Showing papers in "Mathematics of Computation in 2003"
TL;DR: A mixed multiscale finite element method with an over-sampling technique for solving second order elliptic equations with rapidly oscillating coefficients and uses homogenization theory to obtain the asymptotic structure of the solutions.
Abstract: The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large-scale structure of the solutions without resolving all the fine-scale structures. Motivated by the numerical simulation of flow transport in highly heterogeneous porous media, we propose a mixed multiscale finite element method with an over-sampling technique for solving second order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carried out for flow transport in a porous medium with a random log-normal relative permeability to demonstrate the efficiency and accuracy of the proposed method.
502 citations
TL;DR: Korn's inequalities for piecewise H 1 vector fields are established and can be applied to classical nonconforming finite element methods, mortar methods and discontinuous Galerkin methods.
Abstract: Korn's inequalities for piecewise H 1 vector fields are established. They can be applied to classical nonconforming finite element methods, mortar methods and discontinuous Galerkin methods.
285 citations
TL;DR: The present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes, in which the solution is represented and computed in a dynamically evolved adaptive grid.
Abstract: The use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by A. Harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation. In this approach the solution is still represented at each time step on the finest grid, resulting in an inherent limitation of the potential gain in memory space and computational time. The present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes, in which the solution is represented and computed in a dynamically evolved adaptive grid. A crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages. Several solutions to this problem are proposed, analyzed, and compared in terms of accuracy and complexity.
217 citations
TL;DR: The rotational form of the pressure-correction method that was proposed by Timmermans, Minev, and Van De Vosse provides better accuracy in terms of the H1-norm of the velocity and of the L2-normof the pressure than the standard form.
Abstract: In this paper we study the rotational form of the pressure-correction method that was proposed by Timmermans, Minev, and Van De Vosse. We show that the rotational form of the algorithm provides better accuracy in terms of the H1-norm of the velocity and of the L2-norm of the pressure than the standard form.
206 citations
TL;DR: Some recovery type error estimators for linear finite elements are analyzed under O(h 1+α ) (α > 0) regular grids and superconvergence of order O( h 1+ρ ) (0 < p < α) is established for recovered gradients by three different methods.
Abstract: Some recovery type error estimators for linear finite elements are analyzed under O(h 1+α ) (α > 0) regular grids. Superconvergence of order O(h 1+ρ ) (0 < p < α) is established for recovered gradients by three different methods. As a consequence, a posteriori error estimators based on those recovery methods are asymptotically exact.
187 citations
TL;DR: It is shown that this stabilization improves the well-posedness of the continuous problem for small values of the viscosity coefficient and the influence of this stabilization on the accuracy of the finite element solution and on the convergence properties of the inexact Uzawa method.
Abstract: In this paper a stabilizing augmented Lagrangian technique for the Stokes equations is studied. The method is consistent and hence does not change the continuous solution. We show that this stabilization improves the well-posedness of the continuous problem for small values of the viscosity coefficient. We analyze the influence of this stabilization on the accuracy of the finite element solution and on the convergence properties of the inexact Uzawa method.
174 citations
TL;DR: This work considers the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations, and shows results displaying the sharpness of the estimates.
Abstract: We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whose edges are of size of the order of Δx only. For example, when polynomials of degree k are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order k+1/2 in the L2-norm, whereas the post-processed approximation is of order 2k + 1; if the exact solution is in L2 only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order k + 1/2 in L2(Ω0), where Ω0 is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.
170 citations
TL;DR: A new computable a posteriori error estimator is introduced, which relies on the solution of small discrete problems on stars to show a competitive performance, show extremely good effectivity indices, and yield quasi-optimal meshes.
Abstract: A new computable a posteriori error estimator is introduced, which relies on the solution of small discrete problems on stars. It exhibits built-in flux equilibration and is equivalent to the energy error up to data oscillation without any saturation assumption. A simple adaptive strategy is designed, which simultaneously reduces error and data oscillation, and is shown to converge without mesh pre-adaptation nor explicit knowledge of constants. Numerical experiments reveal a competitive performance, show extremely good effectivity indices, and yield quasi-optimal meshes.
