework for the finite volume or covolume schemes developed for second order elliptic problems in mixed form, i.e., written as first order systems via a one-to-one transfer operator between trial and test spaces.
Abstract: We present a general framework for the finite volume or covolume schemes developed for second order elliptic problems in mixed form, i.e., written as first order systems. We connect these schemes to standard mixed finite element methods via a one-to-one transfer operator between trial and test spaces. In the nonsymmetric case (convection-diffusion equation) we show one-half order convergence rate for the flux variable which is approximated either by the lowest order Raviart-Thomas space or by its image in the space of discontinuous piecewise constants. In the symmetric case (diffusion equation) a first order convergence rate is obtained for both the state variable (e.g., concentration) and its flux. Numerical experiments are included.
TL;DR: The two-dimensional Navier-Stokes equations subject to homogeneous (nonslip) Dirichlet boundary conditions and other equations, such as reaction-diffusion systems and the Cahn-Hilliard equations are studied.
Abstract: In a recent paper we have introduced a postprocessing procedure for the Galerkin method for dissipative evolution partial differential equations with periodic boundary conditions. The postprocessing technique uses approximate inertial manifolds to approximate the high modes (the small scale components) in the exact solutions in terms of the Galerkin approximations, which in this case play the role of the lower modes (large scale components). This procedure can be seen as a defect-correction technique. But contrary to standard procedures, the correction is computed only when the time evolution is completed. Here we extend these results to more realistic boundary conditions. Specifically, we study in detail the two-dimensional Navier-Stokes equations subject to homogeneous (nonslip) Dirichlet boundary conditions. We also discuss other equations, such as reaction-diffusion systems and the Cahn-Hilliard equations.