Showing papers in "Mathematics of Computation in 2005"

Adaptive finite element methods for differential equations.

Abstract: Gegenstand des Buches ist die Dual Weighted Residual method (DWR), ein sehr effizientes numerisches Verfahren zur Behandlung einer großen Klasse von variationell formulierten Differentialgleichungen. Das numerische Verfahren ist adaptiv, d.h. es konstruiert eigenständig eine Folge von Approximationen für eine gegebene Fragestellung. Typische Fragestellungen sind die Bestimmung gewichteter Mittelwerte der Lösung oder ihrer Ableitungen, die Bestimmung von Randintegralen über Lösungskomponenten (relevant z.B. für die Berechnung von strömungsmechanischen Kenngrößen) oder die Bestimmung von Spannungsintensitätsfaktoren (z.B. in der Bruchmechanik). Das Verfahren basiert auf Projektionsmethoden wie z.B. der Finiten Elemente Methode (FEM). Dort wird die Approximationsgüte durch die Wahl der Gitter gesteuert. Der Kern jeder adaptiven FEM ist deshalb die Art, wie die Gitter gewählt werden. Typischerweise geschieht dies in einer adaptiven Schleife, in der in mehreren Durchgängen schrittweise das Gitter verbessert wird, bis eine gewünschte Genauigkeit erreicht ist. Bei der DWR wird in jedem Schleifendurchgang ein lineares Hilfsproblem—das sog. duale Problem, welches von der vorliegenden Fragestellung abhängt—(näherungsweise) gelöst. Weiterhin wird eine Approximation der Differentialgleichung bestimmt. Aus diesen nun vorliegenden Daten wird dann herausdestilliert, wo das Gitter verfeinert werden sollte bzw. vergröbert werden kann, um eine genauere Lösung zu erhalten. Ziel eines adaptiven Algorithmus ist, das gewünschte Ergebnis möglichst effizient zu bestimmen, d.h. mit möglichst geringem Bedarf an Resourcen (Rechenzeit, Speicherbedarf etc.). Mit zahlreichen Beispielen belegt das Buch, daß die DWR dieses Ziel erreicht. Es sei hier besonders hervorgehoben, daß eine Kosten-Nutzen-Betrachtung für die DWR besonders bei nichtlinearen Problemen günstig ausfällt, da die Kosten für die Lösung des linearen Hilfsproblems vergleichbar mit denen eines Newtonschrittes sind und somit nur einen kleinen Teil der Gesamtkosten ausmachen. Das Buch gibt einen sehr guten Überblick über die Technik und die Möglichkeiten der DWR. In einleitenden Kapiteln wird die DWR an gewöhnlichen Differentialgleichungen und dann an einfachen linearen, elliptischen partiellen Differentialgleichungen sehr klar und verständlich vorgeführt. Anschließend wird die DWR in einem abstrakten funktionalanalytischen Rahmen vorgestellt. Der Rest des Buches illustriert auf eindrucksvolle Weise die Leistungsfähigkeit und Breite der Anwendungsfähigkeit des Konzeptes an Hand von Fallbeispielen: Es werden Eigenwertprobleme, Optimierungsaufgaben mit Zwangsbedingungen, die durch eine partielle Differentialgleichung gegeben sind, Strukturmechanikprobleme (lineare Elastizität, Plastizität), Strömungsmechanik (hydrodynamische Stabilitätsanalyse, Berechnung von Strömungskennwerten) behandelt. Auch zeitabhängige Probleme wie die Lösung der Wellengleichung werden mit der DWR erfolgreich bearbeitet. Insgesamt wird klar ersichtlich, daß die DWR eine sehr flexible und vielseitig anwendbare Technik ist. Die ausgewählten numerischen Beispiele, die vor allem aus umfangreichen numerischen Untersuchungen der Gruppe von Rolf Rannacher aus den letzten 10 Jahren ausgewählt wurden, sind sehr illustrativ. Die Erläuterungen zu den Beispielen sind auch deshalb interessant, weil eine Menge zusätzlicher Informationen über die numerische Behandlung des vorliegenden Problems quasi nebenbei einfließen. Das Buch entstand aus einer fortgeschrittenen Spezialvorlesung, die an der ETH Zürich gehalten wurde. Einen Lehrbuchcharakter erhält das Buch dadurch, daß Übungsaufgaben (mit detailierten Lösungen im Anhang) jedes Kapitel abschließen. Die Aufgaben enthal-

Automatic sequences. Theory, applications, generalizations.

