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SIAM J. on Numerical Analysis

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1986

Volume 23, Issue 4, pp. 693-912


Folds on the Solution Manifold of a Parametrized Equation

James P. Fink and Werner C. Rheinboldt

SIAM J. Numer. Anal. 23, pp. 693-706 (14 pages)

Online Publication Date: July 14, 2006

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This paper presents a framework for studying singular points on the solution manifold of a parameter-dependent nonlinear equation $F(z,\lambda ) = 0$. The approach is based on a systematic combination of general constrained mappings with the tangent map of differential geometry. This framework is then used to develop a geometrically instructive and coordinate-free treatment of fold points on the solution manifold. The treatment includes a detailed analysis of the types of points that may occur on fold lines.

A Nonmonotone Line Search Technique for Newton’s Method

L. Grippo, F. Lampariello, and S. Lucidi

SIAM J. Numer. Anal. 23, pp. 707-716 (10 pages) | Cited 114 times

Online Publication Date: July 14, 2006

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In this paper a nonmonotone steplength selection rule for Newton’s method is proposed, which can be viewed as a generalization of Armijo’s rule. Numerical results are reported which indicate that the proposed technique may allow a considerable saving both in the number of line searches and in the number of function evaluations.

On the Successive Projections Approach to Least-Squares Problems

J. E. Dennis, Jr. and Trond Steihaug

SIAM J. Numer. Anal. 23, pp. 717-733 (17 pages) | Cited 3 times

Online Publication Date: July 14, 2006

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In this paper, we suggest a generalized Gauss–Seidel approach to sparse linear and nonlinear least-squares problems. The algorithm, closely related to one given by Elfving (1980), uses the work of Curtis, Powell, and Reid (1974) as extended by Coleman and Moré (1983) to divide the variables into nondisjoint groups of structurally orthogonal columns and then projects the updated residual into each column subspace of the Jacobian in turn. In the linear case, this procedure can be viewed as an alternate ordering of the variables in the Gauss–Seidel method. Preliminary tests indicate that this leads quickly to cheap solutions of limited accuracy for linear problems, and that this approach is promising for an inexact Gauss–Newton analog of the inexact Newton approach of Dembo, Eisenstat, and Steihaug (1982).

A Semi-Implicit Method for Hyperbolic Problems with Different Time-Scales

Jaime Guerra and Bertil Gustafsson

SIAM J. Numer. Anal. 23, pp. 734-749 (16 pages) | Cited 8 times

Online Publication Date: July 14, 2006

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Hyperbolic systems with two different time-scales are considered, where the solutions vary on the slow scale only. For this type of problem semi-implicit difference methods are very natural, and in this paper we analyze the leap-frog backwards Euler scheme. In particular it is shown, that when the ratio $\varepsilon $ between the slow and the fast scale tends to zero, the solutions of the approximation converge to solutions of the reduced differential equation. Numerical experiments are included for illustration of the theoretical results.

Finite Element Approximation of the Nonstationary Navier–Stokes Problem, Part II: Stability of Solutions and Error Estimates Uniform in Time

John G. Heywood and Rolf Rannacher

SIAM J. Numer. Anal. 23, pp. 750-777 (28 pages) | Cited 34 times

Online Publication Date: July 14, 2006

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In this paper, assumptions about the stability of a solution are introduced in the numerical analysis of the nonstationary Navier–Stokes problem, for the purpose of extending local a priori error estimates, and local a posteriori error estimates, globally in time.

A Moving Finite Element Method with Error Estimation and Refinement for One-Dimensional Time Dependent Partial Differential Equations

Slimane Adjerid and Joseph E. Flaherty

SIAM J. Numer. Anal. 23, pp. 778-796 (19 pages) | Cited 18 times

Online Publication Date: July 14, 2006

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We discuss a moving finite element method for solving vector systems of time dependent partial differential equations in one space dimension. The mesh is moved so as to equidistribute the spatial component of the discretization error in $H^1 $. We present a method of estimating this error by using $p$-hierarchic finite elements. The error estimate is also used in an adaptive mesh refinement procedure to give an algorithm that combines mesh movement and refinement.
We discretize the partial differential equations in space using a Galerkin procedure with piecewise linear elements to approximate the solution and quadratic elements to estimate the error. A system of ordinary differential equations for mesh velocities are used to control element motions. We use existing software for stiff ordinary differential equations for the temporal integration of the solution, the error estimate, and the mesh motion. Computational results using a code based on our method are presented for several examples.

Mesh-Independent Spectra in the Moving Finite Element Equations

A. J. Wathen

SIAM J. Numer. Anal. 23, pp. 797-814 (18 pages)

Online Publication Date: July 14, 2006

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We derive the moving finite element (MFE) equations for the solution of a scalar evolutionary equation in $d$ space dimensions $(d \geqq 1)$ and introduce the elementwise approach to MFE. This approach yields a decomposition of the mesh- and solution-dependent matrix $A$ in the (semidiscretised) nonlinear system or ordinary differential equations $A(y)\dot y = g(y)$ which forms the basis for proofs of eigenvalue clustering. Previous analysis (A. J. Wathen and M. J. Baines, IMA J. Numer. Anal., 5 (1985), pp. 161–182) for the MFE method is described and extended particularly for $d \geqq 2$, and it is shown that with a simple, specific block diagonal preconditioner, $D$, the eigenvalue spectrum of the preconditioned MFE matrix $D^{ - 1} A$ is $[\frac{1}{2},1 + {d / 2}]$ independently of the mesh configuration, the solution and the number of nodes. A more specific result is established for the case $d = 1$. These results guarantee extremely rapid solution techniques using, for example conjugate gradient methods. We show how the analysis extends to systems of partial differential equations when a separate moving mesh is used for each component.