165 citations
TL;DR: This result can be viewed as a discrete version of the Rado-Kneser-Choquet theorem for harmonic mappings, but is also closely related to Tutte's theorem on barycentric mappings of planar graphs.
Abstract: We call a piecewise linear mapping from a planar triangulation to the plane a convex combination mapping if the image of every interior vertex is a convex combination of the images of its neighbouring vertices. Such mappings satisfy a discrete maximum principle and we show that they are one-to-one if they map the boundary of the triangulation homeomorphically to a convex polygon. This result can be viewed as a discrete version of the Rado-Kneser-Choquet theorem for harmonic mappings, but is also closely related to Tutte's theorem on barycentric mappings of planar graphs.
148 citations
TL;DR: The local discontinuous Galerkin method for the numerical approximation of the time-harmonic Maxwell equations in a low-frequency regime is introduced and analyzed, derived by introducing suitable auxiliary variables and so-called numerical fluxes.
Abstract: The local discontinuous Galerkin method for the numerical approximation of the time-harmonic Maxwell equations in a low-frequency regime is introduced and analyzed. Topologically nontrivial domains and heterogeneous media are considered, containing both conducting and insulating materials. The presented method involves discontinuous Galerkin discretizations of the curl-curl and grad-div operators, derived by introducing suitable auxiliary variables and so-called numerical fluxes. An hp-analysis is carried out and error estimates that are optimal in the meshsize h and slightly suboptimal in the approximation degree p are obtained.
132 citations
TL;DR: In extending the theory of composition, this paper is able to obtain a closed formula for the number of pairs of length 2 k n generated by a primitive pair of length n, and identifies five primitive pairs.
Abstract: In his 1961 paper, Marcel Golay showed how the search for pairs of binary sequences of length n with complementary autocorrelation is at worst a 2 -6 problem. Andres, in his 1977 master's thesis, developed an algorithm which reduced this to a 2 n/2-1 search and investigated lengths up to 58 for existence of pairs. In this paper, we describe refinements to this algorithm, enabling a 2 n/2-5 search at length 82. We find no new pairs at the outstanding lengths 74 and 82. In extending the theory of composition, we are able to obtain a closed formula for the number of pairs of length 2 k n generated by a primitive pair of length n. Combining this with the results of searches at all allowable lengths up to 100, we identify five primitive pairs. All others pairs of lengths less than 100 may be derived using the methods outlined.
TL;DR: A new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic, based on a simple partial differential equation that gives a system of linear equations.
Abstract: A new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic. It is based on a simple partial differential equation that gives a system of linear equations. As in Berlekamp's and Niederreiter's algorithms for factoring univariate polynomials, the dimension of the solution space of the linear system is equal to the number of absolutely irreducible factors of the polynomial to be factored, and any basis for the solution space gives a complete factorization by computing gcd's and by factoring univariate polynomials over the ground field. The new method finds absolute and rational factorizations simultaneously and is easy to implement for finite fields, local fields, number fields, and the complex number field. The theory of the new method allows an effective Hilbert irreducibility theorem, thus an efficient reduction of polynomials from multivariate to bivariate.
TL;DR: Algorithms for its computation are given and a formula is derived allowing the pointwise approximation of Riemann theta functions, with arbitrary, user-specified precision.
Abstract: The Riemann theta function is a complex-valued function of g complex variables. It appears in the construction of many (quasi-)periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation are given. First, a formula is derived allowing the pointwise approximation of Riemann theta functions, with arbitrary, user-specified precision. This formula is used to construct a uniform approximation formula, again with arbitrary precision.
TL;DR: In this work, the bilinear finite element method on a Shishkin mesh for convection-diffusion problems is analyzed in the two-dimensional setting and an e-uniform convergence of order N-3/2ln5/2N + eN-1ln1/ 2N in the L∞ norm is proved for some mesh points in the boundary layer region.