Abstract: What do you do to start reading automatic sequences theory applications generalizations? Searching the book that you love to read first or find an interesting book that will make you want to read? Everybody has difference with their reason of reading a book. Actuary, reading habit must be from earlier. Many people may be love to read, but not a book. It's not fault. Someone will be bored to open the thick book with small words to read. In more, this is the real condition. So do happen probably with this automatic sequences theory applications generalizations.

Structured preconditioners for nonsingular matrices of block two-by-two structures

Abstract: For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to efficient and high-quality preconditioning matrices for some typical matrices from the real-world applications.

Heterogeneous multiscale methods for stiff ordinary differential equations

Abstract: The heterogeneous multiscale methods (HMM) is a general framework for the numerical approximation of multiscale problems It is here developed for ordinary differential equations containing differe

Global optimization of explicit strong-stability-preserving Runge-Kutta methods

Abstract: Strong-stability-preserving Runge-Kutta (SSPRK) methods are a type of time discretization method that are widely used, especially for the time evolution of hyperbolic partial differential equations (PDEs). Under a suitable stepsize restriction, these methods share a desirable nonlinear stability property with the underlying PDE; e.g., positivity or stability with respect to total variation. This is of particular interest when the solution exhibits shock-like or other nonsmooth behaviour. A variety of optimality results have been proven for simple SSPRK methods. However, the scope of these results has been limited to low-order methods due to the detailed nature of the proofs. In this article, global optimization software, BARON, is applied to an appropriate mathematical formulation to obtain optimality results for general explicit SSPRK methods up to fifth-order and explicit low-storage SSPRK methods up to fourth-order. Throughout, our studies allow for the possibility of negative coefficients which correspond to downwind-biased spatial discretizations. Guarantees of optimality are obtained for a variety of third- and fourth-order schemes. Where optimality is impractical to guarantee (specifically, for fifth-order methods and certain low-storage methods), extensive numerical optimizations are carried out to derive numerically optimal schemes. As a part of these studies, several new schemes arise which have theoretically improved time-stepping restrictions over schemes appearing in the recent literature.

A posteriori error estimates for the Crank–Nicolson method for parabolic equations

Abstract: We derive optimal order a posteriori error estimates for time discretizations by both the Crank-Nicolson and the Crank-Nicolson-Galerkin methods for linear and nonlinear parabolic equations. We examine both smooth and rough initial data. Our basic tool for deriving a posteriori estimates are second-order Crank-Nicolson reconstructions of the piecewise linear approximate solutions. These functions satisfy two fundamental properties: (i) they are explicitly computable and thus their difference to the numerical solution is controlled a posteriori, and (ii) they lead to optimal order residuals as well as to appropriate pointwise representations of the error equation of the same form as the underlying evolution equation. The resulting estimators are shown to be of optimal order by deriving upper and lower bounds for them depending only on the discretization parameters and the data of our problem. As a consequence we provide alternative proofs for known a priori rates of convergence for the Crank-Nicolson method.

Stabilized Galerkin approximation of convection-diffusion-reaction equations: Discrete maximum principle and convergence

Abstract: We analyze a nonlinear shock-capturing scheme for H1-conforming, piecewise-affine finite element approximations of linear elliptic problems. The meshes are assumed to satisfy two standard conditions: a local quasi-uniformity property and the Xu-Zikatanov condition ensuring that the stiffness matrix associated with the Poisson equation is an M-matrix. A discrete maximum principle is rigorously established in any space dimension for convection-diffusion-reaction problems. We prove that the shock-capturing finite element solution converges to that without shock-capturing if the cell Peclet numbers are sufficiently small. Moreover, in the diffusion-dominated regime, the difference between the two finite element solutions super-converges with respect to the actual approximation error. Numerical experiments on test problems with stiff layers confirm the sharpness of the a priori error estimates

Mathematical and computational methods for compressible flow.