Boundary Conditions in Chebyshev and Legendre Methods

Claudio Canuto

SIAM J. Numer. Anal. 23, pp. 815-831 (17 pages) | Cited 9 times

Online Publication Date: July 14, 2006

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We discuss two different ways of treating non-Dirichlet boundary conditions in Chebyshev and Legendre collocation methods for second order differential problems. An error analysis is provided. The effect of preconditioning the corresponding spectral operators by finite difference matrices is also investigated.

On the Convergence of Stability Constants

Rolf Dieter Grigorieff

SIAM J. Numer. Anal. 23, pp. 832-836 (5 pages) | Cited 3 times

Online Publication Date: July 14, 2006

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The stability constants of finite difference schemes approximating linear $m$th order integro-differential equations are shown to converge to their continuous counterparts.

Order Results for Implicit Runge–Kutta Methods Applied to Differential/Algebraic Systems

L. R. Petzold

SIAM J. Numer. Anal. 23, pp. 837-852 (16 pages) | Cited 22 times

Online Publication Date: July 14, 2006

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In this paper we study the order, stability and convergence properties of implicit Runge–Kutta methods applied to a relatively simple class of nonlinear differential/algebraic systems. These methods often do not attain the same order of accuracy for differential/algebraic systems as they do for purely differential systems. We derive a set of order conditions which the method coefficients should satisfy in addition to the usual order conditions to ensure a given order of accuracy, and we present results on the stability and convergence properties of these methods.

An Efficient Implementation of a Conformal Mapping Method Based on the Szegö Kernel

Manfred R. Trummer

SIAM J. Numer. Anal. 23, pp. 853-872 (20 pages) | Cited 17 times

Online Publication Date: July 14, 2006

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A fast implementation of a method to compute the Riemann mapping function is presented. The method has been recently introduced by N. Kerzman and the author; it expresses the Szegö kernel as the solution of an integral equation of the second kind. The complexity of this new algorithm is $O(n^2 )$, where $n$ is the number of collocation points on the boundary of the region. Previous algorithms for mapping from the problem domain to the disk require $O(n^2 \log n)$ operations. It is shown how to treat symmetric regions. The algorithm is tested on several examples. The numerical results show that the method is competitive with respect to accuracy, stability, and efficiency.

A Numerically Stable Circular Harmonic Reconstruction Algorithm

W. G. Hawkins and H. H. Barrett

SIAM J. Numer. Anal. 23, pp. 873-890 (18 pages) | Cited 4 times

Online Publication Date: July 14, 2006

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A numerically stable algorithm based on the orthogonal function solution of the 2-D Radon transform is given. This form of image reconstruction is often called circular harmonic transform (CHT) reconstruction. Series of Zernike polynomials are evaluated by solving a system of nonhomogeneous difference equations. The solution is based on a recursion formula for Zernike polynomials. We show that the method is numerically stable for the paraxial case. This algorithm overcomes the loss of spatial and contrast resolution associated with CHT algorithms. When compared with ramp-filtered back-projection, it is more resistant to ringing and it is not subject to systematic errors that seem to be caused by the discretization of the back-projection operator. The execution time is proportional to ${{9N^3 } / {32}}$ for $N^2 $ points on a polar grid, so that it is comparable to back-projection methods in execution time.

Piecewise-Polynomial Quadratures for Cauchy Singular Integrals

Apostolos Gerasoulis

SIAM J. Numer. Anal. 23, pp. 891-902 (12 pages) | Cited 9 times

Online Publication Date: July 14, 2006

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In this paper, we propose piecewise-polynomial methods for the approximation of Cauchy principal value integrals and develop a simple, efficient and numerically stable algorithm for the evaluation of the weights of the resulting piecewise-polynomial quadratures. We present two examples to illustrate the advantages of these quadratures versus the Gauss–Jacobi quadratures.

On the Convergence of Some Cubic Spline Interpolation Schemes

R. K. Beatson

SIAM J. Numer. Anal. 23, pp. 903-912 (10 pages) | Cited 2 times

Online Publication Date: July 14, 2006

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Cubic spline interpolation schemes which require no derivative information at the end points are of great practical importance and have been included in several general purpose software libraries. In this paper optimal order error estimates are developed for three popular schemes of this “derivative free” type. The approximation of $C^1 [a,b]\backslash C^2 [a,b]$ functions by any such “derivative free” method that reproduces cubics, necessarily displays some dependence on the local mesh ratio. However, for the spline interpolants studied here this dependence is restricted to the first and last subintervals.
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