Abstract: In this work, the bilinear finite element method on a Shishkin mesh for convection-diffusion problems is analyzed in the two-dimensional setting. A superconvergence rate O(N-2 ln2 N + eN-1.5lnN) in a discrete e-weighted energy norm is established under certain regularity assumptions. This convergence rate is uniformly valid with respect to the singular perturbation parameter e. Numerical tests indicate that the rate O(N-2ln2 N) is sharp for the boundary layer terms. As a by-product, an e-uniform convergence of the same order is obtained for the L2-norm. Furthermore, under the same regularity assumption, an e-uniform convergence of order N-3/2ln5/2N + eN-1ln1/2N in the L∞ norm is proved for some mesh points in the boundary layer region.
TL;DR: A modification of the Goldfeld-Oesterle work, which used an elliptic curve L-function with an order 3 zero at the central critical point to instead consider Dirichlet L-functions with low-height zeros near the real line, which agrees with the prediction of the Generalised Riemann Hypothesis.
Abstract: The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number N. The first complete results were for N = 1 by Heegner, Baker, and Stark. After the work of Goldfeld and Gross-Zagier, the task was a finite decision problem for any N. Indeed, after Oesterle handled N = 3, in 1985 Serre wrote, No doubt the same method will work for other small class numbers, up to 100, say. However, more than ten years later, after doing N = 5, 6, 7, Wagner remarked that the N = 8 case seemed impregnable. We complete the classification for all N < 100, an improvement of four powers of 2 (arguably the most difficult case) over the previous best results. The main theoretical technique is a modification of the Goldfeld-Oesterle work, which used an elliptic curve L-function with an order 3 zero at the central critical point, to instead consider Dirichlet L-functions with low-height zeros near the real line (though the former is still required in our proof). This is numerically much superior to the previous method, which relied on work of Montgomery-Weinberger. Our method is still quite computer-intensive, but we are able to keep the time needed for the computation down to about seven months. In all cases, we find that there is no abnormally large exceptional modulus of small class number, which agrees with the prediction of the Generalised Riemann Hypothesis.
TL;DR: A kinetic interpretation of upwinding taking into account the source terms of scalar conservation laws with a zeroth order source with low regularity and a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions.
Abstract: We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced CFL condition and refined meshes required because of the lack of accuracy on equilibrium states. The source term should be taken into account in the upwinding and discretized at the nodes of the grid. In order to solve numerically the problem, we introduce a so-called equilibrium schemes with the properties that (i) the maximum principle holds true; (ii) discrete entropy inequalities are satisfied; (iii) steady state solutions of the problem are maintained. One of the difficulties in studying the convergence is that there are no BV estimates for this problem. We therefore introduce a kinetic interpretation of upwinding taking into account the source terms. Based on the kinetic formulation we give a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions. The computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that, in comparison with an usual upwind scheme, the corresponding equilibrium version is far more accurate. Furthermore, numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients.
TL;DR: The local discontinuous Galerkin method is introduced and analyzed for a class of shape-regular meshes with hanging nodes and optimal a priori estimates for the errors in the velocity and the pressure in L 2 - and negative-order norms are derived.
Abstract: We introduce and analyze the local discontinuous Galerkin method for the Oseen equations of incompressible fluid flow. For a class of shape-regular meshes with hanging nodes, we derive optimal a priori estimates for the errors in the velocity and the pressure in L 2 - and negative-order norms. Numerical experiments are presented which verify these theoretical results and show that the method performs well for a wide range of Reynolds numbers.
TL;DR: This paper proves the convergence of the ghost fluid method for second order elliptic partial differential equations with interfacial jumps through discretization of the weak formulation of the problem.
Abstract: This paper proves the convergence of the ghost fluid method for second order elliptic partial differential equations with interfacial jumps. A weak formulation of the problem is first presented, which then yields the existence and uniqueness of a solution to the problem by classical methods. It is shown that the application of the ghost fluid method by Fedkiw, Kang, and Liu to this problem can be obtained in a natural way through discretization of the weak formulation. An abstract framework is given for proving the convergence of finite difference methods derived from a weak problem, and as a consequence, the ghost fluid method is proved to be convergent.