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Hybridized globally divergence-free LDG methods. Part I: The Stokes problem

Abstract: We devise and analyze a new local discontinuous Galerkin (LDG) method for the Stokes equations of incompressible fluid flow. This optimally convergent method is obtained by using an LDG method to discretize a vorticity-velocity formulation of the Stokes equations and by applying a new hybridization to the resulting discretization. One of the main features of the hybridized method is that it provides a globally divergence-free approximate velocity without having to construct globally divergence-free finite-dimensional spaces; only elementwise divergence-free basis functions are used. Another important feature is that it has significantly less degrees of freedom than all other LDG methods in the current literature; in particular, the approximation to the pressure is only defined on the faces of the elements. On the other hand, we show that, as expected, the condition number of the Schur-complement matrix for this approximate pressure is of order h -2 in the mesh size h. Finally, we present numerical experiments that confirm the sharpness of our theoretical a priori error estimates.

Construction algorithms for polynomial lattice rules for multivariate integration

Abstract: We introduce a new construction algorithm for digital nets for integration in certain weighted tensor product Hilbert spaces. The first weighted Hilbert space we consider is based on Walsh functions. Dick and Pillichshammer calculated the worst-case error for integration using digital nets for this space. Here we extend this result to a special construction method for digital nets based on polynomials over finite fields. This result allows us to find polynomials which yield a small worst-case error by computer search. We prove an upper bound on the worst-case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights. We extend the results for the weighted Hilbert space based on Walsh functions to weighted Sobolev spaces. In this case we use randomly digitally shifted digital nets. The construction principle is the same as before, only the worst-case error is slightly different. Again digital nets obtained from our search algorithm yield a worst-case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights. These results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules. We conclude the article with numerical results comparing the expected worst-case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules.

A parameter robust numerical method for a two dimensional reaction-diffusion problem

Abstract: In this paper a singularly perturbed reaction-diffusion partial differential equation in two space dimensions is examined. By means of an appropriate decomposition, we describe the asymptotic behaviour of the solution of problems of this kind. A central finite difference scheme is constructed for this problem which involves an appropriate Shishkin mesh. We prove that the numerical approximations are almost second order uniformly convergent (in the maximum norm) with respect to the singular perturbation parameter. Some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method.

Minimality and other properties of the width-w nonadjacent form

Abstract: Let w ≥ 2 be an integer and let D w be the set of integers that includes zero and the odd integers with absolute value less than 2 w-1 . Every integer n can be represented as a finite sum of the form n = Σa i 2 i , with a i ∈E D w , such that of any w consecutive α i 's at most one is nonzero. Such representations are called width-w nonadjacent forms (w-NAFs). When w = 2 these representations use the digits {0, ±1} and coincide with the well-known nonadjacent forms. Width-w nonadjacent forms are useful in efficiently implementing elliptic curve arithmetic for cryptographic applications. We provide some new results on the w-NAF. We show that w-NAFs have a minimal number of nonzero digits and we also give a new characterization of the w-NAF in terms of a (right-to-left) lexicographical ordering. We also generalize a result on w-NAFs and show that any base 2 representation of an integer, with digits in D w , that has a minimal number of nonzero digits is at most one digit longer than its binary representation.

Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums

Abstract: We show that integrals of the form ∫ 0 1 x m Li p (x)Li q (x)dx (m ≥ -2,p,q ≥ 1) and ∫ 0 1 log r (x)Li p (x)Li q (x) / x dx (p, q,r ≥ 1) satisfy certain recurrence relations which allow us to write them in terms of Euler sums. From this we prove that, in the first case for all m, p, q and in the second case when p + q + r is even, these integrals are reducible to zeta values. In the case of odd p + q + r, we combine the known results for Euler sums with the information obtained from the problem in this form to give an estimate on the number of new constants which are needed to express the above integrals for a given weight p + q + r. The proofs are constructive, giving a method for the evaluation of these and other similar integrals, and we present a selection of explicit evaluations in the last section.