TL;DR: This article shows how to generalize the CM-method for elliptic curves to genus two and describes the algorithm in detail and discusses the results.
Abstract: In this article we show how to generalize the CM-method for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation.
TL;DR: Three mixed linear finite element methods for the numerical simulation of the two-dimensional Signorini problem are studied, and Falk's Lemma and saddle point theory allows us to state the convergence rate of each of them.
Abstract: We study three mixed linear finite element methods for the numerical simulation of the two-dimensional Signorini problem. Applying Falk's Lemma and saddle point theory to the resulting discrete mixed variational inequality allows us to state the convergence rate of each of them. Two of these finite elements provide optimal results under reasonable regularity assumptions on the Signorini solution, and the numerical investigation shows that the third method also provides optimal accuracy.
TL;DR: An algorithm that computes the prime numbers up to N using O(N/log log N) additions and N 1/2+o(1) bits of memory is introduced.
Abstract: We introduce an algorithm that computes the prime numbers up to N using O(N/log log N) additions and N 1/2+o(1) bits of memory. Tie algorithm enumerates representations of integers by certain binary quadratic forms. We present implementation results for this algorithm and one of the best previous algorithms.
TL;DR: It is shown that the number field sieve outperforms the gaussian integer method in the hundred digit range by successfully computing discrete logarithms with GNFS in a large prime field.
Abstract: In this paper, we describe many improvements to the number field sieve. Our main contribution consists of a new way to compute individual logarithms with the number field sieve without solving a very large linear system for each logarithm. We show that, with these improvements, the number field sieve outperforms the gaussian integer method in the hundred digit range. We also illustrate our results by successfully computing discrete logarithms with GNFS in a large prime field.
TL;DR: The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical method, in particular, by focusing on the dependence of the error bounds on the parameter e, known as the measure of the interface thickness.
Abstract: We propose and analyze a fully discrete finite element scheme for the phase field model describing the solidification process in materials science. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical method, in particular, by focusing on the dependence of the error bounds on the parameter e, known as the measure of the interface thickness. Optimal order error bounds are shown for the fully discrete scheme under some reasonable constraints on the mesh size h and the time step size k. In particular, it is shown that all error bounds depend on 1 a only in some lower polynomial order for small e. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Chen, and to establish a discrete counterpart of it for a linearized phase field operator to handle the nonlinear effect. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence of the solution of the fully discrete scheme to solutions of the sharp interface limits of the phase field model under different scaling in its coefficients. The sharp interface limits include the classical Stefan problem, the generalized Stefan problems with surface tension and surface kinetics, the motion by mean curvature flow, and the Hele-Shaw model.
TL;DR: A range of mesh-dependent inequalities for piecewise constant and continuous piecewise linear finite element functions u defined on locally refined shape-regular and non-quasi-uniform meshes are presented.
Abstract: We present a range of mesh-dependent inequalities for piecewise constant and continuous piecewise linear finite element functions u defined on locally refined shape-regular (but possibly non-quasi-uniform) meshes. These inequalities involve norms of the form ∥h α u∥ W s,p (Ω) for positive and negative s and α, where h is a function which reflects the local mesh diameter in an appropriate way. The only global parameter involved is N, the total number of degrees of freedom in the finite element space, and we avoid estimates involving either the global maximum or minimum mesh diameter.
Our inequalities include new variants of inverse inequalities as well as trace and extension theorems. They can be used in several areas of finite element analysis to extend results – previously known only for quasi-uniform meshes – to the locally refined case. Here we describe applications to (i) the theory of nonlinear approximation and (ii) the stability of the mortar element method for locally refined meshes.
TL;DR: This paper shows that the Sinc-Galerkin method is a very effective tool in numerically solving sixth-order boundary-value problems with two-point boundary conditions and a comparison with the modified decomposition method is made.
Abstract: mmThere are few techniques available to numerically solve sixth-order boundary-value problems with two-point boundary conditions. In this paper we show that the Sinc-Galerkin method is a very effective tool in numerically solving such problems. The method is then tested on examples with homogeneous and nonhomogeneous boundary conditions and a comparison with the modified decomposition method is made. It is shown that the Sinc-Galerkin method yields better results.
TL;DR: It is shown that the symmetric (EiCP) is equivalent to finding an equilibrium solution of a differentiable optimization problem in a compact set and similar results apply to the Symmetric Generalized Eigenvalue Complementarity Problem (GEiCP).
Abstract: In this paper the Eigenvalue Complementarity Problem (EiCP) with real symmetric matrices is addressed. It is shown that the symmetric (EiCP) is equivalent to finding an equilibrium solution of a differentiable optimization problem in a compact set. A necessary and sufficient condition for solvability is obtained which, when verified, gives a convenient starting point for any gradient-ascent local optimization method to converge to a solution of the (EiCP). It is further shown that similar results apply to the Symmetric Generalized Eigenvalue Complementarity Problem (GEiCP). Computational tests show that these reformulations improve the speed and robustness of the solution methods.
TL;DR: This work presents efficient algorithms for solving Legendre equations over Q (equivalently, for finding rational points on rational conics) and parametrizing all solutions and shows that no integer factorization is required.
Abstract: We present efficient algorithms for solving Legendre equations over Q (equivalently, for finding rational points on rational conics) and parametrizing all solutions. Unlike existing algorithms, no integer factorization is required, provided that the prime factors of the discriminant are known.
TL;DR: The approximate error between the solution of the simplified problem and that of the full-interaction problem is analyzed so as to answer the question mathematically for a one-dimensional model.
Abstract: In many applications materials are modeled by a large number of particles (or atoms) where any one of particles interacts with all others. Near or nearest neighbor interaction is expected to be a good simplification of the full interaction in the engineering community. In this paper we shall analyze the approximate error between the solution of the simplified problem and that of the full-interaction problem so as to answer the question mathematically for a one-dimensional model. A few numerical methods have been designed in the engineering literature for the simplified model. Recently much attention has been paid to a finite-element-like quasicontinuum (QC) method which utilizes a mixed atomistic/continuum approximation model. No numerical analysis has been done yet. In the paper we shall estimate the error of the QC method for this one-dimensional model. Possible ill-posedness of the method and its modification are discussed as well.
TL;DR: It is shown that some of the quality measures are equivalent, in the sense of displaying the same extremal and asymptotic behavior, and that it is therefore possible to achieve a concise classification of triangle quality measures.
Abstract: Several of the more commonly used triangle quality measures are analyzed and compared. Proofs are provided to verify that they do exhibit the expected extremal properties. The asymptotic behavior of these measures is investigated and a number of useful results are derived. It is shown that some of the quality measures are equivalent, in the sense of displaying the same extremal and asymptotic behavior, and that it is therefore possible to achieve a concise classification of triangle quality measures.
TL;DR: Various operator functions are considered, the operator exponential e -tL, negative fractional powers L -α , the cosine operator function cos(t√L) L-k and, finally, the solution operator of the Lyapunov equation, using the Dunford-Cauchy representation.
Abstract: llIn previous papers the arithmetic of hierarchical matrices has been described, which allows us to compute the inverse, for instance, of finite element stiffness matrices discretising an elliptic operator L. The required computing time is up to logarithmic factors linear in the dimension of the matrix. In particular, this technique can be used for the computation of the discrete analogue of a resolvent (zI - L) -1 , z ∈ C. In the present paper, we consider various operator functions, the operator exponential e -tL , negative fractional powers L -α , the cosine operator function cos(t√L) L-k and, finally, the solution operator of the Lyapunov equation. Using the Dunford-Cauchy representation, we get integrals which can be discretised by a quadrature formula which involves the resolvents (z k I - L) -1 mentioned above. We give error estimates which are partly exponentially, partly polynomially decreasing.