Solving quadratic equations using reduced unimodular quadratic forms

Abstract: Let Q be an n x n symmetric matrix with integral entries and with det Q ¬= 0, but not necesarily positive definite. We describe a generalized LLL algorithm to reduce this quadratic form. This algorithm either reduces the quadratic form or stops with some isotropic vector. It is proved to run in polynomial time. We also describe an algorithm for the minimization of a ternary quadratic form: when a quadratic equation q(x,y,z) = 0 is solvable over Q, a solution can be deduced from another quadratic equation of determinant ±1. The combination of these algorithms allows us to solve efficiently any general ternary quadratic equation over Q, and this gives a polynomial time algorithm (as soon as the factorization of the determinant of Q is known).

Error analysis of variable degree mixed methods for elliptic problems via hybridization

Abstract: A new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree Raviart-Thomas method for second order elliptic problems. The approach gives error estimates for the Lagrange multipliers without using error estimates for the other variables. Error estimates for the primal and flux variables then follow from those for the Lagrange multipliers. In contrast, traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the Lagrange multipliers. The new approach not only gives new error estimates for the new variable degree Raviart-Thomas method, but also new error estimates for the classical uniform degree method with less stringent regularity requirements than previously known estimates. The error analysis is achieved by using a variational characterization of the Lagrange multipliers wherein the other unknowns do not appear. This approach can be applied to other hybridized mixed methods as well.

Hermite methods for hyperbolic initial-boundary value problems

Abstract: We study arbitrary-order Hermite difference methods for the numerical solution of initial-boundary value problems for symmetric hyperbolic systems. These differ from standard difference methods in that derivative data (or equivalently local polynomial expansions) are carried at each grid point. Time-stepping is achieved using staggered grids and Taylor series. We prove that methods using derivatives of order m in each coordinate direction are stable under m-independent CFL constraints and converge at order 2m + 1. The stability proof relies on the fact that the Hermite interpolation process generally decreases a seminorm of the solution. We present numerical experiments demonstrating the resolution of the methods for large m as well as illustrating the basic theoretical results.

The multi-symplecticity of partitioned Runge-Kutta methods for hamiltonian pdes

Abstract: In this article we consider partitioned Runge-Kutta (PRK) methods for Hamiltonian partial differential equations (PDEs) and present some sufficient conditions for multi-symplecticity of PRK methods of Hamiltonian PDEs.

Error estimates on anisotropic ₁ elements for functions in weighted Sobolev spaces

Abstract: In this paper we prove error estimates for a piecewise Q 1 average interpolation on anisotropic rectangular elements, i.e., rectangles with sides of different orders, in two and three dimensions. Our error estimates are valid under the condition that neighboring elements have comparable size. This is a very mild assumption that includes more general meshes than those allowed in previous papers. In particular, strong anisotropic meshes arising naturally in the approximation of problems with boundary layers fall under our hypotheses. Moreover, we generalize the error estimates allowing on the right-hand side some weighted Sobolev norms. This extension is of interest in singularly perturbed problems. Finally, we consider the approximation of functions vanishing on the boundary by finite element functions with the same property, a point that was not considered in previous papers on average interpolations for anisotropic elements. As an application we consider the approximation of a singularly perturbed reaction-diffusion equation and show that, as a consequence of our results, almost optimal order error estimates in the energy norm, valid uniformly in the perturbation parameter, can be obtained.

Reliable a posteriori error control for nonconforming finite element approximation of Stokes flow

Abstract: We derive computable a posteriori error estimates for the lowest order nonconforming Crouzeix-Raviart element applied to the approximation of incompressible Stokes flow. The estimator provides an explicit upper bound that is free of any unknown constants, provided that a reasonable lower bound for the inf-sup constant of the underlying problem is available. In addition, it is shown that the estimator provides an equivalent lower bound on the error up to a generic constant.

The approximation of the Maxwell eigenvalue problem using a least-squares method

Abstract: In this paper we consider an approximation to the Maxwell's eigenvalue problem based on a very weak formulation of two div-curl systems with complementary boundary conditions. We formulate each of these div-curl systems as a general variational problem with different test and trial spaces, i.e., the solution space is L 2 (Ω) ≡ (L 2 (Ω)) 3 and components in the test spaces are in subspaces of H 1 (Ω), the Sobolev space of order one on the computational domain Q. A finite-element least-squares approximation to these variational problems is used as a basis for the approximation. Using the structure of the continuous eigenvalue problem, a discrete approximation to the eigenvalues is set up involving only the approximation to either of the div-curl systems. We give some theorems that guarantee the convergence of the eigenvalues to those of the continuous problem without the occurrence of spurious values. Finally, some results of numerical experiments are given.

Optimal error estimate of the penalty finite element method for the time-dependent Navier-Stokes equations

Abstract: A fully discrete penalty finite element method is presented for the two-dimensional time-dependent Navier-Stokes equations. The time discretization of the penalty Navier-Stokes equations is based on the backward Euler scheme; the spatial discretization of the time discretized penalty Navier-Stokes equations is based on a finite element space pair (X h , M h ) which satisfies some approximate assumption. An optimal error estimate of the numerical velocity and pressure is provided for the fully discrete penalty finite element method when the parameters e, At and h are sufficiently small.

Algorithms for hyperbolic quadratic eigenvalue problems

Abstract: We consider the quadratic eigenvalue problem (or the QEP) (λ 2 A + λB + C)x = 0, where A, B, and C are Hermitian with A positive definite. The QEP is called hyperbolic if (x*Bx) 2 > 4(x*Ax)(x*Cx) for all nonzero x ∈ C R . We show that a relatively efficient test for hyperbolicity can be obtained by computing the eigenvalues of the QEP. A hyperbolic QEP is overdamped if B is positive definite and C is positive semidefinite. We show that a hyperbolic QEP (whose eigenvalues are necessarily real) is overdamped if and only if its largest eigenvalue is nonpositive. For overdamped QEPs, we show that all eigenpairs can be found efficiently by finding two solutions of the corresponding quadratic matrix equation using a method based on cyclic reduction. We also present a new measure for the degree of hyperbolicity of a hyperbolic QEP.

On monotonicity and boundedness properties of linear multistep methods

Abstract: In this paper an analysis is provided of nonlinear monotonicity and boundedness properties for linear multistep methods. Instead of strict monotonicity for arbitrary starting values we shall focus on generalized monotonicity or boundedness with Runge-Kutta starting procedures. This allows many multistep methods of practical interest to be included in the theory. In a related manner, we also consider contractivity and stability in arbitrary norms.

Lower bounds and stochastic optimization algorithms for uniform designs with three or four levels

Abstract: New lower bounds for three- and four-level designs under the centered L 2 -discrepancy are provided. We describe necessary conditions for the existence of a uniform design meeting these lower bounds. We consider several modifications of two stochastic optimization algorithms for the problem of finding uniform or close to uniform designs under the centered L 2 -discrepancy. Besides the threshold accepting algorithm, we introduce an algorithm named balance-pursuit heuristic. This algorithm uses some combinatorial properties of inner structures required for a uniform design. Using the best specifications of these algorithms we obtain many designs whose discrepancy is lower than those obtained in previous works, as well as many new low-discrepancy designs with fairly large scale. Moreover, some of these designs meet the lower bound, i.e., are uniform designs.

Superconvergence of spectral collocation and $p$-version methods in one dimensional problems

Abstract: Superconvergence phenomenon of the Legendre spectral collocation method and the p-version finite element method is discussed under the one dimensional setting. For a class of functions that satisfy a regularity condition (M): ∥u (k) ∥L∞ ≤ cM k on a bounded domain, it is demonstrated, both theoretically and numerically, that the optimal convergent rate is supergeometric. Furthermore, at proper Gaussian points or Lobatto points, the rate of convergence may gain one or two orders of the polynomial degree.

On computing rational Gauss-Chebyshev quadrature formulas

Abstract: We provide an algorithm to compute the nodes and weights for Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary real poles outside [-1,1]. Contrary to existing rational quadrature formulas, the computational effort is very low, even for extremely high degrees, and under certain conditions on the poles it can be shown that the complexity is of order O(n). This method is based on the derivation of explicit expressions for Chebyshev orthogonal rational functions, which are (thus far) the only examples of explicitly known orthogonal rational functions on [-1,1] with arbitrary real poles outside this interval.

A Class of Singularly Perturbed Semilinear Differential Equations with Interior Layers

Abstract: In this paper singularly perturbed semilinear differential equations with a discontinuous source term are examined. A numerical method is constructed for these problems which involves an appropriate piecewise-uniform mesh. The method is shown to be uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented that validate the theoretical results.